Page 193 - Fluid Mechanics and Thermodynamics of Turbomachinery

174 Fluid Mechanics, Thermodynamics of Turbomachinery
2
(ii) The rate of mass flow, Pm D .r 2 r / c x
t h
2
D .0.5 2 0.45 /1.5 ð 136 D 30.4 kg/s.
(iii) The power absorbed by the stage,

P W c DPmU t .c 2t c 1t /
DPmU t c x .tan ˛ 2t tan ˛ 1t /
p p
D 30.4 ð 314.2 ð 136. 3 1/ 3/
D 1.5MW.

(iv) At inlet to the rotor tip,
p
c 1t D c x tan ˛ 1 D 136/ 3 D 78.6 m/s.

The absolute flow is a free-vortex, rc D constant.
Therefore c 1h D c 1t .r t /r h / D 78.6 ð 0.5/0.45 D 87.3 m/s.
At outlet to the rotor tip,
p
c 2t D c x tan ˛ 2 D 136 ð 3 D 235.6 m/s.
Therefore c 2h D c 2t .r t /r h / D 235.6 ð 0.5/0.45 D 262 m/s.
The flow angles at the hub are,
tan ˛ 1 D c 1h /c x D 87.3/136 D 0.642,

tan ˛ 1 D 1.436,
tan ˇ 1 D U h /c x
tan ˛ 2 D c 2h /c x D 262/136 D 1.928,
tan ˛ 2 D 0.152.
tan ˇ 2 D U h /c x
Thus ˛ 1 D 32.75 ° , ˇ 1 D 55.15 ° , ˛ 2 D 62.6 ° , ˇ 2 D 8.64 ° at the hub.
(v) The reaction at the hub can be found by several methods. With eqn. (6.9)

R D 1 k/r 2
and noticing that, from symmetry of the velocity triangles,

2
R D 0.5at r D r t , then k D 0.5r .
t
2
Therefore R h D 1 0.5.0.5/0.45/ D 0.382.
The velocity triangles will be asymmetric and similar to those in Figure 5.4(b).
The simplicity of the flow under free-vortex conditions is, superficially, very
attractive to the designer and many compressors have been designed to conform to
this flow. (Constant (1945, 1953) may be consulted for an account of early British
compressor design methods.) Figure 6.4 illustrates the variation of fluid angles and
Mach numbers of a typical compressor stage designed for free-vortex flow. Charac-
teristic of this flow are the large fluid deflections near the inner wall and high Mach
numbers near the outer wall, both effects being deleterious to efficient performance.