Page 228 - Fluid Mechanics and Thermodynamics of Turbomachinery

Centrifugal Pumps, Fans and Compressors 209
where
0 D p 0 /.RT 0 /.

The absolute Mach number M 1 and the relative Mach number M r1 are defined as

M 1 D c x1 /a 1 D M r1 cos ˇ s1 and w s1 D M r1 a 1 .
Using these relations together with eqn. (7.10)
2 3 3
P m RT 01 M a
r1 1 2
sin ˇ s1 cos ˇ s1
D 1/.y 1/
1 C .
1/M
k p 01 1 2
2 1
1 1/2
Since a 01 /a 1 D 1 C .
1/M 2 and a 01 D .
RT 01 / 1/2 the above equation is
2 1
rearranged to give
P m 2 3 2
M sin ˇ s1 cos ˇ s1
r1 (7.11)
k
p 01 .
RT 01 / 1/2 D 1 C .
2 2 1/.
1/C3/2
1
r1
2 1/M cos ˇ s1
This equation is extremely useful and can be used in a number of different ways.
For a particular gas and known inlet conditions one can specify values of
, R, p 01
2
and T 01 and obtain Pm /k as a function of M r1 and ˇ s1 . By specifying a particular
value of M r1 as a limit, the optimum value of ˇ s1 for maximum mass flow can be
found. A graphical procedure is the simplest method of optimising ˇ s1 as illustrated
below.
Taking as an example air, with
D 1.4, eqn. (7.11) becomes
3 2
2 3 M sin ˇ s1 cos ˇ s1
r1
f.M r1 / DPm /. k 01 a / D . (7.11a)
01
4
1 2 2
1 C M cos ˇ s1
r1
5
The rhs of eqn. (7.11a) is plotted in Figure 7.4 as a function of ˇ s1 for M r1 D 0.8
and 0.9. These curves are a maximum at ˇ s1 D 60 deg (approximately).
Shepherd (1956) considered a more general approach to the design of the
compressor inlet which included the effect of a free-vortex prewhirl or prerotation.
The effect of prewhirl on the mass flow function is easily determined as follows.
From the velocity triangles in Figure 7.5,
c 1 D c x / cos ˛ 1 D w 1 cos ˇ 1 / cos ˛ 1 ,

c 1 w 1 cos ˇ 1 cos ˇ 1
M 1 D D D M r1 .
a 1 a 1 cos ˛ 1 cos ˛ 1
Also,
U 1 D w 1 sin ˇ 1 C c 1 sin ˛ 1 D w 1 cos ˇ 1 .tan ˇ 1 C tan ˛ 1 /,
and

P m D 1 A 1 c x1 ,

k 2 k 1 3 3 2
∴ Pm D 1 U w 1 cos ˇ 1 D w cos ˇ 1 .tan ˇ 1 C tan ˛ 1 / . .7.11b/
1
1
 2  2