Compressor Power and Efficiency Equations

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Gas compression can be expressed in terms of pressure and temperature variation as,
PVn = constant

A compression process typically following this pressure volume relation is known as polytropic process. If n=1, the process is isothermal (constant temperature). If n=γ=CP/CV, the compression is adiabatic compression (no heat exchange with the surrounding). Most gas compressions generally follow the adiabatic curve. Hence compressor equations are also based on adiabatic curve with n=γ,
PVγ = constant

Let subscripts 1 and 2 stand for inlet and outlet process conditions of the compressor. Then the pressure ratio of the compressor is P2/P1.
P2/P1 = (V1/V2)n ... Polytropic compression
P2/P1 = (V1/V2)γ ... Adiabatic compression

The head developed by the compressor is expressed in following compressor equations.

... Adiabatic compressor head equation

... Polytropic compressor head equation

This head when multiplied by the volumetric flow of gas (Q) gives the compressor power equations.

... Adiabatic compressor power equation

... Polytropic compressor power equation

Adiabatic efficiency for the compressor is calculated as,

Adiabatic efficiency, η = (Actual Polytropic work / Adiabatic work)

... Adiabatic efficiency equation

Polytropic efficiency or hydraulic efficiency for a compressor is represented by,

... Polytropic efficiency equation