Environmental Engineering Reference
In-Depth Information
As water passes through the constricted section of the pipe (section
B), we know from continuity of flow that the velocity at section B must
be greater than the velocity at section A because of the smaller flow area
at section B. This means that the velocity head in the system increases
as the water flows into the constricted section; however, the total energy
must remain constant. For this to occur, the pressure head, and there-
fore the pressure, must drop. In effect, pressure energy is converted into
kinetic energy in the constriction. The fact that the pressure in the nar-
rower 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 flow and conservation of energy. The fact that there is a
pressure difference allows measurement of flow rate in the closed pipe.
Example 2.13
Problem:
In Figure 2.12, the diameter at section A is 8 in., and at section B
it is 4 in. The flow rate through the pipe is 3.0 cfs, and the pressure at sec-
tion A is 100 psi. What is the pressure in the constriction at section B?
Solution:
Compute the flow area at each section:
2
π
×
(0.666 ft)
4
2
a
=
=
0.349 ft
(rounded)
a
2
π
×
(
0.333ft)
4
2
a
=
=
0.087 ft
B
From
q
=
a
×
v
or
v
=
q
/
a
, we get:
3.0ft s
0.349 ft
3
v
=
=
8.6ft/s(rounded)
a
2
3
3.0ft s
0.087ft
v
=
=
34.5ft/s(rounded)
B
2
Applying Equation 2.18, we get:
(100 144)
62.4
×
(8.6)
(2
2
(
P
B
144)
62.4
×
34.5)
(2
(
2
+
=
+
×
32.2)
×
32.2)
Note:
The pressures are multiplied by 144 in.
2
/ft
2
to convert from psi to
lb/ft
2
to be consistent with the units for
w
; the energy head terms are in
feet of head.
Continuing, we get:
231 + 1.15 = 2.3
P
B
+ 18.5
and
(232.2
−
18.5)
213.7
2.3
P
B
=
=
=
93 psi(rounde
d)
2.3
