fluid properties and definitions

8/6/2019 Fluid Properties and Definitions

1/31

1.1Fluid properties and definitionsBy: Dr. Rajendra Shrestha

2011-01-05

Definition of fluid and fluid mechanics:

y The matters in the world are classified in three states that are solid liquid and gas.y The gas and the liquid are called fluid.y Most of the matters we come in our daily life are the various forms of the fluid.y Fluid is not only important for our daily life, it also use full for the generation of electricity,

different phenomena of surface and ground water.

Definition of fluid and fluid mechanics:

y we are only specific on the normal conditions of atmosphere.y We will consider the effects of temperature and pressure in compressible flow chapter.Fluid mechanics:

y The study of fluid is one of the important subjects in the study of science and technology.y The study of fluid and different phenomena of it is called Fluid mechanics.y We study theoretical and experimental characterises of fluid.Fluid mechanics:

y In context of Nepal fluid is not only important for the study of science, it equally important ofthe generation of electricity from its rich resource of water.

Idealfluid: Realfluid:

Viscosity no yesSurface

Tension yes yes


2/31

y The fluid of liquid is also classified as an ideal fluid and real fluid.Ideal fluid and Real fluid:

Properties of fluid:

y The study of fluid mechanics urges to identify the liquid in terms of different properties.y We could get the benefits of fluid phenomena until we use the correct types of fluid. For

example, we use different types of liquids as a manometers fluid according to its properties.

Other properties of fluid:2

Density Specific Volume SpecificWeight Specific Gravity Viscosity Newtonian and non-Newtonian fluid

Other properties of fluid:2

Cohesion and adhesion Surface tension Compressibility Vapor pressure Capillarity Thermodynamic Properties (Equation of state of perfect gas)

Fluid statics-1

Fluid pressure

Pressure variation in a fluid ( fundamental equation of fluid static) Pressure head Atmospheric pressure Gauge and absolute pressure

Fluid statics-2

Compressibility no yesDensity yes yes


3/31

Measurement of pressure Barometer Manometer Total force on plane and curved surface area Center of pressure and gravity Centroid (C.G.) and measurement Bouyancy and flotation

Kinematics of fluid flow

Description of fluid motion Line of flow: Path line Stream line Stream tube Different types of displacement of fluid particles Translation Rotation Distortion and deformation

Kinematics of fluid flow

General types of fluid flow Steady and unsteady flow Uniform and non-uniform Laminar flow and Turbulent flow Compressible and incompressible Flow

1.2Continuity of FlowContinuity of Flow

Matter is neither created nor destroyed. This principle conservation of mass can be applied toa flowing fluid.

Considering any fixed region the flow constituting a control volume. Mass of fluid entering per unit time=Mass of fluid leaving per unit time+

Increasing/decreasing of mass of fluid in the control volume per unit time

Continuity of Flow contd.


4/31


For steady flow, the mass of fluid in the control volume remains constant and relation reducesto

Mass of fluid entering per unit time = Mass of fluid leaving per unit time Applying this principle to steady flow in a stream tube having a cross sectional area small

enough for the velocity to be considered as constant over any given cross-section, for the

region between sections I and 2, since there can be no flow through the walls of a stream

tube;


Mass entering per unit time at section 1 = Mass leaving per unit time-at section 2 Suppose that at section 1 the area of the stream tube isHA1, the velocity of the fluid u1 and its

densityV1while at section 2 the corresponding values are HA2, u2 andV2while

Mass entering per unit time at 1 = HA1 u1V1


5/31

Mass leaving per unit time at 2 = HA2 u2V2Continuity of Flow contd.


HA1 u1V1= HA2 u2V2 =Constant This is the equation of continuity for the flow of a compressible fluid through a any fixed

region in stream tube, u1 and u2 being the velocities measured at right angles to the cross-

sectional areas HA1 andHA2.


For the flow of a real fluid through a pipe or other conduit, the velocity will vary from wall towall. However, using the mean velocity, the equation of continuity for steady flow can be

written as

HA1 u1V1= HA2 u2V2 = m dot where HA1 and HA2 are the cross-sectional areas and m dot is the mass rate of flow.


