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I was searching for information about pressure-volume processes when I read, in a heat engineering book, "The process is polytropic with $PV^n = \text{constant}$."

How can I know that a real life process follows this model? Are all processes related to this heat engineering, pressure-volume, etc., in fact polytropic?

And if they aren't then how can I tell when that model is valid?

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No, all thermodynamic processes are not polytropic. As an example, consider a real engine cycle. Any undergraduate engineering thermodynamics text should have a plot showing the true P-V diagram. You also can see examples online. Clearly parts of this cycle can not be modeled with a polytropic process.

A polytropic process is an approximation, much like a linear regression. It is correct to the extent by which the real and approximate curves are similar, however you define "similar".

For ideal gases, sometimes with constant specific heats, you can derive certain polytropic relations, e.g. for an isothermal process of an ideal gas you get $P V =$ constant, for an adiabatic process with no entropy generation you get the isentropic process relations, etc. These are polytropic processes, and their use often follows the validity of their assumptions. That is, if the gas is ideal and the temperature is constant, the (polytropic) isothermal process is a good model. It should be clear that for other equations of state you will not necessarily get a polytropic process keeping the other assumptions the same.

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