Flluiidd MMeecchanniiccss Chhaapptteerr--66 …brijrbedu.org/Brij Data/Brij FM/SM/Chapter-6 Viscous Flow.pdf · 6.5 Flow Through A Concentric Annulus 13 6.6 ... Cengel & Cimbala,

For more information log on www.brijrbedu.org

Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)

Copyright by Brij Bhooshan @ 2013 Page 1

FFlluuiidd MMeecchhaanniiccss

CChhaapptteerr--66 VViissccoouuss FFllooww

PPrreeppaarreedd BByy

BBrriijj BBhhoooosshhaann

AAsssstt.. PPrrooffeessssoorr

BB.. SS.. AA.. CCoolllleeggee ooff EEnngggg.. AAnndd TTeecchhnnoollooggyy

MMaatthhuurraa,, UUttttaarr PPrraaddeesshh,, ((IInnddiiaa))

SSuuppppoorrtteedd BByy::

PPuurrvvii BBhhoooosshhaann

In This Chapter We Cover the Following Topics

S. No. Contents Page No.

6.1 Couette Flow 4

6.2 Fully Developed Laminar Flow Between Infinite Parallel Plates 7

6.3 Flow Through A Pipe 9

6.4 Laminar And Turbulent Regimes 12

6.5 Flow Through A Concentric Annulus 13

6.6 Momentum And Kinetic Energy Correction Factor 17

6.7 Flow Potential and Flow Resistance 19

6.8 Flow Through Branched Pipes 21

6.9 Flow Through Perforated Pipes 22

6.10 Ventilation Network 23

6.11 Hardy Cross Method 23

6.12 Power Transmission By A Pipeline 24

References:

1. Andersion J. D. Jr., Computational Fluid Dynamics “The Basics with applications”,

1st Ed., McGraw Hill, New York, 1995.

2. Frank M. White, Fluid Mechanics, 6th Ed., McGraw Hill, New York, 2008.

3. Frank M. White, Viscous Fluid Flow, 2nd Ed., McGraw Hill, New York, 1991.

4. Fox and McDonald’s, Introduction to Fluid Mechanics, 6th Ed., John Wiley & Sons,

Inc., New York, 2004.

5. Welty James R., Wicks Charles E., Wilson Robert E. and, Rorrer Gregory L.,

Fundamentals of Momentum, Heat, and Mass Transfer, 5th Ed. John Wiley & Sons,

Inc., New York, 2008.

6. Mohanty A. K., Fluid Mechanics, 2nd Ed, Prentice Hall Publications, New Delhi,

2005.

7. Rathakrishnan E., Gas Dynamics, 2nd Ed., Prentice Hall Publications, New Delhi,

2005.

8. Gupta Vijay, and, Gupta S. K., Fluid Mechanics & its Applications, 1st Ed., New Age

International (P) Limited, Publishers, New Delhi 2005.




2 Chapter 6: Viscous Flow

9. Cengel & Cimbala, Fluid Mechanics Fundamentals and Applications, 1st Ed.,

McGraw Hill, New York, 2006.

10. Modi & Seth, Hydraulics and Fluid Mechanics, Standard Publications.

Please welcome for any correction or misprint in the entire manuscript and your

valuable suggestions kindly mail us [email protected].




3 Fluid Mechanics By Brij Bhooshan

This chapter is completely devoted to an important practical fluids engineering problem:

flow in ducts with various velocities, various fluids, and various duct shapes. Piping

systems are encountered in almost every engineering design and thus have been studied

extensively. There is a small amount of theory plus a large amount of experimentation.

The basic piping problem is this: Given the pipe geometry and its added components

(such as fittings, valves, bends, and diffusers) plus the desired flow rate and fluid

properties, what pressure drop is needed to drive the flow? Of course, it may be stated in

alternate form: Given the pressure drop available from a pump, what flow rate will

ensue? The correlations discussed in this chapter are adequate to solve most such piping

problems.

Now that we have derived and studied the basic flow equations in Chap. 5, you would

think that we could just whip off myriad beautiful solutions illustrating the full range of

fluid behavior, of course expressing all these educational results in dimensionless form.

The fact of the matter is that no general analysis of fluid motion yet exists. There are

several dozen known particular solutions, there are some rather specific digital

computer solutions, and there are a great many experimental data. There is a lot of

theory available if we neglect such important effects as viscosity and compressibility,

but there is no general theory and there may never be. The reason is that a profound

and vexing change in fluid behavior occurs at moderate Reynolds numbers.

