Steady Flow through Pipes

Steady Flow through Pipes and Pipe Systems - 12 -

Chapter “10”
When the student finishes studying this chapter, he should be able to:
 Have a deep understanding of laminar and turbulent flow in pipes
and the analysis of fully developed flow,
 Measure the pressure drop in the straight section of smooth, rough,
and packed pipes as a function of flow rate,
 Correlate this in terms of the friction factor and Reynolds number,
and compare results with available theories and correlations,
 Determine the influence of pipe fittings on pressure drop, and
 Analyze pipelines system in series or in parallel and predict how the
flow splits among all of the baths and how the pressure drops
across such systems.

Steady Flow through
Pipes and Pipe systems
By: Dr. Ezzat El sayed G. SALEH

Water Distribution System
Pumping Station
Distribution Storage
Distribution Piping
Components
Pumping Station
Structural
Electrical
Pumping
Piping
Pumps
Power
Transmission
Driver Control
Main-
components
components
Sub-
subcomponents

Distribution Storage
Tanks
Pipes
Valves
Sub-components
components
Reinforcem
ent
Concrete
course
4
Civil.
Distribution Piping
Pipes
Valves
Sub-components
components

Fluid flow in circular and noncircular pipes is commonly
encountered in practice,
 The hot and cold water that we use in our homes is pumped
through pipes.
 Water in a city is distributed by extensive piping networks.
 Oil and natural gas are transported hundreds of miles by large
pipelines. Blood is carried throughout our bodies by arteries
and veins.
Introduction
 The cooling water in an engine is transported by hoses to the
pipes in the radiator where it is cooled as it flows.
 Thermal energy in a hydronic space heating system is
transferred to the circulating water in the boiler, and then it is
transported to the desired locations through pipes.
Introduction

Fluid flow is classified as external and internal, depending on
whether the fluid is forced to flow over a surface or in a conduit.
Internal and external flows exhibit very different characteristics. In
this chapter we consider:
»internal flow where the conduit is completely filled with the
fluid, and flow is driven primarily by a pressure difference.
»This should not be confused with open-channel flow where the
conduit is partially filled by the fluid and thus the flow is partially
bounded by solid surfaces, as in an irrigation ditch, and flow is
driven by gravity alone.
Flow Classification
(a) Pipe Flow (b) Open Channel Flow
What is the Difference Between Pipe Flow and Open Channel Flow??

Typical Pipe System Component
Typical Pipe System Component
Pipe Flow and Pump System
Fig. (1)
 We start this chapter with a general physical description of
internal flow and the velocity boundary layer. We continue with
a discussion of the dimensionless Reynolds number and its
physical significance.
 We then discuss the characteristics of flow inside pipes and
introduce the pressure drop correlations associated with it for
both laminar and turbulent flows. Then we present the minor
losses and determine the pressure drop and the sizes
requirements for real-world piping systems.
Objectives

 The terms pipe, duct, and conduit are usually used
interchangeably for flow sections. In general, flow sections of
circular cross section are referred to as pipes (especially when
the fluid is a liquid), and flow sections of noncircular cross section
as ducts (especially when the fluid is a gas).
 Small diameter pipes are usually referred to as tubes. Given this
uncertainty, we will use more descriptive phrases (such as a
circular pipe or a rectangular duct) whenever necessary to avoid
any misunderstandings.
Objectives

If you have been around smokers, you probably noticed that the
cigarette smoke rises in a smooth plume for the first few
centimeters and then starts fluctuating randomly in all directions
as it continues its rise. Other plumes behave similarly (Fig. 1).
Likewise, a careful inspection of flow in a pipe reveals that the
fluid flow is streamlined at low velocities but turns chaotic as the
velocity is increased above a critical value, as shown in Fig. (2).
The flow regime in the first case is said to be laminar,
characterized by smooth streamlines and highly ordered motion,
and turbulent in the second case, where it is characterized by
velocity fluctuations and highly disordered motion.
Laminar and Turbulent Flow
The transition from laminar to turbulent flow does not occur
suddenly; rather, it occurs over some region in which the flow
fluctuates between laminar and turbulent flows before it becomes
fully turbulent. Most flows encountered in practice are turbulent.
Laminar flow is encountered when highly viscous fluids such as oils
flow in small pipes or narrow passages.
Laminar and Turbulent Flow

Time “t” Dependence of Fluid Velocityat a Point
Is the Flow in the Pipe Laminar or Turbulent??
(b) Typical Dye Streaks
Experiment to Illustrate Types of Flow

Laminar Flow
Transitional Flow
Turbulent Flow
Fig. (1) The Behavior of Colored
Injected into the Flow in Laminar
and Turbulent Flows in Pipes.
Fig. (2) Laminar and Turbulent Flow
Regimes of Candle Smoke.

