SEV5343
Water Resources Engineering
Topic 01
Pipe Flow and Networks
1
Classifying Flow
• Laminar Flow and Turbulent Flow
• Flow in a conduit is classified as being either
laminar or turbulent, depending on the
magnitude of the Reynolds number.
2
Classifying Flow
• Laminar Flow and Turbulent Flow
• When the velocity was low, the streak of dye flowed down the
tube with little expansion
• If velocity was increased, at some paint in the tube, the dye
would all at once mix with the water.
• When the dye exhibited rapid mixing , illumination with an
electric spark revealed eddies in the mixed fluid
3
Classifying Flow
• Laminar Flow and Turbulent Flow
• The Reynolds number is often written as ReD, where the
subscript “D” denotes that diameter is used in the
formula. (eg.Rex, ReL)
• Reynolds number can be calculated with four different
equations.
• Based on Reynolds' experiments, engineers use
guidelines to establish whether or not flow in a conduit
will be laminar or turbulent.
Precise values
of Reynolds
number versus
flow regime do
not exist.
4
Classifying Flow
• Developing Flow and Fully Developed Flow
• As the fluid moves down the pipe, the velocity
profile changes in the streamwise direction as
viscous effects cause the plug-type profile to
gradually change into a parabolic profile. This
region of changing velocity profile is called
developing flow.
• After the parabolic distribution is achieved, the
flow profile remains unchanged in the
streamwise direction, and flow is called fully
developed flow.
5
Classifying Flow
• Developing Flow and Fully Developed Flow
6
Classifying Flow
• Developing Flow and Fully Developed Flow
• The distance required for flow to develop is called the
entry or entrance length (Le).
• The entry length is
defined as the
distance at which
the shear stress
reaches 2% of the
fully developed
value.
7
Classifying Flow
• Developing Flow and Fully Developed Flow
– Valid for flow entering a circular pipe from a reservoir
under quiescent conditions.
• Other upstream components such as valves,
elbows, and pumps produce complex flow fields
that require different lengths to achieve fully
developing flow.
8
Specifying Pipe Sizes
• Standard Sizes for Pipes (DN)
• One of the most common standards for pipe
sizes is called the Diameter Nominal (DN) system.
• Pipe schedule is related to the
thickness of the wall. The original
meaning of schedule was the ability
of a pipe to withstand pressure, thus
pipe schedule correlates with wall
thickness.
9
Specifying Pipe Sizes
• Standard Sizes for Pipes (DN)
Diameter Nominal
10
Pipe Head Loss
• Combined (Total) Head Loss
• Pipe head loss is one type of head loss; the other
type is called component head loss. All head loss
is classified using these two categories:
• Component head loss is associated with flow
through devices such as valves, bends, and tees.
• Pipe head loss is associated with fully developed
flow in conduits, and it is caused by shear
stresses that act on the flowing fluid.
11
Pipe Head Loss
• Derivation of the Darcy-Weisbach Equation
• Apply the momentum equation to the control
volume
12
Pipe Head Loss
• Derivation of the Darcy-Weisbach Equation
• The net efflux of momentum is
zero because the velocity
distribution at section 2 is
identical to the velocity
distribution at section 1. The momentum accumulation
term is also zero because the flow is steady.
13
Pipe Head Loss
• Derivation of the Darcy-Weisbach Equation
• sinα = (Δz/ΔL).
• Apply the energy equation to the control volume
with hp = ht = 0, V1 = V2, and α1= α2
14
Pipe Head Loss
• Derivation of the Darcy-Weisbach Equation
Re-arrange
15
Pipe Head Loss
• Derivation of the Darcy-Weisbach Equation
• Define a new π-group called
the friction factor f that
gives the ratio of wall shear
stress (τ0) to kinetic pressure (ρV2/2):
– Other names: friction factor, Darcy friction factor, DarcyWeisbach friction factor, and the resistance coefficient.
• Another coefficient, the Fanning friction factor, often
used by chemical engineers
16
Pipe Head Loss
• Derivation of the Darcy-Weisbach Equation
Darcy-Weisbach Equation
– To use the Darcy-Weisbach equation, the flow should
be fully developed and steady.
– For either laminar flow or turbulent flow and for
either round pipes or non round conduits such as a
rectangular duct.
17
Stress Distributions in Pipe Flow
• In pipe flow the pressure acting on a plane
that is normal to the direction of flow is
hydrostatic.
• The pressure distribution varies linearly.
