hope this photo posts ok:
That thing has been around for years under a bunch of names,"stack robber" is one.
They work pretty good but the down side is they might lower the stack temp below 300 degrees ,the point where creosote forms.
The air thing,I just use the ceiling fans.The insert has a fan on it.
In a week or so,as soon as the temp gets below twenty degrees I'll fire up the woodstove.It will maintain above 70 degrees in the whole house of about 1800 sg feet,L shaped. Even at zero,if the stove is fired correctly my geothermal seldom comes on.I usually shut it down about mid March .
Much above tweny degrees and it is too hard to maintain the temp other than setting around in your undies and opening a window.It's a brick house with good insulation,windows etc.Around 30,000 btu heat loss per hour at zero,if my calculations are corect.