"Why would you cool down to later compress (and heat up again)?" Because the efficiency of any combustion engine depends on the temperature differential betwee

"Why would you cool down to later compress (and heat up again)?" Because the efficiency of any combustion engine depends on the temperature differential between the input and the output. It cannot produce more energy than that differential (second law of thermodynamics) so cooling the input allows the engine to produce more energy, assuming the output remains the same.