A compressed piston would allow energy recovery even after allowing the entire heat generated by the compression to dissipate down to ambient levels. The pressu

A compressed piston would allow energy recovery even after allowing the entire heat generated by the compression to dissipate down to ambient levels. The pressure gets slightly lower from the distillation, but it remains considerably above ambient. The fraction of the work that went into the pressure delta post-dissipation can be recovered 1:1 and then you'd still have a temperature delta left to hypothetically salvage, in part, using some carnot approximation. Your purely thermal approach would certainly try to salvage the work that went towards pressure right in the heat pump (decompressing through a piston or turbine mechanically connected to its counterpart on the compression side), but all recovery would still be constrained by carnot, which isn't the case for the "pressure part" of the work if you store that instead of immediately reusing it to gain more temperature delta. Thermal storage is the right tool for the job when all you have to begin with its a temperature gradient (e.g. in a mirror collector spat plant) and you want to time-shift conversion to electricity, or when you have a thermal use case and you want to time-shift electricity use from thermal utilization, but not for round trip electricity/electricity. Some quick googling yields a presentation comparing A-CAES and PTES by Xue Haobai at 2019 Offshore Energy and Storage Summit that puts the low end of round trip efficiencies of A-CAES to where the best of the best in PTES/Carnot batteries would max out. I can't tell you exactly where your model goes wrong, but I really believe that you must be missing something.