6/31


If the fluid can be considered as incompressible, so that V1 =V2 equation reduce to, HA1 u1 = HA2 u2 = Q dotContinuity of Flow contd (application):

The continuity equation can also be applied to determine the relation between the flows intoand out of a junction. In Figure, for steady conditions,

Total inflow to junction = Total outflow from junction, V1Q1 =V2Q2 +V3Q3Continuity of Flow contd.


For an incompressible fluid, V1 =V2 =V3 so that Q1 =Q2 +Q3 A1Dv1= A2Dv2+ A3v3


7/31

In general, if we consider flow towards the junction as positive and flow away from thejunction as negative, then for steady flow at any junction the algebraic sum of all the mass

flows must be zero:

7VQ=0


8/31


9/31


10/31

1.2b Continuity Equation for 3D flow using cartesian coordinate

The control volume ABCDEFGH in figure below is taken in the form of a small rectangularprism with sides Hx, Hy and Hz in the x, y and z directions, respectively. The mean values of thecomponent velocities in these directions are vx, vy and vz.


11/31

Considering flow in the x direction, Mass inflow through ABCD in unit time =VvxHyHz In the general case, both mass densityV and velocity vx will change in the x direction and so, Mass out flow through EFGH in unit time =

Net rate of mass flow into the box in the X-direction = -


12/31

Net rate of mass flow into the box in the Y-direction = -

Net rate of mass flow into the box in the Z-direction = -

Total net rate of mass flow into the X,Y and Z direction = -

The sum of the rate of inflow in the 3Ds must equal to the time rate of change of mass in thebox.


13/31

Equation holds for every point in a fluid flow weather steady or U/S, compressible or U/C.However for incompressible flow, the density is constant,

For 2D incompressible flow

1.3Bernoullis Equation of motion along a streamline:2 A total energy in the fluid (hydro) is constant unless external energy is added or extracted from

the system.

3 The energy in the system is expressed in terms of pressure energy, velocity energy and elevationenergy Bernoullis theorem

4 p/Vg + v2/2g + z = constant


14/31

The fluid forces are:

1. Pressure of the fluid all around the body.2. Gravity force is considered.3. Viscosity, surface tension, electricity, magnetism, chemical or nuclear reactions are assumed

negligibly small.

Consider:

1. A stream tube surrounding the streamline and having a cross-sectional area small enough forthe velocity to be considered constant over the cross-section.

2. The force due to pressure at the side of the element have component in the flow direction andperpendicular to it.

3. The fluid is assumed to be inviscid, there will be no shear stresses on the sides of the streamtube.

Conditions:

It is also assumed that the flow is steady.


15/31

Curvature of the streamlines over this distance may be neglected. The pressure, velocity and so on will (in general) vary with Hs. Each variable may be regarded as a function of s only. Hp is negative. At the sides of the element the pressure varies along the length, but a mean value of p + kHp

may be assumed and k is a fraction less than unity.

Total force:

The total force acting on the element in the direction of flow is: = pA -(p + Hp)(A + HA) + (p + kHp)HA - VgAHs cosU = -AHp -VgAHs cosU Newtons IInd Law force = VAHs du/dt and Hs cosU = Hz -AHp -VgAHz = VAHs du/dt D

ividing byVAHs and taking limit Hsp0

1/V dp/ds + du/dt + g dz/ds = 0Total force:

du/dt = u xu/xs +xu/xt But for steady flow the local acceleration xu/xt = 0 and so du/dt = u(du/ds) 1/V dp/ds + u du/ds + g dz/ds = 0 It is called Eulers Equation. For fluid of constant density, and integrating, p/V + u2/2 +gz = constant It is called Bernoullis Equation.Limitation:

Bernoulli's equation applies. The fluid must be frictionless (inviscid) and of constant density; the flow must be Steady. The relation holds in general only for a single streamline and stream tube.Problems:

A pipe line carrying oil (S= 0.8) change in diameter from 300 mm at position 1 to 600 mmdiameter at position 2 which is 5 m at a higher level. If the pressures at position 1 and 2 are100kN/m2and 60kN/m2 respectively and the discharge is 300 L/s. Determine the loss of head

and the direction of the flow.

1.4 ImpulseMomentum

Development. of impulse momentum principle

It is derived from 'Newton's Second Law ofMotion' and which states:


16/31

"The resultant external force acting on any body in any direction is equal to the rate of changeof momentum

What may be the fluid or flow type, i.e. compressible or incompressible, real or ideal, steadyor unsteady the impulse equation is applied to all of them.

