Page 233 - Fluid Mechanics and Thermodynamics of Turbomachinery

214 Fluid Mechanics, Thermodynamics of Turbomachinery
The relative eddy concept

Suppose that an irrotational and frictionless fluid flow is possible which passes
through an impeller. If the absolute flow enters the impeller without spin, then at
outlet the spin of the absolute flow must still be zero. The impeller itself has an
angular velocity  so that, relative to the impeller, the fluid has an angular velocity
of ; this is the termed the relative eddy. A simple explanation for the slip effect
in an impeller is obtained from the idea of a relative eddy.
At outlet from the impeller the relative flow can be regarded as a through-flow
on which is superimposed a relative eddy. The net effect of these two motions is
that the average relative flow emerging from the impeller passages is at an angle to
the vanes and in a direction opposite to the blade motion, as indicated in Figure 7.8.
This is the basis of the various theories of slip.
Slip factor correlations

One of the earliest and simplest expressions for the slip factor was obtained by
Stodola (1927). Referring to Figure 7.9 the slip velocity, c s D c 0 c 2 , is consid-
2
ered to be the product of the relative eddy and the radius d/2 of a circle which
can be inscribed within the channel. Thus c s D d/2. If the number of vanes is
0
denoted by Z then an approximate expression, d ' .2 r 2 /Z/ cos ˇ can be written
2
if Z is not small. Since  D U 2 /r 2 then
U 2 cos ˇ 0 2
c s D . (7.13c/
Z
0
Now as c 0 c r2 tan ˇ the Stodola slip factor becomes
2 D U 2 2
c 2 c s
D 0 D 1 0 (7.14/
c c r2 tan ˇ
2 U 2 2
or,
. /Z/ cos ˇ 0 2
D 1 (7.15/
1 2 tan ˇ 0
2
where 2 D c r2 /U 2 .

FIG. 7.8. (a) Relative eddy without any throughflow. (b) Relative flow at impeller exit
(throughflow added to relative eddy).