Page 55 - Fluid Mechanics and Thermodynamics of Turbomachinery

36 Fluid Mechanics, Thermodynamics of Turbomachinery
From the relation Tds D dh vdp, for a constant pressure process .∂h/∂s/ p 1 D T.
This means that the higher the fluid temperature the greater is the slope of the
constant pressure lines on the Mollier diagram. For a gas where h is a function
of T, constant pressure lines diverge and the slope of the line p 2 is greater than
the slope of line p 1 at the same value of entropy. At equal values of T, constant
pressure lines are of equal slope as indicated in Figure 2.6. For the special case
of a perfect gas (where C p is constant), C p .dT/ds/ D T for a constant pressure
process. Integrating this expression results in the equation for a constant pressure
line, s D C p log T C constant.
Returning now to the more general case, since
h 1 /,
υW Df.h x h 1 / C .h y h x / CÐ Ð Ðg D .h 2
then

h 1 /.
p D [.h xs h 1 / C .h ys h x / C ÐÐÐ]/.h 2
The adiabatic efficiency of the whole compression process is
h 1 /.
c D .h 2s h 1 //.h 2
Because of the divergence of the constant pressure lines
h 1 /,
f.h xs h 1 / C .h ys h x / C ÐÐÐg >.h 2s
i.e.

υW min >W min .
Therefore,

p > c .
Thus, for a compression process the isentropic efficiency of the machine is less than
the small stage efficiency, the difference being dependent upon the divergence of
the constant pressure lines. Although the foregoing discussion has been in terms of
static states it can be regarded as applying to stagnation states if the inlet and outlet
kinetic energies from each stage are equal.

Small stage efficiency for a perfect gas

An explicit relation can be readily derived for a perfect gas (C p is constant)
between small stage efficiency, the overall isentropic efficiency and pressure ratio.
The analysis is for the limiting case of an infinitesimal compressor stage in which
the incremental change in pressure is dp as indicated in Figure 2.7. For the actual
process the incremental enthalpy rise is dh and the corresponding ideal enthalpy
rise is dh is .
The polytropic efficiency for the small stage is
vdp
dh is
p D D (2.31)
dh C p dT’
vdp.
since for an isentropic process Tds D 0 D dh is