Tom
In the air conditioning packs the pneumatic bleed air is passed from the pack flow control valve through the first stage of a dual heat exchanger, where it gives up some of it's heat. This charge air is further reduced in temperature by the ACM. The now cooler air is heated again by virtue of passing through the compressor but gives up some of this heat again as it crosses the second part of the heat exchanger, now the greatest reduction in temperature is achieved by the expansion of the air across the turbine which gives up it's (heat)energy in driving the compressor.(bootstrap cycle)
The ACM also drives a fan in the inlet duct, this is there purely for use on the ground to move air across the heat exchangers and is redundant in flight as ram air is the cooling medium for the heat exchangers.
FWIW the packs are very good at reducing the temperature and if uncontrolled will spit ice. Look at the mix manifold temp!
The moisture removed by the water separators is sprayed onto the surface of the heat ex. to futher reduce the temperature by evaporation and the reheater is there to inject a bit of warm air to prevent freezing as mentioned above.
The cabin temperature control system controls the pack outlets to the cabin zone that requires the least amount of heat and trim air is added to the other zones as required.
Active clearance control is there to reduce tip losses and therefore improve the SFC.
Depending on the engine will dictate how sophisticated the system is but basically for a modern powerplant the engine EEC will control the valve(s) based on inlet temp. and n1 and n2 shaft speeds. For example LPT Acc. the EEC will use n1 speed and open the valve fully when t.o. power is set.
HPT Acc. the EEC uses inlet temp. n1 and n2 speed and the valve will open fully (briefly) once t.o. power is set and then modulate. It's all done by calculation!!
Hope that helps I'm

myself now