Page 58 - Fluid Mechanics and Thermodynamics of Turbomachinery

Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 39
Solution. From eqn. (2.28), substituting h D C p T, the efficiency can be written
as,
.
1//
p 02
1 1/3.5
T 02s T 01 p 01 8 1
D D D D 0.85.
C
T 02 T 01 T 02 /T 01 1 586 Ð 4/300 1
From eqn. (2.33), taking logs of both sides and re-arranging, we get,

1 ln .p 02 /p 01 / 1 ln 8
p D D ð D 0.8865

ln .T 02 /T 01 / 3.5 ln 1.9547

Turbine polytropic efficiency

A similar analysis to the compression process can be applied to a perfect gas
expanding through an adiabatic turbine. For the turbine the appropriate expressions
for an expansion, from a state 1 to a state 2, are
p .
1//
T 2 p 2
D .2.37/
T 1 p 1
" # " #
p .
1//
.
1//

p 2 p 2
t D 1 1 . .2.38/
p 1 p 1
The derivation of these expressions is left as an exercise for the student. “Overall”
isentropic efficiencies have been calculated for a range of pressure ratio and different
polytropic efficiencies and are shown in Figure 2.9. The most notable feature of these
results is that, in contrast with a compression process, for an expansion, isentropic
efficiency exceeds small stage efficiency.

FIG. 2.9. Turbine isentropic efficiency against pressure ratio for various polytropic effi-
ciencies (
D 1.4).