fluid_mechanics

12.5 Dimensionless Parameters and Similarity Laws 669
8
⋅ ⋅
W shaft1 /W shaft2
6
Q 1 /Q 2
4
2
h a1 /h a2
0
0 0.5 1 1.5 2
D 1 /D 2
Pump affinity laws
relate the same
pump at different
speeds or geometrically
similar pumps
at the same speed.
and
W # shaft1
W # shaft2
D5 1
D 5 2
(12.41)
Thus, for a family of geometrically similar pumps operating at a given speed and the same flow
coefficient, the flow varies as the diameter cubed, the head varies as the diameter squared, and the
power varies as the diameter raised to the fifth power. These strong effects of diameter variation
are illustrated in the sketch in the margin. These scaling relationships are based on the condition
that, as the impeller diameter is changed, all other important geometric variables are properly scaled
to maintain geometric similarity. This type of geometric scaling is not always possible due to practical
difficulties associated with manufacturing the pumps. It is common practice for manufacturers
to put impellers of different diameters in the same pump casing. In this case, complete geometric
similarity is not maintained, and the scaling relationships expressed in Eqs. 12.39, 12.40, and 12.41
will not, in general, be valid. However, experience has shown that if the impeller diameter change
is not too large, less than about 20%, these scaling relationships can still be used to estimate the
effect of a change in the impeller diameter. The pump similarity laws expressed by Eqs. 12.36
through 12.41 are sometimes referred to as the pump affinity laws.
The effects of viscosity and surface roughness have been neglected in the foregoing similarity
relationships. However, it has been found that as the pump size decreases these effects more
significantly influence efficiency because of smaller clearances and blade size. An approximate,
empirical relationship to estimate the influence of diminishing size on efficiency is 1Ref. 92
1 h 2
a D 15
1
b
1 h 1 D 2
(12.42)
In general, it is to be expected that the similarity laws will not be very accurate if tests on a model
pump with water are used to predict the performance of a prototype pump with a highly viscous
fluid, such as oil, because at the much smaller Reynolds number associated with the oil flow, the
fluid physics involved is different from the higher Reynolds number flow associated with water.
12.5.2 Specific Speed
A useful pi term can be obtained by eliminating diameter D between the flow coefficient and the
head rise coefficient. This is accomplished by raising the flow coefficient to an appropriate exponent
1122 and dividing this result by the head coefficient raised to another appropriate exponent
1342 so that
1QvD 3 2 1 2
1gh av 2 D 2 2 3 4 v1Q
1gh a 2 3 4 N s
(12.43)
The dimensionless parameter N s is called the specific speed. Specific speed varies with flow coefficient
just as the other coefficients and efficiency discussed earlier do. However, for any pump it
is customary to specify a value of specific speed at the flow coefficient corresponding to peak efficiency
only. For pumps with low Q and high h a , the specific speed is low compared to a pump
with high Q and low h a . Centrifugal pumps typically are low-capacity, high-head pumps, and therefore
have low specific speeds.
Specific speed as defined by Eq. 12.43 is dimensionless, and therefore independent of the system
of units used in its evaluation as long as a consistent unit system is used. However, in the United
States a modified, dimensional form of specific speed, N sd , is commonly used, where
v1rpm2 1Q1gpm2
N sd
3h a 1ft24 3 4
(12.44)
In this case N sd is said to be expressed in U.S. customary units. Typical values of N sd are in the
range 500 6 N sd 6 4000 for centrifugal pumps. Both N s and N sd have the same physical meaning,
but their magnitudes will differ by a constant conversion factor 1N sd 2733 N s 2 when v in
Eq. 12.43 is expressed in rads.
Each family or class of pumps has a particular range of values of specific speed associated with
it. Thus, pumps that have low-capacity, high-head characteristics will have specific speeds that are