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https://youtu.be/YWUhwmmZa7A
This is a pretty long boring dull video about Peltier device refrigeration, but around 20 minutes in, relates some interesting facts.
The Peltier device, without an active cooling system, heat sink and fan on the hot side and insulation on the cold side won't get cold. It will act as a heat pump, so one side will get very hot but on the cold side, without insulation, the heat taken away will be immediately replaced by ambient heat.
It will, though, "pump" maybe 2x more heat than it uses in electricity, so each 1 watt of electricity will transfer about 3 watts heat energy total to the hot side 1+2 = 3 but only provide 0 to 2 watts of cooling.
Apparently for the same wattage, using more Peltier devices at lower voltage increases the COP.
Bottom line is these things throw a LOT of heat. The heat pumped but also the heat from the applied electrical current.
A Stirling engine is actually, as I've deduced from a number of observations and experiments, a kind of gentle heat pump in itself, but in the opposite direction of the imagined "heat flow".
In other words, a Stirling engine acts like a dam holding heat back on the hot side of the engine, forcing the heat /energy to do work on the piston.
So putting a Peltier on a Stirling engine for cooling is just about completely useless, as it will never get cold, but using it to supply heat, you might get somewhere if the Peltier is just used as a heat pump to supply heat to the engine intermittently.
A Stirling engine is too frugal in its heat consumption to keep a Peltier device cool on its hot side. As I said it's a kind of heat pump itself but working in the opposite direction. It will gradually convert the heat into work so that the heat is gradually used up but it will never use up the heat fast enough to cool the Peltier device enough to actually keep it cool
So, sandwiched between two Stirling engines, as seen in the earlier video, the engine on the hot side of the Peltier runs very fast on all the excess heat available, but the engine on the "coldish" side, it's hardly running at all.
In fact for the description for this video, posted again for convenience
https://youtu.be/vsTIWhZ-6HM
It is mentioned that the Peltier gets so hot it could damage both the little LTD engine as well as the Peltier, which is likely why the video is very brief.
From the description:
For best results I would suggest a TEC1-12715 DC12V Heatsink Thermoelectric Cooler Cooling Peltier Plate Module 40x40MM over the TEC1-12706 as the TEC1-12715 can run at a higher amperage. If you only have a TEC1-12706 then you may wish to place some form of resisters or Temperature Controller Module otherwise the heat may be too great for both the Low Heat Sterling Engine & TEC1-12706.
So, really WAY too much heat being pumped to the hot side of the Peltier and you don't want to melt your engine, which is probably part plastic and foam rubber, or burn up your Peltier.
I'm thinking the Peltier could be used in conjunction with some phase change type thermal storage, like Xylitol as an intermediary.
Use the Peltier to pump heat into the xylitol for long term storage then run the engine off the hot xylitol rather than directly on the Peltier.