Stirling engine Thermodynamics

Efficiency of Stirling engine and Carnot's theorem

In an ideal Stirling cycle, the isochoric steps have heat exchanged across an infinitesimal temperature difference, which is maintained by the regenerator having a continuous gradient of temperature between the hot and cold reservoirs. The gas can then cool or heat in alignment with that gradient. This is the very ideal part of the design that enables zero change in entropy during the two isochoric stages. This heat is just shuffled back and forth internally and so the only actual exchange with the outside is in via the hot reservoir and out via the cold. Hence the ideal efficiency. I'm not sure it's correct to call what happens in the regenerator stages isothermal. The temperature is changing continuously but ideally always across an infinitesimal difference. Is there a commonly used term for that? None the less, the isochoric stages are very different to the isothermal stages.

I've noticed in my searches on the internet about the topic of Stirling engines that many sources confuse these ideas. I've often seen efficiency analyses that ignore the effect of the regenerator altogether. This is possibly to do with the fact that isochoric processes are not usually associated with zero change in entropy but in the case of the Stirling engine there is a very special type of this process involved, using a regenerator.

The ideal Stirling engine has the same efficiency as the Carnot cycle, but its advantage is that it enables the building of real engines that, although they may not be able to achieve perfect isothermal and totally smooth regenerator isochoric stages, they do come close and are much more feasible than the possibility of building a practical Carnot engine.

So, in reality, real manufactured Stirling engines do not achieve the full ideal Carnot efficiency, but many do much better than other types of heat engine.

In conclusion, in consideration of the ideal Stirling engine:

(1) The ideal maximum efficiency of the Carnot engine is achieved. (2) Your calculation does not contradict this because it is wrong. You include the heat exchanged in the isochoric stages as part of the cost, whereas the only cost is the external heat input during the isothermal power stroke. (3) This cycle is reversible as there is no change in entropy during the isochoric stages. (4) The diagram on its own is not enough to show this as we need to also know that the ideal regenerator is what enables the third point. That is, if you remove the regenerator the diagram is still the same.

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