PIPE FLOW

HYDRAULICS CVE 304

DR. A.A. BADEJO

Principles of Pipe Flow
• In a pipe flow, fluid fills the entire cross section, and no free surface is formed, since
the fluid has no free surface, it can be either a liquid or a gas. The fluid pressure is
generally greater than the atmospheric pressure but in certain reaches it may be less
than the atmospheric pressure, allowing free flow to continue through siphon action.
• However, if the pressure is much less than the atmospheric pressure, the dissolved
gases in the fluid will come out and the continuity of the fluid in the pipeline will be
hampered and flow will stop.
• As the fluid flows over the solid boundary a shear stress will be developed at the
surface of contact which will oppose fluid motion. This so-called frictional resistance
results in an energy transfer within the system, experienced as a ‘loss’, measurable in
a fluid flow by changes in fluid pressure or head. In addition to the losses attributable
to friction, separation/form losses due to the flow disruption at changes in section,
direction or around valves and other flow obstructions also contribute to the overall
energy transfers to be accounted for.
The first approach to the analysis of bounded systems is to consider the
energy balance between two chosen locations along the flow

For flow across the control volume boundaries represented by the

conditions at A and B, the energy audit may be expressed, in terms of
energy per unit volume, as
• The pressure loss experienced as a result of friction and separation of
the flow from the walls of the conduit has been shown to be defined
by a term of the form , where u is the local flow velocity and K is a
constant dependent upon the conduit parameters, i.e. length,
diameter, roughness or fitting type, utilized here to represent both
frictional and separation losses.
• Traditionally in the study of water conduits the steady flow energy
equation has been cast in its form of energy per unit weight, resulting
in all the terms having the dimensions of head
• Also, for steady flow to be maintained it is necessary that:
Mass per unit time entering the control volume at A = Mass per unit
time leaving the control volume at B.
• For incompressible flow the density remains constant and hence the
continuity of mass flow equation above reduces to:

• Analysis of all steady flow problems in pipes and channels is based on

the application of the steady flow energy equation and the continuity
of volumetric flow equation, applied between suitable points in the
system.
Incompressible flow through ducts and pipes
• For incompressible flow, since there is no change of density with
pressure, the steady flow energy equation reduces to a form of
Bernoulli’s equation with the addition of terms for the energy losses
due to friction and separation, for work done by the fluid in driving
turbines or for work done on the fluid by the introduction of a pump
or fan.
• The pressure loss, Δp, or energy lost per unit volume due to friction,
may be conveniently expressed via the Darcy equation

• For a circular cross-section conduit flowing full. In terms of head this

expression becomes.
• Both forms of the Darcy equation may be applied to either laminar or
turbulent flow provided the correct form of friction factor, f, is
introduced. In laminar flow f =16/Re and hence depends only on flow
velocity, v. This second form of the Darcy equation may also be
utilized in the study of steady, uniform, free surface flow, provided
that the conduit diameter hydraulic mean depth, m = D/4, is replaced
by an appropriate value of m
To Note
• The Darcy-Weisbach equation is equivalent to the Hagen-
Poiseuille equation for laminar flow with the exception that
an empirical friction factor f is introduced
• The f value used in the equation is different from that used
in the American practise with the following relationship

f American  4 f
 Sometimes the f is replaced by the Greek letter  where
  f American  4 f
 Great care must be taken in choosing f with attention given
to the source of the value
Surface Resistance

The coefficient of surface resistance for turbulent flow depends on the

average height of roughness projection, Ɛ, of the pipe wall.
Darcy–Weisbach Equation
The head loss on account of surface resistance is given by the Darcy–
Weisbach equation:
fLV 2 8 fLQ 2
hf   2 5
2 gD  gD
f = friction factor (dimensionless) american
hf = head-loss due to friction (m)
D = internal diameter of pipe (m)
V = mean velocity of flow (ms-1)
Q = Discharge (m3s-1)
L = Length of pipe (m)
NB: f is obtained empirically from developed equations, graphs and
tables
Form Resistance

• The form-resistance losses are due to bends, elbows, valves, enlargers,

reducers etc. Unevenness of inside pipe surface on account of imperfect
workmanship also causes form loss. A form loss develops at a pipe
junction where many pipelines meet. Similarly, form loss is also created
at the junction of pipeline and service connection.
• All these losses, when added together, may form a sizable part of overall
head loss.
• In a water supply network, form losses play a significant role. However,
form losses are unimportant in water transmission lines like gravity
mains or pumping mains that are long pipelines having no off-takes.
 Form Resistance Contd.

