51. Impeller with finite number of vanes

So far we have been investigating the performance of an idealised centrifugal pump with an infinite number of vanes and unity efficien­cy. The physical meaning of these assumptions was examined in Sec. 49.

In going over to real pumps we shall begin with eliminating the first assumption, retaining the second for the time being.

Thus, a pump with a finite number of vanes. Real pumps usually have from six to twelve vanes. The relative flow through the vane passages is not so laminar as assumed before and the velocity distri­bution is not uniform, On the leading surface, of the vane, denoted by a "plus" sign in Fig. 136a, the pressure is higher and the velocity lower, and the velocity distribution in the passage is approximately as shown.

The velocity distribution can be regarded as the resultant of two flows: one with uniform velocity distribution, as was the case when

z=oo (Fig. 1366), and a rotational flow inside the passage in the opposite direction of the rotation of the impeller (Fig. 136c). In pure form the rotational flow is present when the discharge through the impeller is zero (Q = 0).

In view of the nonuniformity of the distribution of the relative and absolute velocities in the vane passages when the number of vanes is unite, the mean velocity for a circle of given radius is in­ troduced. Our interest is the mean value of the tangential component of the absolute exit velocity и_шЧ which determines the head de­veloped by the pump. This component is smaller for a finite number of vanes than for an infinite number because the less the number of vanes the less the whirl imparted to the fluid by the impeller. In the absence of vanes (z = 0) the whirl is zero, i. e., v_%a = 0, and the fluid (in the ideal case) issues from the impeller in a radial direction.

A reduction of the velocity v_2u in passing over to a finite number of vanes is also accounted for by the prerotation mentioned before. This relative motion gives rise to an additional absolute velocity hv_2u at the outer periphery of the impeller (see Fig. 136c), which is directed opposite to v_2u and, hence, is subtracted from the latter.

Owing to this the velocity triangle at the impeller exit changes. In Fig. 137, the solid lines give the velocity vectors when the num­ber of vanes is infinite and the broken lines are the velocity vectors for a finite number. The construction was made for identical values of u₂ and v_2r i. e., for identical rotational speeds and rates of dis­charge. The primed values are for the case of a finite number of

vanes.

A reduction of the tangential component v_2u in transition to a finite number of vanes results in a drop in the pumping head. The head that would have been generated by a pump if there were no head losses inside the pump is called the theoretical, or ideal, head, denoted H_tz. From Eq. (12.8), we have

We shall call the ratio of H_tz to H_z the vane-number coefficient:

whence the head in question Is

The problem now is to determine the numerical value of \x. Ob­viously, the coefficient depends first and foremost on the number of vanes 2, though it is also affected by the length of the Vanes, which

depends on the ratio — and on the angle of inclination of the

vanes p₂- *

Theoretical investigations reveal that*-u-dow not depend on the operating conditions of a pump, i; e., on 0, H_ptttl^ or n. It is wholly determined by impeller geometry and is constant for a given impeller.

Without going into the theory of the effect of the number of vanes on the head, here is the conclusion of this theory as represented by a formula for ix:

where

Here, for example, is the value of μ for β₂ = 30° and = 0.5

Table 6

z	4	6	8	10	12	16	24
μ	0.624	0.714	0.768	0.806	0.834	0.870	0.908

Thus, at

As the ratio between H_tg and H_t(g> is constant for a given pump, the theoretical characteristic curve for a finite number of vanes, like the characteristic curve of an idealised pump with a uniform speed of rotation (n = const), is a straight line. At β₂ 90°, it is parallel to the characteristic curve of an idealised pump, and at β₂ < 90° it intersects the latter on the axis of abscissas, as H_tz = 0 and H_t>= 0 at the same discharge

This follows from Eqs (12.12) and (12.18).