Thermal Electrical Behaviour of Metallurgical Processes


Efficient production of metals requires extremely high temperatures within large furnaces where the heat is obtained through huge amounts of electrical energy. There is a great need for a better understanding of the interdependent mechanisms inside these furnaces, in particular, between the electrical currents, the temperature distribution, the chemical reactions and the movement of the material. Due to the high temperatures within the furnaces, accurate internal measurements are almost impossible to achieve, therefore mathematical models are required to ascertain the behaviour. This insight is vital in understanding how to optimise the process and minimise the energy needed to obtain a satisfactory product. 

Anthracite (see Figure, below), a coal of high carbon content, is an extremely valuable material within the metallurgy industry, particularly for use within furnaces as a vital component of electrodes and lining materials. In order to be used in furnaces, the anthracite must undergo heat treatment to create a product with appropriate properties and characteristics. A furnace known as the calciner (Figure, right) is used for such treatment, where the material is heated via Joule heating, also known as Ohmic heating. 

Figure: CT scan of a collection of anthracite particles.

The goal of this project is to obtain a concise, coupled model to explain the thermal, electrical and mechanical mechanisms taking place within metallurgical processes involving the heat treatment of granular material. 


Trying to understand all the interdependent mechanisms simultaneously is an arduous task, therefore we initially consider each process separately. We begin with the electrical problem; that is, understanding how current propagates through granular material. We model the domain in which we hope to solve the equations for current flow as a collection of carbon particles in contact with each other, along with air gaps, together denoted the carbon bed. The inhomogeneity of particle size and shape, along with the intricacy of the contacts between particles, add a level of complexity that makes the problem extremely difficult to solve. We understand that the scale of a particle is small compared to the scale of the carbon bed, and the scale of particle contacts smaller still. Therefore, we can apply mathematical homogenisation by separating the scales of the problem, thus defining and solving a microscopic problem on the scale of a particle to obtain an effective conductivity. This effective conductivity will be a parameter in the macroscopic problem on the scale of the carbon bed, and allows the simplification of the macroscopic problem while still capturing the important features of the system.

We have derived homogenisation models in two and three dimensions that allow us to solve the electrical problem for simple particle geometries. The models are solved using the finite elements package FEniCS. We have also conducted an asymptotic analysis to obtain an approximate analytical solution to validate our numerical solvers.

This analysis has produced some very interesting results, including the importance of understanding the nature of the particle contacts. We discovered that the current passing through a very thin contact, whose limiting case is a line, behaves very differently than the current passing through a circular contact. From our analysis, we deduce that there is extremely high resistance incurred at particle contacts, and the contribution of such resistance accounts for a considerable amount of the bulk resistivity of the carbon bed.

Future Work

Our immediate next step is to incorporate thermal effects into our existing model. The thermal and electrical problems are heavily coupled, in particular, the electric potential results in Ohmic heating, whereas the temperature effects cause changes to the material resistivity. The thermal problem will naturally lead us to understand the evolution of the material properties, thus encapsulating the chemical problem as the material undergoes the calcination process. Careful study of solid mechanics will allow for the inclusion of forces and stresses within the furnace, and how these affect particle contacts and material flow. 

Following the complete understanding of the Calciner furnace, we can use the coupled model for carbon particles and air as a foundation when considering more complicated furnaces that may include multiple phases and an increased number of electrodes.


This work is supported by the EPSRC Centre for Doctoral Training in Industrially Focused Mathematical  Modelling  (EP/L015803/1) in collaboration with Teknova (Norway). This work is part of the research project “Electrical Conditions and their Process Interactions in High Temperature Metallurgical Reactors” (ElMet- 247791/030), a competence project to establish a detailed understanding of how the conditions in an electrical smelting furnace are influence by 3-phase alternating current. The project is financed by the Research Council of Norway (BIA KPN), Elkem and Eramet. Teknova is the project owner and NTNU, University of Oxford, University of Santiago de Compostela are the academic research partners.