Page 54 - Fluid Mechanics and Thermodynamics of Turbomachinery

Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 35
Small stage or polytropic efficiency

The isentropic efficiency described in the preceding section, although fundamen-
tally valid, can be misleading if used for comparing the efficiencies of turbomachines
of differing pressure ratios. Now any turbomachine may be regarded as being
composed of a large number of very small stages irrespective of the actual number
of stages in the machine. If each small stage has the same efficiency, then the
isentropic efficiency of the whole machine will be different from the small stage
efficiency, the difference depending upon the pressure ratio of the machine. This
perhaps rather surprising result is a manifestation of a simple thermodynamic effect
concealed in the expression for isentropic efficiency and is made apparent in the
following argument.
Compression process

Figure 2.6 shows an enthalpy entropy diagram on which adiabatic compression
between pressures p 1 and p 2 is represented by the change of state between points 1
and 2. The corresponding reversible process is represented by the isentropic line 1
to 2s. It is assumed that the compression process may be divided up into a large
number of small stages of equal efficiency p . For each small stage the actual work
input is υW and the corresponding ideal work in the isentropic process is υW min .
With the notation of Figure 2.6,

υW min h xs h 1 h ys h x
p D D D D ...
υW h x h 1 h y h x

Since each small stage has the same efficiency, then p D .υW min /υW/ is also
true.

FIG. 2.6. Compression process by small stages.