Well, don't think this is that complicated, but still. When we feel hot we (the people in the middle of the society who can afford fans, but not ACs) switch on the fan. Now, heat is just the measure of kinetic energy and as KE increases so does heat. So, if we switch on the fan, aren't we supposed to feel more hot than cool as the KE of air increases. Just a thought.
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The reason the heat you feel is actually the rate at which KE (heat) is leaving your body. Now because the ambient Tem. is usually lower than that of your body's heat will be leaving your body. But as heat leaves your body and enters the air the air will become hotter and heat will flow out of your body slower. So when you turn on the fan the air you're heating up will be replaced by the ambient air, this phenomenon is know as convection. The moving air also helps evaporate your sweat and because of the enthalpy of vapoization of water this causes your body to give of more heat. The heat given of by the fan is negligible.
Sanjay A. , yes,heat is the measure of kinetic energy,but the velocity that matters in this kinetic energy is the root mean square velocity,that doesn't count the velocity of the center of mass. When the air blows on you,it has center of mass velocity. therefore,the reason of feeling cold using fans it's because of the heat conduction body-->air,and the air is a good conductor only when it gains velocity.