Impulse momentum principle:

Also it can be stated as'' the impulse of a force F acting on a fluid mass 'm' in a short intervalof time dt is equal to the change of momentum d(mv) in the direction of force".

Fdt = d(mv)Application ofImpulse momentum principle:

To determine the resultant force acting on the boundary of flow passage. (I) Pipe bends (II) Reducers (III) Moving and stationary vanes (IV) Jet Propulsion To determine the characteristic of flow when there is abrupt change of flow section. (I) Sudden Enlargement in pipe (II) Hydraulic Jump in Channel etc.


17/31

Force on a pipe bend:

Let us consider a flow in a pipe bend as shown below. Flow can be assumed to normal to theinlet and outlet areas.

Let V1,V1= average velocity and density at inlet V 2,V2 = average velocity an density at outlet Again suppose that the mass of fluid in the region 1234 shifts to new position 1'2'3'4' due to

the effect of external forces on the stream after a short interval.

Due to gradual increase in the flow area in the direction of flow, velocity of fluid mass andhence the momentum is gradually reduced.

Since the area 1'2'34 is common to both the regions 1234 and 1'2'3'4' therefore it will notexperience any change in momentum.

Obviously then the changes in momentum of the fluid masses in the sections 122'1' and 433'4'will have to be considered.

According to the principle of mass conservation Fluid mass within the region 122'1' = Fluid mass within the region 433'4 V1A1ds1 =V2A2ds2 Momentum of fluid contained in the region 122'1' = (V1A1ds1)V1 = (V1A1V1dt)V 1 Momentum of fluid contained in the region 433'4' = (V2A2ds2)V2 = (V2A2V2dt)V 2 Change in momentum = (V2A2V2dt)V2 - (V1A1V1dt)V 1 For incompressible flowV1 =V1=V And from continuity equation A1 V 1 = A2 V 2 = Q Change in momentum =VQ (V 2 V1) dt . Using impulse momentum principle, Fdt=VQ (V2 V1) dt F=VQ (V2 V1) Resolving (V 2 and V 1) along x-axis and y -axis, we get Components along X-axis: V1 cosU1 and V2 cosU2 Components along y -axis: V1 sinU1 and V2 sinU2 Where,U1 andU2 are the inclinations with the horizontal of the centerline of the pipe at 1-2

and 3-4

Now, Components of force along X-axis and Y-axis are: Fx =(VQ )(V2 cosU2 - V1 cosU1)


18/31

Fy =(VQ )(V2 sinU2 - V1 sinU1) These are the forces on fluid by the bend, To determine the force on bend by fluid simply change the direction. Fx =(VQ )(V1 cosU1 V2 cosU2) Fy =(VQ )(V1 sinU1 V2 sinU2) The magnitude of the resultant force acting on the pipe bend is given by, F = (Fx2 +Fy2) And the direction of resultant force with x-axis is given by, U = tan -1 Fy / FxDynamic forces and static pressure forces:

Since the dynamic forces must be supplemented by the static pressure forces acting over theinlet and outlet sections, therefore, we have:

Fx =(VQ )(V1 cosU1 V2 cosU2) + p1A1cosU1 p2A2cosU2 Fy =(VQ )(V1 sinU1 V2 sinU2) +p 1A1sinU1 p2A2sinU2

Problems: St pp 120

Find the force exerted by the nozzle on the pipe. The fluid is oil S=0.85 and p1= 0.7 MPa Fig. Book

1.5 STREAMLINESANDTHE STREAM FUNCTION

STREAMLINES :

y The streamline is the one which is a purely theoretical line in space defined as being tangential toinstantaneous velocity vectors.

y From this definition of a streamline, it follows that there can be no flow across it.y The concept of the streamline is very useful, especially in ideal flow, because it enables the fluid

flow to be conceived as occurringin patterns of streamlines.

y These patterns may be described mathematically so that the whole system of analysis may bebased on it.

y The stream function =, based on the continuity principle, is a mathematical expression thatdescribes a flow field (St. Lines).

