fluid_mechanics

12.5 Dimensionless Parameters and Similarity Laws 667
Head, ft
100%
Efficiency
80
60
80
70
40
20
Head
60
50
40
30
20
10
0
Horsepower
0
100
80
60
40
20
0
0 1000 2000 3000 4000 5000
Capacity, gal/min
Brake horsepower Efficiency
0.25
0.20
0.15
C H
0.10
0.05
0
C P (pump)
C H
η
100%
80
60
40
20
0
0 0.025 0.050 0.075 0.100
C Q
0
0.016
0.012
0.008
0.004
η
CP
(a)
(b)
F I G U R E 12.17 Typical performance data for a centrifugal pump: (a) characteristic curves for
a 12-in. centrifugal pump operating at 1000 rpm, (b) dimensionless characteristic curves. (Data from Ref. 8,
used by permission.)
The dimensionless parameter C Q QvD 3 is called the flow coefficient. These three equations
provide the desired similarity relationships among a family of geometrically similar pumps. If
two pumps from the family are operated at the same value of flow coefficient
it then follows that
a Q
vD 3b a Q
1 vD 3b 2
(12.32)
a gh a
v 2 D 2b a gh a
1 v 2 D 2b 2
(12.33)
Pump scaling laws
relate geometrically
similar pumps.
a W# shaft
(12.34)
rv 3 D 5b a W# shaft
1 rv 3 D 5b 2
h 1 h 2
(12.35)
where the subscripts 1 and 2 refer to any two pumps from the family of geometrically similar
pumps.
With these so-called pump scaling laws it is possible to experimentally determine the performance
characteristics of one pump in the laboratory and then use these data to predict the corresponding
characteristics for other pumps within the family under different operating conditions.
Figure 12.17a shows some typical curves obtained for a centrifugal pump. Figure 12.17b shows
the results plotted in terms of the dimensionless coefficients, C Q , C H , C p , and h. From these curves
the performance of different-sized, geometrically similar pumps can be predicted, as can the effect
of changing speeds on the performance of the pump from which the curves were obtained. It is to
be noted that the efficiency, h, is related to the other coefficients through the relationship
h C Q C H C 1 p . This follows directly from the definition of h.
E XAMPLE 12.5
Use of Pump Scaling Laws
GIVEN An 8-in.-diameter centrifugal pump operating at 1200
rpm is geometrically similar to the 12-in.-diameter pump having
the performance characteristics of Figs. 12.17a and 12.17b while
operating at 1000 rpm. The working fluid is water at 60 °F.
FIND For peak efficiency, predict the discharge, actual head
rise, and shaft horsepower for this smaller pump.