18
Stress Distributions in Pipe Flow
• Vin=Vout ---net momentum eflux is zero.
• Steady--- momentum accumulation is also zero.
19
Stress Distributions in Pipe Flow
• Let W = γAΔL, and let
sin α = Δz/ΔL
Divide by AΔL
• Shear-stress distribution varies linearly with r.
• Shear stress is zero at the
centerline, it reaches a
maximum value of τ0 at the
wall, and the variation is
linear in between.
20
Laminar Flow in a Round Tube
• laminar flow occurs when ReD = 2000.
• Laminar flow in a round tube is called Poiseuille flow or
Hagen-Poiseuille flow in honor of researchers who
studied low-speed flows in the 1840s.
• Velocity Profile
– Change variables by letting y = r0 - r, where r0 is pipe
radius and r is the radial coordinate.
21
Laminar Flow in a Round Tube
• Velocity Profile
– The left side of the equation is a function of radius r,
and the right side is a function of axial location s. This
can be true if and only if each side is equal to a
constant. Thus,
– Δh is the change in piezometric head over a length ΔL
of conduit.
22
Laminar Flow in a Round Tube
• Velocity Profile
Integrate
• Apply the no-slip condition, which states that the
velocity of the fluid at the wall is zero.
23
Laminar Flow in a Round Tube
• Velocity Profile
• The maximum velocity occurs at r = 0:
24
Laminar Flow in a Round Tube
• Velocity Profile
• Velocity varies as radius
squared (V ~ r2),
parabolic distribution.
25
Laminar Flow in a Round Tube
• Discharge and Mean Velocity V
• Apply the flow rate equation
Integrate
26
Laminar Flow in a Round Tube
• Discharge and Mean Velocity V
• Apply
• substitute D/2 for r0
27
Laminar Flow in a Round Tube
• Head Loss and Friction Factor f
• Apply the energy equation from section 1 to 2
• Let hL = hf
• Expand mean velocity equation
28
Laminar Flow in a Round Tube
• Head Loss and Friction Factor f
replace ΔL with L.
• assumptions are (a) laminar flow, (b) fully
developed flow, (c) steady flow, and (d)
Newtonian fluid.
29
Laminar Flow in a Round Tube
• Head Loss and Friction Factor f
– head loss in laminar flow varies linearly with velocity.
• Combine with the Darcy-Weisbach equation
– Friction factor for laminar flow depends only on
Reynolds number.
30
Turbulent Flow and the Moody Diagram
• Qualitative Description of Turbulent Flow
• Turbulent flow occurs when Re ≥ 3000.
• Produces high levels of mixing and has a velocity
profile that is more uniform or flatter than the
corresponding laminar velocity profile.
• Model turbulent flow by using an empirical
approach. This is because the complex nature of
turbulent flow has prevented researchers from
establishing a mathematical solution of general
utility.
31
Turbulent Flow and the Moody Diagram
• Equations for the Velocity Distribution
• The time-average velocity distribution is often
described using an equation called the power
law formula.
– m is an empirically determined variable that depends
on Re
– Velocity in the center of the pipe is typically about
20% higher than the mean velocity V.
32
Turbulent Flow and the Moody Diagram
• Equations for the Velocity Distribution
• To use the turbulent boundary-layer equations,
the velocity distribution can be expressed by the
logarithmic velocity distribution
– u*, the shear velocity, is given by
.
33
Turbulent Flow and the Moody Diagram
• Equations for the Friction Factor, f
• After integration, algebra, and tweaking the
constants to better fit experimental data, the
result is
34
Turbulent Flow and the Moody Diagram
• Equations for the Friction Factor, f
Resistance coefficient f versus Reynolds number for sand-roughened pipe
(After Nikuradse).
ks---Sand roughness height
ks/D---relative roughness
35
Turbulent Flow and the Moody Diagram
• Equations for the Friction Factor, f
Effects of Wall Roughness
• In laminar flow, wall roughness does not
influence f.
• In turbulent flow, wall roughness has a major
impact on f.
36
Turbulent Flow and the Moody Diagram
• Moody Diagram
• Colebrook advanced Nikuradse's work by acquiring data for
commercial pipes and then developing an empirical equation,
called the Colebrook-White formula, for the friction factor.
• Moody used the Colebrook, White formula to generate a
design chart, named as the Moody diagram for commercial
pipes.
• ks denotes the equivalent sand roughness.
