Just FYI, hydrogen fuel cells are hydrogen-air batteries. There's no rational reason to reinvent the wheel for what is basically an inferior version of hydrogen fuel cells.
Hydrogen fuel cells are not especially efficient, and are quite expensive. Efficient electrolysis is also expensive. Hydrogen storage is quite expensive. And, mitigating explosion risks around hydrogen is complicated. So, there are sound reasons to prefer other non-problematic storage.
If panels get cheap enough, overprovisioning to account for even a large round-trip loss may become a thing. Then, the cost of the storage and conversion equipment dominates. So, it seems like even after you have other reasons to make hydrogen anyway, a better storage medium seems worth using, too.
Nobody is quoting round-trip efficiency for the iron-air battery, so I would guess that is not very close to as good as lithium tech. Their descriptions of battery installations say they include a fraction of lithium cells; probably the lithium cells are used to smooth off the peaks, falling back to iron when the lithium cells get low. We have lots of other reasons to overprovision panels.
It is possible that, as hydrogen gets more integrated into the energy system, starting with steel production and later aviation, its use for primary storage will increase. That probably depends on developing cheap, volume production of aerogels for LH2 tankage. Cost will always be important.
The round-trip efficiency for iron-air is going to be worse or the same as hydrogen-air. They are both the same idea but iron-air is much less mature.
The other point is that hydrogen electrolysis and storage is cheap and in fact extremely so. In large facilities it is <$1/kWh and is basically unrivallable by anything else.
With iron-air, you charge by splitting rust to iron and oxygen, vent the oxygen, keep the iron in the battery; discharge by oxidizing the iron with oxygen from air. How do you get the iron to rust fast enough? How do you get the oxygen through the membrane and into the electrolyte fast enough? High pressure? Maybe you need thousands of cells in parallel to get much current flowing?
With hydrogen-air, you charge by separating water into hydrogen and oxygen, venting the oxygen, refrigerating and condensing the hydrogen into insulated tanks underground. (It slowly boils off, according as how good your tank insulation is; underground insulation can be very good.) Bank the removed heat? Discharge by boiling and then oxidizing the hydrogen to water vapor, condensing that to a tank for later, maybe using heat from condensation to help boil the LH2?
So, for iron you need to pump air to high pressure, and probably heat the iron. For hydrogen, you have to liquify it after it is hydrolysed, and move heat around a fair bit, and capture both H2 and (probably) H2O.
The Chileans are building a liquified-air (O2, N2) storage system. That charges by refrigerating and condensing air into insulated tanks, and pumping the removed heat into a heat reservoir. Discharge by boiling, using banked heat and latent atmospheric heat, venting through a turbine. The turbine needs much less maintenance than a steam turbine, because nothing gets hot.
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