Water Hydraulics - Mathematics and Statistics for the Environment - page 597

22.4.1 b ernoulli ' s e quation
In a hydraulic system, total energy head is equal to the sum of three individual energy heads. This
can be expressed as

Total Head = Elevation Head + Pressure Head + Velocity Head

This can be expressed mathematically as

2

Ez p
w

V

(22.16)

=+ +

2 g

where
E = Total energy head.
z = Height of the water above a reference plane (ft).
p = Pressure (psi).
w = Unit weight of water (62.4 lb/ft 3 ).
v = Flow velocity (ft/sec).
g = Acceleration due to gravity (32.2 ft/sec 2 ).

Consider the constriction in the section of pipe shown in Figure 22.4. We know, based on the law
of energy conservation, that the total energy head at section A ( E 1 ) must equal the total energy head
at section B ( E 2 ), and using Equation 22.16 we get Bernoulli's equation:

2 2

P
w

V P

w

V

AA BB

z ++=++ z

(22.17)

A B

2 g 2 g

The pipeline system shown in Figure 22.4 is horizontal; therefore, we can simplify Bernoulli's equa-
tion because z A = z B . Because they are equal, the elevation heads cancel out from both sides, leaving

Total energy line

2

2

v B /2 g

v A /2 g

Pressure
drop

P A / w P B / w

E 1 E 2

Q B

A

Constriction

z B

z A

Reference plane

FIGURE 22.4 The law of conservation: Because the velocity and kinetic energy of the water flowing in
the constricted section must increase, the potential energy will decrease. This is observed as a pressure drop
in the constriction. (Adapted from Nathanson, J.A., Basic Environmental Technology: Water Supply, Waste
Management, and Pollution Control , 2nd ed., Prentice-Hall, Upper Saddle River, NJ, 1997, p. 29.)