Flluiidd MMeecchanniiccss Chhaapptteerr--66 …brijrbedu.org/Brij Data/Brij FM/SM/Chapter-6 Viscous...
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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.
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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].
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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.
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4 Chapter 6: Viscous Flow
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
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5 Fluid Mechanics By Brij Bhooshan
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.
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6 Chapter 6: Viscous 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 .
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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
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8 Chapter 6: Viscous Flow
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,
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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
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10 Chapter 6: Viscous Flow
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
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11 Fluid Mechanics By Brij Bhooshan
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.
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12 Chapter 6: Viscous Flow
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.
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13 Fluid Mechanics By Brij Bhooshan
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
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14 Chapter 6: Viscous Flow
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),
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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
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16 Chapter 6: Viscous Flow
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
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17 Fluid Mechanics By Brij Bhooshan
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
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18 Chapter 6: Viscous Flow
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
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19 Fluid Mechanics By Brij Bhooshan
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.
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20 Chapter 6: Viscous Flow
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
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21 Fluid Mechanics By Brij Bhooshan
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
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22 Chapter 6: Viscous Flow
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
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23 Fluid Mechanics By Brij Bhooshan
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]
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24 Chapter 6: Viscous Flow
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,
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25 Fluid Mechanics By Brij Bhooshan
or