53. Pump efficiency

The energy losses in a pump taken into account in rating the over­all efficiency T) are:

1. Hydraulic losses, examined in the previous section and deter­ mined by the hydraulic efficiency [Eq. (12.21)]:

2. Volumetric losses, due to leakage through clearance spaces be tween the impeller and the casing. The impeller drives fluid from the suction pipe to the discharge pipe, but because of the pressure drop it produces some of the fluid leaks back (Fig. 141).

In Sec. 48 we denoted as Q the actual discharge of a pump at the outlet. It follows, then, that the discharge through the impeller is equal to

where q = internal leakage.

Volumetric energy losses are evaluated by the so-called volumetric ejfciency

The values of volumetric losses and efficiency will be discussed in Sec. 56.

3. Mechanical losses, which include energy degradation due to friction in packings and bearings as well as surface friction of the fluid on the impeller. Denoting loss of power due to friction by N_m and total shaft horse­power by N_o, the mechanical efficiency of a pump is

(for method of computing N_m see Sec. 56).

The numerator of this expression repre­sents the so-called internal or hydraulic horsepower and can be expressed by the formula

Now let us write the expression of the overall efficiency of a pump as a ratio of the water horse­power to the shaft horsepower:

and multiply the numerator of this expression by N_h and the denominatorby the same quan­ tity, only as expressed by Eq. (12.28). We have

and after cancelling out and rearranging,

(12.29)

i. e., the overall efficiency of a pump is equal to the product of its hydraulic, volumetric and mechanical efficiencies.

Overall efficiency of centrifugal pumps usually ranges from 0.7 to 0.85; small pumps for auxiliary duty may have lower efficiency values.

'Fig. 142 presents a diagram showing the change in overall and hydraulic efficiency and the characteristic curve of a pump operat­ing at constant rpm.

54. Similarity formulas

Let us investigate similar operating conditions of homologous centrifugal pumps. As mentioned before in Sec. 20, hydrodynamic similarity is provided by geometric, kinematic and dynamic simil­arity. As applied to centrifugal pumps, kinematic similarity means similarity of the velocity triangles constructed for any correspond­ing points of the impellers. Dynamic similarity is ensured by equal­ity of the Reynolds numbers of the flow^Ts through the pumps in question.

When operating conditions of centrifugal pumps are similar a proportionality between heads generated and lost, and between ac­tual delivery and leakage, is observed. It can therefore be assumed that homologous pumps have the same hydraulic and volumetric efficiencies.* The mechanical efficiencies of homologous pumps will generally vary, but the total efficiency can nonetheless be assumed equal without much error.

Let us consider similar operating conditions of two homologous centrifugal pumps. The values referring to the first pump are denoted by the additional subscript I, and to the second, by the subscript II (Fig. 143).

Taking into account that the peripheral velocities of the impel­lers are proportional to the number of rpm times the respective im­peller diameters D, the condition for kinematic similarity at the impeller exit can be written down as follows:

Since, from Eq. (12.10),

and from geometric similarity

we can write, from Eq. (12.30),

♦ In practice the efficiencies tu and ru are not identical in different pumps because of different relative roughness or the inner surfaces, which affect т]_Л (the so-called scale effect), and to disproportions in the relative spacing dimen­sions in pumps of different size, which determine leakage.

This means that the rate of discharge of homologous pumps under similar operating conditions is proportional to the rpm and the cube of the diameters.

From Eq. (12.8), the theoretical heads for an infinite number of vanes are proportional to the product of the peripheral and tangen­tial velocities, while the vane-number coefficient \x is the same for homologous impellers. Consequently,

whence, taking into account (12.30),

The actual head generated by the pump is

(We shall henceforth denote the delivered head by the symbol H without a subscript.)

But, as (т)_л), = (т]л)и, instead of Eq. (12.32) we can write

i.e., the actual heads developed by homologous pumps under simi­lar operating conditions are proportional to the square of the prod­uct of the rpm times the impeller diameter.

