You do that because you don't need a large underground compressed air cavity. Instead, you can store energy as heat in materials, which is among the cheapest ways to store it.
But how much cycle inefficiency would you be willing to take for that advantage? I could totally see that storage setup be a huge success at places where you have continuous or forced-schedule intermittent low intensity heating and cooling demands (data center within insulated pipe distance of a spa resort or something like that), and want to make the best out of their opposite heat demands using intermittent electricity, but for general purpose electricity where the recovery would have to go through one of the famous thermal cycles? I really doubt it.
Um, you are aware that CAES also has these inefficiencies. Adiabatic CAES does compression with the heat transferred to a store. The compressed air is a store of "low entropy". Pumped thermal energy storage stores that low entropy as cold rather than as compressed gas. But in both cases, energy is recovered by exploiting a compressed gas (preexisting in one case, manufactured by compression of a cooled gas in the other) heated and reexpanded. I don't understand why you're apparently ok with one and not the other.
Aren't you completely forgetting the non-thermal part of that firm of energy storage? CAES even works completely without heat recovery/reuse. Not particularly well, the adiabatic/store is certainly added for a reason, but a carnot cycle (theoretical best) would take a non-trivial amount of delta to reach even that, even if you had perfectly lossless on the charging side. You'd have to be well within the domain of molten salt to achieve a delta where the theoretical carnot maximum minus real life losses could be interesting. Heat/cold storage can be very useful if you have direct uses for the heat/cold, but if you want to get back to general purpose electricity, it's not good at all.
When a compressed gas expands and does work, its thermal energy is converted to work, and the temperature declines. There is no "energy of compression" in the gas aside from its thermal energy content. The internal energy of an ideal gas is a function only of temperature, not of pressure.
Adiabatic CAES ideally stores ALL the energy invested during the compression into the thermal store. The compressed gas is just there as an entropy sink, so that this stored thermal energy can be converted back to work efficiently. But that's just what the "cold" in a PTES system is, too. Ideally, the efficiency of the two systems is the same.
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.
I'm curious what you think about my claim in https://news.ycombinator.com/item?id=32046602 that sensible and latent heat storage are not practical for seasonal energy stores, but TCES might be.
It's plausible. But understand that one can imagine putting heat into underground masses with extremely long time constants, artificial geothermal in essence. It's possible that this wouldn't be practical but I'd want to see details. Geothermal heat pumps already do this.
It's true, there are district heating systems using subterranean seasonal sensible heat storage accessed through heat pumps, like Drake Landing. I wonder if the tighter temperature precision requirements of efficient adiabatic CAES would make it difficult — such a reservoir necessarily has enormous thermal mass, and if its temperature has to remain essentially constant during operation you must store vastly more energy in it than the "tidal volume" of heat during a yearly cycle. But maybe you can avoid such a requirement by clever enough engineering using several such reservoirs or a continuum of them (like packed-bed heat exchange media).
Constant temperature in such thermal stores is achieved by some flavor of counterflow heat exchange. One could, for example, send a fluid through a long bed of some granular material, in one direction for storing heat and in the opposite direction for withdrawing it. The temperature will be hot at the charging input end and cool at the discharging input end.
Yes, that's what I meant about packed-bed heat exchange media—but making the bed long of course gives it more surface area to conduct heat to the surrounding rock. And the moving thermal gradient inside the packed bed still gives rise to some temperature variation over time at the hot end and at the cool end. For district heating this doesn't matter.
Possibly this is less of a problem than I imagine it to be for adiabatic CAES; my knowledge of these systems is obviously pretty sketchy.
