Page 81 - Fluid Mechanics and Thermodynamics of Turbomachinery

62 Fluid Mechanics, Thermodynamics of Turbomachinery
magnitude and direction. Thus c 1 D c 2 D c and eqn. (3.24) becomes identical with
the Kutta Joukowski theorem obtained for an isolated aerofoil.

Efficiency of a compressor cascade
The efficiency D of a compressor blade cascade can be defined in the same way
as diffuser efficiency; this is the ratio of the actual static pressure rise in the cascade
to the maximum possible theoretical pressure rise (i.e. with p 0 D 0). Thus,

p 2 p 1
D D
1 2 2
2 .c 1 c /
2
p 0
D 1 .
2
c tan ˛ m .tan ˛ 1 tan ˛ 2 /
x
Inserting eqns. (3.7) and (3.9) into the above equation,

D D 1 . (3.25)
C f tan ˛ m
2
Equation (3.20) can be written as /C f + .sec ˛ m /C D /C L which when substituted
into eqn. (3.25) gives

2C D
D D 1 . (3.26)
C L sin 2˛ m
Assuming a constant lift drag ratio, eqn. (3.26) can be differentiated with respect
to ˛ m to give the optimum mean flow angle for maximum efficiency. Thus,

∂ D 4C D cos 2˛ m
D D 0,
2
∂˛ m C L sin 2˛ m
so that
˛ m opt D 45 deg,

therefore
2C D
D max D 1 . (3.27)
C L
This simple analysis suggests that maximum efficiency of a compressor cascade is
obtained when the mean flow angle is 45 deg, but ignores changes in the ratio C D /C L
with varying ˛ m . Howell (1945) calculated the effect of having a specified variation
of C D /C L upon cascade efficiency, comparing it with the case when C D /C L is
constant. Figure 3.6 shows the results of this calculation as well as the variation of
C D /C L with ˛ m . The graph shows that D max is at an optimum angle only a little less
than 45 deg but that the curve is rather flat for a rather wide change in ˛ m . Howell
suggested that value of ˛ m rather less than the optimum could well be chosen with
little sacrifice in efficiency, and with some benefit with regard to power weight ratio
of compressors. In Howell’s calculations, the drag is an estimate based on cascade