Søderberg electrodes feature prominently in the operation of metallurgical electrical furnaces. The electrode material must bake before entering the furnace; failure to bake will lower the efficiency of the process, and may cause physical harm to the furnace itself through a soft breakage. As such, ensuring that the baking isotherm remains within the region of the electrode outside of the furnace is essential. We propose a mathematical model for a Søderberg electrode taking into account the heat, mass, and current transfer mechanisms at play, along with realistic boundary conditions on the outside of the electrode that are strongly heterogeneous in height. The resulting model describes a strongly heterogeneous cylindrical "thermistor" which moves slowly downward and is acted on by current clamps which provide Joule heating. Although it is often ignored in the literature on thermistor problems, we find that the Péclet number resulting from the downward motion strongly influences the position of the baking isotherm. Aside from some specific reductions leading to analytical solutions, the general form of the model is complicated enough to require numerical simulations. Still, our modeling approach provides us with a qualitative understanding of many aspects of the Søderberg electrode baking process, and permits us to identify three parameters of key importance to the positioning of the baking isotherm. In particular, our results suggest desired ranges for the the lowering rate of the electrode (in terms of a Péclet number), the radius of the electrode, and the strength of the Joule heating due to an applied current, which are the three aspects which may be controlled (to varying degrees) in industrial applications.