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What does the word, "polytropic" mean? Who invented the idea, why, and how? We need not know the answers. Be it sufficient to understand, to start, that systems the substances of which are an ideal gas are highly amenable to analytic analysis. Properties of a gas are "locked together," ruled be the ideal gas equation. That one, a simple equation tells truth of ideal gas behavior, permits grand analysis schemes to be imagined and launched (on paper) - Polytropic Processes.
The properties are: p,v and T. So if a relation of two is defined then (by the ideal gas law) the third is known. At the time thinking happened, the easier properties of the ideal gas, to know and measure, were pressure and volume with volume easier than pressure.
The stratagem "polytropic process," is an "exponent controlled," analytic assumption to define ideal processes of an ideal gas as having continuous (equilibrium path) of ideal gas states. The basic "exponential definition" of path is defined analytically as a product of the ideal gas pressure times its volume with the volume itself enhanced by the "polytropic coefficient", a power to which volume is raised.
|pVn = constant||(1)1|
To use the relation it is made specific to a path started at known State 1 by:
i) Selection of a "polytropic coefficient,(n) so that path passes through a specified second state of the gas. (Used commonly with industrial experimentation).
ii) From the initial state path is selected by specification in accord with constancy of a gas initial state property or interaction condition upon change. Common choices are: n = 0, or 1, or cp/cv , ... n or ∞)
iii) This discussion applies to a unit mass (whereupon V would be written v) or any amount of gas, provided properties of the gas are uniform for at the beginning state, and all between it and the final state.
All students would have less difficulty with thermodynamics had their calculus course caused them to learn to integrate the work expression of an event of a polytropic gas process. The task is completed below. Students should learn the steps thoroughly.
POLYTROPIC PROCESS AND ITS WORK: If a gas is contained in a classical piston/cylinder apparatus and the piston is forced to move in a violent manner, shock waves will reverberate throughout the gas and cause local temperature increases. But were the piston moved ever so slowly, the pressure in the gas would change in an undisturbed, uniform (idealized as frictionless) manner. The ideal gas equation would apply continuously as the gas changed from one state to a next.
Below is written the "polytropic equation." The "n" is called the polytropic coefficient. The constant is never calculated; it is finessed. The polytropic equation form is written below.
The polytropic equation is functional, not dimensional. The constant need never be calculated. To express the work of polytropic processes of an ideal gas will require two integrations. The first integration is for all values of the coefficient except, n = 1. The general expression for compression work is:
To integrate anything, one first determine the dependence of the integrand. In this integral, the pressure of the process (the integrand) varies with volume, the differential. The equation representing that variation is the polytropic equation.
This is substituted into the equation for work.
This is a standard integration:
We apply the limits and multiply above and below by (-1) to obtain:
At this point the constant can be removed by two thoughtful substitutions.
Now multiply the volumes to their powers:
Showing ALL steps:
When one reads an equation, the idea, "How can it fail?" occurs. For the above equation, the "n-1" as denominator indicates a potential equation destruction. Were "n" to approach "1" either the work would approach infinite or (pV)2 must approach (pV)1. Infinite work is impossible. And were (pV)2 to equal (pV)1, there would be no event. Both of these possibilities are bad. If an equation can deconstruct - how can we trust it? Let us look closely.
The answer is explained by ideal gas behavior, that is, by using the ideal gas equation with our result.
The substance, an ideal gas, has the equation of state:
Substitution into the work expression for "pV" yields something that makes sense. Physically, as "n" becomes "1," (putting a zero in the denominator) the temperature of the process becomes constant, putting a zero in the numerator.