Velocity and Pressure!
One reason fan air is cooler is because the increased velocity causes a less matter within an area. If each flowing particle has the same energy as the particles of stagnant air, but there were less particles in an volume => less energy in that volume => lower temperature => it’s cooler!
Whenever you think of heat/temperature, ALWAYS think about what’s happening with pressure!
It’ll usually help explain some of the “missing” things! (Pressure-temperature are deeply connected just like electromagnetism)
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