The variation is in the radial direction, the axial variation is absent since the duct

chosen is of uniform cross-section. Such conditions are satisfied, in practice, far

downstream of the inlet. The flow in this case is also one-dimensional, but realistic.

Since viscous loss is present, Bernoulli's equation cannot be applied without

modification through the addition of loss energy. The situations when the real velocity

profile does not change in the axial direction are called parallel flow. In considering the

one-dimensional viscous flow, i.e. with only one non-zero component of velocity, we are

necessarily confined to parallel flows. Our aim is to estimate the viscous loss by

considering transverse variation of velocity. Once the loss energy is evaluated and

introduced in the modified Bernoulli's equation, the effect of transverse variation can be

tacitly disregarded in the subsequent fluid mechanical calculations.

A complete description of the statistical aspects of turbulence is given in Ref. 1, while

theory and data on transition effects are given in Refs. 2 and 3. At this introductory

level we merely point out that the primary parameter affecting transition is the

Reynolds number. If Re = UL/, where U is the average stream velocity and L is the

“width,” or transverse thickness, of the shear layer, the following approximate ranges

occur:

0 < Re < 1: highly viscous laminar “creeping” motion

1 < Re < 100: laminar, strong Reynolds-number dependence

100 < Re < 103: laminar, boundary-layer theory useful

103 < Re < 104: transition to turbulence

104 < Re < 106: turbulent, moderate Reynolds-number dependence

106 < Re < : turbulent, slight Reynolds-number dependence

These are representative ranges which vary somewhat with flow geometry, surface

roughness, and the level of fluctuations in the inlet stream. The great majority of our

analyses are concerned with laminar flow or with turbulent flow, and one should not

normally design a flow operation in the transition region.





6.1 COUETTE FLOW

Flow between two parallel flat plates, one of which is at rest and the other moving with

a velocity U is generally termed as the Couette flow, after its investigator. The physical

model of a plane Couette flow in which there is no superimposed pressure gradient is

the same as in Diagram 1.2 (Chapter 1), adopted for explaining Newton's law of shear

stress. In fact, the popularity of Couette flow arises from its adoptability to model

several flow conditions in practice through simple mathematics.

One of these is the hydrodynamic lubrication in a journal bearing. In Diagram 1.2 or 6.1,

the upper plate may be considered to be the journal moving at a velocity U = r, and

the lower surface the bearing, the gap between the two filled with a lubricant. is the

angular speed of the journal and r its radius, the curvature and variation of the

lubricant film thickness being considered negligible. An infinitesimal control volume of

size (x y) is chosen in the flow field. The directions of shear stress on the x sides

have been fixed considering that the control surface at y is tending to move at a higher

velocity than its surrounding (lower layers) and that at (y + y) is moving at a slower

speed with respect to the upper layers.

Diagram 6.1 Control Volume Analysis of Couette Flow.

The conservation of momentum for the control volume is then written as

for unit thickness perpendicular to the plane of the flow; dvol = x y 1.

At steady state u/t = 0, and u/x = 0 for parallel flow, yielding thereby Du/Dt = 0.

The momentum conservation then implies simply the balance between the pressure and

the shear force:

By definition, = (du/dy), see Chapter 1; alternative, by kinematics of Chapter 3,

deformation θ = du/dy.

or

The constants of integration C and D are evaluated from the conditions

(a) Physical model (b) Velocity profile

x

p

u y

y

y = h U U

p = 0

p < 0

p > 0

A B





u = 0 at y = 0 on the stationary surface [6.2b]

u = U at y = h on the moving surface [6.2c]

yielding thereby

A pressure gradient parameter

is defined for writing the velocity equation in a condensed form:

Plane Couette Flow

In the absence of a pressure gradient P = 0, we get the linear velocity profile

due solely to the Newtonian shear resistance. This situation, as stated earlier, is termed

as the plane or simple Couette flow. The shear stress

is constant all along the fluid thickness. The external force required to push the upper

plate at U, per unit area of the contacting surface, is also equal to U/h.

Favourable Pressure Gradient

The situation of decreasing pressure in the downstream direction is more descriptively

is referred as of a favourable pressure gradient since the fluid motion is assisted by the

external pressure.

