Page 56 - Fluid Mechanics and Thermodynamics of Turbomachinery

Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 37

FIG. 2.7. Incremental change of state in a compression process.

Substituting v D RT/p in eqn. (2.31), then
R T dp
p D
C p p dT

and hence
dT .
1/ dp
D .2.32/
T
p p
as C p D
R/.
1/.

Integrating eqn. (2.32) across the whole compressor and taking equal efficiency for
each infinitesimal stage gives,
.
1// p
T 2 p 2
D . (2.33)
T 1 p 1
Now the isentropic efficiency for the whole compression process is

c D .T 2s T 1 //.T 2 T 1 / (2.34)

if it is assumed that the velocities at inlet and outlet are equal.
For the ideal compression process put p D 1 in eqn. (2.32) and so obtain

.
1//
T 2s p 2
D (2.35)
T 1 p 1

which is also obtainable from pv = constant and pv D RT. Substituting eqns. (2.33)
and (2.35) into eqn. (2.34) results in the expression
" # " #
.
1//
.
1// p
p 2 p 2
c D 1 1 . (2.36)
p 1 p 1