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

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

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Compressors, Gas Turbines and Jet Engines
For a two stage reciprocating compressor, compression from p₁ to p₃ is with perfect intercooling and no pressure losses. If compression in both the cylinders follows the same polytropic process and the atmospheric pressure is pa, then the intermediate pressure p₂ is given by

p₂ = p₁. p₃
p₂ = Pa p₃/p₁
P₂ = Pa × p₃/p₁
p₂ = (p₁ + p₃)/2

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