Compressors, Gas Turbines and Jet Engines
The assumption made in two stage compression with inter-cooler is that

There is no pressure drop in the inter-cooler
The compression in both the cylinders is polytropic
The suction and delivery of air takes place at constant pressure
All of the listed here

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Compressors, Gas Turbines and Jet Engines
The overall isothermal efficiency of compressor is defined as the ratio of

Work to compress air isothermally to work for actual compression
Isothermal h.p. to the BHP of motor
Isothermal h.p. to adiabatic h.p.
Power to drive compressor to isothermal h.p.

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