fundamentals of fluid mechanics chapter 3 …taiwan921.lib.ntu.edu.tw/mypdf/fluid03.pdf · 1...
TRANSCRIPT
1
FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS
Chapter 3 Fluids in Motion Chapter 3 Fluids in Motion -- The Bernoulli EquationThe Bernoulli Equation
JyhJyh--CherngCherng ShiehShiehDepartment of BioDepartment of Bio--Industrial Industrial MechatronicsMechatronics Engineering Engineering
National Taiwan UniversityNational Taiwan University0928200909282009
2
MAIN TOPICSMAIN TOPICS
NewtonNewtonrsquorsquos Second Laws Second LawF=ma Along a StreamlineF=ma Along a StreamlineF=ma Normal to a StreamlineF=ma Normal to a StreamlinePhysical Interpretation Physical Interpretation of of Bernoulli Equation
Static Stagnation Dynamic and Total PressureStatic Stagnation Dynamic and Total PressureApplication of the Bernoulli EquationApplication of the Bernoulli EquationThe Energy Line and the Hydraulic Grade LineThe Energy Line and the Hydraulic Grade LineRestrictions on Use of the Bernoulli EquationRestrictions on Use of the Bernoulli Equation
Bernoulli Equation推導出推導出
解釋解釋Bernoulli equationBernoulli equation
3
NewtonNewtonrsquorsquos Second Law s Second Law 1616
As a fluid particle moves from one location to another it As a fluid particle moves from one location to another it experiences an acceleration or decelerationexperiences an acceleration or deceleration
According to According to NewtonNewtonrsquorsquos second law of motions second law of motion the net force acting the net force acting on the fluid particle under consideration must equal its mass tion the fluid particle under consideration must equal its mass times mes its accelerationits acceleration
F=maF=ma In this chapter we consider the motion of In this chapter we consider the motion of inviscidinviscid fluidsfluids That is That is
the fluid is assumed to have the fluid is assumed to have zero viscosityzero viscosity For such case For such case it is it is possible to ignore viscous effectspossible to ignore viscous effects
The forces acting on the particle Coordinates used The forces acting on the particle Coordinates used
作用在質點的合力等於質點質量乘以加速度
本章仍以非黏性流體為對象即假設流體沒有黏性忽略其黏性效應本章仍以非黏性流體為對象即假設流體沒有黏性忽略其黏性效應討論作用在質點上的力使用的座標討論作用在質點上的力使用的座標
4
NewtonNewtonrsquorsquos Second Law s Second Law 2626
The fluid motion is governed byThe fluid motion is governed byF= Net pressure force + Net gravity forceF= Net pressure force + Net gravity force
To apply NewtonTo apply Newtonrsquorsquos second law to a fluid s second law to a fluid an appropriate an appropriate coordinate system must be chosen to describe the coordinate system must be chosen to describe the motionmotion In general the motion will be three In general the motion will be three--dimensional dimensional and unsteady so that and unsteady so that three space coordinates and timethree space coordinates and timeare needed to describe itare needed to describe it
作用在質點上的作用在質點上的ForceForce
5
NewtonNewtonrsquorsquos Second Law s Second Law 3636
The most often used coordinate The most often used coordinate systems are systems are rectangular (xyz) rectangular (xyz) and cylindrical (rand cylindrical (rz) systemz) system選用合適的座標系統使用哪一種座標系統選用合適的座標系統使用哪一種座標系統
6
StreamlineStreamline Streamline coordinateStreamline coordinate
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity V which fluid particle is described in terms of its velocity V which is defined as the time rate of change of the position of the is defined as the time rate of change of the position of the particleparticle
The particleThe particlersquorsquos velocity is a vector quantity with a s velocity is a vector quantity with a magnitude and directionmagnitude and direction
As the particle moves about it follows a particular path As the particle moves about it follows a particular path the shape of which is governed by the velocity of the the shape of which is governed by the velocity of the particleparticle
7
StreamlineStreamline Streamline coordinateStreamline coordinate
The location of the particle along the path is a function of The location of the particle along the path is a function of where the particle started at the initial time and its velocity where the particle started at the initial time and its velocity along the pathalong the path
If it is If it is steady flowsteady flow each successive particle that passes each successive particle that passes through a given point will follow the same paththrough a given point will follow the same path
8
StreamlineStreamline Streamline coordinateStreamline coordinate
For such cases the path is a fixed line in the For such cases the path is a fixed line in the xx--zz planeplaneNeighboring particles that pass on either side of point (1 Neighboring particles that pass on either side of point (1
following their own paths which may be of a different following their own paths which may be of a different shape than the one passing through (1)shape than the one passing through (1)
The entire The entire xx--zz plane is filled with such pathsplane is filled with such paths
9
StreamlineStreamline Streamline coordinateStreamline coordinate
For steady flows each particle slides along its path and its For steady flows each particle slides along its path and its velocity is everywhere tangent to the pathvelocity is everywhere tangent to the path
The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlinesstreamlines
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
2
MAIN TOPICSMAIN TOPICS
NewtonNewtonrsquorsquos Second Laws Second LawF=ma Along a StreamlineF=ma Along a StreamlineF=ma Normal to a StreamlineF=ma Normal to a StreamlinePhysical Interpretation Physical Interpretation of of Bernoulli Equation
Static Stagnation Dynamic and Total PressureStatic Stagnation Dynamic and Total PressureApplication of the Bernoulli EquationApplication of the Bernoulli EquationThe Energy Line and the Hydraulic Grade LineThe Energy Line and the Hydraulic Grade LineRestrictions on Use of the Bernoulli EquationRestrictions on Use of the Bernoulli Equation
Bernoulli Equation推導出推導出
解釋解釋Bernoulli equationBernoulli equation
3
NewtonNewtonrsquorsquos Second Law s Second Law 1616
As a fluid particle moves from one location to another it As a fluid particle moves from one location to another it experiences an acceleration or decelerationexperiences an acceleration or deceleration
According to According to NewtonNewtonrsquorsquos second law of motions second law of motion the net force acting the net force acting on the fluid particle under consideration must equal its mass tion the fluid particle under consideration must equal its mass times mes its accelerationits acceleration
F=maF=ma In this chapter we consider the motion of In this chapter we consider the motion of inviscidinviscid fluidsfluids That is That is
the fluid is assumed to have the fluid is assumed to have zero viscosityzero viscosity For such case For such case it is it is possible to ignore viscous effectspossible to ignore viscous effects
The forces acting on the particle Coordinates used The forces acting on the particle Coordinates used
作用在質點的合力等於質點質量乘以加速度
本章仍以非黏性流體為對象即假設流體沒有黏性忽略其黏性效應本章仍以非黏性流體為對象即假設流體沒有黏性忽略其黏性效應討論作用在質點上的力使用的座標討論作用在質點上的力使用的座標
4
NewtonNewtonrsquorsquos Second Law s Second Law 2626
The fluid motion is governed byThe fluid motion is governed byF= Net pressure force + Net gravity forceF= Net pressure force + Net gravity force
To apply NewtonTo apply Newtonrsquorsquos second law to a fluid s second law to a fluid an appropriate an appropriate coordinate system must be chosen to describe the coordinate system must be chosen to describe the motionmotion In general the motion will be three In general the motion will be three--dimensional dimensional and unsteady so that and unsteady so that three space coordinates and timethree space coordinates and timeare needed to describe itare needed to describe it
作用在質點上的作用在質點上的ForceForce
5
NewtonNewtonrsquorsquos Second Law s Second Law 3636
The most often used coordinate The most often used coordinate systems are systems are rectangular (xyz) rectangular (xyz) and cylindrical (rand cylindrical (rz) systemz) system選用合適的座標系統使用哪一種座標系統選用合適的座標系統使用哪一種座標系統
6
StreamlineStreamline Streamline coordinateStreamline coordinate
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity V which fluid particle is described in terms of its velocity V which is defined as the time rate of change of the position of the is defined as the time rate of change of the position of the particleparticle