The term dp/dx is negative and P > 0. Consequently, the fluid velocity at a given y is

higher than the corresponding plane flow value. This is shown in Diagram 6.1(b).

By differentiating Eq. (6.5), we can write the general expression for shear stress as

The shear stress at the two walls are

and

In other words, the static wall shear stress is increased whereas that at the moving wall

is decreased over the plane flow value. The external force applied at the moving wall is

smaller for the favourable pressure gradient condition.

Adverse Pressure Gradient

The reverse situation of pressure increasing in the upstream direction is adverse for the

flow. P is negative, and the velocity everywhere is decreased compared to the plane flow.





For some values of P, part of the velocity near the solid wall may be in the direction

opposite to the motion of the upper plate. One such situation is shown as curve B in

Diagram 6.1(b). This is likened to the rolling back of the fluid particle discussed in

connection with the diverging section of the venturimeter in Chapter 5.

The curve A in Diagram 6.1(b), in this context, is the limiting or neutral one. We note

from the shape of this curve that the velocity does not change with y in the

neighbourhood of the solid wall.

Mathematically, this means that

du/dy = 0 at y = 0. [6.8]

From Eq. (6.7b), the zero gradient condition arises when

P = 1 [6.9]

The static wall shear stress is zero and flow is said to have 'separated' there. On the

other hand, the shear resistance at the upper wall, for P = 1.

is double the plane flow value. In other words, the external force required is

considerably increased notwithstanding the zero shear stress value on the solid wall.

In the case of a still higher adverse pressure gradient, such as for curve B, the force

requirement is further increased.

Volume flow rate

The volume flow rate is given by

Thus volume flow rate per unit depth is

Average Velocity

The average velocity is given by

Point of maximum velocity

There is no simple relation between maximum velocity umax and mean velocity .





6.2 FULLY DEVELOPED LAMINAR FLOW BETWEEN INFINITE PARALLEL

PLATES

Both Plates Stationary

Fluid in high-pressure hydraulic systems (such as the brake system of an automobile)

often leaks through the annular gap between a piston and cylinder. For very small gaps

(typically 0.005 mm or less), this flow field may be modeled as flow between infinite

parallel plates. To calculate the leakage (low rate, we must first determine the velocity

field.

Consider two parallel fixed plates kept at a distance h apart as shown in Diagram 6.2.

let us suppose a fluid element of length dx and thickness dy at a distsnce y from the

lower fixed plate. For our analysis we select a differential control volume of size dVol =

dxdydz,

Diagram 6.2 Control volume for analysis of laminar flow between stationary infinite parallel plates.

The conservation of momentum for the control volume is then written as

At steady state Du/Dt = 0. The momentum conservation then implies simply the

balance between the pressure and the shear force:

Integrating this equation, we obtain

By definition, yx = (du/dy), see Chapter 1; alternative, by kinematics of Chapter 3,

deformation θ = du/dy.

or

The constants of integration C and D are evaluated from the conditions

u = 0 at y = 0, then D = 0.

u = 0 at y = h.

Hence

y

h

dx

dy umax





This gives

and hence,

At this point we have the velocity profile. What else can we learn about the flow?

Shear Stress Distribution

The shear stress distribution is given by

Shear stress is zero at y = b/2, and maximum at y = 0.

Volume Flow Rate

The volume flow rate is given by

For a depth l in the z direction,

Thus volume flow rate per unit depth is

Flow Rate as a Function of Pressure Drop

Since p/x is constant, the pressure varies linearly with x and

Substituting into the expression for volume flow rate gives

Since

Then,





Average Velocity

The average velocity magnitude, , is given by

Point of Maximum Velocity

To find the point of maximum velocity, we set du/dy equal to zero and solve for

corresponding y. From Eq. 6.16

Thus, du/dy = 0, at y = h/2.

Transformation of Coordinates

In deriving the above relations, the origin of coordinates, y = 0, was taken at the bottom

plate. We could just as easily have taken the origin at the centerline of the channel. If

we denote the coordinates with origin at the channel centerline as x, y', the boundary

conditions are u = 0 at y' = h/2.

To obtain the velocity profile in terms of x, y', we substitute y = y' + h/2 into Eq. 6.16.

The result is

Equation 6.22 shows that the velocity profile for laminar flow between stationary

parallel plates is parabolic, as shown in Diagram 6.3.

