fluid_mechanics

666 Chapter 12 ■ Turbomachines
curve applies2. However, for a relatively flat system curve, as shown in Fig. 12.16b, a significant increase
in flowrate can be obtained as the operating point moves from point 1A2 to point 1B2.
12.5 Dimensionless Parameters and Similarity Laws
Dimensionless pi
terms and similarity
laws are important
pump considerations.
As discussed in Chapter 7, dimensional analysis is particularly useful in the planning and execution
of experiments. Since the characteristics of pumps are usually determined experimentally, it
is expected that dimensional analysis and similitude considerations will prove to be useful in the
study and documentation of these characteristics.
From the previous section we know that the principal, dependent pump variables are the actual
head rise, h shaft power, W # a ,
shaft, and efficiency, h. We expect that these variables will depend
on the geometrical configuration, which can be represented by some characteristic diameter, D, other
pertinent lengths, / i , and surface roughness, e. In addition, the other important variables are flowrate,
Q, the pump shaft rotational speed, v, fluid viscosity, m, and fluid density, r. We will only consider
incompressible fluids presently, so compressibility effects need not concern us yet. Thus, any one
of the dependent variables h a , W # shaft, and h can be expressed as
dependent variable f 1D, / i , e, Q, v, m, r2
and a straightforward application of dimensional analysis leads to
dependent pi term fa / i
D , e D , Q rvD2
vD3, m
b
(12.28)
The dependent pi term involving the head is usually expressed as C H gh av 2 D 2 , where
gh a is the actual head rise in terms of energy per unit mass, rather than simply h a , which is energy
per unit weight. This dimensionless parameter is called the head rise coefficient. The dependent
pi term involving the shaft power is expressed as C p W # shaftrv 3 D 5 , and this standard
dimensionless parameter is termed the power coefficient. The power appearing in this dimensionless
parameter is commonly based on the shaft 1brake2 horsepower, bhp, so that in BG units,
W # shaft 550 1bhp2. The rotational speed, v, which appears in these dimensionless groups is expressed
in rads. The final dependent pi term is the efficiency, h, which is already dimensionless.
Thus, in terms of dimensionless parameters the performance characteristics are expressed as
C H gh a
v 2 D f 2 1 a / i
D , e D , Q rvD2
vD3, m
b
C p W# shaft
rv 3 D f 5 2 a / i
D , e D , Q rvD2
vD3, m
b
h rgQh a
W # f 3 a / i
D , e D , Q rvD2
shaft
vD
3, m
b
The last pi term in each of the above equations is a form of Reynolds number that represents
the relative influence of viscous effects. When the pump flow involves high Reynolds numbers, as
is usually the case, experience has shown that the effect of the Reynolds number can be neglected.
For simplicity, the relative roughness, eD, can also be neglected in pumps since the highly irregular
shape of the pump chamber is usually the dominant geometric factor rather than the surface
roughness. Thus, with these simplifications and for geometrically similar pumps 1all pertinent dimensions,
/ i , scaled by a common length scale2, the dependent pi terms are functions of only
QvD 3 , so that
gh a
v 2 D 2 f 1 a Q
vD 3b
W # shaft
rv 3 D 5 f 2 a Q
vD 3b
h f 3 a Q
vD 3b
(12.29)
(12.30)
(12.31)