Glenn Herbert wrote:40 cm (16") deep is really tight. What is the material of the wall behind? If it is masonry, your dimensions might work. If wood framing, you need at least about 10 cm clear for air circulation to cool the wall. For heat circulation inside the mass cavity to work well, you need around 20 cm or more clear. Brick walls need another 10 cm thickness in front, more for large flat spans.
Your idea of a mass that is higher at the sides than at the center can work fine, using a "bell" or stratification chamber, which is essentially a hollow box for hot gases to fill, give up their heat to the container, and fall as they cool to exit near the bottom.
Exiting below floor level, traveling several meters, and rising on the outside wall is problematic. It might work if conditions are right, but may well not work and want to backdraft smoke into the room. Rising to near the ceiling and exiting on the outside wall has a much better chance of working. What is the structure like? How many floors is it and where are you located in that? What is the prevailing wind in relation to your space?
Glenn Herbert wrote:I don't have time for a full reply right now, but the structure sounds good. I would want to put a layer of fireproof material spaced a bit off the existing wall to make sure the other side of it doesn't get too hot... something like cement board in moderate heat areas, sheet metal in high heat areas behind the J-tube and riser and up. Plasterboard with its paper surfaces would need to be stripped off the brick, which would add a tiny bit to the available working depth. What size system are you thinking of? 150 or 200 mm? The mass space you have may not be enough to extract and store the heat output of a 200 mm system. I can make a sketch of possibilities and post it, but that will take a few days until I have the spare time and tools accessible. The column of space on the right should be usable.