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232 Chapter 7 Water Distribution Systems: Modeling and Computer Applications
7.3 ENERGY LOSSES AND GAINS

The hydraulic theory behind friction losses is the same for pressure piping as it is for open
channel hydraulics. The most commonly used methods for determining head losses in
pressure piping systems are the Hazen-Williams equation and the Darcy-Weisbach equa-
tion (see Chapter 5). Many of the general friction loss equations can be simplified and re-
vised because of the following assumptions that can be made for a pressure pipe system:
1. Pressure piping is almost always circular, so the flow area, wetted perimeter, and
hydraulic radius can be directly related to diameter.
2. Pressure systems flow full (by definition) throughout the length of a given pipe, so
the friction slope is constant for a given flow rate. This means that the energy grade
line and the hydraulic grade line drop linearly in the direction of flow.
3. Because the flow rate and cross-sectional area are constant, the velocity must also
be constant. By definition, then, the energy grade line and hydraulic grade line are
2
parallel, separated by the constant velocity head (v /2g).
These simplifications allow for pressure pipe networks to be analyzed much more
quickly than systems of open channels or partially full gravity piping. Several hydraulic
components that are unique to pressure piping systems, such as regulating valves and
pumps, add complexity to the analysis.
Pumps are an integral part of many pressure systems and are an important part of
modeling head change in a network. Pumps add energy (head gains) to the flow to coun-
teract head losses and hydraulic grade differentials within the system. Several types of
pumps are used for various purposes (see Chapter 8); pressurized water systems typically
have centrifugal pumps.
To model the behavior of the pump system, additional information is needed to ascer-
tain the actual point at which the pump will be operating. The system operating point is the
point at which the pump curve crosses the system curve—the curve representing the static
lift and head losses due to friction and minor losses. When these curves are superimposed
(as in Fig. 7.2), the operating point is easily located.
As water surface elevations and demands throughout the system change, the static
head and head losses vary. These changes cause the system curve to move around, whereas
the pump characteristic curve remains constant. These shifts in the system curve result in a
shifting operating point over time (see Chapter 8).
A centrifugal pump’s characteristic curve is fixed for a given motor speed and im-
peller diameter, but can be determined for any speed and any diameter by applying the
affinity laws. For variable-speed pumps, these affinity laws are presented in Eq. 7.1:
= and H n 2
n
Q 1 1 1 = a 1 b (7.1)
Q 2 n 2 H 2 n 2
where
3
3
Q pump flow rate, m /s (ft /s)
H pump head, m (ft)
n pump speed, rpm.
Thus, pump discharge rate is proportional to pump speed, and the pump discharge
head is proportional to the square of the speed. Using this relationship, once the pump curve
is known, the curve at another speed can be predicted. Figure 7.3 illustrates the affinity laws
applied to a variable-speed pump. The line labeled “Best Efficiency Point” indicates how
the best efficiency point changes at various speeds.