You'd be actively cooling your wood stove by much more than the power you'd extract, and you won't get more than enough to charge a phone or laptop. TECs are just terrible.
You'd get much better result by making a small steam plant with the same setup: boil water on stove, drive plant (turbine or piston), condense outside, repeat.
However, if you really want to do this with TECs, stack them to lower the per-unit temperature differential, or distribute the heat energy over a larger area and run them in parallel.
I doubt there's enough heat to extract there. Low-pressure exhaust gas is useless, and I don't think a metal chimney exiting out through a wall would conduct enough heat on the outside on its own.
The good ol' Stirling engine was tried out -- at a few locations in the US Southwest 10 years back -- to generate megawatts of solar-powered electricity (25kW per engine) at the focus of parabolic mirrors.
It got beat out by subsidized Chinese-produced PV panels (SES was forced into bankruptcy).
My senior year project ten years ago was exactly this - solar thermal + a Stirling engine. When we started the project, we estimated it 20% or so cheaper than solar PV. In that one year solar prices halved or so, leaving us at a dead end.
I ended up consulting with leather plant for their hot water needs via solar thermal. At least that is still viable.
Interesting. I just learned about that energy combo, and Stirling also makes a lot of sense for someone with a non-solar heat source!
Amazing how China came along at the time it did. The hardball politics aimed against US solar startups since Carter has been a fascinating tale with little mainstream coverage.
You'd get much better result by making a small steam plant with the same setup: boil water on stove, drive plant (turbine or piston), condense outside, repeat.
However, if you really want to do this with TECs, stack them to lower the per-unit temperature differential, or distribute the heat energy over a larger area and run them in parallel.