Encyclopedia of Environmental Science and Engineering, Volume I and II

1280 WATER FLOW

in which CC 1 / l^12 / is the discharge coefficient. Equation (35)

can be readily extended to multiple conduits in parallel.

Pipe Networks

Introduction The Hardy Cross method is most suitably

adapted to the resolution of pipe networks. The statement of

the problem resolves itself into:

1) the method of balancing heads is directly appli-

cable if the discharges at inlets and outlets are

known,

2) the method of balancing flows is very suitable if

the heads at inlets and outlets are known.

It is assumed that:

a) sizes, lengths and roughness of pipes in the system

are given,

b) law governing friction loss and flow for each pipe

is known,

c) equations for losses in junctions, bends, and other

minor losses are known. These relations are most

conveniently expressed in terms of equivalent

lengths of pipes.

The objectives of the analysis are:

a) to determine the flow distribution in the individual

pipes of the network,

b) to compute the pressure elevation heads at the

junctions.

In applying the Hardy Cross Method, two sets of condi-

tions have to be satisfied:

a) the total change in pressure head along any closed

circuit is zero:

∑H0, (36)

b) the total discharge arriving at any nodal point

equals the total flow leaving it:

∑Q0. (37)

For the pressure head change in any closed path, the clock-

wise positive sign convention is used.

For the discharge continuity requirement at a nodal point,

the inward flow positive sign convention is adopted.

The friction head loss equation is used in the form:

HrQ

(^2). (38)
Using Darcy’s formula:
H
fLV
D
fL
D
 Q
2
2 25 2
8
ggp
⎛
⎝⎜
⎞
⎠⎟
and
r
fL
D

8
gp^25.
Method of Balancing Heads Based on the condition required
by Eq. (36), the following equations for any closed pipe loop
results (Figure 4):
∑∑HrQQ(), 0 2 0 (39)
where Q 0  assumed flow in the circuit for any one pipe,
Q  required flow correction. Expanding Eq. (39) and
approximating by retaining only the first two terms, the flow
correction Q, can be expressed as:
Q
rQ
rQ

 02
(^20)
∑
∑
.
(40)
Method of Balancing Flows Utilising the continuity require-
ment at a pipe junction as given by Eq. (37), the head
correction, H, at anodal point is given by the equation:
H
H
r
H
H
r

⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
∑
∑
12
12
1
2
/
/. (41)
In both Eq. (40) and (41), the proper sign conventions must
be used in the numerators.
OPEN CHANNEL FLOW
Introduction
Open channel flow refers to that class of water discharge in
which the water flows with a free surface. The stream flow
is said to be steady if the discharge does not vary with time.
If the discharge is time dependent, the water flow is termed
unsteady. Uniform flow refers to the case in which the mean
velocities at any cross-section of the stream are identical; if
these mean velocities vary from one cross-section to another,
the flow is considered non-uniform. Steady uniform flow
requires the conveyance section of the stream channel to
be prismatic. Where the water surface profile is controlled

• 
  
  
  
  • Q Q
    FIGURE 4 Pipe network.
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