Page 237 - Fluid Mechanics and Thermodynamics of Turbomachinery

218 Fluid Mechanics, Thermodynamics of Turbomachinery
The above equation is applicable to a limiting mean radius ratio for the impeller
given by the empirical expression:
0
r 1 8.16 cos ˇ 2
ε D D exp . (7.19c/
r 2 Z
lim
For values of r 1 /r 2 >ε the slip factor is determined from the empirical expression:
" #
ε 3
0
D w 1 r 1 /r 2 . (7.19d/
w
1 ε
Head increase of a centrifugal pump

The actual delivered head H measured as the head difference between the inlet
and outlet flanges of the pump and sometimes called the manometric head, is less
than the ideal head H i defined by eqn. (7.3b) by the amount of the internal losses.
The hydraulic efficiency of the pump is defined as

H gH
h D D . (7.20/
H i U 2 c 2
From the velocity triangles of Figure 7.2

c r2 tan ˇ 2 .
c 2 D U 2
2
Therefore H D h U .1 2 tan ˇ 2 //g .7.20a/
2
where 2 D c r2 /U 2 and ˇ 2 is the actual averaged relative flow angle at impeller
outlet.
0
With the definition of slip factor, D c 2 /c , H can, more usefully, be directly
2
related to the impeller vane outlet angle, as
2
0
H D h U .1 2 tan ˇ //g. (7.20b/
2 2
In general, centrifugal pump impellers have between five and twelve vanes inclined
backwards to the direction of rotation, as suggested in Figure 7.2, with a vane tip
0 0
angle ˇ of between 50 and 70 deg. A knowledge of blade number, ˇ and 2
2
2
(usually small and of the order 0.1) generally enables to be found using the
Busemann formula. The effect of slip, it should be noted, causes the relative flow
0
angle ˇ 2 to become larger than the vane tip angle ˇ .
2
3
EXAMPLE 7.3 A centrifugal pump delivers 0.1m /s of water at a rotational speed
of 1200 rev/min. The impeller has seven vanes which lean backwards to the direction
0
of rotation such that the vane tip angle ˇ is 50 deg. The impeller has an external
2
diameter of 0.4 m, an internal diameter of 0.2 m and an axial width of 31.7 mm.
Assuming that the diffuser efficiency is 51.5%, that the impeller head losses are
10% of the ideal head rise and that the diffuser exit is 0.15 m in diameter, estimate
the slip factor, the manometric head and the hydraulic efficiency.