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I realize there would not be any phase change in this system, but as long as the compressed air get hot enough above the outside ambient temp it seems like you could extract heat (and even some humidity) from it. In some cases couldn't this end up being more practical than using a refrigerant and then needing a heat exchanger to indirectly cool the air?

Why does it seem to never be used?

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what you describe is called an air-cycle refrigeration system. they are less efficient at moving heat, as pointed out by @johnrennie, but they are simple and can be made light in weight. this favors their use in passenger planes, where weight savings are extremely important and compressed air is plentiful because it can be bled off the compressor sections of the gas turbine engines that propel the plane. This reduces the power output of the engines slightly but the weight savings make up for this effect.

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In general the latent heat of vaporisation is far greater than the specific heat of the gaseous phase. So if you're trying to move heat around it is far better to use a vaporisation/condensation cycle that a simple compression/expansion cycle.

If you look at Freon, which is a common refrigerant, then the specific heat of the vapour is around $0.6$ kJ/kg/K while the latent heat of vaporisation is around 160 kJ/kg.

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