Irreversible heat engines strictly less efficient than reversible ones 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.
 A: 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?

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*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.

A: 
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.
