fluid_mechanics

668 Chapter 12 ■ Turbomachines
SOLUTION
As is indicated by Eq. 12.31, for a given efficiency the flow coefficient
has the same value for a given family of pumps. From Fig.
12.17b we see that at peak efficiency Thus, for the
8-in. pump
Q C Q vD 3 C Q 0.0625.
10.062521120060 revs212p radrev21812 ft2 3
Q 2.33 ft 3 s
(Ans)
and
W # shaft C p rv 3 D 5
W # shaft
10.014211.94 slugsft 3 21126 rads2 3 1812 ft2 5
7150 ft # lbs
7150 ft # lbs
550 ft #
13.0 hp
lbshp
(Ans)
or in terms of gpm
(Ans)
The actual head rise and the shaft horsepower can be determined
in a similar manner since at peak efficiency C H 0.19
and C p 0.014, so that with v 1200 revmin11 min60 s2
12p radrev2 126 rads
h a C Hv 2 D 2
g
Q 12.33 ft 3 s217.48 galft 3 2160 smin2
1046 gpm
10.1921126 rads22 1812 ft2 2
32.2 fts 2 41.6 ft
(Ans)
COMMENT This last result gives the shaft horsepower,
which is the power supplied to the pump shaft. The power actually
gained by the fluid is equal to gQh a , which in this example is
p f gQh a 162.4 lbft 3 212.33 ft 3 s2141.6 ft2 6050 ft # lbs
Thus, the efficiency, h, is
p f
h
W # shaft
6050
7150 85%
which checks with the efficiency curve of Fig. 12.17b.
Effects of changes
in pump operating
speed and impeller
diameter are often
of interest.
8
6
4
2
⋅ ⋅
W shaft1 /W shaft2
h a1 /h a2
Q 1 /Q 2
0
0 0.5 1 1.5 2
w 1 /w 2
12.5.1 Special Pump Scaling Laws
Two special cases related to pump similitude commonly arise. In the first case we are interested
in how a change in the operating speed, v, for a given pump, affects pump characteristics. It follows
from Eq. 12.32 that for the same flow coefficient 1and therefore the same efficiency2 with
D 1 D 2 1the same pump2
(12.36)
The subscripts 1 and 2 now refer to the same pump operating at two different speeds at the same
flow coefficient. Also, from Eqs. 12.33 and 12.34 it follows that
and
(12.37)
(12.38)
Thus, for a given pump operating at a given flow coefficient, the flow varies directly with speed,
the head varies as the speed squared, and the power varies as the speed cubed. These effects of angular
velocity variation are illustrated in the sketch in the margin. These scaling laws are useful in
estimating the effect of changing pump speed when some data are available from a pump test obtained
by operating the pump at a particular speed.
In the second special case we are interested in how a change in the impeller diameter, D, of
a geometrically similar family of pumps, operating at a given speed, affects pump characteristics.
As before, it follows from Eq. 12.32 that for the same flow coefficient with v 1 v 2
Similarly, from Eqs. 12.33 and 12.34
Q 1
Q 2
v 1
v 2
h a1
h a2
v 1 2
v 2
2
W # shaft1
W # shaft2
v3 1
v 3 2
Q 1
Q 2
D 1 3
D 2
3
(12.39)
h a1
h a2
D2 1
D 2 2
(12.40)