From the expression for the water horsepower of a pump, Eq. (12.2), and the developed equations (12.31) and (12.32'), we can write the relationship between the power generated by homologous pumps under similar operating conditions:

If we wish to consider similar operating conditions of the same pump at different rotational speeds n_x and n₂, the Eqs (12.31), (12.32') and (12.33) become simpler, as D and у are the same all

through, and take the form

(the subscripts 1 and 2 denoting the different rpm values).

Equations (12.34) and (12.35) are used tocompute pump characteris­tics for different rotative speeds. If the relationship between H and Q at n_x = const is given, a similar curve for n₂ = const can be obtained by computing the abscissas of the points of the former curve (the rates of discharge) proportionally to the rpm ratio, and the ordinates (the heads), proportionally to the square of that ratio (Fig. 144).

Thus it is possible to calculate and plot the characteristics of a pump for any desired rotative speed and to draw a series of charac­teristic curves for the same pump at different values of n (Fig. 144).

The points A₁ A₂, A₃ A₄ on these curves joined by the coordinate relationships given in (12.34) and (12.35) represent similar operating conditions. The other rows of points (B₁, B₂,B₃, B₄), (C₁ C₂, C₃, C₄), etc., give a second, third, etc., row of similar operating conditions.

It is easy to develop the equations of the curves joining the points of similar operating conditions. According to Eqs (12.34) and (12.35), for any row of points we can write

Hence, for a row of similar operating conditions we have

For another row

Consequently, the points representing similar operating condi­tions in an H-Q coordinate system are located on parabolas of the second power through the point of origin,* shown in dashed lines in Fig. 144.

*It should be noted that the Reynolds number Re changes somewhat alonga parabola of similar operating conditions. Experiments show, however, that

On the basis of the afore-men­tioned considerations concerning the hydraulic and volumetric effi­ciency of pumps under similar operating conditions, the parabolas for similar operating conditions are seen to be curves of constant val­ues of η_h and η_v. It can also be assumed, with a fair degree of ac­curacy, that the overall efficiency is constant along a parabola of similar operating conditions.

In connection with similar operating conditions of centrifugal pumps the following question arises: How long are operating condi­tions similar when the running speed changes, and when does the similarity end?

Pump operating conditions are determined by the point of inter­section of the characteristic curves of the pump and the pipeline. Hence, for a constant pipeline characteristic (uniform throttle conditions) a change in pump rpm shifts the operating point along the pipeline characteristic. If the latter is a parabola of the second power through the origin of the coordinate system it will coincide with one of the similarity parabolas, which means that similarity holds through the rpm changes. Thus, a looping pipeline with tur­bulent flow has a characteristic curve very like a parabola of the second power through the origin of the coordinate system. It can be assumed, therefore, that in this case the change in the running speed of a pump will not affect the similarity of operating condi­tions.

In the case of a fluid pumped from a lower reservoir to a higher one, when the elevation difference , the pipe characteristicwill be of the type shown in Fig. 145. A change in pump speed from rc_t to n₂ will shift the operating point from A to B. But points A and В belong to different similarity parabolas, hence the similarity is violated.

It follows from the foregoing that there are two ways of regulat­ing a centrifugal pump: by throttling, as mentioned in Sec. 52, and by changing the running speed. In the case of throttling the pipeline characteristic changes and the operating point moves along the uniform pump curve (see Fig. 140); when the rpm is changed the pump characteristic varies and the operating point moves along the uniform pipe curve (see Fig. 144).

flow regimes in centrifugal pumps approach the rough-law regime, when thevalue of Re has no perceptible effect (see Sec. 26). Of decisive importance in this case is kinematic similarity.

The second method is more expedient since in changing the rota­tional speed the efficiency of the pump can be maintained more or less uniform (if the pipeline curve is through the origin of the coor­dinate axes). On the other hand, running speed adjustment usually involves auxiliary equipment, and throttling is simpler. In throt­tling the efficiency of the pump varies along the curve shown in Fig. 142 and, consequently, when delivery is reduced considerably pump efficiency falls accordingly.