"We found that the only way to keep the temp constant was a combination of the compressor and a heater "

Correct. For any fine control of temperature within a close band, this kind of dual-ended ( heating+cooling) was required normally. But now with the advanced PID controllers, atleast for heating applications we get away with single ended control (only heaters). We have achieved +/- 0.5 deg C accuracy in controlling the temperature of a large oil volume of about 600 liters. In such cases we have a safety back up with a cooler that cuts in when the temperature rises on account of a process problem even with the heaters switched off.

However in the current application, I am required to start the cooling compressor when the temperature crosses 35 deg C and stop it when the temperature just falls below 31 deg C. The thermal inertia of the controlled system takes care of the overshoots and undershoots.

Raghu