The air from a fan is not a cooler temperature. Increasing the air flow increases the amount of thermal energy that can be moved away. Heat will always “want” to equalize at the lowest temperature available. Normal body temperature is 98.6°f. So if the air temp is lower. Then the heat from the body can be pulled away into the air. The more airflow there is. the more heat transfer can occur. Sweating increases this effect as sweat helps to move away more heat. This process results in a cooling feeling. Until the air is at a higher temperature.
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