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First the name of the process is nothing but evaporation. Evaporation is the change of water into vapor and it results in the cooling of water. The water particles gain energy in the form of heat and hence they evaporate and mix with the gases outside.clay pot has small microscopic holes which allows the water to seep out.hence the water particles gain energy , they turn into a gas , that is, they get evaporated and this causes cooling.
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It is known that evaporation causes cooling. 
In this case, the small pores on the pot helps in removing heat from the pot & is released to surroundings. 
Hence water inside a clay pot become cool after sometime. 
It is not based on any theories.
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