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
In the cross compounding of the gas turbine plant

All are connected in series
H.P. compressor is connected to L.P. turbine and L.P. compressor is connected to H.P. turbine
H.P. compressor is connected to H.P. turbine and L.P. compressor to L.P. turbine
Both the arrangements can be employed

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