37
Turbulent Flow and the Moody Diagram
• Moody Diagram
38
Turbulent Flow and the Moody Diagram
• Moody Diagram
• To find f, given Re and ks/D, one goes to the right
to find the correct relative roughness curve.
Then one looks at the bottom of the chart to find
the given value of Re and, with this value of Re,
moves vertically upward until the given ks/D
curve is reached. Finally, from this point one
moves horizontally to the left scale to read the
value of f.
39
Turbulent Flow and the Moody Diagram
• Moody Diagram
• The top of the Moody diagram presents a scale
based on the parameter Re f1/2, This parameter is
useful when hf and ks/D are known but the
velocity V is not. Using the Darcy-Weisbach
equation and the definition of Reynolds number,
one can show that
40
Turbulent Flow and the Moody Diagram
• Moody Diagram
• In the Moody diagram, curves of constant Re f1/2 are
planed using heavy black lines that slant from the left to
right.
• When using computers to carry out pipe-flow
calculations, it is much more convenient to have an
equation for the friction factor as a function of Reynolds
number and relative roughness. By using the ColebrookWhite formula, Swamee and Jain developed an explicit
equation for friction factor, namely
– predicts friction factors that differ by less than 3% from those on
the Moody diagram for 4 X 103 < ReD < 108 and
10-4 < ks/D < 2 X 10-2
41
Turbulent Flow and the Moody Diagram
• Strategy for Solving Problems
• To solve a turbulent flow problem using the
traditional approach, one classifies the problems
into three cases:
• Case 1 to find the head loss, given the pipe
length, pipe diameter, and flow rate. This
problem is straightforward because it can be
solved using algebra;
42
Turbulent Flow and the Moody Diagram
• Strategy for Solving Problems
• Case 2 to find the flow rate, given the head loss
(or pressure drop), the pipe length, and the pipe
diameter. This problem usually requires an
iterative approach;
– An iterative approach can sometimes be avoided by
using an explicit equation developed by Swamee and
Jain
43
Turbulent Flow and the Moody Diagram
• Strategy for Solving Problems
• Case 3 to find the pipe diameter, given the flow
rate, length of pipe, and head loss (or pressure
drop). This problem usually requires an iterative
approach.
– can sometimes use an explicit equation developed by Swamee
and Jain and modified by Streeter and Wylie
44
Combined Head Loss
• The Minor Loss Coefficient, K
• When fluid flows through a component such as a
partially open value or a bend in a pipe, viscous
effects cause the flowing fluid to lose mechanical
energy. To characterize component head loss,
engineers use a π-group called the minor loss
coefficient K
– the head loss across a single component or transition
is hL = K(V2/2g), where K is the minor loss coefficient
for that component or transition.
45
Combined Head Loss
• The Minor Loss Coefficient, K
• Most values of K are found by experiment.
• To find K, flow rate is measured and mean velocity is
calculated using V = (Q/A).
• Pressure and elevation measurements are used to
calculate the change in piezometric head.
• Then, values of V and Δh are used to calculate K.
46
• Data for the
Minor Loss
Coefficient
47
Combined Head Loss
• Combined Head loss Equation
Combined head loss equation
48
Nonround Conduits
• When a conduit is noncircular, then engineers
modify the Darcy-Weisbach equation, to use
hydraulic diameter Dh in place of diameter.
• The hydraulic diameter that emerges
from this derivation is
– where the “wetted perimeter” is that portion of the
perimeter that is physically touching the fluid.
49
Nonround Conduits
• The wetted perimeter of a rectangular duct of dimension
L × w is 2L + 2w. Thus, the hydraulic diameter of this
duct is:
• The resistance coefficient f is found using a Reynolds
number based on hydraulic diameter.
• Introduces an uncertainty of 40% for laminar flow and
15% for turbulent flow.
50
Nonround Conduits
• In addition to hydraulic diameter, engineers also
use hydraulic radius, which is defined as
– Notice that the ratio of Rh to Dh is 1/4 instead of 1/2.
Although this ratio is not logical, it is the convention
used in the literature and is useful to remember.
51
Pumps and Systems of Pipes
• Modeling a Centrifugal Pump
• A centrifugal pump is a machine that uses a rotating set
of blades situated within a housing to add energy to a
flowing fluid.
• The amount of energy that is added is
represented by the head of the pump hp and
the rate at which work is done on the flowing
fluid is P = ṁghp.
• Engineers commonly use a graphical solution
involving the energy equation and a pump curve.
