There are several different ways that water is used in air conditioning. The most common use is in cooling towers. A cooling tower can be any size, from really gigantic down to the size of an automobile.
In a cooling tower, air blows over water to create evaporative cooling, and this cooler water is used in the heat exchanger of, say, an air conditioner. A more direct form of evaporative cooling is often found in greenhouses, farmers' markets and chicken houses. In these kinds of systems, water flows over a mesh, and a fan blows air through the wet mesh into the building. Humid, cooler air is the result.
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