Silicon is produced in submerged arc furnaces which are heated by electric currents passing through the furnace. It is important to understand the distribution of heating within the furnace in order to accurately model the silicon production process, yet many existing studies neglect aspects of this current flow. In the present paper, we formulate a model that couples the electrical current to thermal, material flow and chemical processes in the furnace. We then exploit disparate timescales to homogenise the model over the timescale of the alternating current, deriving averaged equations for the slow evolution of the system. Our numerical simulations predict a minimum applied current that is required in order to obtain steady-state solutions of the homogenised model and show that for high enough applied currents, two spatially heterogeneous steady-state solutions exist, with distinct crater sizes. We show that the system evolves to the steady state with a larger crater radius and explain this behaviour in terms of the overall power balance typically found within a furnace. We find that the industrial practice of stoking furnaces increases the overall rate of material consumption in the furnace, thereby improving the efficiency of silicon production.

Evaporation within porous media is both a multiscale and interface-driven process, since the phase change at the evaporating interfaces within the pores generates a vapour flow and depends on the transport of vapour through the porous medium. While homogenised models of flow and chemical transport in porous media allow multiscale processes to be modelled efficiently, it is not clear how the multiscale effects impact the interface conditions required for these homogenised models. In this paper, we derive a homogenised model, including effective interface conditions, for the motion of an evaporation front through a porous medium, using a combined homogenisation and boundary layer analysis. This analysis extends previous work for a purely diffusive problem to include both gas flow and the advective–diffusive transport of material. We investigate the effect that different microscale models describing the chemistry of the evaporation have on the homogenised interface conditions. In particular, we identify a new effective parameter, $\mathcal{L}$ , the average microscale interface length, which modifies the effective evaporation rate in the homogenised model. Like the effective diffusivity and permeability of a porous medium, $\mathcal{L}$ may be found by solving a periodic cell problem on the microscale. We also show that the different microscale models of the interface chemistry result in fundamentally different fine-scale behaviour at, and near, the interface.

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The decontamination of hazardous chemical agents from porous media is an important and critical part of the clean-up operation following a chemical weapon attack. Decontamination is often achieved through the application of a cleanser, which reacts on contact with an agent to neutralise it. While it is relatively straightforward to write down a model that describes the interplay of the agent and cleanser on the scale of the pores in the porous medium, it is computationally expensive to solve such a model over realistic spill sizes.In this paper, we consider the homogenisation of a pore-scale model for the interplay between agent and cleanser, with the aim of generating simplified models that can be solved more easily on the spill scale but accurately capture the microscale structure and chemical activity. We consider two situations: one in which the agent completely fills local porespaces and one in which it does not. In the case when the agent does not completely fill the porespace, we use established homogenisation techniques to systematically derive a reaction–diffusion model for the macroscale concentration of cleanser. However, in the case where the agent completely fills the porespace, the homogenisation procedure is more in-depth and involves a two-timescale approach coupled with a spatial boundary layer. The resulting homogenised model closely resembles the microscale model with the effect of the porous material being incorporated into the parameters. The two models cater for two different spill scenarios and provide the foundation for further study of reactive decontamination.

Following the release of a chemical warfare agent, it is crucial for public health that the affected environment is entirely decontaminated. If the agent has seeped into a porous building material, the decontamination is achieved by applying a cleanser solution to the surface of the porous material, and allowing it to react in, neutralising the agent. Typically, the agent and cleanser solution are immiscible fluids and so the reaction occurs at the fluid–fluid interfaces within the pores. Previous studies have shown that the rate of decontamination of the porous material can depend on both the chemical reaction rate and the transport of cleanser to the reacting interface. These studies have all assumed that the two fluids have the same densities, so that diffusion is the only cleanser-transport mechanism. In this paper, we relax this assumption and investigate the effect of a fluid flow—generated by a change in density of the material (a swelling, or contraction) during the chemical reaction—on the decontamination process. This flow of fluid results in advection as well as diffusion of chemicals. Buoyancy effects are neglected. In particular, we show that when the agent is more dense than the reaction product, the decontamination process is slower, due to the adverse advection effect.