Diagram 6.3 Dimensionless velocity profile for fully developed laminar flow between infinite parallel

plates.

6.3 FLOW THROUGH A PIPE

Flow of a fluid through a duct or a pipe is perhaps the most common physical

arrangement. Exact analysis is possible for such internal flows when the

(i) geometry is simple, uniform and symmetric, (ii) flow rate is moderate and (iii) the

flow section of interest is far downstream of the inlet.

The cross-sectional momentum of a fluid in internal flow changes for some distance from

the inlet; the variation ceases far downstream in a duct of uniform cross-section. By

invoking condition (iii), we are considering the flow in such downstream sections where

it is said to be 'fully developed', essentially meaning parallel.

1

y’/a

0

1/2

1/2

y

x

y’

a u





The physical model for the fully developed flow through a uniform pipe is shown in

Diagram 6.4.

Diagram 6.4 Control Volume Analysis of a Fully Developed Pipe Row.

The pipe is shown to be horizontal for convenience. However, the result shall be equally

applicable to arbitrary orientation if it is remembered that the hydrostatic pressure is

balanced by the body force, and the hydrodynamic pressure differential alone is

responsible for the velocity head.

A control volume 1-2-3-4 of radius r and length dx is chosen. (r) is the frictional shear

stress at radius r. Since momentum change is zero between 1-2 and 3-4, we can write

so that

The Newtonian shear stress is given by

since the directions of increasing y and r are opposing, Diagram 6.4. Thus

or

u = 0 at r = R gives

leading to

as the fluid velocity at a radius r. The maximum velocity occurs at the pipe centre, r = 0,

and is

The parabolic velocity obtained, when the shear stress is Newtonian, is referred as the

Haggen-Poiseulle profile. A similar solution is obtained for flow between parallel walled

Fully developed region — of

unchanging momentum

Entry region of

changing momentum

r R

y 1

p

r

4

3 2

dx

R

dr

r





ducts. The volumetric flow rate is obtained by integration of flow through an

infinitesimal width dr at radius r,

or

The area averaged velocity is

such that

In fully developed flow, the pressure gradient, p/x, is constant. Therefore,

Substituting into the expression for volume flow rate gives

Then,

for laminar flow in a horizontal pipe. Note that Q is a sensitive function of D.

The shear stress at the pipe wall is written following Eq. (6.23) as

For a circular geometry, the ratio A/P = D/4.

The concept of a 'diameter is used for flow geometries of different shapes, circular and

non-circular, by defining

where Aw and Pw are respectively the net flow area and the wetted perimeter.

The generalized characteristic dimension Dh is termed as the hydraulic diameter that

equals D for a circular pipe.

The expression for shear stress in fully developed flow through arbitrary geometry is,

thus,

The work done in overcoming the shear force acting over a unit area, through a unit

distance is w•12•1 = w. This dissipation is expressed as a fraction of the kinetic energy

of a fluid element of unit volume at the average velocity. The fraction is named as

'friction factor'.

is known as the Fanning’s friction factor.





The pressure loss due to friction is obtained by combination of (6.30) and (6.31) as

A Darcy-Weisbach friction factor f = 4 is often defined, so that the head lost due to

friction over a pipe length L is expressed as

The hf so estimated is then used in the modified Bernoulli's equation, disregarding

thereafter the cross-sectional variation of velocity.

Note that expressions (6.30) to (6.33) are kinematic descriptions of steady, fully

developed flows and are not restricted by the law of shear stress.

Combination of Eqs. (6.30) and (6.31) give, for a pipe, the equation

Substituting for uavg, from Eq. (6.27a), we get

or

it will be noted by substitution of the dimensions of the terms involved that (/ uavgD)

is dimensionless, and is defined as Reynolds number

Hence

or

are the values of friction factor for fully developed flow at 'moderate' rates through a

circular pipe.

6.4 LAMINAR AND TURBULENT REGIMES

In laminar flow the fluid particles move along straight parallel path in layers or lamina,

such that the path individual fluid particles do not cross these of neighbouring particles.

Laminar flow is possible only at low velocity and when fluid is highly viscous. But when

the velocity increased or fluid is less, viscous the fluid particles do not move straight

paths. The fluid particle moves in random manner resulting in general mixing of

particles. This type of fluid is termed as turbulent flow.

