Details on this combined cycle :


1st step : the heat pump based on Carnot thermodynamic cycle


Heat pump carnot cycle
the heat transfer to the hot reservoir is

Qh = Qc + W where Qh is the heat that will be used in second step, Qc is the free heat taken from environment and W is the energy bought by the compression via electrical energy.

The important thing is that Qc is typically 4 time more important than W so that global efficiency is Eff = Qc (heat caught) / W (energy spent) = 400%

This is a typical reverse "air conditioning" arrangement that is used to cold or warm an house:
Heat pump devices

One solution to be explored for industrial purpose is to increase tremendously the efficiency of the system is to put a turbine instead of an evaporator in order to recover the energy provided by the gas expansion and to transfert it directly to the compressor. this system is sold under the name of turbo-expander having a typical efficiency of 75%. Nevertheless, the 400% efficiency of the cycle will not be multiplied by 4 (75% of saving on electrical energy spent W) because such expansion through turbine is not fully adiabatic (A to D) and so energy recovered Qc is reduced but the global gain remain far above 400% of efficiency.

2nd step : the stirling cycle to convert the heat caught into electricity:


Stirling cycle

The complete theoretical explaination is very well done on this site : link so i will not detail it again

Efficiency of this system = 1 - (cold side temperature/hot side temperature)

Example with cold side 0C and hot side 200C : Efficiency = 1 - (273.15/200+273.15) = 42 %