Most of the wall behind is brick. There is an attic space above the back patio behind the lower right half of the cabinets. The pitch of the patio roof makes it impossible to go straight out from the top of the cabinet. If I want to dump heat into that space, the duct would follow the same path but go straight back instead of up at the corner. The duct woud be shorter but a problem is I'd have to install an outlet for the fan in the patio attic. There is already one in the attic over the room.
The top of the rack is about 2 feet above where the roof meets the fourth stud (from the left) in this picture. The right corner of the room is about another 3 feet to the LEFT on this pic. The horizontal board is the floor sill.