Science of Water : Concepts and Applications

Water Hydraulics 63
E 1
smaller fl ow area at section B. This means that the velocity head in the system increases as the water
fl ows into the constricted section. However, the total energy must remain constant. For this to occur,
the pressure head, and therefore the pressure, must drop. In effect, pressure energy is converted into
kinetic energy in the constriction.
The fact that the pressure in the narrower pipe section (constriction) is less than the pressure in
the bigger section seems to defy common sense. However, it does follow logically from continuity
of fl ow and conservation of energy. The fact that there is a pressure difference allows measurement
of fl ow rate in the closed pipe.
Example 3.13
v A 2 /2g
P A w
z A
FIGURE 3.12 The result of the law of conservation. Since the velocity and kinetic energy of the water fl owing
in the constricted section must increase, the potential energy may 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.)
Problem:
In Figure 3.12, the diameter at section A is 8 in., and at section B it is 4 in. The fl ow rate through the
pipe is 3.0 cfs and the pressure at section A is 100 psi. What is the pressure in the constriction at section
B?
Solution:
Step 1: Compute the fl ow area at each section, as follows:
Step 2: From Q = AV or Q/A or, we get
A
A
A
B
V
V
2
(
0. 666ft)
0.
394 (
4
2
ft rounded)
2
(
0. 333ft)
2
0.
087ft ( rounded)
4
A
B
Total energy line
Pressure
drop
A Q B
Constriction
Reference plane
3
30 . ft s 86
. ft/s (rounded)
2
0. 349ft
3
30 . ft s 345
. ft/s (rounded)
2
0. 087ft
v B 2 /2g
P B w
z B
E 2