Page 242 - Fluid Mechanics and Thermodynamics of Turbomachinery

Centrifugal Pumps, Fans and Compressors 223

1 2 2 1 2 2
h 1 D .U U / C .w w /,
h 2 2 1 1 2
2 2
1 2 1 2 1 2 2 1 2 2
∴ h 2 D .h 1 C w 1 U / C .U 2 w / D h 01 C .U 2 w /
1
2
2
2 2 2 2
hence,

2
2
U w w 2
T 2 2 2 2 2
D 1 C 2 D 1 C .
1/M u 1 2 , .7.27/
T 01 a / .
1/ U
01 2
2
since h 01 D C p T 01 D a /.
1/.
01
From the exit velocity triangle, Figure 7.7,
2 2 2 2 0 2
w D c C .U 2 c 2 / D c C .U 2 c /
2 r2 r2 2
2 0 2
D c C [U 2 .U 2 c r2 tan ˇ /] ,
r2
2
2
w 2 2 0 2
1 D 1 [1 .1 2 tan ˇ /] . .7.28/
2 2
U 2
Substituting eqns. (7.26), (7.27) and (7.28) into eqn. (7.25), we get:
h 2 i
M 2 2 1 2 tan ˇ 0 C 2
2 u 2 2
M D . (7.29)
2
2
1
2
0
1 C .
1/ M f1 2 1 .1 2 tan ˇ / g
2 u 2 2
Although eqn. (7.29) at first sight looks complicated it reduces into an easily
managed form when constant values are inserted. Assuming the same values used
0
previously, i.e.
D 1.4, D 0.9, 2 D 0.375 and ˇ D 0, 15, 30 and 45 deg, the
2
solution for M 2 can be written as
AM u
, (7.29a)
M 2 D q
2
.1 C BM /
u
where the constants A and B are as shown in Table 7.1, and, from which the curves
of M 2 against M u in Figure 7.14 have been calculated.
According to Whitfield and Baines (1990) the two most important aerodynamic
parameters at impeller exit are the magnitude and direction of the absolute Mach
number M 2 .If M 2 has too high a value, the process of efficient flow deceleration
within the diffuser itself is made more difficult leading to high friction losses as well
as the increased possibility of shock losses. If the flow angle ˛ 2 is large the flow
path in the vaneless diffuser will be excessively long resulting in high friction losses
TABLE 7.1. Constants used to evaluate M 2
0
ˇ (degrees)
2
Constant 0 15 30 45
A 0.975 0.8922 0.7986 0.676
B 0.2 0.199 0.1975 0.1946