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Pipe Network Analysis
Hardy-Cross Method
Pipe Network Analysis
Analysis of water distribution system includes determining
quantities of flow and head losses in the various pipe lines, and
resulting residual pressures. In any pipe network, the following two
conditions must be satisfied:
1. The algebraic sum of pressure drops around a closed loop
must be zero, i.e. there can be no discontinuity in pressure.
2. The flow entering a junction must be equal to the flow leaving
that junction; i.e. the law of continuity must be satisfied.
Based on these two basic principles, the pipe networks are
generally solved by the methods of successive approximation. The
widely used method of pipe network analysis is the Hardy-Cross
method.
Hardy-Cross Method
This method consists of assuming a distribution of flow in the
network in such a way that the principle of continuity is satisfied at
each junction. A correction to these assumed flows is then
computed successively for each pipe loop in the network, until the
correction is reduced to an acceptable magnitude.
If Q a is the assumed flow and Q is the actual flow in the pipe, then
the correction d is given by
d=Q-Q a; or Q=Q a+d
Now, expressing the head loss (HL) as
HL=K.Q x
we have, the head loss in a pipe
=K.(Q a+d)x
=K.[Q ax + x.Q ax-1d + .........negligible terms]
=K.[Q ax + x.Q ax-1d]
Now, around a closed loop, the summation of head losses must be
zero.
\
or

SK.[Q ax + x.Q ax-1d] = 0
SK.Q ax = - SKx Q ax-1d

Since, d is the same for all the pipes of the considered loop, it can
be taken out of the summation.
\

SK.Q ax = - d. SKx Q ax-1

or

d =-SK.Q ax/ Sx.KQ ax-1

Since d is given the same sign (direction) in all pipes of the loop,
the denominator of the above equation is taken as the absolute

sum of the individual items in the summation. Hence,
or

d =-SK.Q ax/ S l x.KQ ax-1 l

or

d =-SHL / x.S lHL/Q al
where HL is the head loss for assumed flow Q a.

The numerator in the above equation is the algebraic sum of the
head losses in the various pipes of the closed loop computed with
assumed flow. Since the direction and magnitude of flow in these
pipes is already assumed, their respective head losses with due
regard to sign can be easily calculated after assuming their
diameters. The absolute sum of respective KQ ax-1 or HL/Q a is then
calculated. Finally the value of d is found out for each loop, and the
assumed flows are corrected. Repeated adjustments are made until
the desired accuracy is obtained.
The value of x in Hardy- Cross method is assumed to be constant
(i.e. 1.85 for Hazen-William's formula, and 2 for Darcy-Weisbach
formula)
Worked-out Example