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

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

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Compressors, Gas Turbines and Jet Engines
The following is true for an open cycle gas turbine having exhaust heat exchanger. Atmospheric air before entering the compressor is

Heated
Compressed air before entering the combustion chamber is heated
Bled gas from turbine is heated and readmitted for complete expansion
Exhaust gases drive the compressor

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

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

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