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I am new to thermodynamics applied to turbines and compressors and I am trying to get my head around what is represented when calculating the work of a compression/expansion process using the polytropic exponent as oppose to the specific heat ratio of 1.4 (I'm working with air).

The attached image ( from Çengel, Y. a. (2004). Thermodynamics: An Engineering Approach.

*McGraw-Hill*.)

Shows the equations for polytropic compression.

So I have two questions:

1) is the polytropic equation taking into account any heat losses compared to the following isentropic equation:

At the moment I am using the isentropic formulae and dividing it by the isentropic effciency which gives me a work value slightly higher accounting for irreversibilities.

2) What is the difference between using the polytropic exponent, n. Like in the first equation. And using the polytropic efficiency as below:

The reason I am asking is because using the polytropic exponent, n gives me greater efficiencies for my compression and expansion system than the ratio of specific heats, k. i.e lower compressor work and greater turbine work. Which I wouldn't expect if the polytopic exponent accounted for losses.

Thank you for your help.