Research

 Freezing and melting problems

Freezing of a liquid is a classical free-boundary problem, known as the Stefan problem. Various mechanisms can cause instabilities in a freezing front, including the presence of impurities or solutes, which are typically rejected from the solid phase. This instability results in ice dendrites or crystals; the shape and speed at which they grow may depend on the transport of both latent heat and impurities away, and on the dendrite geometry.

As the crystals grow, they too become unstable, growing secondary and tertiary dendrites. Eventually, rather than a sharp freezing front separating pure solid and pure liquid a multi-scale mushy mixture of dendritic solid and liquid forms. We have used homogenisation analysis to systematically derive a model for the freezing mush, arguing that the ice-crystal (pore) length-scale is set by the balance of macroscale heat diffusion and microscale solute diffusion. This has implications for industrial metallurgy in the solidification of binary alloys, and also in the geo-dynamo (solidification in the Earth’s core).

Another important freezing and melting application is in latent-heat energy storage. In this developing green-energy technology, heat energy is stored and released by the melting and freezing of a material with a high latent heat of fusion. Such materials are typically poor thermal conductors, so a high-conductivity copper mesh is added, forming a multi-scale porous structure. We have used asymptotic analysis methods to derive new, efficient models for phase change in these composite materials. In particular, we have showed the fastest battery charging and discharging behaviour occurs in a parameter regime captured by our new model but not by existing models in the literature.

Related papers:

Homogenised models for composite phase-change materials EK Luckins, F Brosa Planella. (under review Proc. Roy. Soc. A)

Modelling mushy zones in binary alloys  EK Luckins, AC Fowler. Geophys. Astrophys. Fluid Dyn. 1–39. doi:10.1080/03091929.2025.2502907 (2025)

 Detection and decontamination of hazardous chemicals in porous substrates

Following the spill or malicious release of a hazardous chemical agent, it is vital that all of the chemical is located and removed from the environment. Both chemical detection and decontamination are particularly difficult if the chemical has soaked into a porous substrate such as brick, concrete, plaster, or wood. The new technology of sampling coatings is a promising route for detecting and quantifying contaminant inside of a porous substrate; our models and analysis provide insight into appropriate sampling timescales and a method to estimate the contamination level.

To decontaminate a porous substrate, a cleanser fluid is typically applied at the surface, and left to react into the material, neutralising the agent in a chemical reaction. We have used asymptotic methods (homogenisation and boundary-layer analysis) to derive reduced decontamination models capturing the multiscale processes and investigated the effect of agent distribution on the decontamination timescales and efficiency. Take a look at our Story on decontamination using VisualPDE.

Related papers:

Mathematical modelling of chemical detection in porous substrates using sampling coatings. A Murray, EK Luckins, Y Sun, S Notman, L Hetherington, M Etzold. (under review Appl. Math. Model.)

The effect of pore-scale contaminant distribution on the reactive decontamination of porous media EK Luckins, CJW Breward, IM Griffiths, CP Please.  Eur. J. Appl. Math. 35(2):318-358. doi:10.1017/S0956792523000219 (2024)

Fluid-flow effects in the reactive decontamination of porous materials driven by chemical swelling or contraction Y Geng, AA Kamilova, EK Luckins. J. Eng. Math.141, 11. https://doi.org/10.1007/s10665-023-10283-6 (2023)

Homogenisation problems in reactive decontamination EK Luckins, CJW Breward, IM Griffiths, Z Wilmott.  Eur. J. Appl. Math. 31(5), 782-805 (2020) 

 Metal production: reduction of silicon dioxide

Silicon is made by reducing quartz rock (containing silicon dioxide) with carbon in a furnace. The energy for the highly-endothermic chemical reactions is provided by an electric arc.

Within the bed of granular material in the furnace, the solid raw materials are consumed in an endothermic reaction, producing gases (from which silicon later forms). The flow of the gas out of the furnace is in the opposite direction to the flow of the raw materials down into the hearth of the furnace. The heat for the reaction is provided by both the flow of hot gases, and by heat radiation onto the free-boundary of the domain. The resulting system is an endothermic, reacting counter-current flow. Asymptotic analysis of this system reveals intricate boundary- and transition-layer structures in a variety of different parameter regimes. This analysis identifies the industrially relevant parameter regime, and highlights how the heating in the majority of the furnace is due to heat transfer from the gas, rather than heat conduction.  

More broadly, all of the electrical, thermal, and chemical processes in the furnace are tightly coupled and occur over a variety of timescales, from the timescale of the alternating electric current (milliseconds) to the slow evolution of the material bed furnace (hours). We used a multiple-timescales homogenisation to incorporate processes on both these timescales into an averaged model for the slow furnace evolution. The averaged model is able to predict the slow growth of the crater region of the furnace towards a stable steady state, unless the furnace is stoked, as seen in practise by furnace operators.

Related papers:

Modelling and analysis of an endothermic reacting counter-current flow EK Luckins, JM Oliver, CP Please, BM Sloman, RA Van Gorder.  J. Fluid Mech.  949, A21.  doi:10.1017/jfm.2022.702 (2022)

Homogenised model for the electrical current distribution within a submerged arc furnace for silicon production EK Luckins, JM Oliver, CP Please, BM Sloman, RA Van Gorder.   Eur. J. Appl. Math. 33(5), 828-863 (2022)

 Electric arcs

Electric arcs are regions of hot plasma which carry an electric current between two electrodes. They occur in circuit breakers, and are used to convert electrical energy to heat in several types of furnace, including silicon furnaces. The flow of current within the arc generates heat by Ohmic heating, and the ionisation of the plasma (and so its electrical conductivity) increases with temperature. The region of plasma that is hot enough to conduct electricity is therefore self-determined by the coupled thermal and electromagnetic problem. The large electric current also generates a fluid flow due to the Lorentz force exerted on the charged particles, and convection is an important heat transfer mechanism, so that the fluid flow is also tightly coupled with the electrical and thermal problems. 

This fluid-flow, electromagnetism, and heat-transfer problem can be modelled with a magnetohydrodynamics (MHD) framework, and simulations of a radially-symmetric arc are shown in the figure to the right. MHD models are complicated and expensive to simulate. We have used dimensional analysis to motivate a much simpler arc model consisting of a single ODE for the arc temperature, with heat lost from the arc solely by radiation. This simple radiation model essentially characterises the arc as a non-linear resistor, and it can be coupled into an equivalent circuit model for the overall electrical system, comparing favourably with empirical arc models currently used in practice. 

Related publication:

Modelling alternating current effects in a submerged arc furnace EK Luckins, JM Oliver, CP Please, BM Sloman, AM Valderhaug, RA Van Gorder.  IMA J. Appl. Math. 87(3), 492-520 (2022)