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Other commenters correctly describe the cost analysis for using evaporative cooling, but I'll add one more reason why it's the preferred method when water is available: evaporating water can dissipate truly outlandish amounts of heat with very few moving parts.
Harkening back to high school physics class, water -- like all other substances -- has a certain thermal capacity, meaning the energy needed to increase the temperature of 1 kg of water by 1 degree C. The specific thermal capacity of water is already quite high, at 4184 J/(kg*C), besting all the common metals and only losing to lithium, hydrogen, and ammonia. In nature, this means that large bodies of water are natural moderators of temperature, because water can absorb an entire day's worth of sunlight energy but not substantially change the water temperature.
But where water really trounces the competition is its "heat of vaporization". This is the extra energy needed for liquid water to become vapor; simply bringing water to 100 C is not sufficient to make it airborne. Water has a value of 2146 kJ/kg. Simplifying to where 1 kg of water is 1 liter of water, we can convert this unit into something more familiar: 0.596 kWh/L.
What these two physical properties of water tell us is that if our city water comes out of the pipe at 20 C, then to get it to 100 C to boil, we need the difference (80) times the thermal capacity (4184 J/kg*C), which is 334,720 J/kg . Using the same simplification from earlier, that comes out to be 0.093 kWh/L. And then to actual make the boiling liquid become a vapor (so that it'll float away), we then need 0.596 kWh/L on top of that.
Let that sink in for a moment: the energy to turn water into vapor (0.596 kWh/L) is six times higher than the energy (0.093 kWh/L) to raise liquid water from 20 C to 100 C. That's truly incredible, for a non-toxic, life-compatible substance that we can (but should we?) safely dump into the environment. If you total the two values, one liter of water can dissipate 0.69 kWh of energy per liter. Nice!
In the context of a 100 megawatt data center (which apparently is what the industry considers as the smallest "hyperscale data center"), if that facility used only evaporative cooling, the water requirement would be 144,927 L/hour. That is an Olympic-size swimming pool every 6.9 ~~seconds~~ hours. Not nice!
And AI datacenters are only getting larger, with some reaching into the low single-digits of gigawatts. But what is the alternative to cooling the more-modest data center from earlier? The reality is that the universe only provides for three forms of heat transfer: conduction, convection, and radiation. The heat from data centers cannot be concentrated into a laser and radiated into space, and we don't have some sort of underground granite mountain that the data centers can conduct their heat into. Convection is precisely the idea of storing the heat into a substance (eg water, air) and then jettisoning the substance.
So if we don't want to use water, then we have to use air. But for the two qualities of water that make it an excellent substance for evaporative cooling, air doesn't come close -- 1003 J/(kg*C) and no heat of vaporization, because air is already gaseous. That means we need to move ungodly amounts of air to dissipate 100 megawatts. But humanity has already invented the means to do this, by a clever structure that naturally encourages air to flow through it.
The only caveat is that the clever structure is a cooling tower, and is characteristic of nuclear power stations. It's also used for non-nuclear power station cooling, but it's most famous in the nuclear context, where generators are well into the gigawatt range. Should AI datacenters use nuclear-sized air cooling towers instead of water evaporation? It would work, but even as someone that's not anti-nuclear, the optics of raising a cooling tower in rural America just to cool a datacenter would be untenable. And that's probably why no AI datacenter has done that.
To be abundantly clear, I'd rather not have AI datacenters at all. But since the question was why water consumption is such a big deal, it might be best to say that it's a physics problem: there isn't any other readily-available way to provide cooling for 100+ megawatts, without building a 100+ meter tower. Water is always going to be cheaper and more on-hand than concrete.
So is air cooling actually feasible but we don't do it cause it would make data centers look like nuclear reactors? Or is it just not feasible?
Air cooling is feasible, as evidenced by existing power stations that use air cooling. A lot of newer nuclear generation use water cooling, being sited along the ocean and in the multi gigawatt range. But we can also find examples of inland power stations that have no water connection, and therefore need some massive cooling towers. Here is one in Germany that has a 2.2 GW rating and a 200 meter tall tower: https://en.wikipedia.org/wiki/Niederaussem_Power_Station
This is, as you can imagine, rather expensive to build, but it's doable. Cooling a coal fire is not substantially different than cooling compute loads in a data center, as it's all just a matter of moving heat around. Will there be differences due to the base temperature of coal versus GPUs? Yes, since the ratio of input to ambient temperature matters. But on the flip side, this should make it easier to construct, as the plumbing for lower temperatures is simpler.
Mechanical engineers can chime in on feasibility for AI data centers, but seeing as it hasn't been done, it's probably still cost related.