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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.

sample math of compressor sample math of compressor

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1 Answer 1

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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.

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  • $\begingroup$ What about method-2? $\endgroup$ Jul 25 at 4:56
  • $\begingroup$ Method 2 is basically Method 3 so probably rounding makes the answer different. Notice that the PV terms have units of energy or work (Joules in SI) and then since you want power you assume the answer is Watts. $\endgroup$
    – W H G
    Jul 25 at 14:57

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