Limit disproved years ago, 1848
The importance of the work at right, is that it is showed how to completely use all available heat energy, until absolute zero is reached.
It clings to the notion of a cold reservoir, but the "hot reservoir" is the exhaust of the previous heat engine, so is determined solely by the remaining heat in the exhaust. There is no cold reservoir.
The texts from that era use the term ratio, and efficiency, but never limit.
And last, it is noted that this is used today, successive expansions is the primary means for generating experiments with temperatures near absolute zero. If there were a "limit" defined by a cold reservoir, it would be impossible to create temperatures below ambient. Clearly, it is not only possible but common.
How do you do better? One can "stack" Carnot cycles, such that the heat generated by work of compression of cycle A, is used as the heat of expansion in Cycle B, and B can source C, and so on, until absolute zero is (almost) reached. The bottom most expansion would eventually result in a liquid, which requires no work, hence no energy, to compress. He describes a 100% conversion of heat.
At right, A3 to A4 would provide heat to another separate volume's cycle, A to A1. Isn't this the same as just a bigger expansion? Yes and no. Two 1 to 10 expansion cycles would equate to one 1 to 100 expansions, but could be built in a total volume of 20, instead of 100. 3 cycles of 1 to 10 in volume 30, instead of 1000, and so on. Much easier to build, and much easier to keep outside air pressure from interfering with operation.
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Josiah William Gibbs
"Josiah Willard Gibbs, the History of a Great Mind, written by a student of Dr. Gibbs, Lynde Phelps Wheeler. It was originally published by Yale University Press and is used with their permission."
Page 68: On the basis of this relation Thomson showed (in 1848) that by
choosing a series of heat reservoirs of uniformly descending
temperatures and by supposing that reversible engines each doing the
same amount of external work are operated between them in such a
manner that the heat rejected by one becomes the heat received by the
next in the series, we would ultimately arrive at a point where there
would no longer remain any heat to be rejected. The temperature of this
last reservoir would then be the lowest conceivable -- the absolute zero
of temperature.
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