Section 2.1.1 - Centrifugal Pump Theory - Sizing The Pump

Sizing the Pump   The next step is to determine the approximate size of a pump or
pump installation. Beginning with the impeller, the agent of the energy transfer to the
fluid in a pump, one utilizes the results of the pump hydraulic geometry optimization
process. This also yields the proportions of the velocity diagrams that correspond to the
geometry associated with the desired value of specific speed. These in turn lead to two
major sizing factors for an impeller, which are plotted for typical commercial pumps in
Figure 12; namely the head coefficient ψ (originally defined in Eq. 31) and the outlet flow

coefficient φ_i defined on this figure⁵. Equations in the figure show how these factors are
used to determine the exit radius r₂ (or diameter D₂) and the width b₂ (Figure 8) respectively.
The flow coefficient φ_i is obviously a velocity component ratio. For the head coefficient
of a pump with zero inlet prewhirl to the impeller (that is, V_{θ, 1} = 0), the relevant
velocity ratio (see Figure 3) is V_{θ, 2} /U₂, which in view of Eqs. 15b and 31 is equal to the
ideal head coefficient ψ_i, as this corresponds to the ideal head H_i. Referring to Eq. 15c, it
is obvious that the actual head coefficient ψ = η_HY × ψ_i, or, in the (common) case of zero
prewhirl, ψ_i = V_{θ, 2} /U₂.

Overall pump dimensions are conveniently viewed in terms of factors times the
impeller diameter. The overall diameter exceeds that of the impeller due to the surrounding
casing. For single stage pumps, this casing will include a volute and perhaps a set of
diffuser vanes immediately surrounding the impeller; so, the casing diameter can be 50%
greater than D₂ or more. On multistage pumps (many with radial-outflow diffusers and
return vanes), this excess is often less than 50%.

Single-stage pump axial length includes provision for inlet passageways and bearing
housings and is therefore approximately the same as the overall pump diameter. On the
other hand, the axial length of a stage in a multistage pump is often less than half of the
impeller diameter. Minimizing this “stage length” (and diameter) is a goal of competitive
pump designers in the quest to create a machine of light weight and low cost. But too
small a stage length is accompanied by inferior hydraulic performance because not
enough room is provided for the passages around and within the impeller to turn the fluid
with minimum loss. Further, the bearing housings and the suction and discharge “heads”
or end pieces must be considered in arriving at the overall length of a multistage pump.

This sizing discussion so far has focused on pumps with a radial-outflow geometry. For
the axial-flow geometries of higher specific speeds, the situation is simpler. The diameters
of the propeller (Figure 9) and any stationary vanes downstream (or upstream) are essentially
the same. The approach and discharge piping is of about the same size. A simple
guideline for the maximum radius of the propeller is found from the following approximate
extrapolation of the ψ-curve of Figure 12:

(46)

where the outer radius of the propeller r_t,2 should be substituted for the mean exit radius
r₂ in the ψ-definition. Thus the ψ-curve is in reality a ψ_t-curve for non-radial-flow pumps.
For small departures from radial flow, as illustrated in Figure 8, the ψ-curve of Figure 12
can be used either way.

These guidelines are often all that pump users or fluid system designers need to plan
their installations. To get beyond this overview, one must pursue the hydraulic design in
detail—as afforded by the following development and examples.

Sizing the Pump   The next step is to determine the approximate size of a pump or
pump installation. Beginning with the impeller, the agent of the energy transfer to the
fluid in a pump, one utilizes the results of the pump hydraulic geometry optimization
process. This also yields the proportions of the velocity diagrams that correspond to the
geometry associated with the desired value of specific speed. These in turn lead to two
major sizing factors for an impeller, which are plotted for typical commercial pumps in
Figure 12; namely the head coefficient ψ (originally defined in Eq. 31) and the outlet flow

coefficient φ_i defined on this figure⁵. Equations in the figure show how these factors are
used to determine the exit radius r₂ (or diameter D₂) and the width b₂ (Figure 8) respectively.
The flow coefficient φ_i is obviously a velocity component ratio. For the head coefficient
of a pump with zero inlet prewhirl to the impeller (that is, V_{θ, 1} = 0), the...