# How to find work done by compressors

Which one is correct method to find work done by a isentropic compressor?

Method-1:
Compressor power = mCpdT
Where m = mass flow rate
Cp = specific heat at constant pressure
dT = Temperature difference at inlet and outlet

Method-2:
Compressor work = (P2V2 - P1V1)/(1 - γ)
Where γ = specific heat ratio
As this formula represent work done by gas so inversely we can say work done by compressor.

Method-3:
compressor power = mCvdT
Where Cv = specific heat at constant volume
We know dq = du + dw. Here dq = 0 since it is isentropic process. So dw= -du and du = mCvdT. It seems that if precise calculation is done method-2 and method-3 give same result.

I applied three method with a sample math. Three method gives three result but I am not sure which one is to be accepted. Please see the photos below for clear understanding.

Method 1 is correct. The power equation has $$c_p \Delta T$$ which is enthalpy change.
Method 3 ignores that this is a flow process and therefore winds up with change in internal energy (hence use of $$c_v$$). Multiply the answer by $$\gamma = 1.4$$, so $$c_p$$ instead of $$c_v$$, to get enthalpy change and you are back to Method 1 and its answer.