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I added a new room with a basement. The basement is half under ground. The basement has one heat duct but no cold air return. I cannot add any more heat ducts. The temperature in the room is about 55. Will it help to add a cold air return? Thanks
The warm air must have some way to work its way back to the furnace, if no direct return, at least a reasonable path through open doorways to where the air return is. Perhaps a louvered door rather than a solid door.
The basement door is probably acting as a return- this is common and with at least 1/2" gap at the bottom usually enough. There are other things to consider though. First is the basement itself- if it's uninsulated you may be admitting more cold by increasing the return size and the HVAC system may not be sized for that. Insulating the basement walls may or may not give you better results but it's a coinsideration. The flow path matters here too; increasing the basement door ventilation may create a cold draft in places you don't want that as it heads toward the return. If it passes the thermostat along the way it may alter how well that works for the rest of the house, creating hot and cold rooms unless the thermostat is moved. Where the conditioned air is ducted to in the basement also matters, and that should be as far away from the door (or return) as is practical. In a basement I'd also want it low, not in the ceiling, so that you don't have all the heat at the top with a closed layer of cold underneath. This will also help alleviate basement humidity more than a ceiling mounted diffuser. Most important here is to not add a return unless you can balance the system properly afterward- simply tossing in a return might again cause hot and cold rooms. The basement ceiling/first floor should be insulated unless the basement can be kept to nearly the same temperature as the house to prevent conductive transference.
Another consideration is your basement usage. If you're down there a lot you want it well-conditioned. If not then the level you speak of is not something to worry about. If you're usage is somewhere in between then you may be better off adding a room heater of some sort, using it only when you're going to be down there. If you choose the latter, be sure that you keep the basement door closed when the axillary heat is in use or it might all go upstairs making little difference in the basement.
It's not always a simple answer that's best for simple problems and each situation varies. If the best answers are not apparent to you, have a local HVAC contractor drop by to advise you on what to do, and remember to tell them how you use the basement.