A laminar flow changes to turbulent flow when

Velocity is increased,

Diameter of a pipe is increased, and,

Viscosity of fluid is decreased.





In case of laminar flow, the loss of pressure head to be directly proportion to velocity but

in case of turbulent flow loss of head is approximately proportional to square of velocity.

We have been scrupulously specifying the flow rate to be moderate, since experiments

have indicated that when the rate exceeds a limit, Eq. 6.35(a) or 6.35(b) is violated.

These experiments were first performed by Osborne Reynolds. The limit of validity of

6.35(a) or 6.35(b) is set in terms of a 'critical value of Reynolds number'. For a pipe, the

region of validity of 6.35(b) is below an approximate value of Re = 2000. The flow is then

said to be in the 'Laminar regime' where the flow is orderly, the kinetic energy being

moderately higher than the viscous resistance.

At higher Re the kinetic energy is much in excess and the flow develops random

fluctuations over and above its directed motion. The regime is called 'Turbulent'; (fRe)

does not equal 64 anymore.

Transition to turbulent conditions is accelerated if disturbances are present in the flow

field. For Re < 2000, however, even strong disturbances do not cause a permanent shift

to turbulence. The range of Re = 2000 to 2300 is known as the transition range. In most

cases, the flow is turbulent when Re exceeds the upper limit of 2300. By careful

experimentation, however, the transition can by delayed.

The fully developed flow, whether in the laminar or in the turbulent regime, being

characterized by unchanging cross-sectional momentum, has a velocity profile that is

same at all downstream sections. Consequently, the shear stress remains constant at all

downstream locations. In other words the pressure drop per unit length is also constant.

In fact the constancy of pressure gradient is taken as the experimental confirmation of

fully developed condition. The velocity profile is then said to be 'similar'.

6.5 FLOW THROUGH A CONCENTRIC ANNULUS

Fully developed flow through a concentric annulus is one of the common occurrences in

practice. The physical model and plausible control volumes are shown in Diagram 6.5.

Diagram 6.5 Axial Flow through a Concentric Annulus.

Control Volume I

In Diagram 6.5(b), we have chosen a control volume to coincide with the interior of the

outer cylinder and the exterior of the inner. We proceed by considering the force balance

on a strip at radius r, width dr.

or

(a) Physical model (b) Control volume I (c) Control volume II

= 0

dr

r

p





neglecting (dr)2 in comparison to dr. Thus

or

where C is the constant of integration. Note that Eq. (6.37) is valid for fully developed

flow both in laminar and turbulent regimes.

In order to obtain an analytical solution, we confine to laminar flow of a Newtonian

fluid, for which r = (du/dr) in the region of increasing velocity, see Diagram 6.5(a).

or

The constants of integration C and D are evaluated from the fact that u = 0 at r = r1, and

r = r2, i.e. on the solid surfaces. Thus

The velocity profile between the two cylinders forming the annulus is parabolic. The

radius of maximum velocity is obtained by seeking du/dr = 0.

From Eq. (6.39b),

At du/dr = 0,

or

Since du/dr = 0, r0 also defines the circle of zero shear stress. Knowledge of the zone of

zero shear stress affords many conveniences in fluid mechanical calculations, especially

in complex flows. The advantages stem basically from choosing a control volume to

coincide with zero shear surface.

We shall demonstrate the principle in the case of an annulus through the control

volumes shown in Diagram 6.5(c).

Control Volume II

Consider the control volume A between r1 and r0. From Eq. (6.37),





Setting u = 0 at r = r1, in Eq. (6.38), we obtain

and

or

is the expression for velocity in terms of the radius r0 of the zero shear circle. It would be

readily recognised that Eq. (6.42) results by substituting for r0 in Eq. (6.42).

An expression for the velocity profile involving r2 and r0 could be similarly obtained by

considering the control volume named B.

When the law of shear stress is known such as in laminar flow, or when the geometry is

a convenient one like the concentric annulus, there is no practical reason for choosing

one control volume in preference to the other.

On the other hand, in turbulent flow where the law of wall shear stress is determined by

experiments, control volume II is preferable since the measured values at any one wall

will suffice.

Similarly when a complex duct is involved, even in laminar flow, the method of control

volume II merits over the other. In Diagram 6.6, the cross-section of an equilateral

triangular duct is shown as an example. The duct is divided into six identical sections by

drawing perpendicular bisectors to the opposite sides. Such sections are more

technically called sub-channels.