The particleThe particlersquorsquos velocity is a vector quantity with a s velocity is a vector quantity with a magnitude and directionmagnitude and direction
As the particle moves about it follows a particular path As the particle moves about it follows a particular path the shape of which is governed by the velocity of the the shape of which is governed by the velocity of the particleparticle
7
StreamlineStreamline Streamline coordinateStreamline coordinate
The location of the particle along the path is a function of The location of the particle along the path is a function of where the particle started at the initial time and its velocity where the particle started at the initial time and its velocity along the pathalong the path
If it is If it is steady flowsteady flow each successive particle that passes each successive particle that passes through a given point will follow the same paththrough a given point will follow the same path
8
StreamlineStreamline Streamline coordinateStreamline coordinate
For such cases the path is a fixed line in the For such cases the path is a fixed line in the xx--zz planeplaneNeighboring particles that pass on either side of point (1 Neighboring particles that pass on either side of point (1
following their own paths which may be of a different following their own paths which may be of a different shape than the one passing through (1)shape than the one passing through (1)
The entire The entire xx--zz plane is filled with such pathsplane is filled with such paths
9
StreamlineStreamline Streamline coordinateStreamline coordinate
For steady flows each particle slides along its path and its For steady flows each particle slides along its path and its velocity is everywhere tangent to the pathvelocity is everywhere tangent to the path
The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlinesstreamlines
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
3
NewtonNewtonrsquorsquos Second Law s Second Law 1616
As a fluid particle moves from one location to another it As a fluid particle moves from one location to another it experiences an acceleration or decelerationexperiences an acceleration or deceleration
According to According to NewtonNewtonrsquorsquos second law of motions second law of motion the net force acting the net force acting on the fluid particle under consideration must equal its mass tion the fluid particle under consideration must equal its mass times mes its accelerationits acceleration
F=maF=ma In this chapter we consider the motion of In this chapter we consider the motion of inviscidinviscid fluidsfluids That is That is
the fluid is assumed to have the fluid is assumed to have zero viscosityzero viscosity For such case For such case it is it is possible to ignore viscous effectspossible to ignore viscous effects
The forces acting on the particle Coordinates used The forces acting on the particle Coordinates used
作用在質點的合力等於質點質量乘以加速度
本章仍以非黏性流體為對象即假設流體沒有黏性忽略其黏性效應本章仍以非黏性流體為對象即假設流體沒有黏性忽略其黏性效應討論作用在質點上的力使用的座標討論作用在質點上的力使用的座標
4
NewtonNewtonrsquorsquos Second Law s Second Law 2626
The fluid motion is governed byThe fluid motion is governed byF= Net pressure force + Net gravity forceF= Net pressure force + Net gravity force
To apply NewtonTo apply Newtonrsquorsquos second law to a fluid s second law to a fluid an appropriate an appropriate coordinate system must be chosen to describe the coordinate system must be chosen to describe the motionmotion In general the motion will be three In general the motion will be three--dimensional dimensional and unsteady so that and unsteady so that three space coordinates and timethree space coordinates and timeare needed to describe itare needed to describe it
作用在質點上的作用在質點上的ForceForce
5
NewtonNewtonrsquorsquos Second Law s Second Law 3636
The most often used coordinate The most often used coordinate systems are systems are rectangular (xyz) rectangular (xyz) and cylindrical (rand cylindrical (rz) systemz) system選用合適的座標系統使用哪一種座標系統選用合適的座標系統使用哪一種座標系統
6
StreamlineStreamline Streamline coordinateStreamline coordinate
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity V which fluid particle is described in terms of its velocity V which is defined as the time rate of change of the position of the is defined as the time rate of change of the position of the particleparticle
The particleThe particlersquorsquos velocity is a vector quantity with a s velocity is a vector quantity with a magnitude and directionmagnitude and direction
As the particle moves about it follows a particular path As the particle moves about it follows a particular path the shape of which is governed by the velocity of the the shape of which is governed by the velocity of the particleparticle
7
StreamlineStreamline Streamline coordinateStreamline coordinate
The location of the particle along the path is a function of The location of the particle along the path is a function of where the particle started at the initial time and its velocity where the particle started at the initial time and its velocity along the pathalong the path
If it is If it is steady flowsteady flow each successive particle that passes each successive particle that passes through a given point will follow the same paththrough a given point will follow the same path
8
StreamlineStreamline Streamline coordinateStreamline coordinate
For such cases the path is a fixed line in the For such cases the path is a fixed line in the xx--zz planeplaneNeighboring particles that pass on either side of point (1 Neighboring particles that pass on either side of point (1
following their own paths which may be of a different following their own paths which may be of a different shape than the one passing through (1)shape than the one passing through (1)
The entire The entire xx--zz plane is filled with such pathsplane is filled with such paths
9
StreamlineStreamline Streamline coordinateStreamline coordinate
For steady flows each particle slides along its path and its For steady flows each particle slides along its path and its velocity is everywhere tangent to the pathvelocity is everywhere tangent to the path
The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlinesstreamlines
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
4
NewtonNewtonrsquorsquos Second Law s Second Law 2626
The fluid motion is governed byThe fluid motion is governed byF= Net pressure force + Net gravity forceF= Net pressure force + Net gravity force
To apply NewtonTo apply Newtonrsquorsquos second law to a fluid s second law to a fluid an appropriate an appropriate coordinate system must be chosen to describe the coordinate system must be chosen to describe the motionmotion In general the motion will be three In general the motion will be three--dimensional dimensional and unsteady so that and unsteady so that three space coordinates and timethree space coordinates and timeare needed to describe itare needed to describe it
作用在質點上的作用在質點上的ForceForce
5
NewtonNewtonrsquorsquos Second Law s Second Law 3636
The most often used coordinate The most often used coordinate systems are systems are rectangular (xyz) rectangular (xyz) and cylindrical (rand cylindrical (rz) systemz) system選用合適的座標系統使用哪一種座標系統選用合適的座標系統使用哪一種座標系統
6
StreamlineStreamline Streamline coordinateStreamline coordinate
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity V which fluid particle is described in terms of its velocity V which is defined as the time rate of change of the position of the is defined as the time rate of change of the position of the particleparticle
The particleThe particlersquorsquos velocity is a vector quantity with a s velocity is a vector quantity with a magnitude and directionmagnitude and direction
As the particle moves about it follows a particular path As the particle moves about it follows a particular path the shape of which is governed by the velocity of the the shape of which is governed by the velocity of the particleparticle
7
StreamlineStreamline Streamline coordinateStreamline coordinate
The location of the particle along the path is a function of The location of the particle along the path is a function of where the particle started at the initial time and its velocity where the particle started at the initial time and its velocity along the pathalong the path