STREAM FUNCTION:


19/31

y In figure are shown two adjacent streamlines of a two-dimensional flow field. Let= (x, y)represent the streamline nearest the origin. Then= + d =is representative of the secondstreamline.

y Since there is no flow across a streamline, we can let = be indicative of the flow carried throughthe area from the origin O to the first streamline.

y d = represents the flow carried between the two SLs of the figure.y For continuity referring to the triangular fluid element ofincompressible fluid,y d == -vdx +udyy d == (x=/ xx) dx + (x=/ xy) dyy Comparing the last 2 Eq.y u= x=/ xy, v= -x=/ xxy Thus, = can be expressed as function of x and y.


20/31

y We can find the velocity components (u and v) at any point of 2D flow field.y If u and v are expressed as functions of x and y we can find = by integrating.y The derivation of= is based on the principle of continuity.y The continuity be satisfied for= to exist.y The vorticity was not considered in the derivation of=, the function need not be irrotational

for = to exist.

Show that: continuity to be satisfied for= to exist.

y According to 2D continuity eq. (x u/ xx) + (x v/ xy) =0y Then from above, u= x=/ xy, v= -x=/ xxy [x (x=/ xy )/ xx] - [x (x=/ xx) / xy] =0y Flow described by = automatically satisfies the continuity equation

1.6 BASIC FLOW FIELDS

Introduction:

` In this section several basic flow fields that are commonly encountered will be discussed.` Though these fields imply an ideal fluid, they closely depict the flow of a real fluid, if viscous

influence provided.

Straight streamlines flow:

` The simplest of all flows is that where the streamlines are straight, parallel. and evenly spaced .` In this case v = 0 and u = constant , U=velocity of flow` a= distance between stream lines` d == -vdx +udy or d == udy or d == Uy


21/31

Source or a sink flow:

` Another flow field ofgeneral interest is that of a source or a sink. In the case of a source, theflow field consists of radial streamlines symmetrically spaced as shown in figure.

` If q is the source strength, or rate of flow from the source` ] = qU/2T` Customarily, for this case, the ] = 0, streamline is defined as that coincident with the direction

of the x axis.

` For sink (inward flow), the stream function is:` ] = - qU/2T


22/31

Combined by superposition Flow:

` Flow fields may be combined by superposition to give other fields ofimportance.` For example, let us combine a source and sink of equal strength with a rectilinear flow.` Let 2a be the distance between the source and sink. Referring to the figure and definingU1 and

U2 as shown, we can write for the combined field.

` ] = Uy + qU1/2T -qU2/2T


23/31

Combined by superposition Flow:

` Transforming the last two terms or this equation to cartesian coordinates,` ] = Uy + q/2T[tan-1 {y/ (x+a)}- tan-1 {y /(x-a)}]` This equation will permit one to plot streamlines by determining values of] at various points in

the flow field having coordinates (x,y). Lines of constant] are streamlines.

Problems:


24/31


25/31

1.7 Velocity potential

Introduction:

The flow takes place in a pipe line if there is a difference of pressure. The direction of flow will take place from higher to the lower pressure. Thus the velocity of flow in a certain direction will depend upon the potential difference which

is known as velocity potential and denoted by N (phi).

The velocity potential is scalar function of position and time. Velocity potential function is a function of x, y and z such that its partial derivative with respect

to displacement in any direction is equal to the velocity component.

u = xN/ xx , v= xN/ xy, w= xN/ xz u, v and w are the velocity components in x, y and z directions respectively.

Relationship between stream function and velocity function

For a two dimensional, steady, irrotational and incompressible flow, is: u = xN/ xx = x=/ xy v= xN/ xy = -x=/ xx IfN and = the real and imaginary parts of an analytic function, then the above equations are

called Cauchy-Riemann Equations.


26/31

Orthogonality ofN and= lines

Potential Line is a line along which the velocity potential is constant. When a velocity potential function is equated to a series of constants, a family of curves is

obtained.

These curves are at right angles to streamlines at every point i.e. they are orthogonal tostreamlines.

N = f (x, y) for steady and 2D dN = (xN/ xx) dx + (xN/ xy) dy But at the potential line, N=0 0 = (xN/ xx) dx + (xN/ xy) dy dy/dx = - (xN/ xx) / (xN/ xy) = -u/ v The slope of the potential line =-u/ v = = f (x, y) for steady and 2D d == (x=/ xx) dx + (x=/ xy) dy But at the stream line,= =0 dy/dx = - (x= / xx) / (x= / xy) = v/u The slope of the stream line = v/u The slope of the stream line v slope of the potential line = -1 Which shows that potential line and stream line are orthogonal to each other.

