I think glass doors are a great idea. Often, people will enjoy a fire in the evening. The fire is not completely out before they turn in for the night, so they have to leave the fireplace damper in the open position all night long. This is going to allow a whole lot of heated home air to travel up the chimney all night long. Furthermore, if you forget to close the damper in the morning, it will continue; you might as well be throwing money out the window. Glass doors are a nice alternative to help restrict the air flow up the chimney until you remember to close the damper. Yes, I think glass doors are a good idea for your fireplace if you do not have them already.
General advice. Contact licensed chimney expert for your particular situation.
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