If it is If it is steady flowsteady flow each successive particle that passes each successive particle that passes through a given point will follow the same paththrough a given point will follow the same path
8
StreamlineStreamline Streamline coordinateStreamline coordinate
For such cases the path is a fixed line in the For such cases the path is a fixed line in the xx--zz planeplaneNeighboring particles that pass on either side of point (1 Neighboring particles that pass on either side of point (1
following their own paths which may be of a different following their own paths which may be of a different shape than the one passing through (1)shape than the one passing through (1)
The entire The entire xx--zz plane is filled with such pathsplane is filled with such paths
9
StreamlineStreamline Streamline coordinateStreamline coordinate
For steady flows each particle slides along its path and its For steady flows each particle slides along its path and its velocity is everywhere tangent to the pathvelocity is everywhere tangent to the path
The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlinesstreamlines
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
5
NewtonNewtonrsquorsquos Second Law s Second Law 3636
The most often used coordinate The most often used coordinate systems are systems are rectangular (xyz) rectangular (xyz) and cylindrical (rand cylindrical (rz) systemz) system選用合適的座標系統使用哪一種座標系統選用合適的座標系統使用哪一種座標系統
6
StreamlineStreamline Streamline coordinateStreamline coordinate
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity V which fluid particle is described in terms of its velocity V which is defined as the time rate of change of the position of the is defined as the time rate of change of the position of the particleparticle
The particleThe particlersquorsquos velocity is a vector quantity with a s velocity is a vector quantity with a magnitude and directionmagnitude and direction
As the particle moves about it follows a particular path As the particle moves about it follows a particular path the shape of which is governed by the velocity of the the shape of which is governed by the velocity of the particleparticle
7
StreamlineStreamline Streamline coordinateStreamline coordinate
The location of the particle along the path is a function of The location of the particle along the path is a function of where the particle started at the initial time and its velocity where the particle started at the initial time and its velocity along the pathalong the path
If it is If it is steady flowsteady flow each successive particle that passes each successive particle that passes through a given point will follow the same paththrough a given point will follow the same path
8
StreamlineStreamline Streamline coordinateStreamline coordinate
For such cases the path is a fixed line in the For such cases the path is a fixed line in the xx--zz planeplaneNeighboring particles that pass on either side of point (1 Neighboring particles that pass on either side of point (1
following their own paths which may be of a different following their own paths which may be of a different shape than the one passing through (1)shape than the one passing through (1)
The entire The entire xx--zz plane is filled with such pathsplane is filled with such paths
9
StreamlineStreamline Streamline coordinateStreamline coordinate
For steady flows each particle slides along its path and its For steady flows each particle slides along its path and its velocity is everywhere tangent to the pathvelocity is everywhere tangent to the path
The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlinesstreamlines
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
6
StreamlineStreamline Streamline coordinateStreamline coordinate
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity V which fluid particle is described in terms of its velocity V which is defined as the time rate of change of the position of the is defined as the time rate of change of the position of the particleparticle
The particleThe particlersquorsquos velocity is a vector quantity with a s velocity is a vector quantity with a magnitude and directionmagnitude and direction
As the particle moves about it follows a particular path As the particle moves about it follows a particular path the shape of which is governed by the velocity of the the shape of which is governed by the velocity of the particleparticle
7
StreamlineStreamline Streamline coordinateStreamline coordinate
The location of the particle along the path is a function of The location of the particle along the path is a function of where the particle started at the initial time and its velocity where the particle started at the initial time and its velocity along the pathalong the path
If it is If it is steady flowsteady flow each successive particle that passes each successive particle that passes through a given point will follow the same paththrough a given point will follow the same path
8
StreamlineStreamline Streamline coordinateStreamline coordinate
For such cases the path is a fixed line in the For such cases the path is a fixed line in the xx--zz planeplaneNeighboring particles that pass on either side of point (1 Neighboring particles that pass on either side of point (1
following their own paths which may be of a different following their own paths which may be of a different shape than the one passing through (1)shape than the one passing through (1)
The entire The entire xx--zz plane is filled with such pathsplane is filled with such paths
9
StreamlineStreamline Streamline coordinateStreamline coordinate
For steady flows each particle slides along its path and its For steady flows each particle slides along its path and its velocity is everywhere tangent to the pathvelocity is everywhere tangent to the path
The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlinesstreamlines
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
7
StreamlineStreamline Streamline coordinateStreamline coordinate
The location of the particle along the path is a function of The location of the particle along the path is a function of where the particle started at the initial time and its velocity where the particle started at the initial time and its velocity along the pathalong the path
If it is If it is steady flowsteady flow each successive particle that passes each successive particle that passes through a given point will follow the same paththrough a given point will follow the same path
8
StreamlineStreamline Streamline coordinateStreamline coordinate
For such cases the path is a fixed line in the For such cases the path is a fixed line in the xx--zz planeplaneNeighboring particles that pass on either side of point (1 Neighboring particles that pass on either side of point (1
following their own paths which may be of a different following their own paths which may be of a different shape than the one passing through (1)shape than the one passing through (1)
The entire The entire xx--zz plane is filled with such pathsplane is filled with such paths
9
StreamlineStreamline Streamline coordinateStreamline coordinate
For steady flows each particle slides along its path and its For steady flows each particle slides along its path and its velocity is everywhere tangent to the pathvelocity is everywhere tangent to the path
The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlinesstreamlines
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
8
StreamlineStreamline Streamline coordinateStreamline coordinate
For such cases the path is a fixed line in the For such cases the path is a fixed line in the xx--zz planeplaneNeighboring particles that pass on either side of point (1 Neighboring particles that pass on either side of point (1
following their own paths which may be of a different following their own paths which may be of a different shape than the one passing through (1)shape than the one passing through (1)
The entire The entire xx--zz plane is filled with such pathsplane is filled with such paths
9
StreamlineStreamline Streamline coordinateStreamline coordinate
For steady flows each particle slides along its path and its For steady flows each particle slides along its path and its velocity is everywhere tangent to the pathvelocity is everywhere tangent to the path
The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlinesstreamlines
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
9
StreamlineStreamline Streamline coordinateStreamline coordinate
