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Polytropic Process

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Nomenclature

P	pressure
V	volume
n	may be any possible value from −∞ to +∞, depending on the particular process

Details

A polytropic process is the second of two classes of problems defined here at the bottom. The polytropic process is a process where the relationship between the pressure P and volume V makes it possible to fit an analytical relationship between them. Then it may be integrated directly.

A polytropic process is one in which:

PV ⁿ = constant

throughout the process. For this type of process, Eq2 from the lesson Work at the Moving Boundary of a Simple Compressible System can be integrated:

PV ⁿ = constant = P₁V₁ⁿ = P₂V₂ⁿ

P =

constant V n

=

P1V1n V n

=

P2V2n V n

∫

2   1

P dV = constant

∫

2   1

dV V n

= constant

(

V  −n + 1 −n + 1

)

2   1

(Eq1)

∫ 2   1 P dV = constant 1 − n (V21 − n − V11 − n ) = P2V2nV21 − n − P1V1nV11 − n 1 − n  =  P2V2 − P1V1 1 − n

Note that Eq1 is valid for any exponent n, except n = 1. Where n = 1:

PV = constant = P₁V₁ = P₂V₂

and:

(Eq2)

∫ 2   1 P dV = P1V1 ∫ 2   1 dV V = P1V1 ln V2 V1

In Eq1 and Eq2 it was not stated that the work is equal to the expressions given in these equations. These expressions give the value of a certain integral, that is, a mathematical result. Whether or not that integral equals the work in a particular process depends on the result of a thermodynamic analysis of that process. It is important to keep the mathematical result separate from the thermodynamic analysis, for there are many situations in which work is not given by Eq2 from the lesson Work at the Moving Boundary of a Simple Compressible System.

The polytropic process as described demonstrates one special functional relationship between P and V during a process. There are many other possible relations.