Laplace Eq.

Writing u, v and w in terms ofN, the continuity equation for steady incompressible flow- xu/ xx + xv/ xy xw/ xz =0 x(xN/ xx )/ xx + x (xN/ xy)/ xy x(xN/ xz) / xz =0 This is called the Laplace Eq.

1.8 Flow Net


27/31

y Since lines ofN and = are required to form an orthogonal network, a flow net can only exist ifirrotationality (the condition for the existence ofN) and continuity (the condition for theexistence of

]) are satisfied.

y Such flows which satisfy continuity and irrotationality are referred to as potential flow.y The Lalplace equation was derived assuming the existence of velocity potentials and the

satisfaction or continuity.

y Thus, if a given flow satisfies the Laplace equation, a flow net can be constructed for that flow.Existence of Laplace equation, a flow net potential flow.

y Because of the irrotationality requirement such flows are usually those ofideal fluids.y An exception where a real fluid satisfies the conditions for potential flow is that of laminar

through porous media.

For a a real fluid:

y In such a case the velocity head is negligible and the energy equation may be written as:y [p/Vg + z]1 + [p/Vg + z]2 = hL


28/31

y Where the hL is directly proportional to velocity for laminar flow.1.9 Force on a stationary blade

9.Force on a stationary blade or vane

In this case fluid is in contact with atmosphere so that the pressure forces disappear. The velocities are often very high, so that we cannot neglect the friction, which result in sizeable

error.

9.Force on a stationary blade

Let us consider a vane as shown in fig below:

9.Force on a stationary blade

The force exerted by the blade on water in x direction is given by, F(b/w)x =VQ(V 2x -V 1x) The force exerted by the blade on water in y direction is given by, F(b/w)y =VQ(V 2y -V 1y) Therefore the resultant force is given by F = (F(b/w)x2 + F(b/w)y2) The angle of resultant with X-direction is U = tan-1 (F(b/w)y / F(b/w)x)

1.10 Relation between absolute and relative velocity:

Absolute velocity V of a body is the velocity relative to the earth. Relative velocity V of a body is the velocity relative to second body.


29/31

Fig. Relative and absolute velocity relation

vector V= Absolute velocity of 1st

body

vector v=Relative velocity of 1st

body wrt 2nd

body

vector u =Absolute velocity of 2nd

body

vector V = vector v+ vector u

Vu =Component of absolute velocity V of the fluid in the direction of u.

Vu = V cosE = u+ v cos F

1.11 FORCE UPON AMOVING VANE OR BLADE

There are two principal differences between the action upon a stationary and a moving object. For the case of a moving object, it is necessary to consider both absolute and relative velocities.

Which makes the determination of(V more difficult.

The other is that the amount of fluid that strikes a single moving object in any time interval ( t isdifferent from that which strikes a stationary object.

If the cross-sectional area of a stream is A1 and its velocity is V1 then the rate at which fluid isemitted from the nozzle is:

The amount of fluid which strikes a single object the body per unit time: m =VgQ =VgA1V1


30/31

But for the case of a single object the amount of fluid which strikes the body per unit time willbe less than this if the body is moving away from the nozzle. (and more than this ifit is moving

toward the nozzle)

If m' = the weight or fluid per second striking a single object moving with a velocity u in the samedirection as V1 then,

m =VgQ =VgA1(V1-u) =VgA1v1 For a single object movingin the same direction as the stream initially, the component of the

force exerted by the vane on the fluid is:

Fu = [m / g] (Vu =VA1v1( Vu It can be prove that ( Vu = ( vu Fu = m / g( Vu =VA1v1( vu For the series of vanes, Fu = m / g( Vu =VA1V1( vu Because one vane come another like a stationary vanes.

1.12 TORQUE IN ROTATINGMACHINES

When a fluid flows through a rotor its radius usually varies along its path. Hence it is desirable to compute torque rather than a force.


31/31

The resultant torque is the summation of the torques produced by all the elementary forces. But it has been shown that the latter may be considered as equivalent to two single forces

one concentrated at the entrance to and the other at the exit from any device.

fluid properties and definitions

Documents