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Before you turn on a fan, the air is essentially hot because it is still and there is no circulation byconvection of the air currents. When you turn on the fan, it sparks a circulation where the hot air at the top flows to the opposite side of the room, and then eventually becomes cooler and more dense, falls to the floor, and is pushed back towards the fan by the incoming hot air. So, when you turn on the air, the fan is just cirulating hot air, not making it colder. The air only becomes cooler when it flows away from the fan.

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Q: Explain the following statement you feel cooler on a hot day when you turn on a fan even though the air being blown around is hot?
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