Air con pumps don't run continuously but kick in and out using their clutch depending on system demand - ie not just when you punch the dashboard button. When engaged it's quite usual for the engine to lift the idle revs a wee bit so you can listen for this which will "tip you off" to when it's actually driving the pump, but this may not happen on a cold start as an elevated idle speed will already be applied. On ours, with the engine running, the centre of the pulley can be seen to be stationary when disengaged and turning as a whole when driving the pump. My point is that when not engaged the pulley is spinning on a ball race in the pulley and the pump input shaft is stationary. When engaged the pump input shaft is locked to the pulley so this bearing will be stationary. Noisy with pump disengaged? probably the pulley bearing. Noisy with pump engaged? Probably something internal to the compressor itself. Noisy all the time, ie whether pump is being driven or not, maybe it's not the compressor at all but could be something else? Try an automotive stethoscope (long handled screwdriver if you're a cheapskate like me) applied, with great care - don't get it in the drive - to the casings of compressor, alternator, water pump (if you can get access) and engine block, with the other end stuck in your ear, until you find the noisy component. - Good luck.
