1
$\begingroup$

There are already many questions on Physics SE asking for a proof of the maximum possible efficiency of Carnot's engine. However, none of them (at least the ones which I looked up) contain a mathematically rigorous proof involving variational calculus.

When I felt like proving it, the natural thought was to use variational calculus, but I am just not sure how to do this. Also, is it even possible to prove it using calculus of variations? If yes, then how?

Improve this question
$\endgroup$
12
  • 1
    $\begingroup$ Maybe the following can help: pdfs.semanticscholar.org/7a18/… $\endgroup$
    – Bob D
    Commented Dec 27, 2019 at 18:15
  • 1
    $\begingroup$ My point is that (unless someone else is rather more clever than I'm being) your approach will allow you to prove that the Carnot cycle is optimal of the ideal gas (hopefully for all values of $\gamma$ in one go), then separately for the van der Waals gas, then for model X, then model Y , then... But an exhaustive search is impractical and the more general you try to make the models the more intricate each calculation becomes. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Dec 27, 2019 at 18:39
  • 1
    $\begingroup$ If you know the calculus of variations then you have all the mathematical tools you need for classical thermodynamics. If you also know about thermal expansion, heat capacities, and latent heats at the level of high school physics class then you have all the foundational physical knowledge that you need to begin. The classical treatment develops eveything you need from there (because that is where the founders of the discipline were when they started). It's not easy (and rigor is usually skipped at first and added the second (or third) time round), but it is within your grasp. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Dec 27, 2019 at 18:43
  • 1
    $\begingroup$ see such an attempt in ijee.ie/articles/Vol14-6/ijee1031.pdf $\endgroup$
    – hyportnex
    Commented Dec 27, 2019 at 19:50
  • 1
    $\begingroup$ @dmckee for T-S plane analysis see this and also Tobin's original paper in AJP, physics.stackexchange.com/questions/300347/…. The Wang paper reference is from Pirx. I prefer Tobin's way over Wang's especially because Wang seemingly hides the problems "under the rug" of his function $\psi(dS)$ (Eq. 19). $\endgroup$
    – hyportnex
    Commented Dec 28, 2019 at 13:33

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.