Does Exhaust Need to be Connected to Cool Tube?

If I have a booster fan blowing air through the cool tube. does the other end need to be directly connected to the ducting? I exhaust out the roof of the tent and when the cool tube is connected to the ducting it can't be centered or level.

I really wish Mac computers had the paint application so please, please, forgive the artwork:

Inside tent you have cool tube, one end connected going out the top of the tent into a fan, then muffler, then filter. The other end of the cool tube is open in the tent. You seal off the second duct hole. Idea being to pull air through the passive vents, into the cool tube and out the top.

If I got that right, yes it will work.

i want to go from booster fan pulling from inside tent>cool tube> hot air rises to top of tent>centrifugal fan exhaust.

You will be blowing the hot air around in your tent, moving air doesn't follow the hot on top rule, that is stagnant air. For your solution you should just ditch the second fan and remove the glass on your tube or better, switch reflectors to an open design. Otherwise, suck the air through the light and out of the box. That fixture/reflector/cooltube makes no sense unless you pump the air from it immediately out of your grow space.

The centrifugal fan mounted up top along with the passive intake does create negative pressure. It's id a 20 square foot room and I'm running the 167 CFM fat 1/2 speed. Also, only a 250w HID lamp.

It may work, just a silly use of that light fixture, wasteful, take the glass out for better light output if you aren't actually using it as intended. Monitor temps, and if you stay in range stick with it.

You are basically running a more expensive fixture with worse output if you run a cooltube without using it's key advantage, less heat entering your grow space. You can at least minimize the light loss by getting rid of the glass that will be doing you no good anyway at this point.