I had a similar issue with my e39. I caught the "culprit" by pure accident - after I parked the car, I returned about 1/2 hour later or so to pick up some stuff from the car and the fan was blowing at max speed!!!!!
To pull the fuse -
here is a linky that shows different fuses and relay locations. It's for the e39, but e 53 is similar in certain aspects. If it's the FSU, make sure you buy the OEM part - if you want to save some $$$, get the Valeo/Sitronic brand, and make sure it's not made in CPR.