The problem with barometric dampers is, the reduced updraft might adversely affect the secondary burn, reducing efficiency and increasing emissions. Further, the intrusion of room-temperature air into the flue cools the flue gases, causing increased creosote formation. Finally, if the increased formation of creosote leads to a chimney fire, the resulting extreme updraft will pull the barometric damper WIDE open, and could allow the chimney fire to rage out of control. The other technique is the manual stovepipe damper. A manual damper is a metal disc inside the stovepipe, attached to a handle on the outside.
Manual dampers reduce air flow through the stove by mechanically blocking the flue.
The price paid is reduced efficiency, increased emissions and excessive creosote formation in the flue. None of the woodstove manufacturers we know of recommend the use of stovepipe dampers, and some specifically forbid their use. The Chimney Sweep, Inc.