**Assuming**

a) uniform stagnation enthalpy per unit mass along the radius of the blades, and

b) Adiabatic conditions (heat transfer effects are negligible),

_{}

from which we get:

_{}

For the stator, since no work is done,

_{}

Thus,
the changes in air properties through a compressor stage are related to the
velocity vectors through the rotor and
stator.

**Stage Efficiency**

Stage efficiency is defined as the ratio of the ideal work to the actual work.

_{}

Thus,
the pressure stage pressure ratio is related to the stage temperature rise
using the isentropic relations as before:

_{}

__Limits on
Stage Pressure Ratio__

*1.
Compressibility:*

As the relative Mach number becomes supersonic, shock losses can become substantial. In earlier compressors, the blade tip Mach number was kept below 1.0 to avoid the transonic drag rise. This imposed a severe limit on compressor shaft rpm, blade radius, and stage pressure ratio. In modern compressors, the rotor operates in the transonic regime, with shocks present in the rotor. While this causes some loss in stagnation pressure, much more work can be done by each rotor stage, and the shock provides a convenient way of increasing static pressure. As a result, stage pressure ratio is higher, and the overall weight of the compressor is reduced.

*2. Flow
Separation:*

Usually, this is what limits the stage
pressure ratio. Note that the flow in the compressor stage is moving against an
adverse pressure gradient. Boundary layers thicken, and may separate if the
pressure gradient becomes too large. This can be expressed using the pressure
coefficient

_{} for the rotor, and

_{}for the stator.

The
limiting value of the pressure coefficient is usually around 0.6 to avoid flow
separation.

_{}

This
shows why the stage pressure ratio is usually limited to about 1.4 for subsonic
rotors. Under extreme conditions,
transonic stages can reach stage pressure ratios as high as 2.2.

The
effect of limiting stage pressure ratio on the number of stages in a compressor
can be seen from the following:

Given
a compressor pressure ratio of _{
}, and a stage pressure ratio of _{}, the number of stages is:

_{}