 For the case of steady, incompressible flow,
h
2
1
2
1
L
R
g
)
L
L
(
h





giving a simple relation between resistance stress  and the head
loss caused by the action of resistance.
Velocity Distribution and its Significance
Velocity Profiles for Parabolic and Non-Parabolic Velocity Distribution
through a Cylindrical Pipe of Radius R.
r
Fig. (2)
The Mean Velocity
Q
r
V
A
d
D
 For  = o, Rh = r/ 2, hL 12 = hL
and L1 = L2 = L, Eq. (1) yields
Fig. (3)
……………. (1)
r
R
2
h
g
h
L








 

 (2)
 The kinetic energy of fluid moving in the differentially small stream
tube of Fig. (3) is (dQ) x g V2/2g, or  V3 dA/2.
 The momentum flux is the differentially small stream tube of Fig. (3)
is (dQ) x V, or  V3 dA/2.
 The total kinetic energy and momentum flux are the respective
integrals of these differential quantities, thus
Total kinetic energy
Momentum flux
 


A
3
dA
v
2
 


A
2
dA
v
2

 These differential equations are usefully expressed in terms of mean
velocity V and total flow rate Q, using the same forms of the
equations,
 Thus total kinematic energy.. )
g
2
/
V
(
g
Q
)
g
2
/
V
(
g
Q 2
2







 Momentum flux ……………. V
Q 



In which  and  are dimensionless and represent correction
factors to the conventional velocity head V2/2g and momentum
flux  Q V, respectively.
 For a uniform velocity distribution  = = 1.0,
 For a non-uniform velocity distribution  >  > 1.0.
 Expressions for  and  may be given as






A
A
3
3
dA
v
dA
v
V
1






A
A
2
dA
v
dA
v
V
1
and

 The transition from laminar to turbulent flow depends on the
geometry, surface roughness, flow velocity, surface temperature,
and type of fluid, among other things. After exhaustive
experiments in the 1880s, Osborne Reynolds discovered that the
flow regime depends mainly on the ratio of inertial forces to
viscous forces in the fluid. This ratio is called the Reynolds
number and is expressed for internal flow in a circular pipe as
Reynolds Number
 Obsorn Reynolds was the first to demonstrate that laminar or
turbulent flow can be predicted if the magnitude of a dimensionless
number, now called the Reynolds number, Re is known. The basic
definition of the Reynolds number is,
How to Determine the Type of Flow???








D
V
D
V
Re
Critical Reynolds Number
 For practical applications in pipe flow, it was found that if:
…………………. (3)

Critical Reynolds Number
 The flow is turbulent
Re is greater than 4000
 It is impossible to predict which type of
flow exists
 Re is between 2000 and 4000
 The flow will be laminar
 Re is less than 2000
 It is impossible to predict which type of flow that of Reynolds
number in the range between 2000 and 4000
 It is impossible to predict which type of flow that of Reynolds
number in the range between 2000 and 4000
Darcy’s Equation for Energy Loss
 For the case of flow in pipes and tubes, Darcy’s equation for energy
loss is expressed mathematically as,
g
2
V
D
L
f
h
2
L 


where
hL : energy loss due to friction (m or ft),
L : length of pipe flow (m or ft),
D : diameter of pipe flow (m or ft), and
V : the average velocity of flow (m/s or ft/s), and
f : friction factor (dimensionless).
……………………..… (4 )

In fully developed laminar flow, each fluid particle moves at a
constant axial velocity along a streamline and the velocity profile
v(r) remains unchanged in the flow direction. There is no motion in
the radial direction, and thus the velocity component in the
direction normal to flow is everywhere zero.
Laminar Flow in Pipes
Friction Loss in Laminar Flow
 When laminar flow exists, the fluid seems to flow as several
layers, one on another. The relationship between the energy
loss (head loss) and the measurable parameters of the flow
system is known as the Hagen - Poiseuille equation:
2
L
D
g
V
L
32
h



where
 ,  : the properties of the fluid, viscosity and density,
L, D : the geometric features of length and diameter pipe,
D : diameter of pipe flow (m or ft), and
V : the dynamics of the flow characterized by the average velocity.
………………….… (5)

