As in the picture, we have a room with vents at the bottom to let cold air in and at the top to let hot air out. Normally the air flow is limited by the temperature gradient (correct me if I'm wrong). If we place a hot object in the vent on top, won't that create a faster draft? That way we could get very high "wind" speed indoors to replace fans.
Is this possible, and if yes, what's the limit for air flow per second?
Yes, an air flow may be created your way although a fan, designed specifically for that purpose, may be more efficient. In any case, we can make rough estimates of amount of the air flow created.
In natural convection, the scale for flow velocity is $v\sim \sqrt{g\beta \Delta T d}$ (see here), where $g$ is gravitational acceleration, $\beta$ is the fluid's thermal expansion coefficient, $\Delta T$ is the temperature difference between the heating element and the far-off ambient, and $d$ is the length scale associated with the heating element (e.g. diameter of a horizontally placed heating wire). For air, $\beta=3\times 10^{-3} $/deg-C. For $\Delta T=100$ deg-C, we have the velocity scale, $v\sim\sqrt{9.81\times 3\times 10^{-3}\times 100}\approx 1.7$ m/s.
Of course, the updraft of hot air thus created in the chimney may be compensated by a downdraft of ambient air along the sides of the same chimney (the more likely scenario, I believe) in which case this contraption will not contribute to air circulation in the room. To avoid this, the heating element must cover the cross-section of the chimney. Even then there are probably other conditions and it may be difficult to guarantee the kind of flow you want. Just use a fan!