For steady flows each particle slides along its path and its For steady flows each particle slides along its path and its velocity is everywhere tangent to the pathvelocity is everywhere tangent to the path
The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlinesstreamlines
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
10
StreamlineStreamline Streamline coordinateStreamline coordinate
For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the ldquoldquostreamlinestreamlinerdquordquo coordinate based on the coordinate based on the streamlines The particle motion is described in terms of streamlines The particle motion is described in terms of its distance its distance s = s = s(ts(t)) along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline streamline R = R = R(sR(s))
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
11
NewtonNewtonrsquorsquos Second Law s Second Law 4646
In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motion
As is done in the study of dynamics the motion of each As is done in the study of dynamics the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV
As the particle moves As the particle moves it follows a particular pathit follows a particular pathThe location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocityof its initial position and velocity
以兩維流動為對象以兩維流動為對象
質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是
初始位置與速度的初始位置與速度的函數函數
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
12
NewtonNewtonrsquorsquos Second Law s Second Law 5656
For steady flowsFor steady flows each particle slides along its path and each particle slides along its path and its velocity vector is everywhere tangent to the path The its velocity vector is everywhere tangent to the path The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines
For such situation the particle motion is described in For such situation the particle motion is described in terms of its distance s=s(t) along the streamline from terms of its distance s=s(t) along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline R=R(s)of the streamline R=R(s)
流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline
在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切
質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
13
NewtonNewtonrsquorsquos Second Law s Second Law 6666
The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamline
The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle
The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection
nRVs
dsdVVnasa
dtVda
2
ns
RVa
dsdVVa
2
ns CHAPTER 04 CHAPTER 04 再討論再討論
由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt
由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip
加速度的兩個分量加速度的兩個分量
先應用先應用
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
14
StreamlinesStreamlines
Streamlines past an airfoilStreamlines past an airfoil
Flow past a bikerFlow past a biker
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
15
F=ma along a Streamline F=ma along a Streamline 1414
Isolation of a small fluid particle in a flow fieldfluid particle in a flow field
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
16
F=ma along a Streamline F=ma along a Streamline 2424
Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure
For steady flow the component of Newtonrsquos second law along the streamline direction s
sVVV
sVmVmaF SS
Where represents the sum of the s components of all the force acting on the particle
SF
沿著沿著ss方向的合力方向的合力
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
17
F=ma along a Streamline F=ma along a Streamline 3434
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction
The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection
sinVsinWWs
Vspynp2ynppynppF SSSps
VspsinFWF psss
sasVV
spsin
Equation of motion Equation of motion along the streamline along the streamline directiondirection
2s
sppS
單位體積
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
18
F=ma along a Streamline F=ma along a Streamline 4444
A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamlineweight along the streamline
For fluid For fluid static situationstatic situation the balance between pressure the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis producedis produced
sasVV
spsin
0spsin
IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight
pressure gradientpressure gradient
造成質點速度改變的兩個因素造成質點速度改變的兩個因素
質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
19
IntegrationIntegrationhelliphellip
sasVV
spsin
Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip
CgzV21dp
0dzVd21dp
dsdV
21
dsdp
dsdz
2
22
along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamlinethe streamline
In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and therefore because the density may not be constant and therefore cannot be removed from under the integral signcannot be removed from under the integral sign
除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
20
Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
ttanconsz2
Vp2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzV21dp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics powerful tool in fluid mechanics published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738(1700~1782) in 1738NONO 剪力剪力
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
21
Example 31 Pressure Variation along A Example 31 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid incompressible steady flow incompressible steady flow along the along the
horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a as B in front of the sphere of radius a as shown in Figure E31a From a more advanced theory of flow past shown in Figure E31a From a more advanced theory of flow past a a sphere the fluid velocity along this streamline issphere the fluid velocity along this streamline is
Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)
3
3
0 xa1VV 已知流體沿著已知流體沿著streamlinestreamline的速度的速度
求沿著求沿著streamlinestreamline的的壓力變化壓力變化
sinsin=0=0 s s xx
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
22
Example 31 Example 31 SolutionSolution1212
4
3
3
32
04
30
3
3
0 xa
xa1V3
xaV3
xa1V
xVV
sVV
The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)
The acceleration term
sVV
sp
(1)(1) sasVV
spsin
The pressure gradient along the streamline is
4
3320
3
xxa1Va3
xp
sp
(2)(2)
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
23
Example 31 Example 31 SolutionSolution2222
The pressure gradient along the streamline
4
3320
3
xxa1Va3
xp
sp
(2)(2)
The pressure distribution along the streamline
2)xa(
xaVp
632
0
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
24
Example 32 The Bernoulli EquationExample 32 The Bernoulli Equation
Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00 as is shown in Figure E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)
求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
25
Example 32 Example 32 SolutionSolution
2V
2Vpp
20
21
12
The Bernoullirsquos equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)
z1=z2
2
22
21
21
1 z2
Vpz2
Vp
(1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclistrsquos nose V2=0
將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
26
F=ma Normal to a StreamlineF=ma Normal to a Streamline1212
For steady flow the For steady flow the component of Newtoncomponent of Newtonrsquorsquos s second law in the second law in the normal normal direction ndirection n
RVV
RmVF
22
n
Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle
nF
法線方向的合力法線方向的合力
HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
27
F=ma Normal to a StreamlineF=ma Normal to a Streamline2222
The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection
The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection
cosVcosWWn
Vnpysp2ys)pp(ysppF nnnpn
VRVV
npcosFWF
2
pnnn
RV
npcos
2
Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction
2n
nppn
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
28
IntegrationIntegrationhelliphellip
RV
dndp
dndz 2
RV
npcos
2
RearrangedRearranged
Normal to Normal to the streamlinethe streamline
In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and the density may not be constant and therefore cannot be removed from therefore cannot be removed from under the integral signunder the integral sign
IntegratedIntegratedhelliphellip
A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline
CgzdnRVdp 2
Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed
ParticleParticle weightweight
pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素
