E.g. 100 GWh (4 days of 1GW thermal power) with a 100C temperature swing (you can't go down all the way to room temperature) will need around 3.6*10^4 tons of sand. That's a lot.
Honestly, power-to-gas sounds like the only reasonable way for now. In theory, with SOFC (Solid Oxide Fuel Cells) for power generation, it can beat the steam cycle.
View another way: if we hook several piles in series, then lukewarm air from a first mostly discharged pile can be directed into later piles for more heating. In the continuous limit this is the concept I described above.
No, because your steam will be at a lower temperature as you draw down the heat, bringing down efficiency.
Over the season, your sand drops from ~1000 to ~600 (which is fine since your power needs drop in the spring)
At 1000C steel will react with _water_.
To be clear, I'm not talking about cooling a small crucible but a structure that is at least 20 meters in size.
But I have no reference for the cost of setting up such a storage and how it compares to battery or hydro.
So the idea is not practical for storage for an individual home, but could work on the scale of a village or small industrial enterprise. It would allow a solar microgrid to be completely disconnected from the larger grid, even at high latitude (assuming one could get a sufficiently good heat engine to work at that scale; it's a bit small for steam turbines.)
If you look at very large piles of dirt, like large earthen dams, if arranged in a suitable geometry they would have thermal time constants (assuming no fluid flow through them) in the thousands of years.
A computation for you: compute the cost of a given R value per m^2 for various materials. Dirt is very cheap, so even if it's not the best insulator it comes out on top. You need the thing being insulated to be large enough that you can fit the thick dirt layer around it, of course.