Diagram 6.6 Sub-channel Analysis of an Equilateral Triangular Duct

Consider the sub-channel BGD. Only one solid surface is involved at BD. BG and DG

can be assumed to be zero shear stress lines due to geometrical symmetry.

The friction factor result for one sector is then suitably combined to generate the value

for the entire duct, treating that the pressure gradient for each sub-channel is the same

as for the whole duct. Since the uniformity of pressure gradient is deviated in

undeveloped flows, the sub-channel method is not recommended for those situations.

Average Velocity and Friction Factor

The volumetric flow rate through the annulus is obtained by integrating

3

2

6

1

5 4

C D B

G

E F

A





and by substituting for u from Eq. (6.39b).

Define

Then

but,

or

The average velocity is

By differentiating Eq. (6.39a), the shear stress at r, is

On the inside of the outer wall,

The negative sign indicates decrease of velocity with radius as the outer wall is

approached.

The total shear force

and





or

The expression (6.44c) is the same as the one obtained for the simple pipe flow, and can

be obtained simply by a control volume balance of the pressure gradients and the

average shear force.

Fanning's friction factor for the annulus is obtained from the definition

or

Defining the terms within the brackets as [B], we get

or

For the annulus, Dh =2(r2 r1)

and

or

where = r1/r2 is the radius ratio of the annulus.

The laminar friction factor value for the annulus increases from Re = 16 to 24 in the

limits varying from zero to unity. Typically, Re = 23.81 for

= 05; Re can be

assumed to be 24 without much error for > 05.

6.6 MOMENTUM AND KINETIC ENERGY CORRECTION FACTOR

So far we have considered that the velocity of flow at a section in a pipe is uniform.

However, in reality velocity at a section is not uniform for flow through a pipe.

Therefore, to apply the energy and momentum equations between any two sections in

the pipeline, the variation in the velocity distribution takes care of by introducing two

dimensionless parameters known as kinetic energy correction factor and momentum

correction factor.

The kinetic energy correction factor at any section is defined as the ratio of kinetic

energy of flow based on actual velocity to the kinetic energy of flow based on average

velocity and is found as follows





or

For a constant density flow, Eq. (6.46a) reduces to

The energy equation (5.29) can be written with the consideration of kinetic energy

correction between two sections of a pipeline as

The momentum correction factor at any section is defined as the ratio of momentum of

flow based on actual velocity to the momentum of flow based on average velocity and is

obtained as follows

or

For a constant density flow, Eq. (6.47a) reduces to

Application 6.1: Determine the momentum correction factor and kinetic energy

correction factor for laminar incompressible, fully developed flow through a circular

pipe.

Diagram 6.7

Solution: The velocity distribution for laminar, incompressible, fully developed flow

through a circular pipes is given by

Consider an differential area dA in the form of a ring at a radius r and of width dr, then

dA = 2rdr

the rate of fluid flowing through a ring

dQ = u.dA = u. 2rdr

Momentum of fluid through ring per second = mdQ u

= u22rdr

The actual momentum of the fluid/sec across the section

R r u dA = 2r.dr

dr





The momentum of the fluid/sec based on the average velocity

= m uavg = A um um = A

Now, um = umax/2, then

The momentum correction factor is given by Eq. (6.47b) as

Now, we know that kinetic energy is

= mu2/2 = .dQ.u2/2 = (u2rdr)u2/2

= r u3dr

The actual kinetic energy of the fluid/sec across the section

The kinetic energy of the fluid/sec based on the average velocity

= A um um/2 = A /2

Now, A = R2, then

The kinetic energy correction factor is given by Eq. (6.46b) as

6.7 FLOW POTENTIAL AND FLOW RESISTANCE

In the light of the modified Bernoulli's equation, fluid flow through a system takes place

due to the difference of the total head between an upstream and a downstream station,

against the losses. The flow resistance, in general, is composed of that due to friction

and due to change of geometry such as sudden expansion, contraction, bends or presence

of throttling devices like valves.





Consider flow of water from a reservoir A to a reservoir B at different levels through a

pipe, Diagram 6.8.

