I think that taking a different approach based on changing energy economics makes this a soluble problem; here's an outline of how to scale direct air capture large enough to reverse climate change, using existing technology, for a cost of well under US$100/tonne, at which point it's economically feasible for an international organization.

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If you had unlimited free energy, you might choose to use cheaper and slower absorption materials than things like triethanolamine. Quicklime, for example, maybe doped with a little soda as a catalyst, like in a rebreather. You can make individual quicklime beds very large indeed; a million tonnes seems feasible, and three million-tonne beds would contain only US$500M of quicklime. The energy cost to regenerate them is staggering (roughly 900 K × .8343 J/g/K = 750 J/g of the calcium carbonate, which is 101 g/mol, capturing 44 g/mol of CO2, so 1.7 kJ/g of CO2, 480 kWh/tonne CO2 = US$10/tonne at US$0.02/kWh), but then, so is the energy available from sunlight.

We probably ought to include the enthalpy of formation of the carbonatation reaction in case it's overwhelming: CaCO3 is -1207 kJ/mol, CaO is -635 kJ/mol, CO2 is -394 kJ/mol, so calcination is endothermic to the tune of 178 kJ/mol = 4 kJ/g of CO2. So actually that about triples the total amount of heat involved to 5.7 kJ/g CO2 = 1600 kWh/tonne CO2 = US$30/tonne.

You might be able to cut down the energy cost of regenerating your soda lime by an order of magnitude with sufficiently careful attention to regenerative heat conservation; you get back that enthalpy of formation when the air recarbonates your soda lime, after all. Also note that my thermodynamics calculation there is extremely crude and could easily be way off. For example I'm pretending the specific heat of calcite stays the same all the way up to calcining temperatures, which is very unlikely. But I'm guessing it's in the ballpark.

Three million tonnes of quicklime would hold 2.3 million tonnes of CO2. Operating regenerator-style with a 20-minute cycle time (perhaps a bit optimistic) that would remove 60 billion tonnes of CO2 from the atmosphere per year. If we amortize the quicklime at 5% per year, it costs us US$25M/year in financing costs, which is US$0.0004/tonne CO2. This low price means that even with year-plus cycle times on the quicklime, rather than 20 minutes, the energy cost would still dominate.

(It won't take a year. https://youtu.be/pFG-nXUw6Ts?t=336 purports to show successful lime-burning in 15 minutes; catalyzed with water instead of soda, whitewash or lime cement mostly carbonates within a few days; and, in anesthesia, soda-lime canisters are changed every 6-14 hours. Not sure if you can get the cycle time down to 20 minutes, though.)

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The Project Vesta folks are touting olivine beaches as an all-in-one solution, with capture, transport, and storage. I'm not convinced it'll be cheaper.

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Onsite sequestration would eliminate the pipeline network and the subsurface struggles. The simplest solution, again assuming unlimited free energy, would be to operate an onsite chlor-alkali plant to produce lye from sea salt, then carbonate the lye into sodium bicarbonate. (At scale this produces surplus chlorine, which ought to be disposed of properly, but I think that's reasonably easy to do, and it's a sort of pollution that won't go very far if it does escape.) This is scalable to terraforming levels: one electron at 2.7 volts gives you one NaOH, and that captures one CO2, so at 100% Faraday efficiency you capture one mole of CO2, 44 grams, per mole of electrons, 96.5 kilocoulombs, which works out to 5.9 MJ/kg CO2, another 1600 kWh/tonne = US$30/tonne.

And of course that's the maximum possible efficiency for the chlor-alkali process; I'd be surprised if it reached half that in practice.

Also, though, I think there are an awful lot of places where you don't have to drill very far to find olivine-rich basalt (or peridotite, or pyroxenite) that you can frack, which will then delightedly slurp up all the CO2 you can feed it. Of course, you have to keep fracking as your basalt undergoes talc-carbonate alteration, but that's clearly a thing you can do.

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Even this US$70/tonne is super low compared to the US$600/tonne or US$100/tonne numbers being bandied about in articles like https://www.science.org/content/article/cost-plunges-capturi..., or the US$1000/tonne Climeworks is apparently charging https://astralcodexten.substack.com/p/carbon-costs-quantifie.... If we use vkou's number in https://news.ycombinator.com/item?id=29327683 of 30 billion tonnes per year (note, this is half of my 60-billion-tonne ballpark above), US$70/tonne would cost US$1.4 trillion of energy per year, which is still not something you could do as a hobby project but definitely within the reach of an international organization.

And if the energy is free, you're left with US$0/tonne, which is to say with all the capex and opex costs that aren't energy, which I didn't even attempt to calculate above, except for the cost of quicklime.

In terms of sunlight, we're talking about 1.7 + 5.7 + 5.9 = 13.3 kJ/g CO2, which is 12.6 TW at 30 billion tonnes CO2 per year. Dividing by 21% PV panel efficiency (even though the lime regeneration could be driven by solar thermal energy) and 30% capacity factor (since we can site the plant anywhere there's seawater), we need 350 000 km² of solar panels to run it, about a 670-kilometer-diameter circle, a totally manageable size. That's US$7 trillion of solar panel modules at today's prices, again, a multi-year project for an international consortium, until solar energy prices resume their decline.

If we do the same 5% financing on the solar panels, plus another US$7 trillion for balance of plant, it costs US$700 billion per year. This is US$23/tonne CO2, replacing the US$70/tonne figure above. It turns out that energy is cheaper if you don't have to generate it in cloudy places, store it for use at night, step it up for long-distance transmission, lose some of it in transmission, lose more in distribution, maintain a distribution network, build gas peakers to prevent blackouts during load peaks, and so on.

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(We'd actually only need to remove two thirds of that amount, 20 billion tonnes CO2 per year, to counteract the current 2.5 ppm/year rise in atmospheric CO2. But we do need to at some point kick it into reverse.)

Of course there's no reason this has to be built as a single world-climate-control machine. And it shouldn't be. Instead of three million tonnes of quicklime and a humongous chlor-alkali plant sucking up 13 terawatts on the coast of Bahrain, powered by 500 kilometers of solar panels in every non-sea direction, you could have a thousand plants each with three thousand tonnes of quicklime and a normal-sized chlor-alkali plant, sucking up 13 gigawatts, powered by 15 kilometers of solar panels in every non-sea direction, only costing a few billion dollars each. And you should probably start at even smaller scales to prove out the concepts.

There's probably cheaper process options available that use fancier chemicals and more process steps to get a lower overall cost.

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I don't know, maybe I'm being too blase about all this. There's clearly a lot of huge challenges between here and there, but fundamentally I just think there are a lot of process avenues open that become very cheap as the renewable transition drives down energy costs to previously-unheard-of levels. And that means that atmospheric carbon capture is an entirely manageable problem.

What are the biggest issues I'm overlooking here? Did I totally flub some of these calculations?