除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開
除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
29
Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline
For the special case of For the special case of incompressible flowincompressible flow
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline
CzdnRVp
2
BERNOULLI EQUATIONBERNOULLI EQUATION
CgzdnRVdp 2
不可壓縮流體不可壓縮流體
一再提醒每一個結論(推導出來的方程式)都有它背後假設一再提醒每一個結論(推導出來的方程式)都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
30
If gravity is neglectedhellipFree vortex
A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motionmotion
If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal
RV
dndp
dndz 2
RV
dndp 2
Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature Thus the pressure outside a tornado is of curvature Thus the pressure outside a tornado is larger than it is near the center of the tornadolarger than it is near the center of the tornado
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
31
Aircraft wing tip vortexAircraft wing tip vortex
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
32
Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b are two flow fields with circular r
streamlines The streamlines The velocity distributionsvelocity distributions are are
)b(r
C)r(V)a(rC)r(V 21
Assuming the flows are steady inviscid and incompressible with streamlines in the horizontal plane (dzdn=0)
已知速度分佈已知速度分佈求壓力場求壓力場
RV
npcos
2
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
33
Example 33 Example 33 SolutionSolution
02
022
1 prrC21p
rV
rp 2
For flow in the horizontal plane (dzdn=0) The streamlines are circles n=-rThe radius of curvature R=r
For case (a) this gives
rCrp 2
1
3
22
rC
rp
For case (b) this gives
0220
22 p
r1
r1C
21p
RV
dndp
dndz 2
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
34
Physical InterpreterPhysical Interpreter1212
Under the basic assumptions Under the basic assumptions the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressibleand incompressible
Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in
To To produce an acceleration there must be an unbalance of produce an acceleration there must be an unbalance of the resultant forcethe resultant force of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important Thus there are three process Thus there are three process involved in the flow involved in the flow ndashndash mass times acceleration (the mass times acceleration (the VV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the z term)z term)
CzdnRVp
2
Cz2
Vp2
基本假設基本假設
過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量z z 」」
作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生
RVa
dsdVVa
2
ns
如如何何解解讀讀
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
35
Physical InterpreterPhysical Interpreter2222
The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particlerdquordquoWork done by force Work done by force FFtimestimesddWork done by weight Work done by weight z z Work done by pressure force pWork done by pressure force p
Kinetic energy Kinetic energy VV2222
CdnRVzp
2
C2
Vzp2
能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變
外力有二外力有二一為壓力一為壓力一為重力一為重力
你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
36
HeadHead
An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by
czg2
VP 2
Pressure HeadPressure Head
Velocity HeadVelocity Head
Elevation HeadElevation Head
The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads
沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
37
Example 34 Kinetic Potential and Example 34 Kinetic Potential and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe
shown in Figure E34 A force applied to the shown in Figure E34 A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe atmospheric at point (1) within the syringe The water flows from the needle point (2) The water flows from the needle point (2) with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory Discuss point (3) at the top of its trajectory Discuss the energy of the fluid at point (1) (2) and (3) the energy of the fluid at point (1) (2) and (3) by using the Bernoulli equation by using the Bernoulli equation
求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係
停停止止移移動動
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
38
Example 34 Solution
The sum of the three types of energy (kinetic potential and prThe sum of the three types of energy (kinetic potential and pressure) essure) or heads (velocity elevation and pressure) must remain constanor heads (velocity elevation and pressure) must remain constant t
The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle Gravity acting on the form the needle Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flighta momentary stop at the top of its flight
streamlinethealongttanconszV21p 2
The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
39
Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure
E35 From section A to B the streamlines are straight while frE35 From section A to B the streamlines are straight while from C om C to D they follow circular paths to D they follow circular paths Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)
求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
40
Example 35 Example 35 SolutionSolution1212
1221221 rhp)zz(rpp
ttanconsrzp
R= for the portion from for the portion from A to BA to B
Using p2=0z1=0and z2=h2-1
Since the radius of curvature of the streamline is infinite the pressure variation in the vertical direction is the same as if the fluids were stationary
Point (1)~(2)
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
41
4
3
z
z
2
343 dzRVrhp
With pp44=0 and z=0 and z44--zz33=h=h44--33 this becomes
334z
z
2
4 rzprz)dz(R
Vp4
3
Example 35 Example 35 SolutionSolution2222
Point (3)~(4)
For the portion from For the portion from C to DC to D
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
42
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515
Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform of pressure
pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows To measure this pressure one must move along flows To measure this pressure one must move along with the fluid thus being with the fluid thus being ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves
C2
Vzp2
Each term can be interpreted as a form of pressure
能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力
pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
43
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure2525
The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe
hhhphp 34133131 The static pressureThe static pressurezz is termed the is termed the hydrostatic hydrostatic
pressurepressure It is not actually a It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changeschanges
利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」
zz不是真的壓力它代表著位能(高度)改變所產生的壓力變化不是真的壓力它代表著位能(高度)改變所產生的壓力變化
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
44
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure3535
VV2222 is termed the is termed the dynamic pressuredynamic pressure It can be interpreted It can be interpreted as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream flow and pointing upstream After the initial transient After the initial transient motion has died out the liquid will fill the tube to a height motion has died out the liquid will fill the tube to a height of Hof H
The fluid in the tube including that at its tip (2) will be The fluid in the tube including that at its tip (2) will be stationary That is Vstationary That is V22=0 or point (2) is a stagnation point=0 or point (2) is a stagnation point
2112 V
21pp Stagnation pressure
Static pressure Dynamic pressureDynamic pressure
VV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在
管端管端((22))衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於
液柱高液柱高HH所產生的壓力所產生的壓力
假設(假設(11)與)與((22))高度相同高度相同
兩者和兩者和稱為稱為sstagnationtagnation pressurepressure
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
45
Stagnation pointStagnation point
Stagnation point flowStagnation point flow
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
46
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure4545
There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is placed into a flowing fluid Some of the fluid flows placed into a flowing fluid Some of the fluid flows ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe object
The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the body
