The compressor makes a gas pretty much at room temperature have much higher pressure. The gas gets hotter because pV = nRT. As p(ressure) goes up, T(emperature) goes up.
Because “=” .
You cool the pressurized gas a bit with a fan.
Then you release the pressurized gas into an unpressurized space. This greatly reduces the temperature because pV = nRT ( see above ) . You capture the cold and use a fan to send it somewhere.
For “drier” , when air with water in it gets colder, the water in it condenses – changes from gas state to a liquid state. It becomes water and gets drained off by gravity. You can experience condensation by putting ice water out in a humid place. Water forms on the outside of the glass, condensed from the air at the edge of the glass.
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