Stove fans are used to improve the circulation of heat which in turn reduces hot and cold spots throughout the home.

It doesn’t have any batteries so how does it work?

The stove fan is powered by the Peltier effect. This effect is due to a temperature difference by transferring heat between two electrical junctions.

The thermoelectric circuit is generated since it is composed of materials with different Seebeck coefficients. This is due to the junctions being made of doped* semiconductors.

In a Peltier fan the configuration is that of a  thermoelectric generator. 

*Doping is the intentional introduction of impurities to modulate electrical properties