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As I understand it, the explanation of the windchill factor goes like this:

Your body is warmer than the air around it, therefore it loses a certain amount of heat to the air. This warms up the air immediately around you first. The heat will then gradually be transferred to the air further away through convection, but there's so much more of the air than there is of you that the air essentially acts as a perfect heat sink, and your body heat isn't likely to raise the overall temperature of the air in even a moderately-sized room by any measurable degree.

However, you're still warming the air immediately around your body first. When the air is moving, it blows away the air immediately around your body, which is slightly warmer than the rest of the air, and replaces it with the slightly cooler air that hasn't been warmed up yet. Since the speed of convective heat transfer is based on the difference between the temperatures of the bodies involved, this means that you lose heat faster, and feel chilled more than you would at the same temperature if there was no wind.

However, I have memories of days when the temperature was well over 100 degrees, and breezes were still a relief. By the explanation above, wind should have made the heat even more oppressive because the convective heat transfer is flowing in the opposite direction, into my body rather than out of it. What's the principle here that makes this possible?

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Your body removes heat by coating the skin with a thin layer of moisture (sweat). As this layer evaporates, the heat of vaporization comes from you and you feel cooler. A breeze speeds the movement of the vapor away from your skin, and therefore speeds the evaporation. This is why very hot, very humid days are so oppressive.

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This illustrates why the wet-bulb temperature can be more useful sometimes. Above a wet-bulb temperature of $35^\circ C$, humans cannot survive prolonged exposure. – Jim May 16 '14 at 21:29
To expand both of these — you are losing heat to the atmosphere, even at temperatures above body temperature. If you're gaining heat instead of losing it, then physics dictates that your temperature is going up. If your temperature goes up too much, you will soon pass out and die. A 400-degree wind is not better than 400-degree still air; it's actually worse. But within the range of temperatures considered habitable, wind chill is a valid concept. – hobbs May 17 '14 at 0:04
@Hobbs A 400-degree gust might be better than 400 degree still air. – Hal May 17 '14 at 23:24

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