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Carnot’s theorem tells us that "the most efficient heat engines operating between 2 reservoirs have the same efficiency as a Carnot engine".

This theorem assumes that the engines are operating between reservoirs; in essence, it assumes that the 2 temperatures stay constant. However, this is not the case when an engine is operating between materials with finite heat capacity, because the temperatures no longer stay constant when heat is transferred.

I've heard of explanation to treat the temperature as constant so that the materials behave as reservoirs instantaneously. However, if the heat transfer over one cycle is large enough then the temperature will not be constant, so I think that this approximation will not hold.

Why does Carnot’s theorem hold even when the temperatures are changing?

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  • $\begingroup$ Talking about fixed temperature reservoirs is not important; you can rephrase Carnot's theorem as referring to entropy transfer at a given pair of maximum and minimum temperatures. The reservoirs come into play when you discuss how one could realize or approximate such transfer in practice. $\endgroup$ – hyportnex May 12 '18 at 16:23
  • $\begingroup$ @hyportnex How would you then prove that the instantaneous efficiency is given by 1-T1/T2, where T1 and T2 is the instantaneous temperature of the low and high temperature reservoir respectively? $\endgroup$ – user148792 May 12 '18 at 16:28
  • $\begingroup$ "instantaneity" does not play here, only the temperatures at which entropy is transferred in or out. If the temperature is time varying then Carnot's efficiency theorem refers to the maximum and the minimum temperatures. $\endgroup$ – hyportnex May 12 '18 at 16:38
  • $\begingroup$ @hyportnex Let's say there's 2 bodies with heat capacity C, one at temperature T1 and the other at a higher temperature T2. I would like to extract as much work as possible from this system. How would you prove that the work extracted by operating a Carnot engine between them will be the maximum possible work? $\endgroup$ – user148792 May 12 '18 at 16:41
  • $\begingroup$ that is a different question and I think @Chemomechanics answered that previously $\endgroup$ – hyportnex May 12 '18 at 16:53
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Carnot's theorem still applies for finite reservoirs. The reservoir temperature simply changes with time, and therefore the Carnot efficiency changes with time.

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