I have a small portable air compressor for tires. The finned area heats up pretty quick. It is not taking ALL of the energy away, but it takes some.
Oh, and I forgot, using the ideal gas law doesn't exactly work for air compressors
Tout = [Tin] + [(Tin) x ((
PR)^0.263 - (1)) / (CE)]
Reference
https://www.physicsforums.com/threads/gas-turbine-engine-compressor-discharge-air-temperature-calculation.677829/
Assume here a CE of 100% ~ 1
Tin = 70F = 530R
PR = 27.2/14.7=1.85
PR^0.263-1=0.1757
Tout =530R+530*.1757R=623.1R=163.1F
So, the air entering the ball is 163.1F, at the END of the cycle. The air temp at the beginning is much lower since the
PR is near 1. Some the air temp in the ball is somewhere between 70F and 163F. If someone wants to integrate the equation from a
PR of 1 to 1.85, go ahead. If even 1/2 way it would be 116F. And if the compressor efficiency is lower, then the temp goes UP.