Pumps that follow the above relations are called similar or homologous pumps. In particular, when the n variable CH, which involves force are equal in the series of pumps, the pumps are said to be dynamically similar. When the n variable Cq, which relates only to the motion of the fluid are equal in the series of pumps, the pumps are said to be kinematically similar. Finally, when corresponding parts of the pumps are proportional, the pumps are said to be geometrically similar. The relationships of Eqs. (4.30) and (4.31) are called similarity, affinity, or scaling laws.
Considering the power P and the efficiency n as the dependent variables, similar dimensional analyses yield the following similarity relations:
pw3aD5a pooD paldl na = n = nc = ••• = n (4.33)
where CP is called the power coefficient. Note that the efficiencies of similar pumps are equal. The similarity relations also apply to the same pump which, in this case, the subscripts a, h, c,—, and n represent different operating conditions of this same pump.
The power P is the power given to the fluid. In plots of characteristic curves such as Figure 4.8, however, the brake power is the one plotted. Because P bears a ratio to that of the brake power in the form of the efficiency n, the similarity laws that we have developed also apply to the brake power, and figures such Figure 4.8 may be used for scaling brake powers of pumps.
Equation (4.17) expresses efficiencies in terms of heads. Letting Phrake represent brake power, hhrake may be obtained from Phrake as follows:
Equation (4.20) is a special case of this equation.
From the equations derived, the following simplified scaling laws for a given pump operated at different speeds, o, are obtained:
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