Diagram 6.8 Water Flow between Two Reservoirs

Application of Bernoulli's equation between points A and B in the two reservoirs results

in

The lost head

hf = loss at entry to the pipe at reservoir A

+ friction loss in the pipe

+ lost kinetic energy at entry to reservoir B

i.e.

or

The velocity V is more generally written in terms of the flow rate Q which is constant,

and the cross-sectional area

V = Q/a

Thus,

or

hf = RQ2 [6.49b]

defining the square bracketted terms as the flow resistance R.

The difference in water level between A and B is, on the other hand, written in terms of

the heights measured from the datum as

h = (hA + zA) (hB + zB) = HA HB [6.49c]

Combining (6.48), (6.49b) and (6.49c), we can write

h = RQ2 [6.50]

Equation (6.50) is comparable to a purely resistive electrical circuit:

V = ri [6.51]

where V is the voltage or potential difference, r the resistance in ohms, and i the

current.

The difference, however, is that while the voltage drop in an electrical circuit is linearly

proportional to the current, the head differential in a fluid circuit is proportional to the

square of the flow rate. This non-linearity imposes restrictions for direct use of electrical

Datum

f l

D V

h patm

patm

B

A

p

at

m

zB

zA

hB

hA





network analyser for solving fluid-flow problems, although there are ways to

approximate the procedure.

Now that we recognize the nature of Eq. (6.51) in Eq. (6.50), we can represent the flow

between the stations A and B through an equivalent circuit, Diagram 6.9.

Diagram 6.9 Equivalent Fluid Network

For simplicity, we neglected variation of cross-section, or existence of bends or valves in

the pipe connecting the two reservoirs. Inclusion of these effects only modify the value of

the flow resistances R; the nature of (6.49b) is not affected. Similarly, the network

concept is not restricted for flow between two reservoir stations A and B where the fluid

is stagnant. What is required is to account the total head for estimating h, Eq. (6.49c).

6.8 FLOW THROUGH BRANCHED PIPES

In practice, flow of a fluid under a given total head differential takes place through

several pipes joined in series and parallel. The network analogy is applicable in all such

cases.

Consider, for example, the flow system indicated in Diagram 6.10. Water enters a pipe

at A that branches into two at B. The two branches again meet at C, and water is

discharged at D through the pipe CD.

The total head at A is HA and that at D is HD. The total head is the sum of the height

above a common datum, the pressure head and the velocity head HA and HD are shown

as batteries with their negative terminals at the common ground potential. The

grounded terminals can be joined together as shown by the dotted line, in which case

(HA HD) is the potential difference. The flow rate Q divides into Q1 and Q2 at B which

recombine at C, A, B, C and D are the nodal points.

Conservation of volume or mass flow rate at each of the nodes, and the pressure head

equation in the form of the modified Bernoulli's equation for a closed loop are the two

relationships used for estimating the flow parameters. These two are equivalent to the

two laws of Kirchoff for an electrical network

Diagram 6.10 Flow through Branched Pipes.

Choosing the flow to a node as positive and that from the node as negative, we can write

the mass conservation as

HB HA

R

Q2

(HA HB)

R

Q2

(a) Physical model (b) Equivalent network

HD HA

D C B

A

Q1

1

2

Q2

Q

Q

l3, d3, f3

l4, d4, f4

C

D

A

B l2, d2, f2 Q2 Q1 l1, d1, f1





ṁ = 0 [6.52a]

or, for incompressible flow,

Q = 0 [6.52b]

Thus in Diagram 6.10 at B,

Q Q1 Q2 = 0

and at C,

Q1 + Q2 Q = 0

Qs are the volumetric flow rates.

The pressure balance for a loop is

h + RQ2 = 0 [6.53a]

In writing RQ2 in (6.53a) it is tacitly assumed that the true flow is in the same direction

as shown in the figure. There is no a priori knowledge to ensure this, however, in all

situations.

In order to account for the direction of flow, and consequently the pressure gradient,

RQ2 is written as R|Q|Q, where Q without the modulus sign sets the direction.

Thus it is more appropriate to write

h + R|Q|Q = 0 [6.53b]

In deriving the resistance R, we had assumed the friction factor f to be constant. The

laminar derivations have shown that this indeed is not true: f = 64/Re. In general, f is

dependent on flow rate, and pressure drop should better be expressed as RQn, n being an

experimentally determined index.

