Mean-line calculation through a compressor rotor
Calculation of the flow through a row of rotor blades is similar to that through a stationary cascade, as described in Chapter 3. The minor complication is the use of relative rather than absolute properties.
Compressible case
Consider the transonic compressor rotor shown in Figure 5.4. The velocity triangle at inlet has been scaled so that it is a Mach number triangle, which is often a useful transformation for high-speed stages.
If the conditions at inlet to the rotor are known, the nondimensional mass flow rate at inlet can be determined from compressible flow tables:
where A1n is the area normal to the flow at inlet, and the projected frontal area of the rotor (or annulus area) is Hs and taken to be constant through the rotor. To find the conditions at exit, the nondimensional mass flow rate at exit can be written in terms of the preceding, using the fact that through a rotor blade with constant mean radius, T01,rel 5 T02,rel:
Once the exit relative Mach number and flow angle from the rotor blade are known, the other properties at exit from the rotor can be determined (via compressible flow relations and the velocity triangle) in order to fully specify the conditions at inlet to the stator. This is demonstrated in Example 5.1.
Incompressible case
In the low speed, incompressible case the equivalent calculations are more straightforward. The continuity equation reduces to
Once the exit static pressure is known all other quantities at rotor exit can be found since the density is fixed and the velocities are known.
EXAMPLE 5.1
A single-stage transonic compressor operates with axial flow at inlet. The inlet absolute stagnation temperature is 288 K and the inlet absolute stagnation pressure is 101 kPa. The relative flow angle at inlet to the rotor is 45o and the inlet relative Mach number is 0.9.
a. Calculate the rotor blade speed and the inlet relative stagnation pressure.
b. The mean radius and the mass flow rate per unit annulus area are constant through the rotor.
If the rotor loss coefficient is 0.068 and the rotor exit relative Mach number is 0.5, find the rotor exit relative flow angle and determine the static pressure ratio across the rotor.
c. Show that the absolute stagnation temperature and pressure at entry to the stator are 322 K and 145 kPa, respectively. Determine the total-to-total isentropic efficiency of the compressor stage if the stagnation pressure loss coefficient for the stator is 0.04.
Solution
a. T01 5 288 K, p01 5 101 kPa. Given that the flow is axial at inlet, the absolute inlet Mach number can be calculated (using the Mach number triangles shown in Figure 5.4):
