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

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

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Compressors, Gas Turbines and Jet Engines
Ratio of compression is the ratio of

Stroke volume and clearance volume
Absolute discharge pressure to the absolute intake pressure
Gauge discharge pressure to the gauge intake pressure
Pressures at discharge and suction corresponding to same temperature

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