Neglecting the elevation Neglecting the elevation effects effects the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline
stagnation pointstagnation point
將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point
忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
47
Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure5555
The The sum of the static pressure dynamic pressure and sum of the static pressure dynamic pressure and hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure
The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline
ttanconspz2
Vp T
2
Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數
三者和三者和稱為稱為TTootaltal pressurepressure
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
48
The The PitotPitot--static Tube static Tube 1515
)pp(2V
2Vpp
pppzz
2Vppp
43
243
14
41
232
Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculatedfluid speed can be calculated
This is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic tube is based
Static pressureStatic pressure
Stagnation pressureStagnation pressure
PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressurevelocity into pressure
利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具
量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
49
Airplane Airplane PitotPitot--static probestatic probe
Airspeed indicatorAirspeed indicator
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
50
The The PitotPitot--static Tube static Tube 2525
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
51
The The PitotPitot--static Tube static Tube 3535
The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressuremeasure the static and stagnation pressure
An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurement
This requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections
Incorrect and correct design of static pressure tapsIncorrect and correct design of static pressure taps
TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免
造成不必要的壓造成不必要的壓升或壓降升或壓降
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
52
The The PitotPitot--static Tube static Tube 4545
Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic tube
The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressuremay be less than free stream static pressure
It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
53
The The PitotPitot--static Tube static Tube 5555
Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder fitted with small tubes and connected to three cylinder fitted with small tubes and connected to three pressure transducers pressure transducers The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal thus indicating thus indicating that the center hole points directly upstreamthat the center hole points directly upstream
21
12
31
PP2V
PP
Directional-finding Pitot-static tube
If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33
可以自動對位的可以自動對位的PitotPitot--static tubestatic tube
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
54
Example 36 Example 36 PitotPitot--Static TubeStatic Tube
An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 Determine the pressure at point (1) far ahead of the airplane point (2) the pressure at point (1) far ahead of the airplane point (2) and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage probe attached to the fuselage 求第一點第二點的壓力求第一點第二點的壓力
機身下貼附機身下貼附PitotPitot--static tubestatic tube
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
55
Example 36 Example 36 Solution Solution 1212
2Vpp
21
12
psia1110)abs(ftlb1456p 21
The static pressure and density at the altitude
If the flow is steady inviscid and incompressible and elevation changes are neglected The Bernoulli equation
3ftslug0017560
With V1=100mihr=1466fts and V2=0
)abs(ftlb)9181456(
2)sft7146)(ftslugs0017560(ftlb1456p2
222322
高度高度10000ft10000ft處的壓力與密度處的壓力與密度
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
56
psi13130ftlb918p 22
In terms of gage pressure
psi131302Vpp
21
12
The pressure difference indicated by the Pitot-static tube
Example 36 Example 36 Solution Solution 2222
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
57
Application of Bernoulli Equation Application of Bernoulli Equation 1212
The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied The result isare satisfied The result is
2
22
21
21
1 z2Vpz
2Vp
Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline
Bernoulli equationBernoulli equation的應用的應用
沿著沿著streamlinestreamline任意兩點任意兩點
要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
58
Application of Bernoulli Equation Application of Bernoulli Equation 2222
Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement
Bernoulli equationBernoulli equation的應用的應用
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
59
Free Jets Free Jets 1313
Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline
gh2h2V
2Vh
2
At point (5)
)Hh(g2V
2
22
21
21
1 z2Vpz
2Vp
pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0
pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
60
Free JetsFree Jets
Flow from a tankFlow from a tank
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
61
Free Jets Free Jets 2323
For the For the horizontal nozzlehorizontal nozzle the the velocity at the centerline Vvelocity at the centerline V22 will be greater than that at the will be greater than that at the top Vtop V11
In general dltlth and use the VIn general dltlth and use the V22
as average velocityas average velocity
For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccurs
The effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner
2
22
21
21
1 z2Vpz
2Vp
VV22gtVgtV11若若
dltlthdltlth則以則以VV22代表平均速度代表平均速度
出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
62
Free Jets Free Jets 3333
Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfiguration
The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flowshole from which it flows
Define Cc = contraction coefficient Define Cc = contraction coefficient
h
jc A
AC
AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole
2
22
21
21
1 z2Vpz
2Vp
不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數
定義縮流係數定義縮流係數
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
63
Example 37 Flow From a TankExample 37 Flow From a Tank-- GravityGravity
A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a Determine the Determine the flowrateflowrate Q Q needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant h = 20m h = 20m
求求flowrateflowrate QQ
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
64
Example 37 Example 37 SolutionSolution1212
22
221
2
11 zV21pzV
21p
22
21 V
21ghV
21
The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis
(1)(1)
With With pp11 = p= p22 = 0 z= 0 z11 = h and z= h and z22 = 0= 0
(2)(2)
For steady and incompressible flow For steady and incompressible flow conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22 where Q = AV Thus A where Q = AV Thus A11VV11 =A=A22VV2 2 or or
22
12 Vd
4VD
4
2
22
21
21
1 z2Vpz
2Vp
22
1 V)Dd(V (3)
Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間
Point(1)Point(1)與與(2)(2)的已知條件的已知條件
加上質量守恆條件加上質量守恆條件
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
65
Example 37 Example 37 SolutionSolution2222
sm266)m1m10(1
)m02)(sm819(2)Dd(1
gh2V4
2
42
Combining Equation 2 and 3
Thus
sm04920)sm266()m10(4
VAVAQ 322211
VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))
4
4
2
2
0 )(11
2])(1[2
DdghDdgh
VV
D
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
66
Example 38 Flow from a TankExample 38 Flow from a Tank--PressurePressure
Air flows steadily from a tank through a hose of diameter Air flows steadily from a tank through a hose of diameter D=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure pressure Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hosethe hose
求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
67
Example 38 Example 38 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
2212
13 V
21ppandp2V
For steady For steady inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamlinealong the streamline
With zWith z11 =z=z22 = z= z33 V V11 = 0 and p= 0 and p33=0=0
(1)(1)
The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law
33
2
1
mkg261K)27315)(KkgmN9286(
kNN10]mkN)10103[(RTp
Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間
Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件
理想氣體的密度理想氣體的密度
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
68
Example 38 Example 38 SolutionSolution2222
sm005420Vd4
VAQ 33
233
sm069
mkg261)mN1003(2p2V 3
231
3
Thus Thus
oror