The general expression for pressure balance is therefore:

h + R|Q|n 1 Q = 0 [6.53c]

6.9 FLOW THROUGH PERFORATED PIPES

In agricultural practices, perforated pipes are frequently used to irrigate the crop.

Similar situation arises in domestic water supply connections in which a large number

of tappings are provided on a trunk line to distribute water at different locations.

It would be desirable to have an uniform water tapping rate along the length of the

trunk pipe.

Consider in Diagram 6.11 a trunk pipe with uniform draw-off compared with a pipe of

the same diameter and length but discharging at the end.

Diagram 6.11 Perforated Pipe Flow.

At a given section x of the perforated pipe, let the flow rate be q for a length dx. Then

the factional loss for dx is

For constant draw-off rate C,

q = Q Cx

Hence

(a) Uniform draw-off (b) End-discharge

dQ/dx = C

Q Q

hf2 hf1

Q = 0 Q





The total head loss for pipe length l is

or

Since the discharge at the free end is zero,

Q = Cl,

Substituting in (6.54a)

where hf2, corresponds to the solid pipe in Diagram 6.11(b).

If the pipe diameter varied along the length or if the drawn off was not constant,

accounting for such situations present no difficulty in principle. The variable are to be

then included within the integral.

6.10 VENTILATION NETWORK

Distribution of air in a ventilation system through ductings is made much the same

manner as water through pipelines. The pressure and temperature variation in

ventilation designs being not significant, the air is considered incompressible.

The seeming difference, however, arises due to the presence of both ford and induced

draft fans, creating respectively positive and negative pressures, as design

requirements. Though suction and delivery pumps could as well be present in water

lines, such arrangements are relatively rare.

Furthermore, leakage of the fluid in a ventilating system is difficult to insulate

completely, and hence is taken as a design parameter. We shall demonstrate the

influence of positive and negative pressure fans, as well as the leakage in a ventilation

network solution in the following problem.

6.11 HARDY CROSS METHOD

We have noted that the distribution of flow in a fluid network is estimated on the basis

of

Q = 0 at a node [6.55]

h + R|Q|n 1 Q = 0 in a loop [6.56]

The calculations are begun by assuming a plausible distribution of flow which satisfies

the continuity Eqn. (6.55) at each node. The assumed distributions, however, would not

ordinarily satisfy the pressure Eq. (6.56). The distribution is then altered and iterations

are carried out until both the equations are satisfied simultaneously. The procedure

proposed by Hardy Cross for such iterations results in a quicker convergence.

In each path let the correct flow be Q0, whereas the assumed flow is Q, and the error Q,

i.e.

Q = Q0 + Q [6.57]





Hence

Qn = + n

Q + higher order terms

The error in the pressure equation is then

ERQ = h + R|Q|n 1 Q

= ( h + R|Q0|n 1 Q0) + nR|Q0|n 1 Q [6.58]

The terms within the parentheses equal to zero, Q0 being the correct solution.

Therefore,

The correction to the assumed flow in a branch is Q, by Eq. (6.57). If a flow path is

common to several loops, the correction derived for each of the loops will have to be

added to the concerned path.

The method of Hardy Cross is convenient for solving fluid network problems using a

computer.

6.12 POWER TRANSMISSION BY A PIPELINE

The fluid conveyed by a pipeline is sometimes used for generating mechanical power.

For example, water from a reservoir under high hydrostatic head is often conveyed

through a large pipeline to operate hydraulic turbines, generally the impulse type. The

difference of water level in the reservoir and the turbine centre line is the available head

H. Because of friction, however, the head available at the pipe exit is less, and say HP at

a given flow rate Q. Denoting the frictional head hf, we have

HP = H hf [6.60]

The fluid power at inlet is QgH and that at outlet of the pipe is QgHP. The efficiency of

power transmission is then

or

Describing the friction head as RQ2, the available power

P = g (HQ – RQ3)

The optimum flow rate for maximum power is obtained as

i.e. when the friction head

hf = RQ2 = H/3 [6.62]

Consequently,

HP opt = 2H/3

and

Pmax = 2QgH/3

max = 2/3 [6.63]

In general,





or

Documents

Flluiidd MMeecchanniiccss Chhaapptteerr--66 …brijrbedu.org/Brij Data/Brij FM/SM/Chapter-6 Viscous Flow.pdf · 6.5 Flow Through A Concentric Annulus 13 6.6 ... Cengel & Cimbala,