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The people saying Maxwell's demon/"cannot separate" are mistaken. They've pattern-matched to the wrong concept. This is possible and is the way swamp coolers operate. They exploit the difference between ambient temperature and dew temperature of the liquid, say water. As long as relative humidity is below 100%, some water will evaporate, leaving your vat colder than environment. The separation is simple - the wind carries away the 100% humid water vapors, replacing it with fresh low-humidity air. You can then use the temperature difference to drive a stirling engine or something.
The point where your free energy ends is when you run out of water. You need to take your 100% humid air and cool it down somewhere else to get the water back by condensing it. In nature this happens automatically at nighttime when the heat radiates into the cold of space and air temperatures drop. If you are running a closed-loop cycle though, like on a spaceship, you need to provide your own source of environmental heat (the Sun) and your own heat sink (the cold of space), and at that point you should just be using a regular high-efficiency heat engine instead of this swamp cooler.
Evaporative cooling doesn't actually create free energy though. The cooling effect comes at the cost of the latent heat of vaporization - you're just moving heat from one place to another, not creating usable energy. A Stirling engine would need both a hot and cold resevoir, and the cooling effect alone dosn't create a temperature gradient that could produce net work.
Yep, that right here is the kind of pattern-matching you have to be careful about! Read what you wrote carefully:
The vat is literally cooler. You put a thermometer in it, will show a lower temperature than thermometer in air. This is not a fake effect "only shows lower because it's wet", it's a real temperature. You put your stirling engine cool coil in the vat and hot coil in the air, you got yourself a temperature gradient. A small one, maybe 10 degrees C, but more than zero.