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We know that on increasing the kinetic energy of a gas its temperature increases:
$$\text{Kinetic Energy} = \frac32 kT$$ where $k$ is the Boltzmann constant. This equation is derived from the kinetic theory of gases.

When we switch on a fan we increase the kinetic energy of the gas molecules increasing their temperature, but we generally feel cold.
This happens even if we are not sweating.
What am I missing here?

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You are confusing the fluid motion of ~10^23 particles per mole, with the average kinetic energy within a Δ(V) of that fluid in its center of mass, whence the formula for temperature comes.

The feeling of coolness comes mainly from the evaporation of moisture from the skin, the skin is moist always, sweating is an extreme condition. Of course a fan cools electronics too, as the contact with cooler air will transfer energy from the hot electronics to the moving air, convection.

Convective heat transfer, often referred to simply as convection, is the transfer of heat from one place to another by the movement of fluids. Convection is usually the dominant form of heat transfer in liquids and gases. Although often discussed as a distinct method of heat transfer, convective heat transfer involves the combined processes of unknown conduction (heat diffusion) and advection (heat transfer by bulk fluid flow).

(Also note that the energy in the bulk motion will eventually end up in increasing a bit the kinetic energy of molecules in the room by multiple scatterings ).

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The still air that surrounds you is warmer than the air in the room. Your body heat is constantly heating up the air that's touching your skin. When you turn on the fan, this warm air is removed and constantly replaced by room temperature air (room temperature meaning the room you are in) which feels cold only in comparison to the air you previously had the chance to heat up. Bigger temperature difference between the air around you and your skin = bigger cold sensation.

If you were to stay in the room with the fan running with no contact outside of the room/fan system, the air would continue to get warmer until it was at equilibrium with your body (which would try to disperse more heat and on and on probably to heat stroke).

Any heat you would feel due to the kinetic energy of a few grams of air at a couple of M/s is completely insignificant.

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  • If a colder air molecule hits you, it will absorb some heat, thus cooling you down and make you feel cold.

  • If a warmer air molecule hits you, it will deliver some heat to you, thus heating you up and making you feel warmer.

The air molecules surrounding you will under normal circumstances heat up quickly and then not absorb any more heat from you. You will quickly not be cooled down any more in any significant amount, since the thermal conduction of air is low and the natural convection happening is not that powerful.

But imagine having the touching air molecules replaced constantly. Each will absorb heat from you and then be moved away, giving place for a new one to absorb heat from you etc. The ceiling fan causes this forced convection.

This means that

  • if the surrounding air is colder than your skin temperature, then the fan cools you down, while
  • if the surrounding air is warmer than your skin temperature, then the fan heats you up.

If you have ever driven through Death Valley in California on a hot summer day, you will agree - one does not roll down the car window, since that only heats up the passengers.

If you sweat, the heat transfer between body and air molecule is faster, so this whole process will be more effective and you will feel even colder. But even without sweat, that effect is still taking place.

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