Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

156 Fluid Mechanics, Thermodynamics of Turbomachinery
The pressure rise in a real stage (involving irreversible processes) can be determined
if the stage efficiency is known. Defining the stage efficiency hs as the ratio of the isentropic
enthalpy rise to the actual enthalpy rise corresponding to the same finite pressure
change, (cf. Figure 2.7), this can be written as
Thus,
(5.27)
If c1 = c3, then hs is a very close approximation of the total-to-total efficiency htt.
Although the above expressions are derived for incompressible flow they are, nevertheless,
a valid approximation for compressible flow if the stage temperature (and pressure)
rise is small.
Pressure ratio of a multistage compressor
It is possible to apply the preceding analysis to the determination of multistage compressor
pressure ratios. The procedure requires the calculation of pressure and temperature
changes for a single stage, the stage exit conditions enabling the density at entry
to the following stage to be found. This calculation is repeated for each stage in turn
until the required final conditions are satisfied. However, for compressors having identical
stages it is more convenient to resort to a simple compressible flow analysis. An
illustrative example is given below.
EXAMPLE 5.1. A multistage axial compressor is required for compressing air at 293
K, through a pressure ratio of 5 to 1. Each stage is to be 50% reaction and the mean
blade speed 275m/s, flow coefficient 0.5, and stage loading factor 0.3, are taken, for
simplicity, as constant for all stages. Determine the flow angles and the number of stages
required if the stage efficiency is 88.8%. Take Cp = 1.005kJ/(kg°C) and g = 1.4 for air.
Solution. From eqn. (5.14a) the stage load factor can be written as
From eqn. (5.11) the reaction is
Solving for tan b 1 and tan b 2 gives
Calculating b1 and b 2 and observing for R = 0.5 that the velocity diagram is
symmetrical,
Writing the stage load factor as y = CpDT0/U 2 , then the stage stagnation temperature
rise is