"Hey, why does your computer sound and feel like a hair dryer? You wouldn't happen to be mining, would you?"
That just made my day :-)
Yeah, the bad news, it's going to be 4x 7970.
The good news, sound is not a problem as the pc would be in another room (not air conditioned).
Are there any recommendations on the case for a closed one?
I'm using a Cooler Master HAF X case with 3x 7950 with good results. I think you might be able to fit 4x 7970 in there with creative use of risers. You would want to remove the dust covers from the various grills to improve airflow.