I have to agree with jammor. You probably want something more like the hose for a dryer vent. That's easy to get hold of and is very cheap. Also, I think you want a fan right next to the projector box, either inside or just outside. The reason has to do with the hose reducing your air flow. If you put the fan in the middle of the hose, you have the air flow reduction before the fan and after the fan which will cut down on the overall efficiency.
So, from a DIY perspective, use steps. Start with the fan in the projector box and attach the dryer hose from the fan to the window and do a couple hour heat test. If this doesn't work, you could try adding a fan in the middle of the hose which will act as a kind of booster.
On the plus side, I've seen this used on commercial projectors both for heat dissapation and for noise reduction. So, you're thinking right.
Thanks,
Zendance