 Eq. (5) is valid only for laminar flow (Re < 2000). Darcy equation
could also be used to calculate the friction loss for laminar flow.
 Equating Eq. (4) and Eq. (5) we get,
2
2
L
D
g
V
L
32
g
2
V
D
L
f
h






or
e
R
64
D
V
64
f 




 ………………….… (6)
The head loss due to friction in laminar flow can be calculated either
from Darcy’s equation
In Summary
g
2
V
D
L
f
h
2
L 


where f = 64 / Re
Or from the Hagen – Poiseuille’s equation
2
L
D
g
V
L
32
h




 For steady flow and incompressible fluid, the mean velocity is
defined by V = Q/A, in which:
dA
V
A
1
V
V
A
r
mean 

 
Q is the flow rate obtained from the differential flow rate, dQ,
passing through the differential area, dA
V is the uniform velocity that will transport the same amount of
mass through the cross section as will the actual velocity
distribution,
……………… (7)
 From which the mean velocity can be obtained by performing the
indicated integration.
The Mean Velocity
 Fig. (4) represents an axisymmetric velocity distribution in a
cylindrical pipe of radius R. The mean velocity, V, can be calculated
in terms of the maximum velocity, Vmax as
Fig. (4 )
R
r
r
d
(Parabolic)
R
max
V
r
V
r
A
d
Ax symmetric Parabolic Velocity Distribution through a Cylindrical
Pipe of Radius R.
Q
dQ

 Taking “r” as the radial distance to any local velocity, Vr and the
element of area dA = 2  r dr, while the equation of the parabolic is
 
 
2
max
r R
/
r
1
V
V 

 Then,
 
  dr
r
2
R
/
r
1
V
R
1
V 2
R
0
max
2
max 



 
 Integration (Vmax = constant) shows that for the parabolic
axisymmetric profile, the mean velocity is one-half the maximum
(centerline) velocity, Vmax, or
2
/
V
V max
mean 
 The relationship between the maximum velocity and the
average velocity is a function of the shape of the velocity
profile.
 For the parabolic profile (as proved before), the average
velocity, V, is one-half the maximum velocity, Vmax, and
the minimum velocity at that section of r = R.
 If the velocity profile is a different shape (non-parabolic),
the average velocity is not necessarily one-half the
maximum velocity (see Fig. 5 ).
We conclude that:
……………… (8)

(Parabolic)
R
r
Velocity Profiles for Parabolic and Non-Parabolic Velocity Distribution
through a Cylindrical Pipe of Radius R.
max
V
2
/
V
V max

max
V
max
V
V 
(non-Parabolic)
Fig. (5)


2
L
2
2
1
D
g
V
L
32
h
D
V
L
32
p
p
p
flow
ar
min
La








 

g
2
V
D
L
/
D
V
64
g
2
V
D
L
f
h
2
2
L 






Pressure Drop and Head Loss
e
R
64
V
D
64
f 



 The friction factor for fully developed laminar flow in a circular pipe
is
 This equation shows that in laminar flow; the friction factor is a
function of Reynolds number only and is independent of the
roughness of the pipe
 The pressure losses and head loss are commonly expressed as

 Friction losses & pressure drop in Fully-developed regions of
Laminar flow in circular pipe was easily analyzed by analytical &
dimensional methods.
Summary of Case of Laminar Pipe Flow
Summary of Case of Laminar Pipe Flow
 Surface roughness is found to be of no significance in laminar-
pipe-flows.
 Viscosity & Reynolds number are found to be very important
in all laminar flows.
Flow in the Sub-layer Near Rough and Smooth Walls



