A Convergence Study of the Numerical Solution of Two Bi-directionally Coupled Partial Differential Equations in Thermoelectricity

Abstract

Thermal and electrical aspects in the complex multiphysics of Aluminum Reduction Cells (ARC), the major worldwide industrial component to produce primary aluminum, represent issues that define, to a great extent, the performance of the ARC’s and the efficiency in energy consumption of the system. Given the intense energy usage of ARC’s there is a continuous wide interest in accurate and reliable modeling of these systems [1]. In this regard, and more specifically, we present the three dimensional mathematical model that describes the thermoelectric behavior of ARC’s consisting of two, nonlinear, bi-directionally coupled partial differential equations (PDE’s) for both the cell voltage field and the cell temperature field. The non-linearity arises as a result of the thermal dependence of both the electrical and thermal physical properties of the cell materials. A numerical solution of the stated system of PDE’s was developed and founded on the Finite Elements Method in an doubly iterative scheme that resolves both the non-linearity and the coupling of the PDE’s. The so called side ledge position, the ARC solid-liquid interface formed between both the liquid aluminum and liquid electrolytic bath with the solidified bath [2], is determined through a procedure framed in the fixed mesh scheme (FMS) in order to computationally track the position of the stated interface that strongly affects the ARC’s performance. The particular form of the FMS was initially described in the work reported in [3], and represents an alternative scheme to procedures that adjust the position of an initial estimate for the location of the solid-liquid interface without changing the element properties. Lastly, in this work, we present actual results of a convergence study for the described solution of the stated system of PDE’s showing the convergence characteristics of the proposed scheme for the stated iterative solution strategy.