Form loss is expressed in the following form:

V2 8Q 2
hm  k f  kf 2 4
2g  gD
where
hm  Local head loss
k f  Constant for a particular fitting (e.g. junction, valve etc)
V  Average velocity of flow in pipe where the local losses occur
Q  Flow rate
Pipe bend

In the case of pipe bend, kf depends on bend angle α and bend radius R
    0.5
D
3.5

k f  0.0733  0.923  
 (Swamee,
 R   1990)
Elbows

Elbows are used for providing sharp turns in pipelines. The loss
coefficient for an elbow is given by
2.17
k f  0.442

where α = elbow angle in radians

Valves

Valves are used for regulating the discharge by varying the head loss
accrued by it. For a 20% open sluice valve, loss coefficient is as high as
31. Even for a fully open valve, there is a substantial head loss
Kf for partly closed valves (Swamee, 1990)
• Valve Type Form-Loss Coefficient k
f

Sluice valve 0.15

Switch valve 2.4
Angle valve 5.0
Globe valve 10.0
Transitions

• Transition is a gradual expansion (called enlarger) or gradual contraction

(called reducer). In the case of transition, the head loss is given by:

V1  V2 2 8D22  D12 Q 2

hm  k f  kf
2g  2 gD14 D24

where the suffixes 1 and 2 refer to the beginning and end of the transition,
respectively.
• The loss coefficient depends on how gradual or abrupt the transition is.
Losses at Sudden Contraction
Pipe Junction

• Little information is available regarding the form loss at a pipe

junction where many pipelines meet. The form loss at a pipe junction
may be taken as:
2
V
hm  k f max
2g
• where Vmax is maximum velocity in a pipe branch meeting at the
junction
Pipe flow problems

In pipe flow, there are three types of problems pertaining to

determination of:
(a) The nodal head;
(b) The discharge through a pipe link; and
(c) The pipe diameter.
In the nodal head problem, the known quantities are L, D, hL, Ɛ, ɤ, and
kf. The nodal head h2 is obtained from:
2
 fL  8Q
h2  h1  z1  z 2   k f   2 4
 D   gD
Equivalent Pipe

• Fitting losses may be included by the addition of an equivalent length of pipe or

duct that would generate the same friction loss as the separation of flow around
the fitting generates; this extra equivalent length is simply added to the conduit
length and is normally expressed as so many conduit diameters
• In the water supply networks, the pipe link between two nodes may consist of a
single uniform pipe size (diameter) or a combination of pipes in series or in parallel.
2
8 fLQ
hL  2 5
 gD
• The discharge Q flows from node A to B through a pipe of uniform diameter D and
length L. The head loss in the pipe can simply be calculated using Darcy–Weisbach
equation, considering hL = hf as:
Equivalent Pipe Contd.

• The set of pipes arranged in parallel and series can be replaced with a
single pipe having the same head loss across points A and B and also
the same total discharge Q.
• Such a pipe is defined as an equivalent pipe.
• Equivalent pipe is an alternative way to express the minor losses in
pipe fittings
• Length of a straight pipe of the same nominal diameter as the
diameter of the pipe where the valve is fitted that would have same
resistance (due to friction) as the valve
Pipes in Series

• When pipes of different diameters are connected end-to-end to form

a pipeline, so that the fluid flows through each in turn, the pipes are
said to be in series. The total loss of energy, or pressure loss, over the
whole pipeline will be the sum of the losses for each pipe together
with any separation losses such as might occur at the junctions,
entrance or exit.
Pipes in Series Contd.
Pipes in Series
• In case of a pipeline made up of different lengths of different
diameters the following head loss and flow conditions should be
satisfied:
hL = hL1 + hL2 + hL3 + ….
Q = Q1 = Q2 = Q3 = ….
• Using the Darcy–Weisbach equation with constant friction factor f,
and neglecting minor losses, the head loss in N pipes in series can be
calculated as: N
8 fL Q 2
hL   2
i
5
i 1  gDi
Pipe in Series Contd.

Denoting equivalent pipe diameter as De, the head loss can be

rewritten as:
8 fQ 2 N
5  i
hL  2 L
 gDe i 1

Equating the 2 equations

0.2
 N

  Li 
De   Ni 1 
 Li 
 5 
 i 1 Di 
Pipes in Parallel

• When two reservoirs are connected by two or more pipes in parallel,

the fluid can flow from one to the other by a number of alternative
routes.
• The difference of head h available to produce flow will be the same
for each pipe. Thus, each pipe can be considered separately, entirely
independently of any other pipes running in parallel.
• For incompressible flow, the steady flow energy equation can be
applied for flow by each route and the total volume rate of flow will
be the sum of the volume rates of flow in each pipe.
Pipes in Parallel

Notes:
 Fluid may flow down any of the available pipes at different rates
 Head difference over each pipe will always be the same
 The total volume flow rate will be the sum of the flow in each pipe
Pipes in Parallel

• If the pipes are arranged in parallel, the following head loss and flow
conditions should be satisfied:
hL = hL1 = hL2 = hL3 = ….
Q = Q1 + Q2 + Q3 + ….

• The pressure head at nodes 1 and 2 remains constant, meaning

thereby that head loss in all the parallel pipes will be the same.
• Using the Darcy–Weisbach equation and neglecting minor losses, the
0 .5
discharge Qi in pipe i can be calculated as: 2  gDi hL 
Q1  Di  
 8 fLi 
Pipes in Parallel Contd.
Thus for N pipes in parallel 0.5
N
 gDi hL 
Q    D 
i
2

i 1  8 fLi 
The discharge Q flowing in the equivalent pipe is
0.5
 gDe hL 
Q  D  2
e

 8 fLi 

where L is the length of the equivalent pipe. This length may be different
from any of the pipe lengths L1, L2, L3, etc.
Pipes in Parallel Contd.

• Equating these two equations of discharge

0.4
N L 0.5

De     Di2.5 
 i 1  Li  