5
2
2
2
L
2
2
1
D
g
LQ
f
8
g
2
V
D
L
f
h
2
V
D
L
f
p
p
p
flow
Turbulent










Pressure Drop and Head Loss
D
Vavg.
hL
 P
 
The following relation for pressure
loss (and head loss) is one of the
most general relations in fluid
mechanics, and it is valid for:
 laminar or turbulent flows,
 circular or noncircular pipes, and
 pipes with smooth or rough
surfaces.
……………… (9)
Equivalent Roughness for New Pipes
0.9 – 9
0.3 – 3.0
0.18 – 0.9
0.26
0.15
0.045
0.0015
0.0 (smooth)
0.003 – 0.03
0.001 – 0.01
0.0006 – 0.003
0.000085
0.0005
0.00015
0.000005
0.0 (smooth)
Rivet steel
Concrete
Wooden stave
Cast iron
Galvanized steel
Commercial steel
or wrought iron
Drawn tubing
Plastic, glass
Millimeters
Feet
Pipe
Equivalent Roughness, 
After Moody (1944) and Colebrook (1981)

Friction Factors as a Function of Reynolds Number and Relative Roughness
 From fluid mechanics point of view, the main idea in
designing piping system is to apply the energy equation
between appropriate locations within the flow system, with
head/pressure loss written in terms of friction factor and
minor loss coefficients. Those locations are usually between
pumps/power machineries and/or tanks/reservoirs.

 Piping systems containing a single pipe (whose length may
be interrupted by various components such as fittings,
valves, flow-meters, ….etc.),
 Piping systems containing multiple pipes interconnected in-
parallel, in-series, or in various network configurations.
Analysis of Piping Systems Considers two Classes of Pipe Systems:
Multi-pipe Systems

A General Pipe Network
Pipe network can include:
 Series pipes
 Parallel pipes, and
 Branching pipes

3
2
1 Q
Q
Q
Q 



3
3
2
2
1
1 V
A
V
A
V
A
Q 


3
L
2
L
1
L
f
L h
h
h
h B
A



 
Pipes in Series
f1 & D1 & L1
f2 & D2 & L2 f3 & D3 & L3


  
EGL
B
A



 L
h
H
B
A
3
2
1 Q
Q
Q
Q 



3
3
2
2
1
1 V
A
V
A
V
A
Q 



3
,
L
2
,
L
1
,
L
f
L h
h
h
h B
A



 
Pipes in Parallel


1
1
1
1 f
D
V 



2
2
2
2 f
D
V 



3
3
3
3 f
D
V 



A
B
1
D
2
D
3
D
1
Q
2
Q
3
Q


1
1
1
1 f
D
V 



2
2
2
2 f
D
V 



3
3
3
3 f
D
V 



A
B
1
D
2
D
3
D
1
Q
2
Q
3
Q


1
1
1
1 f
D
V 



2
2
2
2 f
D
V 



3
3
3
3 f
D
V 



A
B
1
D
2
D
3
D
1
Q
2
Q
3
Q

Pipes in Parallel
and
3
2
1 Q
Q
Q 

3
,
L
1
L
2
,
L
1
L
B
A
f
L h
,
h
or
h
,
h
h
B
A 



  
Branching pipes

Three Reservoir System
Junction


HGL
HGL
HGL
Z1
Z2
Z3
ZJ
PJ / g
V3
V2
V1
Reference Datum



Branching Pipe System

Minor Losses
some valves + flow meters + pipe
fittings (elbows, bends, ..etc).
Valves + Pipe fittings (elbows, bends, ..etc).

Flow Through a Valve
where:
VV = flow velocity in valve throat,
VP = flow velocity in pipeline, and
K = resistance coefficient which is found
experimentally by the valve manufacturer for
different valve openings.
Note: K depends greatly on the %age of valve
opening (as shown in the next Table)
where:
VV = flow velocity in valve throat,
VP = flow velocity in pipeline, and
K = resistance coefficient which is found
experimentally by the valve manufacturer for
different valve openings.
Note: K depends greatly on the %age of valve
opening (as shown in the next Table)

(a) Reentrant KL= 0.8, (b) Sharp-edged KL= 0.5,
(c) Slightly Rounded KL= 0.2, (d) Well Rounded KL= 0.04
(a) Reentrant KL= 0.8, (b) Sharp-edged KL= 0.5,
(c) Slightly Rounded KL= 0.2, (d) Well Rounded KL= 0.04
Entrance Flow Conditions and Loss Coefficient

