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The reason the water at the bottom is coldest is for one reason: energy is drawn from the coolant as it evaporates. As a portion of the droplet goes up in smoke, it takes energy with it as the water becomes a more excited state: steam. This leaves the water droplet colder than it was before, and the steam is warmer than the droplet was. No energy is lost: some is just transferred during the phase change.
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Actually both the water and air end up colder (though the air may not be significantly so on such a small evaporator). If the air ended up warmer how could evaporative coolers work so well. Here in tucson the watervapor typically leaves the cooler ("bong") a good 10C below ambinet!
This change in energy is called enthalpy of vaporization. Basically you are moving the water from a more orderly state (liquid) into a more chaotic state (gas). This change requires heat, and in so doing transforms the heat into potiential energy in the coolant. Thus both the drop of water as well as the evaporating fluid lose some of their heat energy.