kuliah viii - ix
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KULIAH VIII - IX. MEKANIKA FLUIDA II Nazaruddin Sinaga. Entrance Length. Shear stress and velocity distribution in pipe for laminar flow. Typical velocity and shear distributions in turbulent flow near a wall: (a) shear; (b) velocity. Solution of Pipe Flow Problems. Single Path - PowerPoint PPT PresentationTRANSCRIPT
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KULIAH VIII - IXKULIAH VIII - IX
MEKANIKA FLUIDA IIMEKANIKA FLUIDA II
Nazaruddin SinagaNazaruddin Sinaga
Entrance LengthEntrance Length
2
Shear stress and velocity distribution in pipe for laminar flow
Typical velocity and shear distributions in turbulent flow near a wall: (a) shear; (b) velocity.
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Solution of Pipe Flow ProblemsSolution of Pipe Flow Problems
• Single Path– Find p for a given L, D, and Q
Use energy equation directly
– Find L for a given p, D, and Q Use energy equation directly
Solution of Pipe Flow ProblemsSolution of Pipe Flow Problems
• Single Path (Continued)– Find Q for a given p, L, and D
1. Manually iterate energy equation and friction factor formula to find V (or Q), or
2. Directly solve, simultaneously, energy equation and friction factor formula using (for example) Excel
– Find D for a given p, L, and Q1. Manually iterate energy equation and friction factor
formula to find D, or2. Directly solve, simultaneously, energy equation and
friction factor formula using (for example) Excel
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Example 1Example 1 Water at 10C is flowing at a rate of 0.03 m3/s through a pipe. The
pipe has 150-mm diameter, 500 m long, and the surface roughness is estimated at 0.06 mm. Find the head loss and the pressure drop throughout the length of the pipe.
Solution: From Table 1.3 (for water): = 1000 kg/m3 and =1.30x10-3 N.s/m2
V = Q/A and A=R2
A = (0.15/2)2 = 0.01767 m2
V = Q/A =0.03/.0.01767 =1.7 m/sRe = (1000x1.7x0.15)/(1.30x10-3) = 1.96x105 > 2000 turbulent
flowTo find , use Moody Diagram with Re and relative roughness (k/D).
k/D = 0.06x10-3/0.15 = 4x10-4
From Moody diagram, 0.018The head loss may be computed using the Darcy-Weisbach equation.
The pressure drop along the pipe can be calculated using the relationship: ΔP=ghf = 1000 x 9.81 x 8.84ΔP = 8.67 x 104 Pa
.m84.881.9x2x15.0
7.1x500x018.0
g2
V
D
Lh
22
f
8
Example 2Example 2 Determine the energy loss that will occur as 0.06 m3/s water
flows from a 40-mm pipe diameter into a 100-mm pipe diameter through a sudden expansion.
Solution: The head loss through a sudden enlargement is given by;
Da/Db = 40/100 = 0.4From Table 6.3: K = 0.70Thus, the head loss is
g2
VKh
2a
m
smA
QV
aa /58.3
)2/04.0(
06.02
m47.081.9x2
58.3x70.0h
2
Lm
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ExExample 3ample 3
Calculate the head added by the pump when the water system shown below carries a discharge of 0.27 m3/s. If the efficiency of the pump is 80%, calculate the power input required by the pump to maintain the flow.
Solution:Applying Bernoulli equation between section 1 and 2
(1)
P1 = P2 = Patm = 0 (atm) and V1=V2 0 Thus equation (1) reduces to:
(2)
HL1-2 = hf + hentrance + hbend + hexit
From (2):
21L
22
22
p
21
11 H
g2
Vz
g
PH
g2
Vz
g
P
21L12p HzzH
g2
V4.39
14.05.04.0
1000x015.0
g2
VH
2
2
21L
81.9x2
V4.39200230H
2
p
The velocity can be calculated using the continuity equation:
Thus, the head added by the pump: Hp = 39.3 m
Pin = 130.117 Watt ≈ 130 kW.
s/m15.2
2/4.0
27.0
A
QV
2
in
pp P
gQH
8.0
3.39x27.0x81.9x1000gQHP
p
pin
EGL & HGL for a Pipe System
• Energy equation
• All terms are in dimension of length (head, or energy per unit weight)
• HGL – Hydraulic Grade Line
• EGL – Energy Grade Line
• EGL=HGL when V=0 (reservoir surface, etc.)
