Compressors, Gas Turbines and Jet Engines
For slow speed large capacity compressor, following type of valve will be best suited

Poppet valve
Mechanical valve of the Corliss, sleeve, rotary or semi rotary type
Any of the above
Disc or feather type

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

All of the listed here
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

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