The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq 1 1and the continuity equationand the continuity equation
sm677AVAVHenceVAVA 23323322
22
23232212
mN2963mN)1373000(
)sm677)(mkg261(21mN1003V
21pp
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
69
Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe
Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity SG less than onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h
reading hreading h
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
70
Example 39 Example 39 SolutionSolution1212
2211 VAVAQ
For steady For steady inviscidinviscid incompressible flow the Bernoulli equation along incompressible flow the Bernoulli equation along the streamlinethe streamline
The continuity equationThe continuity equation
Combining these two equations Combining these two equations
(1)(1)
22221
211 zpV
21pzpV
21p
])AA(1[pV21)zz(pp 2
122
21221
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
71
Example 39 Example 39 SolutionSolution2222
h)SG1()zz(pp 1221
2121 phSGh)zz(p
2
1
222 A
A1pV21h)SG1(
This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2
oror(2)(2)
SG1g2)AA(1AQh
2122
2
Since VSince V22=QA=QA22
uarr- darr+
be independent of be independent of θθ
Point(1)(2)Point(1)(2)的壓力差的壓力差
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
72
Confined Flows Confined Flows 1414
When the fluid is physically constrained within a device When the fluid is physically constrained within a device its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jetthe free jet
Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to anotheris different from one section to another
For such situations it is necessary to use the concept of For such situations it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equationthe Bernoulli equation
Tools Bernoulli equation + Continuity equation
受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe
解解「「受限流受限流」」的工具的工具
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
73
Confined Flows Confined Flows 2424
Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outlet
Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is
222111 VAVA
212211 QQVAVA 加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
74
Confined Flows Confined Flows 3434
If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decrease
This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressurevapor pressure
Pressure variation and cavitationin a variable area pipe
受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低
當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現
VenturiVenturi channelchannel
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
75
Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation
A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will result
The water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively large
With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe produced
The sound is a result of The sound is a result of cavitationcavitation
水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加
出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
76
Damage from Damage from CavitationCavitation
Cavitation from propeller
前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
77
Example 310 Siphon and Example 310 Siphon and CavitationCavitation
Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E310 Determine the maximum diameter hose as shown in Figure E310 Determine the maximum height of the hill H over which the water can be siphoned withheight of the hill H over which the water can be siphoned without out cavitationcavitation occurring The end of the siphon is 5 ft below the bottom occurring The end of the siphon is 5 ft below the bottom of the tank Atmospheric pressure is 147 of the tank Atmospheric pressure is 147 psiapsia
The value of H is a function of both the The value of H is a function of both the specific weight of the fluid specific weight of the fluid γγ and its and its vapor pressure vapor pressure ppvv
Max HMax H才能避免才能避免cavitationcavitation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
78
Example 310 Example 310 SolutionSolution1212
32332
2221
211 zV
21pzV
21pzV
21p
For ready For ready inviscidinviscid and incompressible flow the Bernoulli equation and incompressible flow the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)
With With zz1 1 = 15 ft z= 15 ft z22 = H and z= H and z33 = = --5 ft Also V5 ft Also V11 = 0 (large tank) p= 0 (large tank) p11 = 0 = 0 (open tank) p(open tank) p33 = 0 (free jet) and from the continuity equation A= 0 (free jet) and from the continuity equation A22VV22 = = AA33VV33 or because the hose is constant diameter V or because the hose is constant diameter V22 = V= V33 The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq 1 to be 1 to be
(1)(1)
22
313 Vsft935ft)]5(15)[sft232(2)zz(g2V (1)(3)(1)(3)
(1)(2)(1)(2)
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
79
Example 310 Example 310 SolutionSolution2222
22212
221
2112 V
21)zz(zV
21zV
21pp
233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414(
Use of Use of EqEq 1 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as
The The vapor pressure of water at 60vapor pressure of water at 60 is 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi
(2)(2)
ftH 228
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
80
FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15
Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations
2211
222
211
VAVAQ
V21pV
21p
])AA(1[)pp(2AQ 212
212
The theoretical The theoretical flowrateflowrate
Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes
Flow metersFlow meters的理論基礎的理論基礎
管流中量測流率的裝置管流中量測流率的裝置
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
81
Example 311 Example 311 VenturiVenturi MeterMeter
Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates
Known Q Determine pKnown Q Determine p11--pp22
求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
82
Example 311 Example 311 SolutionSolution1212
33O2H kgm850)kgm1000(850SG
22
2
A2])AA(1[Q
pp2
12
21
For steady For steady inviscidinviscid and incompressible flow the relationship between and incompressible flow the relationship between flowrateflowrate and pressure and pressure
The density of the flowing fluidThe density of the flowing fluid
The area ratioThe area ratio
360)m100m0060()DD(AA 221212
])AA(1[)pp(2AQ 212
212
EqEq 320 320
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
83
Example 311 Example 311 SolutionSolution2222
The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis
The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis
22
22
21 ])m060)(4[(2)3601()850)(050(pp
kPa116Nm10161 25
kPa161Nm1160])m060)(4[(2
)3601()kgm850()sm0050(pp2
22
2323
21
kPa116-ppkPa161 21
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
84
FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25
The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channelan open channel
The The flowrateflowrate Q Q is function of the water depth upstream zis function of the water depth upstream z11 the the width of the gate b and the gate opening awidth of the gate b and the gate opening a
212
212 )zz(1
)zz(g2bzQ
With pWith p11=p=p22=0 the =0 the flowrateflowrate
22221111
22221
211
zbVVAzbVVAQ
zV21pzV
21p
水閘門調節與量測流率水閘門調節與量測流率
理論值理論值
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
85
FlowrateFlowrate Measurement Measurement sluice gate 35sluice gate 35
In the limit of In the limit of zz11gtgtzgtgtz22 this result simply becomes this result simply becomes
This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio zratio z11zz22 is large is large the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately
ZZ2 2 是求是求QQ的關鍵的關鍵
12 gz2bzQ
12 gz2V
特殊情況特殊情況zz11遠大於遠大於zz22
z2lta(束縮效應) Z2 = Cc a縮流係數 Cc
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
86
FlowrateFlowrate Measurement Measurement sluice gate 45sluice gate 45
As we discussed relative to flow from an orifice As we discussed relative to flow from an orifice the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1
Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly
縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
87
Example 312 Sluice Gate
Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
88
Example 312 Example 312 SolutionSolution1212
sm614)m05m4880(1
)m4880m05)(sm819(2)m4880(bQ 2
2
2
212
212 )zz(1
)zz(g2zbQ
For steady inviscid incompreesible flow the flowerate per uniFor steady inviscid incompreesible flow the flowerate per unit widtht width