• EGL slopes in the direction of flow
22
22
211
21
1 22z
p
g
Vhz
p
g
VL
zp
HGL
g
VHGLz
p
g
VEGL
22
22
EGL & HGL for a Pipe System
• A pump causes an abrupt rise in EGL (and HGL) since energy is introduced here
EGL & HGL for a Pipe System
• A turbine causes an abrupt drop in EGL (and HGL) as energy is taken out
• Gradual expansion increases turbine efficiency
EGL & HGL for a Pipe System
• When the flow passage changes diameter, the velocity changes so that the distance between the EGL and HGL changes
• When the pressure becomes 0, the HGL coincides with the system
EGL & HGL for a Pipe System
• Abrupt expansion into reservoir causes a complete loss of kinetic energy there
EGL & HGL for a Pipe System
• When HGL falls below the pipe the pressure is below atmospheric pressure
FLOW MEASUREMENTFLOW MEASUREMENT• Direct Methods
– Examples: Accumulation in a Container; Positive Displacement Flowmeter
• Restriction Flow Meters for Internal Flows– Examples: Orifice Plate; Flow Nozzle; Venturi; Laminar
Flow Element
Definisi tekanan pada aliran di sekitar sayap
Flow Measurement
• Linear Flow Meters– Examples: Float Meter
(Rotameter); Turbine; Vortex; Electromagnetic; Magnetic; Ultrasonic
Float-type variable-area flow meter
Flow Measurement• Linear Flow Meters
– Examples: Float Meter (Rotameter); Turbine; Vortex; Electromagnetic; Magnetic; Ultrasonic
Turbine flow meter
Flow Measurement
• Traversing Methods– Examples: Pitot (or Pitot Static) Tube; Laser Doppler
Anemometer
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The measured stagnation pressure cannot of itself be used to determine the fluid velocity (airspeed in aviation).
However, Bernoulli's equation states:
Stagnation pressure = static pressure + dynamic pressure
Which can also be written
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Solving that for velocity we get:
Note: The above equation applies only to incompressible fluid.where:
V is fluid velocity;pt is stagnation or total pressure;ps is static pressure;and ρ is fluid density.
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The value for the pressure drop p2 – p1 or Δp to Δh, the reading on the manometer:
Δp = Δh(ρA-ρ)g
Where:ρA is the density of the fluid in the manometerΔh is the manometer reading
EXTERNAL EXTERNAL INCOMPRESSIBLE INCOMPRESSIBLE VISCOUS FLOWVISCOUS FLOW
Main TopicsMain Topics• The Boundary-Layer Concept• Boundary-Layer Thickness• Laminar Flat-Plate Boundary Layer: Exact Solution• Momentum Integral Equation• Use of the Momentum Equation for Flow with Zero
Pressure Gradient• Pressure Gradients in Boundary-Layer Flow• Drag• Lift
The Boundary-Layer Concept
The Boundary-Layer Concept
Boundary Layer Thickness
Boundary Layer Thickness
• Disturbance Thickness, where
Displacement Thickness, *
Momentum Thickness,
Boundary Layer LawsBoundary Layer Laws
Laminar Flat-PlateBoundary Layer: Exact Solution
• Governing Equations
Laminar Flat-PlateBoundary Layer: Exact Solution
• Boundary Conditions
Laminar Flat-PlateBoundary Layer: Exact Solution
• Equations are Coupled, Nonlinear, Partial Differential Equations
• Blassius Solution:– Transform to single, higher-order, nonlinear, ordinary
differential equation
Laminar Flat-PlateBoundary Layer: Exact Solution
• Results of Numerical Analysis
Momentum Integral Equation
• Provides Approximate Alternative to Exact (Blassius) Solution
Momentum Integral Equation
Equation is used to estimate the boundary-layer thickness as a function of x:
1. Obtain a first approximation to the freestream velocity distribution, U(x). The pressure in the boundary layer is related to the freestream velocity, U(x), using the Bernoulli equation
2. Assume a reasonable velocity-profile shape inside the boundary layer
3. Derive an expression for w using the results obtained from item 2
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Simplify Momentum Integral Equation(Item 1)
The Momentum Integral Equation becomes
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Laminar Flow– Example: Assume a Polynomial Velocity Profile (Item 2)
• The wall shear stress w is then (Item 3)
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Laminar Flow Results(Polynomial Velocity Profile)
Compare to Exact (Blassius) results!
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Turbulent Flow– Example: 1/7-Power Law Profile (Item 2)
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Turbulent Flow Results(1/7-Power Law Profile)
Pressure Gradients in Boundary-Layer Flow
Drag
• Drag Coefficient
with
or
Drag
• Pure Friction Drag: Flat Plate Parallel to the Flow
• Pure Pressure Drag: Flat Plate Perpendicular to the Flow
• Friction and Pressure Drag: Flow over a Sphere and Cylinder
• Streamlining
Drag• Flow over a Flat Plate Parallel to the Flow: Friction
Drag
Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available
Drag• Flow over a Flat Plate Parallel to the Flow: Friction
Drag (Continued)
Laminar BL:
Turbulent BL:
… plus others for transitional flow
Drag Coefficient
Drag• Flow over a Flat Plate Perpendicular to the
Flow: Pressure Drag
Drag coefficients are usually obtained empirically
Drag• Flow over a Flat Plate Perpendicular to the
Flow: Pressure Drag (Continued)
Drag• Flow over a Sphere and Cylinder: Friction and
Pressure Drag
Drag• Flow over a Sphere and Cylinder: Friction and
Pressure Drag (Continued)
Streamlining• Used to Reduce Wake and Pressure Drag
Lift• Mostly applies to Airfoils
Note: Based on planform area Ap
Lift
• Examples: NACA 23015; NACA 662-215
Lift• Induced Drag
Lift• Induced Drag (Continued)
Reduction in Effective Angle of Attack:
Finite Wing Drag Coefficient:
Lift
• Induced Drag (Continued)
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