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38 Fluid Mechanics, Thermodynamics of Turbomachinery
FIG. 2.8. Relationship between isentropic (overall) efficiency, pressure ratio and small
stage (polytropic) efficiency for a compressor (
D 1.4).
Values of “overall” isentropic efficiency have been calculated using eqn. (2.36) for a
range of pressure ratio and different values of p , and are plotted in Figure 2.8. This
figure amplifies the observation made earlier that the isentropic efficiency of a finite
compression process is less than the efficiency of the small stages. Comparison of
the isentropic efficiency of two machines of different pressure ratios is not a valid
procedure since, for equal polytropic efficiency, the compressor with the highest
pressure ratio is penalised by the hidden thermodynamic effect.
The term polytropic used above arises in the context of a reversible compressor
compressing a gas from the same initial state to the same final state as the irreversible
n
adiabatic compressor but obeying the relation pv = constant. The index n is called
the polytropic index. Since an increase in entropy occurs for the change of state in
both compressors, for the reversible compressor this is only possible if there is a
reversible heat transfer dQ R D Tds. Proceeding farther, it follows that the value of
the index n must always exceed that of
. This is clear from the following argument.
For the polytropic process,
dQ R D du C pdv.
C v
D d.pv/ C pdv.
R
n
Using pv = constant and C v D R/.
1/, after some manipulation the expres-
sion dQ R D .
n//.
1/pdv is derived. For a compression process dv < 0 and
dQ R > 0 then n>
. For an expansion process dv > 0, dQ R < 0 and again n>
.
EXAMPLE 2.1. An axial flow air compressor is designed to provide an overall
total-to-total pressure ratio of 8 to 1. At inlet and outlet the stagnation temperatures
are 300 K and 586.4 K, respectively.
Determine the overall total-to-total efficiency and the polytropic efficiency for the
compressor. Assume that
for air is 1.4.
