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Am I missing something? It occurs to me that if heat from a condenser
were dissipated into a Stirling engine a couple of good things would
happen. The refrigerant would give up its heat faster. Less
refrigerant would be needed and a smaller compressor would be required
for the same refrigeration effect.

Being that temperature equals pressure and a refrigeration system
typically had a COP of 4 to 1 would it be possible to generate enough
electricity by having the Stirling turn a generator to run the
compressor motor as soon as the refrigeration system gets into the
operating zone?

What am I missing?

Hi
I thought about the point too
It seems to be perpetum mobile
If the eficienty of Some Stirling could be about 50%
and COP of refrigarator is 2.2 means - 0.5 * 2.2 =110%eficienty 8-()
!!!???
The way to check is - lets build it and test.

Yes, I'm afraid so. Make a block diagram of the Refrigerator with the mechanical power input, the heat input from the refrigerated space and the heat rejected to the condenser.
Input the heat rejected to the condenser as the input to a Stirling engine, and the heat rejected by the Stirling to atmosphere to see what the power out put of the Stirling would be.
Much less than the power required to run the refrigerator.

Second Law and all that! Then that devil entropy

COP of a heat pump can be 5 to 1. That means that it uses 100 watts to run the compressor but it produces 500 watts because it is extracting energy/heat from a fluid. In the case of a an air conditioner it is rejecting heat into the air. The condenser temp maybe 180F and the ambient may be 80F. A Stirling would not run with such a low delta T but if the condenser temp could be brought up to 300F by using a new type of refrigerant and extremely high pressures.

When a Stirling is driven one side heats and one side cools, right?