Is Carnot cycle the only "most efficient" cycle?

Question

In all the books, that I have studied so far, they say that Carnot cycle is the most efficient cycle. But why isn't any other reversible process as efficient as Carnot cycle? Can somebody please provide mathematical explanation for why that is or isn't the case?

Answer 1

A nuance is that the Carnot cycle is the only reversible cycle between only two reservoirs. There are other idealized cycles that are also reversible (i.e., that generate no entropy) but require more reservoirs.

See da Silva, "Some considerations about thermodynamic cycles", Eur. J. Phys. 33 (2012) 13–42 for the math framework of evaluating engine reversibility. Note the statement "This property makes the Carnot engine a very special engine: it is the unique reversible heat engine functioning with only two reservoirs. (Any other cycle could be performed in a reversible way, but that would require a virtually infinite number of thermal reservoirs.)"

See also Leff, "Thermal efficiency at maximum work output: New results for old heat engines," Am. J. Phys. 55, 602 (1987) and Leff, "Reversible and irreversible heat engine and refrigerator cycles," Am. J. Phys. 86, 344 (2018) for mathematical analyses of non-Carnot reversible cycles, such as the reversible Otto cycle.

Answer 2

The Carnot cycle efficiency is given by

$$ζ=1-\frac{T_L}{T_H}$$

It is the most efficient because all of the heat added to the cycle occurs at a single temperature (high temperature reservoir, $T_H$) and all the heat rejected occurs at a single low temperature (low temperature reservoir, $T_L$). This minimizes the ratio $T_{L}/T_{H}$ and maximizes $ζ$.

For any other reversible cycle that takes in heat from a series of variable temperature reservoirs and/or rejects heat to a series of variable temperature reservoirs, you would have to use the mean temperature of the reservoirs in place of $T_L$ and/or $T_H$. That would result in a higher value of the ratio $T_{L}/T_{H}$ and lower value of $ζ$. That is because the mean value of the series of low temperature reservoirs would be a number greater than the lowest temperature reservoir of the series, and the mean value of the series of high temperature reservoirs would be a number lower than the highest temperature reservoir of the series.

Hope this helps.