Modelling the feeding process for aluminium production
Keywords: Asymptotic analysis
Mathematical Modelling
Porous Media
Stefan problem

In this thesis, we build and solve various mathematical models to describe the interplay between alumina particles and a cryolite bath that takes place in a Hall-Héroult cell, which is the foundational process for aluminium production. We aim to understand the heat and mass transfer mechanisms taking place in order to gather insights into the aluminium production process and use these insights to suggest process improvements.
First, we examine the problem of adding a cold alumina particle to a bath of hot cryolite. We develop a spherically symmetric model to describe heat and mass transfer between the particle and the surrounding material, allowing for freezing and melting of the cryolite close to the particle and its subsequent dissolution. We solve this model numerically using the method of lines using the early-time asymptotic solution to initiate the numerical scheme. We explore the behaviour as we vary the key dimensionless parameters. We also consider the model in the limits of small superheat and of small Stefan number. Using these asymptotic solutions, we find approximate expressions for the freezing, melting, and dissolution timescales, with freezing being the fastest effect and dissolution the slowest. Also, using the early-time solution, we determine constraints on the parameters for which shell growth is possible. Our results predict that increasing the superheat or decreasing the particle size should decrease the time it takes for a particle to dissolve, in accordance with practical experience. Furthermore, we note that the temperature changes are localised to a 5--10 particle radii.
Second, we formulate a one-dimensional model for the phase-changing infiltration into a porous alumina raft placed on top of the cryolite. We simplify our model in the limit of small superheat. We solve the equations using the same approach as in Chapter 2 with the method of lines, using the early-time asymptotic solution to initiate the numerical scheme. We find that the early-time problem can exhibit a stable and an unstable solution and in certain parameter regions, these solutions disappear altogether. Furthermore, depending on the heat loss through the top of the raft, the model can undergo a discontinuous change in behaviour which corresponds to the clogging of the pore space. Additionally, near clogging, we find power-law behaviour of the frozen cryolite fraction and the location of the infiltration and solidification fronts.
Third, we formulate a model for the disintegration of a spherical clump of alumina particles. We solve the problem numerically with the method of lines and see that, for certain parameters, the solution behaves like a travelling dissolution front, while in other regimes there is bulk dissolution. We investigate the travelling dissolution front by using the method of matched asymptotic expansions to reduce the full problem. The asymptotic expressions match well with the numerical solutions up to the order that would be expected.
In Chapter 5 we formulate a model for the phase-changing infiltration into a spherical clump of alumina particles, which physically corresponds to a larger batch that is entirely enclosed by the cryolite. We find that many of the results in Chapter 3 carry over, such as the multiple solutions for early time, the late time clogging, or the fact that the phase-changing front is slower than the isothermal one. The presence of clogging in this model is especially interesting as it is caused by the slowing down of the front, as opposed to the cooling at the top, which was the case in Chapter 3.
Finally, we summarize the findings and offer possible extensions to our models in Chapter 6 and present our mathematical findings in an approachable way in Chapter 7. The alumina feeding is a process with many interacting parts producing complex behaviour. In this thesis we shed light on a few problems, but the research is not over.