(a) Reentrant KL=1.0, (b) Sharp-edged KL=1.0,
(c) Slightly Rounded KL=1.0, (d) Well Rounded KL=1.0
(a) Reentrant KL=1.0, (b) Sharp-edged KL=1.0,
(c) Slightly Rounded KL=1.0, (d) Well Rounded KL=1.0
Exit Flow Conditions and Loss Coefficient
Loss Coefficient for a Contraction

Pipe friction pressure losses is sensitive to changes in
diameter and roughness of pipe.
For given rate of flow and fixed friction factor,
pressure drop per meter of pipe length varies inversely
with the fifth power of the pipe diameter.
Therefore, a 2% reduction of diameter due to age
causes an 11% increase in pressure drop; a 5%
reduction of diameter causes an 29% increase in
frictional pressure drop.
Effect of Age and Use on Pipe Friction
Any piping system must include at least some of the
following parts:
1- some lengths of straight pipes (can be different materials and
/ or diameters). All pressure losses, ΔP, or head losses, Δh =
ΔP/ρg , in straight pipes parts are called the major losses.
2- some valves + flow meters + pipe fittings (elbows, bends,
..etc). Minor losses are any pressure or head losses in any
thing which is not a pipe !
Note: In many cases minor losses >> major losses (the names
are just for fluid terminology !).
Summary of Some Design Basic Fundamentals

»Total Losses:
To get the total pressure or head losses in a piping system you
do algebraic sum:
Total losses = Σ Major losses + Σ Minor losses
»Average velocity:
To do design calculations you need to known the average flow
velocity in the pipes.
(from the required flow rate and area of the pipe
V= Q/A = Q / ( D2/4)
 If the pipe is not circular, use the hydraulic diameter,
Dh= 4*(the area/wetted perimeter)=4 P.
 You get Reynolds number in the pipe, Re = ρVD/μ,
 So you can tell if the flow is:
laminar (if Re < 2000)
or
Turbulent in the pipe (if Re > 4000).

a
Square Tube Rectangular Tube Circular Tube
D
a
a
4
a
4
D
2
(squ.)
h 


)
b
a
(
b
a
2
D (Rect.)
h

 D
D (Circular)
h 
b
a
a
x (A/P)
4
=
h
Hydraulic Diameter D
»All losses are usually calculated with respect to the available
kinetic energy of the fluid which is = (½ ρV2). The friction factor
or resistance coefficient, K (or KL or f) is defined as:
K = ΔP /(½ ρV2)
Calculations of Major Losses in Straight Pipes
»Get K (or KL or f ) from the very well known Moody chart for
straight pipes. Note that the turbulent part of the chart is in fact
an empirical or experimental part.
»The value of K is found (both analytically and experimentally) as
a function of the Reynolds number, Re, and the surface
roughness, ε/d, of the pipe material K = L/d Φ{Re, ε/d},
where L is pipe length and d is the internal pipe diameter.

 We use the same analogy as for a straight pipe. That is we
define a friction factor or resistance coefficient, K (or KL or f ) for
any thing that is not a pipe.
K = ΔP / (½ ρ V2)
 Therefore, we have infinite number of values for K. These
values of K can not be found in one chart like for a straight pipe.
But we get these K from all experimental or empirical relations
given by the manufacturers of the valves, flow-meters, pipe
fittings, elbows, bends, …etc.
Calculations of Minor Losses
Note: some manufacturers define the equivalent length, Le ,
of the valve or pipe fittings so that you can consider it as an
additional length of the actual straight pipe parts so that the
total length of your pipes shall be increased by all the
additional equivalent lengths of any thing in your systems that
is not a straight pipe.
Calculations of Major Losses in Straight Pipes

Given 
Given 
Determined ?
D: Pressure drop and
head loss
Given 
Determined ?
Given 
C: Flow rate and average
velocity
Determined ?
Given 
Given 
Given 
Given 
Given 
Given 
Given 
Given 
B: Pipe
Diameter
Length
Roughness
Given 
Given 
Given 
Given 
Given 
Given 
A: Fluid
Density
Viscosity
Type 3
Type 2
Type 1
Variable
Pipe Flow Problem Types
El. 50
El. 40
El. 150 ft
1000 ft &
D1=8 in
2
0
0
0
f
t
&
D
2
=
6
i
n
P


A
B
HP
hL(2)
hL(1)
p
)
2
(
L
)
1
(
L H
)
B
A
(
h
h 



2
5
2
2
L Q
D
g
L
f
8
g
2
V
D
L
f
h 



c
/
H
Q
g
power p


Neglecting all losses except
main losses (friction losses)
EGL
Rise in EGL and HGL due to Pump
HGL