52
Pumps and Systems of Pipes
• Modeling a Centrifugal Pump
• Apply Energy Equation
• For a system with one size of pipe,
this simplifies to
• Hence, for any given discharge, a certain head hp must be
supplied to maintain that flow. A head-versus-discharge
curve is constructed and called the system curve.
53
Pumps and Systems of Pipes
• Modeling a Centrifugal Pump
• Now, a given centrifugal pump has a head-versusdischarge curve that is characteristic of that pump. This
curve, called a pump curve, can be acquired from a
pump manufacturer, or it can be measured.
54
Pumps and Systems of Pipes
• Modeling a Centrifugal Pump
• As the discharge increases in a pipe, the head required
for flow also increases.
• However, the head that is produced by the pump
decreases as the discharge increases.
55
Pumps and Systems of Pipes
• Modeling a Centrifugal Pump
• The two curves intersect, and the operating point is at
the point of intersection--that point where the head
produced by the pump is just the amount needed to
overcome the head loss in the pipe.
56
Pumps and Systems of Pipes
• Pipes in Parallel
• A pipe that branches into two parallel pipes and
then rejoins.
• The pressure difference between the two
junction points is the same.
• The elevation difference between the two
junction points is the same.
57
Pumps and Systems of Pipes
• Pipes in Parallel
• Because hL = (p1/γ + z1) - (p2/γ + z2), it follows that hL
between the two junction points is the same in both of
the pipes of the parallel pipe system. Thus,
58
Pumps and Systems of Pipes
• Pipe Networks
• The most common pipe networks are the water
distribution systems for municipalities.
• Engineer is often engaged to
design the original system or
to recommend an economical
expansion to the network.
• An expansion may involve
additional housing or
commercial developments,
or it may be to handle
increased loads within the existing area.
59
Pumps and Systems of Pipes
• Pipe Networks
• Have to estimate the future loads for the system
and will need to have sources (wells or direct
pumping from streams or lakes) to satisfy the
loads, layouts, pipe sizes and cost.
• The design process usually
involves a number of iterations
on pipe sizes and layouts
before the optimum design
(minimum cost) is achieved.
60
Pumps and Systems of Pipes
• Pipe Networks
• Must determine pressures throughout the
network.
• For various combinations of pipe sizes, sources,
and loads. The solution of a problem for a given
layout and a given set of sources and loads
requires that two conditions be satisfied:
• 1. The continuity equation must be satisfied.
That is, the flow into a junction of the network
must equal the flow out of the junction. This
must be satisfied for all junctions.
•
ρQin=ρQout
61
Pumps and Systems of Pipes
• Pipe Networks
• 2. The head loss between any two junctions
must be the same regardless of the path in the
series of pipes taken to get from one junction
point to the other. This requirement results
because pressure must be continuous
throughout the network (pressure cannot have
two values at a given point). This condition leads
to the conclusion that the algebraic sum of head
losses around a given loop must be equal to
zero. Here the sign (positive or negative) for the
head loss in a given pipe is given by the sense of
the flow with respect to the loop, that is,
whether the flow has a clockwise or
counterclockwise direction.
62
Pumps and Systems of Pipes
• Pipe Networks
• Hardy Cross method
• For each loop of the network, a discharge correction is
applied to yield a zero net head loss around the loop
• Consider the isolated loop, the loss of head in the
clockwise direction will be given by
• The loss of head for the loop in the
counterclockwise direction is
63
Pumps and Systems of Pipes
• Pipe Networks
• Hardy Cross method
• For a solution, the clockwise and
counterclockwise head losses have to be equal,
or
• A correction in discharge, ΔQ, will
have to be applied to satisfy the head
loss requirement.
If the clockwise head loss is
greater than the
counterclockwise head loss
64
Pumps and Systems of Pipes
• Pipe Networks
• Hardy Cross method
• Expand the summation on either side of equation and
include only two terms of the expansion:
65
Pumps and Systems of Pipes
• Pipe Networks
• Hardy Cross method
• Thus if ΔQ as computed is positive, the
correction is applied in a counterclockwise sense
(add ΔQ to counterclockwise flows and subtract
it from clockwise flows).
• A different ΔQ is computed for each loop of the
network and applied to the pipes.
• Some pipes will have two ΔQs applied.
66
Pumps and Systems of Pipes
• Pipe Networks
• Hardy Cross method
• For most loop configurations, applying ΔQ as
computed produces too large a correction.
• Fewer trials are required to solve for Qs if
approximately 0.6 of the computed ΔQ is used.
67