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I understand how Carnot's theorem implies that irreversible heat engines must be no more efficient than reversible one's, but it is less clear why they need to be less efficient, as I have seen stated in some places.

If they could be equally efficient then an irreversible engine engine could be used to drive a reversible engine operating between the same heat reservoirs, without any net energy transfer between the reservoirs. It would then be unclear what is irreversible about the irreversible engine. Does that constitute an actual contradiction though? If so can the argument be stated more tightly? It feels a little sloppy as is.

It could also be a question of how an irreversible engine is defined in Carnot's theorem. I understood it to mean one that cannot be run in reverse as a heat pump, which could presumably include a Carnot engine with a one way ratchet attached. If it really means an engine whose thermodynamic effects can not be undone, the implication would be trivial.

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I understand how Carnot's theorem implies that irreversible heat engines must be no more efficient than reversible one's, but it is less clear why they need to be less efficient

Irreversibility means that entropy is generated in the engine (rather than just transferred, as with the reversible heat engine).

Entropy can't be destroyed, although it is transferred by heating.

The heat engine is assumed to operate cyclically, so entropy can't accumulate there; the engine returns to its original state every cycle.

The work output can't carry entropy.

We can't shift entropy to the hot reservoir through heating, as the hot reservoir is itself heating the engine as an energy source.

The only remaining possibility is that the generated entropy must be transferred to the cold reservoir through heating (in addition to the entropy that arrived in the engine from the hot reservoir's heating).

This additional required heating of the cold reservoir is energy that's now not available to output as work. Therefore, for a given amount of input energy from the hot reservoir, the work output is lower, as is the efficiency, QED.

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May be, we have to refresh our mindset about an irreversible engine.

The very idea of reversibility is made associated always with quasi-static (QS) process. Otherwise the system can never be close to equilibrium. So a QS process is necessary for the reversibility. But it is not sufficient. Even in a QS process if there is friction (which is always present in reality) that also forbids reversibility. From experimental point of view, if one tries to go back in the reverse QS process to achieve the former state by resetting his/her experimental parameters, say pressure, volume, temperature etc. (which he/she can certainly do), the effect of irreversibility through frictional losses will be reflected in the internal energy (this will not return back to it's former value).

Whenever we discuss irreversibility in regards to the second law of thermodynamics, it is not a non-QS process that is of main concern - rather it is concerned to be due to the presence of dissipation (or friction or whatever equivalent one may say) in a QS process.

Now, for a given temperature of the cold and the hot reservoir, a reversible engine can extract maximum amount of work from a given amount of heat taken from the hot reservoir. That defines it's efficiency. For an irreversible engine as mentioned above, a part of this work will be lost through dissipation. Hence it has a lower efficiency.

Now what about the case, where friction is absent and irreversibility is caused by a non-QS process?

  • Well, I think that kind of irreversibility cannot describe an engine as it cannot define a cycle in the P-V (or T-S) plane, which the engine is going to follow over and over.
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  • $\begingroup$ This answer doesn't seem to address the question to completion; if the heat engine is turning heat into work, why can't the dissipative heat itself also be turned into work? Please see my answer, which introduces the entropy flow, along with the fact that entropy can't be destroyed. $\endgroup$ Jul 3, 2023 at 4:33
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I understand how Carnot's theorem implies that irreversible heat engines must be no more efficient than reversible one's, but it is less clear why they need to be less efficient, as I have seen stated in some places.

The answer is that it is always possible to construct an incredibly inefficient engine, which is to say, it is always possible to obtain a lower efficiency than the reversible engine. This, coupled with the fact that it is very difficult to construct a truly reversible engine, places real engines at efficiencies below that of the Carnot limit.

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  • $\begingroup$ This does not seem to answer the question. They are asking why an irreversible engine must be less efficient than a reversible one. Just because it's possible to have a lower efficiency than a reversible engine doesn't explain why an irreversible one must be less efficient. $\endgroup$
    – JMac
    Dec 5, 2018 at 12:32
  • $\begingroup$ I don't understand your objection. If an engine has the Carnot efficiency, it is reversible. By definition then, irreversible engine means lower efficiency than Carnot. Are you asking why it is not possible to construct a truly reversible engine? $\endgroup$
    – Themis
    Dec 5, 2018 at 18:58

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