Looking for a recommendation for a company that might have damper for a 4 sided fireplace. We found one that will build a custom damper but would prefer to find one already manufactured.
If you want a damper with this, you will have to use a top of flue damper. Note that choking down the opening sizes, especially on the long sides, will help.
I would push for a 2 sided though, they still don't draw well, but you at least have a chance. Built to this plan they sometimes work without doors, other types I have seen do not:
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