Fan size to cool a 600W HPS Cool Tube

Okay, so here's my story. I have a closet I'm growing in which is about 200 cubic feet (5x5x. I need a light that will cover the whole closet, and I'm thinking of getting a 600W HPS.

Noise is a problem where I live, so I have an ultra silent 120 cfm fan blowing through a DIY carbon filter, and I think I'm getting about 40 cfm after the filter. I don't want to upgrade my exhaust fan because of the noise.

So I'm thinking of hooking up an inline fan to a 600W Cool tube that exhausts into the closet, just to keep the light itself cool, while the exhaust fan will move the heat out of the room.

What cfm fan do I need, just to keep the light cool enough that I can touch the tube without getting burned?

Just to clarify, are currently pushing air through your filter as opposed to pulling air through it? I'm a little unclear on your setup.

I'm pushing air through my carbon filter using a 120cfm Panasonic Whisperline fan. It runs constantly, so I think I should be able to vent the air from the cool tube just into the cab, and the whisperline should take care of it. Just not sure how big of a fan to use for the light.