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

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

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Compressors, Gas Turbines and Jet Engines
If p₁ is the pressure of air entering the L.P, cylinder and p₃ is the pressure of air leaving the H.P. cylinder, then the ratio of cylinder diameters for a single acting, two stage reciprocating compressor with complete intercooling is given by

D₁/D₂ = (p₁ p₃)1/2
D₁/D₂ = (p₁ p₃)1/4
D₁/D₂ = (p₃/p₁)1/4
D₁/D₂ = (p₁/p₃)1/4

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Compressors, Gas Turbines and Jet Engines
In n₁ and n₂ are the indices of compression for the first and second stage of compression, then the ratio of work-done on the first and second stages (W₁/W₂) with perfect intercooling is given by

W₁/W₂ = n₁(n₂ - 1)/n₂(n₁ - 1)
W₁/W₂ = n₁/n₂
W₁/W₂ = n₂(n₁ - 1)/n₁(n₂ - 1)
W₁/W₂ = n₂/n₁

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