# Centrifugal compressor with relief valve formulas

1. Dec 30, 2016

### Colandor

Hi folks,

I have a question regarding centrifugal compressors when a relief valve is included in the system.
Allow me to explain...
There is a centrifugal compressor supplying air to an aircraft engine. As altitude varies, the compressor provides enough pressure up to altitude "X" for the engine to operate at full manifold pressure.
Now, in order for the pressure not to rise above the engine limits, a relief valve opening at max supported pressure.

My questions are:
1.- At lower altitudes, when the relief valve is letting air escape due to the higher inlet pressure, is the power required to operate the compressor lower (because of the lower compression) or the same (because the air flow escaping through the relief valve) ?
2.- Similarly, is the compressor discharge temperature (which depends on compression ratio) lower or the same?

Basically my doubt is if, internally in the compressor, the compression ratio and air flow remains the same making the required power and discharge temperature also remain the same as if the relief valve was not there.

Hope that explains my question. Thanks in advance!

2. Dec 31, 2016

### Andrew Mason

Welcome to PF Colandor!

There must be a sufficient mass flow of air to the engine to ensure complete combustion of fuel. In order for sufficient air to flow into the engine cylinders, that air must be at a sufficient pressure. The compressor appears to create this pressure by taking in a fixed volume of air per cycle and adiabatically compressing it to a smaller fixed volume i.e. Vi/Vf is constant.

The work done by the compressor per cycle is $W = \int_{V_i}^{V_f} PdV$, where Vi and Vf are the initial and final volumes. and $PV^\gamma = \text{ constant} = P_iV_i^\gamma$ so $P = P_iV_i\left(\frac{1}{V}\right)^\gamma$. Since Vi/Vf is fixed, the work done per cycle is proportional to Pi. So the power required to operate the compressor is higher at lower altitudes.

If I understand your system correctly, the relief valve is on a tank that the compressor supplies. If that is the case, the relief valve will not affect the work done by the compressor. The compressor still compresses air from Vi to Vf each cycle but the tank that it supplies then lets it out to limit the pressure.
The temperature will depend on the final pressure in the compressor. It will be proportional to Pf i.e. Tf = PfVf/mR

AM

Last edited: Dec 31, 2016