With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate
212
212 )zz(1
)zz(g2bzQ
Eq321Eq321
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
89
Example 312 Example 312 SolutionSolution1212
If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have
sm834m05sm8192m4880gz2zbQ 22
12
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
90
FlowrateFlowrate Measurement Measurement weirweir 5555
For a typical rectangular sharpFor a typical rectangular sharp--crested the crested the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height Pthe weir height Pww the width of the the width of the channel b and the head Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir
23111 Hg2bCgH2HbCAVCQ The The flowrateflowrate
Where CWhere C11 is a constant to be determinedis a constant to be determined
堰 待由實驗來決定待由實驗來決定
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
91
Example 313 WeirExample 313 Weir
Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based on a simple analysis using the Bernoulli equation determiBased on a simple analysis using the Bernoulli equation determine ne the dependence of the dependence of flowrateflowrate on the depth H on the depth H If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00 estimate the estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00 HH提高提高33倍流率增加倍流率增加
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
92
Example 313 Example 313 SolutionSolution
gH2
For steady For steady inviscidinviscid and incompressible flow the average speed and incompressible flow the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( 2)]2)]The The flowrateflowrate
where Cwhere C11 is an unknown constant to be determined experimentallyis an unknown constant to be determined experimentally
251
211 Hg2
2tanC)gH2(
2tanHCVACQ
An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of
615Hg22tanCH3g22tanC
2501
2501
H
H3
0
0
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
93
EL amp HGL EL amp HGL 1414
For For steady steady inviscidinviscid incompressible flow incompressible flow the total energy the total energy remains constant along a streamlineremains constant along a streamline
gp
g2V 2
zH
The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)
The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)
The elevation head ( potential energy )The elevation head ( potential energy )
The total head for the flowThe total head for the flow
Httanconszg2
VP 2
每一項化成長度單位-每一項化成長度單位-HEADHEAD
沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
94
EL amp HGL EL amp HGL 2424
Energy Line (EL) represents the total head heightEnergy Line (EL) represents the total head height
Hydraulic Grade Line (HGL) height Hydraulic Grade Line (HGL) height represents the sum of the represents the sum of the elevation and static pressure headselevation and static pressure heads
The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head
zg2
VP 2
zP
g2V2
沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線
沿著沿著streamlinestreamline的的「「壓力頭壓力頭」+「」+「高度頭高度頭」」連結成一條線連結成一條線
ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
95
EL amp HGL EL amp HGL 3434
Httanconszg2
VP 2
「「壓力頭壓力頭」+「」+「高度頭高度頭」」
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
96
EL amp HGL EL amp HGL 4444
Httanconszg2
VP 2
Httanconszg2
VP 2
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
97
Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line
Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 through a hose of constant diameter A small hole is found in the hose at hose of constant diameter A small hole is found in the hose at location (1) as indicate When the siphon is used will water lelocation (1) as indicate When the siphon is used will water leak out ak out of the hose or will air leak into the hose thereby possibly caof the hose or will air leak into the hose thereby possibly causing using the siphon to malfunctionthe siphon to malfunction
空氣跑進去或水溢出來空氣跑進去或水溢出來
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
98
Example 314Example 314 SolutionSolution1212
Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric Which happens can be easily determined Which happens can be easily determined by using the energy line and hydraulic grade line concepts Withby using the energy line and hydraulic grade line concepts With the the assumption of steady incompressible assumption of steady incompressible inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus the energy line is horizontalthus the energy line is horizontal
Since the hose diameter is constant it follows from the continuSince the hose diameter is constant it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout Thus throughout Thus the hydraulic grade line is constant distance Vthe hydraulic grade line is constant distance V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
99
Example 314Example 314 SolutionSolution2222
Since the pressure at the end of the hose is atmospheric it folSince the pressure at the end of the hose is atmospheric it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet outlet The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressureline will be at less than atmospheric pressure
Thus Thus air will leak into the hose through the hole at point (1) air will leak into the hose through the hole at point (1)
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
100
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14
The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flows
In certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gases
To account for compressibility effectsTo account for compressibility effects
CgzV21dp 2
考慮壓縮效應考慮壓縮效應
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
101
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 24compressibility effects 24
For isothermal flow of perfect gas
For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P P ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio
22
2
11
21 z
g2V
PPln
gRTz
g2V
ttanconsgzV21dPPC 2k
1k1
ttanconsgzV21dpRT 2
2
22
2
21
21
1
1 gz2
VP1k
kgz2
VP1k
k
理想氣體等溫流理想氣體等溫流
理想氣體等熵流理想氣體等熵流
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
102
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 34compressibility effects 34
To find the pressure ratio as a function of Mach number
1111a1 kRTVcVM The upstream Mach number
Speed of sound
1M
21k1
ppp 1k
k
21a
1
12Compressible flow
Incompressible flow21a
1
12 M2k
ppp
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
103
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 44compressibility effects 44
21a
1
12 M2k
ppp
1M
21k1
ppp 1k
k
21a
1
12
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
104
Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 at an altitude of 10 km in a
standard atmosphere Determine the stagnation pressure on the standard atmosphere Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible and if tleading edge of its wing if the flow is incompressible and if the he flow is incompressible isentropicflow is incompressible isentropic
kPa512pp
4710M2k
ppp
12
21a
1
12
For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow
kPa714pp
5501M2
1k1p
pp
12
1kk
21a
1
12
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
105
Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects
For unsteady flow V = V ( s t )For unsteady flow V = V ( s t )
To account for unsteady effects To account for unsteady effects
sVV
tVaS
0dzVd21dpds
tV 2
Along a streamline
+ Incompressible condition+ Incompressible condition
2222
S
S12
11 zV21pds
tVzV
21p 2
1
考慮考慮非穩定效應非穩定效應
Oscillations Oscillations in a Uin a U--tubetube
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
106
Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash UU--TubeTube
An incompressible An incompressible inviscidinviscid liquid liquid is placed in a vertical constant is placed in a vertical constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E316 When released from Figure E316 When released from the the nonequilibriumnonequilibrium position shown position shown the liquid column will oscillate at a the liquid column will oscillate at a specific frequency Determine this specific frequency Determine this frequencyfrequency
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation
107
Example 316Example 316 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )
dtdVds
dtdVds
tV 2
1
2
1
S
S
S
S
The total length of the liquid colum
g20zg2dt
zd
gdtdzV
zdtdVz
2
2
Liquid oscillationLiquid oscillation