Heat lost on toilet window shaft

My toilet window opened into the 3×4′ shaft, which has skylight and vents on the roof. The window is not closed and I suspect there is a lot of heat should leave house though toilet/window/shaft/vent. It look like one big chimney. Am I correct? what is the general consensus about such derelict constructions? Should I replace window with a toilet fan?

Actually shaft made out of drywall and has all the plambing and wires go though it. There are a lot of holes in the walls of the shaft. So I suspect there is a lot of heat lost though the warn air escaping. Should I rebuild walls? Or maybe I can knock down the walls, extent bathrooms and equip them with modern fans?

what do you think.
PS on the other side the exposed pipes in the shaft sounds like a big advantage when I will need to work on plumbing and I can though as many BX cables as I want though it as well.

By bobjohn |