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Ammonia is currently considered as one of the more promising options for future shipping fuels.

There are a number of projects planning to create green ammonia at scale, e.g. this: https://asianrehub.com/

Ammonia is already made from hydrogen today, making that green is pretty straightforward, you just need enough clean electricity. Just get the hydrogen from electrolysis, the ammonia synthesis process itself is well established technology.

Ammonia is one of the less pleasant chemicals one could use. I'd rather have Hydrogen-exuding paste than Ammonia in an accident. The Hydrogen might explode. But the Ammonia definitely will hurt you.
Ammonia is quite ok. You can smell it :-) You can put it into water, where it is quite stable. We know how to handle it at industrial scale because of fertilizers.
Yes, you can smell it, when drastically diluted in water (which, as you say, is stable) such as for a household cleaner. But in a concentrated form such as that required for use as fuel, if you smell it, it will kill you. Quickly.
Also, water-free Ammonia is gaseous at room temperature, you have to transport it in pressurized containers. Should one of those rupture, all the Ammonia will boil off, creating a nice cloud of fun for all involved. Neutralizing it works by spraying acetic or hydrochloric acid, which is not quite as bad, but still quite ugly.
Why not just use a closed loop hydrocarbon from electricity system as a shipping fuel if you have excess power and a need for transit?

The main theoretical point of hydrogen is very high specific energy per mass but it is essentially purest acidic gas and a pain. Bonding it to something else mitigates it so why 3 H per N instead of 4 H per C? Both are toxic gases at this point leaving ammonia's main advantage in the context being its own oderant.

The problem with every hydrocarbon-based fuel is that you need the carbon.

You get that from CO2. But then you need to get the CO2. Where do you get it from? From a fossil-based plant? Well, ideally you'd want to get rid of those, not exactly smart to create incentives to keep them running. The alternative is either biomass (problematic) or direct air capture (expensive and inefficient). (Some insightful discussion on green methanol: https://www.youtube.com/watch?v=jXACyUxxBts )

With non-carbon based fuels like hydrogen or ammonia you skip that problem (air is 78% nitrogen, much easier to extract).

or direct air capture (expensive and inefficient)

The big question in my mind is whether direct air capture is inherently expensive and inefficient...or if this is just a chicken/egg problem where we haven't invested time and money in making it cheaper because it's expensive, and it's expensive because we haven't invested time and money in making it cheaper.

I don't know enough about physics and chemistry to answer the question on what the theoretical lower bound on cost might be.

From some research into the feasibility of an indoor CO2 scrubber that targets pre-industrial concentrations (100~200 ppm), using a sodium hydroxide solution in a simple counter- or cross-flow packed-bed wet scrubber does a good job at scavenging CO2 from the air down to <100 ppm on the exhaust, while the hygroscopic sodium hydroxide has an equilibrium with ambient humidity at all relevant indoor living room temperatures and humidities.

Regeneration is easy in another counter-flow packed bed reactor, this time reacting with a CaOH bed to exchange the carbonate ion. The output is mostly CaCO3, with some NaOH contamination. This can probably be washed for home-scale disposal (and recuperation of the NaOH), while the industrial scale process follows up with thermally decomposing the CaC03 into CaO and CO2. This can be very pure CO2 suitable for direct sequestration, if the thermal energy is provided electrically or by combusting a hydrocarbon with purified oxygen.

So the lower cost would seem to be that of calcinating the limestone (at 900~1050°C), and a trade-off between cap-ex and op-ex for the scrubbers. The lower the flow rate, the less energy is needed to force the solution and air through the packed bed.

But afaik freezing the CO2 out of the exhaust from fossil fuel power plants and industrial processes requires less energy than the calcination, and is therefore economically favored until all easy opportunities have been converted.

The calcination seems to require about 800 Wh/kg of CO2. At typical electricity rates in favorable locations of 10 ct/kWh, this makes 1 kg DAC-CO2 cost >~8 ct. If you want the carbon out of this, you're looking at 1.25 $/kg of DAC carbon. Assuming perfect electrolyzation of the CO2.

It's inherently expensive, because you have to undo the entropy loss of letting the CO2 diffuse into the atmosphere. It basically means running an expensive molecule sorting operation, whose cost has a floor set by the laws of thermodynamics, in advance of whatever else you wanted to do with the CO2.
That entropy loss is not that large (it's logarithmic in the dilution). A bigger problem is simply the cost of handling a lot of air.
Direct air capture is inherently expensive unless you are doing it in the airstream of some industrial process that produces large amounts of CO2, and even then its still somewhat expensive.

Even as a waste product from separating Nitrogen, Oxygen and Argon from the air, its still expensive (retail its ~$1/lb of liquid CO2).

> The big question in my mind is whether direct air capture is inherently expensive

Have a look at a tree...

I suppose it's possible that billon years of evolution has ended on a local optimum for low energy input (direct sunlight), and we might revolutionize it with high energy (eg: high voltage electricity, fusion etc) - but I doubt it.

> low energy input (direct sunlight)

Direct sunlight is actually quite powerful, around 1kW/m^2 at sea level.

Although I'd guess most trees aren't particularly efficient at absorbing CO2 vs the energy they consume, which makes sense, since they only have to be as efficient as necessary to survive.

2% efficient, at best. 1000J in, 20J worth of wood and leaves out. Burning it, you get back only a fraction of that, with much of it carried away in the smoke.
It seems pretty obvious that extracting a gas that is less than a tenth of a percent in the air is inherently difficult.

That said: I'm all for investing in DAC technology. We will definitely need it for some sectors. But you need to consider the costs and if there are alternatives they will in many cases make more sense.

We need to deploy direct air capture of some form anyway because we're not reducing our carbon output quickly enough.
According to Wikipedia, CH4 is actually non-toxic (which is what I'd expect for something so ubiquitous in biological systems).