Explain why the Carnot cycle efficiency is unrealistically high for a real engine. Introducing the concept of external irreversibility, evaluate the efficiency of an endoreversible engine at maximum power output. Consider a heat engine connected to high-temperature reservoir at \(T_{\mathrm{H}}=1200 \mathrm{~K}\) and a low temperature one at \(T_{\mathrm{C}}=300 \mathrm{~K}\). If the heat transfer conductance from the reservoirs to the engine are in the ratio \((U A)_{\mathrm{H}} /(U A)_{\mathrm{C}}=C_{\mathrm{H}} / C_{\mathrm{C}}=2\), evaluate the following:

1. the maximum Carnot cycle efficiency;

2. the work output of the Carnot cycle;

3. the engine efficiency at maximum power output;

4. the maximum power output, \(\dot{W} / C_{\mathrm{H}}\) and 5. the maximum and minimum temperatures of the working fluid at maximum power output.

Derive all the necessary equations, but assume the Carnot efficiency, \(\eta=1-T_{\mathrm{C}} / T_{\mathrm{H}}\). \([0.75 ; 0 ; 0.5 ; 100 ; 1000 \mathrm{~K} ; 500 \mathrm{~K}]\)