It might be worthwhile to try and forget or put aside all the "Carnot limit" nonsense, (and IMO it is just that, utter nonsense) and consider what is really going on in a Stirling heat engine.
BTW, just as an historic factoid I found somewhat interesting. Stirling invented his engine (patented 1816) before Carnot published his theories.(1824). Carnot in his writings, having heard rumors of hot air engines, expressed doubt that they actually existed.
Also, the so-called Carnot efficiency formula often erroneously attributed to Carnot did not originate with him. It was worked out much later, perhaps by Kelvin, but the idea of an absolute zero did not exist at the time. Any zero temperature refered to by Carnot had to be that of the Celsius scale.
Anyway, 100% conversion of heat into work is not considered impossible In fact, the process of heating a gas in a cylinder to drive a piston to do work, as a single process, is considered a complete conversion of heat into work, neglecting loses to friction.
It is only when making reference to a cyclic process that the so-called 2nd law comes into play. The supposed problem is, how to get the piston back to the starting point.
To site one reference:
All standard heat engines (steam, gasoline, diesel) work by supplying heat to a gas, the gas then expands in a cylinder and pushes a piston to do its work. So it’s easy to see how to turn heat into work, but that’s a one shot deal. We need it to keep repeating to have a useful engine. The heat and/or the gas must therefore be dumped out of the cylinder before the next cycle begins, otherwise all the work the gas delivered on expanding will be used up compressing it back!
https://galileo.phys.virginia.edu/class ... Engine.htm
The rationale here, however, is not only illogical, it is a violation of the first law of thermodynamics.
If in pushing the piston out, all of the added heat was converted to work, then how can any of the heat that was added still be left in the cylinder to require removal?
If heat were a fluid, like water, then filling the cylinder to push out the piston would result in a cylinder full of water/fluid, and the water/fluid/heat would have to be removed. But heat is not a fluid, it is energy, and conservation of energy dictates that heat cannot be converted to work and still remain to be transfered to a sink. It's a quite blatant and obvious contradiction, so how people continue to write nonsense like the above quoted paragraph in this day and age, when it is well established that heat is not a fluid, I can't understand.
Either the heat is energy and was converted to work (really the kinetic energy was transfered), or heat is a fluid and entered the cylinder under some unknown form of hydraulic pressure associated with differences in temperature, or some such thing (entropy?)
Both ideas are compelling in their own way, but both cannot be right, I don't think, unless we are dealing with some weird quantum partical/wave duality phenomenon.
Anyway, let's imagine this:
A little heat is introduced to the cylinder by raising the displacer to expose the surface of the hot heat exchanger. The gas "expands" a little, that is, some molecules gain some kinetic energy. Soon those hot molecules bouncing around at a higher rate of speed strike the piston, the energy is transfered to the piston and the piston moves a tiny bit, the molecules that hit the piston lose energy and cool off, but, the piston, having been set in motion tends to stay in motion. If no more heat were added at this point, and if the piston and cylinder were perfectly frictionless, the fact that the volume of the gas had expanded would mean that the gas pressure inside the engine would be lower than it started, lower than outside, because the piston was driven out, but the gas did work and cooled back down, having transfered it's energy to the piston.
So, at this point the outside atmosphere, the energetic air molecules on the other side of the piston, would push the piston back to re-establish equilibrium, but...
At this point, the displacer moves a tiny bit more, and so more air becomes hot, and bouncing around, ends up striking the piston, adding a bit more to the pistons already established momentum.
Again those molecules cool down, having transfered their energy to the piston, and so the outside atmosphere would now push the piston back (with a bit more vigor) to re-establish equalibrium, but ..
The displacer moves a bit more, introducing a little more heat to energize a few more molecules that work to keep the piston moving and building up momentum.
This process continues until the displacer has gone through it's motion and introduced as much heat as it is going to, but, by this time, the piston has built up quite a bit of momentum and so keeps traveling, though the gas that propelled it has long since transfered all of it's kinetic energy gained from the added heat to the piston.
The gas, because the piston keeps moving, is now being mechanically expanded due to the momentum of the piston and so becomes colder colder even than what it was before.heat was added. The displacer is now, once again, covering the heat source, so no new heat is being added, but the gas continues to expand and cool resulting in a sharp drop in internal pressure.
At this point, the outside atmospheric pressure, having been for a long time, delayed, now drives the piston back inward with great force and without any resistance from internal pressure.
At the point when the piston is nearly driven all the way back down the cylinder by atmospheric pressure, the internal pressure and temperature being re-established, the piston, having again picked up momentum, but in the opposite inward direction, starts compressing the gas, transferring energy back to the gas while the displacer again rises, again, introducing a blast of heat, the result is a sudden intense rise in temperature and pressure again driving the piston outward... A bit more vigorously this time
As this oscillation continues, the engine picks up speed and strength
Notice, there is no "left over WASTE HEAT" in this cyclic process, no violation of the 1st law of thermodynamics, no contradiction of heat having been converted entirely to work but still requiring removal,
As far as I can see, the "Carnot limit" is nothing but a fairy tale It is completely bogus
Nobody has to look very far to find an engine that violates the "Carnot limit". Any off the shelf, common model Stirling engine on the market will do that.
"Carnot efficiency" is pure mythology. It should have been scraped as any kind of theory 100 years ago, or more like 200 years ago.