Mechanics of Fluids

630 Fluid machines
Fig. 13.28
We assume for the moment that there is no discrepancy between the direction
of the relative velocity R2 and the outlet edge of a blade. Consequently
the blade outlet angle φ2 is identical with the angle β2 in the diagram.
Figure 13.27 shows the differences in the outlet vector diagrams of the
three types of impeller for the same blade velocity u2.
Ideally, that is, if the fluid were frictionless, the increase of total energy
across the pump divided by mass would be u2vw2. Now, from Fig. 13.27.
vw2 = u2 − vr2 cot β2 where vr represents the radial component of the fluid
velocity (sometimes termed velocity of flow). If Q represents the volume
rate of flow through the pump and A2 the outlet area perpendicular to vr2
(i.e., the peripheral area of the impeller less the small amount occupied by
the blades themselves) then for uniform conditions vr2 = Q/A2. The ideal
increase of energy divided by mass therefore equals
�
u2vw2 = u2(u2 − vr2 cot β2) = u2 u2 − Q
�
cot β2
A2
(13.20)
The blade velocity u2 is proportional to the rotational speed ω, so the ideal
energy increase gH equals
C1ω 2 − C2ωQ
where C1 and C2 are constants. Thus for a fixed speed ω the variation of H
with Q is ideally linear as shown in Fig. 13.28.
In practice, however, energy losses occur and some of the assumptions
on which eqn 13.20 rests are not fulfilled. Consider the flow in the volute.
Apart from frictional effects, no torque is applied to a fluid particle once
it has left the impeller. The angular momentum of the particle is therefore
constant, that is it follows a path along which vwr = constant. Ideally, the
radial velocity from the impeller does not vary round the circumference. The
combination of uniform radial velocity with the free vortex (vwr = constant)
gives a pattern of spiral streamlines which should be matched by the shape
of the volute. The latter is thus an important feature of the design of the
pump. At maximum efficiency about 10% of the energy increase produced
by the impeller is commonly lost in the volute. Even a perfectly designed
volute, however, can conform to the ideal streamline pattern at the design
conditions only. At rates of flow greater or less than the optimum there are
increased losses in the volute (Fig. 13.29); in addition there are variations of
pressure and therefore of radial velocity vr2 round the impeller.