El. 50
El. 40
El. 150 ft
1000 ft &
D1=8 in
2
0
0
0
f
t
&
D
2
=
6
i
n
T


A
B
HT
hL(1)
hL(2)
)
B
A
(
H
h
h T
)
2
(
L
)
1
(
L 



2
5
2
2
L Q
D
g
L
f
8
g
2
V
D
L
f
h 



c
/
H
Q
g
power p


Neglecting all losses except
main losses (friction losses)
EGL
Drop in EGL and HGL due to Pump
HGL
Branching Pipe Systems



A
B
junction
1
1
1 D
&
f
&
L
2
2
2 D
&
f
&
L
3
3
3 D
&
f
&
L
B
A
h
h )
2
(
L
)
1
(
L 


)
3
(
L
)
2
(
L h
h 
3
2
1 Q
Q
Q
. 

)
1
(
L
h
)
3
(
L
)
2
(
L h
h 
2
Q
c 


 2
5
2
L Q
D
g
L
f
8
h

c
/
h
Q L

where



Pipe and pipe
are connected in Series

 
Pipe and pipe
are connected in Series
 
and 5
2
D
g
/
L
f
8
c 





A
B
C
junction
1
1
1 D
&
f
&
L
2
2
2 D
&
f
&
L
3
3
3 D
&
f
&
L
B
A
h
h )
B
J
(
L
)
J
A
(
L 

 

C
A
h
h )
C
J
(
L
)
J
A
(
L 

 

3
2
1 Q
Q
Q 

)
J
A
(
L
h 
)
B
J
(
L
h 
)
C
J
(
L
h 
2
2
5
2
L
Q
c
Q
D
g
L
f
8
h





c
/
h
Q L

where







A
B
C
junction
1
1
1 D
&
f
&
L
2
2
2 D
&
f
&
L
3
3
3 D
&
f
&
L
B
A
h
h )
B
J
(
L
)
J
A
(
L 

 

C
A
h
h )
C
J
(
L
)
J
A
(
L 

 

3
1 Q
with
Q
Compare
)
J
A
(
L
h 
0
h )
B
J
(
L 

)
C
J
(
L
h 
2
2
5
2
L
Q
c
Q
D
g
L
f
8
h





c
/
h
Q L

where
= 0








A
B
C
junction
1
1
1 D
&
f
&
L
2
2
2 D
&
f
&
L
3
3
3 D
&
f
&
L
C
A
h
h )
C
J
(
L
)
J
A
(
L 

 

C
B
h
h )
C
J
(
L
)
J
B
(
L 

 

3
2
1 Q
with
Q
Q
Compare 
)
J
A
(
L
h 
0
h )
J
B
(
L 

)
C
J
(
L
h 
2
2
5
2
L
Q
c
Q
D
g
L
f
8
h





c
/
h
Q L

where




Equivalent size of compound pipe
¤ Equivalent size of compound pipe can be determined by using,
5
3
3
3
5
2
2
2
5
1
1
1
5
D
L
.
f
D
L
.
f
D
L
.
f
D
L
.
f



Where:
L = Length of equivalent pipe,
D = Diameter of equivalent pipe,
f = Friction factor of equivalent pipe,
3
2
1 L
,
L
,
L = Length of pipes 1, 2, and 3 respectively,
3
2
1 D
,
D
,
D = Diameter of pipes 1, 2, and 3 respectively,
3
2
1 f
,
f
,
f = friction factors of pipes 1, 2, and 3 respectively,
Equivalent Size of Compound Pipe
f1 & D1 & L1
f2 & D2 & L2 f3 & D3 & L3


  
EGL
B
A



 L
h
H
B
A
F1 & D1 & L1
& V1
F2 & D2 & L2
& V2
F3 & D3 & L3
& V3
Equivalent Size of Compound Pipe
3
,
L
2
,
L
1
,
L
L h
h
h
h 



 If the friction factor of all the pipes and the equivalent pipe is the
same
5
3
3
5
2
2
5
1
1
5
D
L
D
L
D
L
D
L




81#
Any Questions???

82#
Any Questions till Now ???

Steady Flow through Pipes

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