It's not for cooling the engine it's for cooling the condenser.

On start up the car sends a pulse off power to the aux fan to make sure it's there and won't let ac engage if no fan.

You can however activate the ac compressor with no fan with the right scanner. I did that for a week or two until i got my new fan

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2011 E70 • N55 (me)
2012 E70 • N63 (wife)