Page 367 - Civil Engineering Formulas

296 CHAPTER TWELVE

FUNDAMENTALS OF FLUID FLOW

For fluid energy, the law of conservation of energy is represented by the
Bernoulli equation:
2 2
p 1 V 1 p 2 V 2
Z 1 Z 2 (12.4)
w 2g w 2g
where Z elevation, ft (m), at any point 1 of flowing fluid above an arbitrary datum
1
Z elevation, ft (m), at downstream point in fluid above same datum
2
2
p pressure at 1, lb/ft (kPa)
1
2
p pressure at 2, lb/ft (kPa)
2
3
3
w specific weight of fluid, lb/ft (kg/m )
V velocity of fluid at 1, ft/s (m/s)
1
V velocity of fluid at 2, ft/s (m/s)
2
2
2
g acceleration due to gravity, 32.2 ft/s (9.81 m/s )
The left side of the equation sums the total energy per unit weight of fluid at
1, and the right side, the total energy per unit weight at 2. The preceding equa-
tion applies only to an ideal fluid. Its practical use requires a term to account
for the decrease in total head, ft (m), through friction. This term h , when added
f
to the downstream side, yields the form of the Bernoulli equation most fre-
quently used:
2 2
p 1 V 1 p 2 V 2
Z 1 Z 2 h f (12.5)
w 2g w 2g
The energy contained in an elemental volume of fluid thus is a function
of its elevation, velocity, and pressure (Fig. 12.3). The energy due to eleva-
tion is the potential energy and equals WZ , where W is the weight, lb (kg),
a
of the fluid in the elemental volume and Z is its elevation, ft (m), above
a
some arbitrary datum. The energy due to velocity is the kinetic energy. It
2
equals WV a /2g , where V is the velocity, ft/s (m/s). The pressure energy
a
2
equals Wp /w, where p is the pressure, lb/ft (kg/kPa), and w is the specific
a a
3
3
weight of the fluid, lb/ft (kg/m ). The total energy in the elemental volume
of fluid is
2
E WZ a Wp a WV a (12.6)
w 2g
Dividing both sides of the equation by W yields the energy per unit weight of
flowing fluid, or the total head ft (m):
2
p a V a
H Z a (12.7)
w 2g
2
p /w is called pressure head; V /2g, velocity head.
a a
As indicated in Fig. 12.3, Z p/w is constant for any point in a cross sec-
tion and normal to the flow through a pipe or channel. Kinetic energy at the