We present a coupled system of elliptic equations describing the steady state of the thermoelectrical behaviour of an aluminium electrolytic cell. The thermal model is a free boundary problem which consists of the heat equation with Joule heating as a source. We neglect the Joule heating in the ledge, and allow for temperature-dependent electrical conductivity. We also formulate a numerical approximation using a finite element method. An iterative algorithm and numerical results are presented. The existence of a weak solution is also proved.