Silicon carbide is a widely used material due to its unique combination of physical and chemical properties. However, existing Acheson furnaces, which cause direct CO_2 emissions, are expensive and energy inefficient. Rotary kilns are a promising alternative, but their use for silicon carbide production is still under development. In this article, we present a mathematical model for the consumption of quartz and carbon, and the formation of silicon carbide in a rotary kiln. We focus on the interplay between reaction kinetics, solid and gas transport, and thermal effects. Assuming radial mixing within the bed, we derive a simplified one-dimensional model that captures the dominant physics of the system. The model tracks the evolution of quartz, carbon, and silicon carbide, as well as gas-phase species and temperature, down the length of the kiln. We nondimensionalise the model and identify key parameter groupings—relating supplied heat to kiln fill level, initial carbon particle size, and the relative speed of the two chemical reactions we consider. Then, we examine the model’s behaviour via asymptotic analysis, before presenting numerical simulations. Our analysis shows that there is only one dimensionless parameter group that strongly influences reactor performance, with silicon carbide yield increasing monotonically with the value of this parameter. These findings offer broad-stroke insights into design principles and parameter operating regimes that favour efficient silicon carbide reactors.

Over the long timescale of many charge/discharge cycles, gas formation can result in large bulging deformations of a Lithium-ion pouch cell, which is a key failure mechanism in batteries. Guided by recent experimental X-ray tomography data of a bulging cell, we propose a homogenised mechanical model to predict the shape of the deformation and the stress distribution analytically. Our model can be included in battery simulation models to capture the effects of mechanical degradation. Furthermore, with knowledge of the bending stiffness of the cathode electrodes and current collectors, and by fitting our model to experimental data, we can predict the internal pressure and the amount of gas in the battery, thus assisting in monitoring the state of health (SOH) of the cell without breaking the sealed case.

In a pouch cell battery, the intercalation of lithium ions into the active particles means the electrodes want to expand. However, since the electrodes are attached to stiff current collectors, this expansion is constrained, leading to a macro-scale deformation and a residually stressed state. This stress state affects the electrochemistry and can also lead to mechanical degradation, causing a reduction in performance. We model the mechanical state of stress in the battery assuming a known compositional expansion of the electrodes, and use asymptotic techniques to generate reduced order models by exploiting the thin aspect ratio, as well as the large stiffness of the current collectors. We obtain analytic expressions for the stress in the bulk of the electrodes and at the interface between electrodes and current collectors, and a reduced-order equation whose solution describes the tension in the current collectors. We compare our results with full 3D finite element simulations with excellent agreement, and use our results with the battery simulation package PyBaMM to predict a realistic stress state in a discharging battery.

Non-invasive parametrisation of physics-based battery models can be performed by fitting the model to electrochemical impedance spectroscopy (EIS) data containing features related to the different physical processes. However, this requires an impedance model to be derived, which may be complex to obtain analytically. We have developed the open-source software PyBaMM-EIS that provides a fast method to compute the impedance of any PyBaMM model at any operating point using automatic differentiation. Using PyBaMM-EIS, we investigate the impedance of the single particle model, single particle model with electrolyte (SPMe), and Doyle-Fuller-Newman model, and identify the SPMe as a parsimonious option that shows the typical features of measured lithium-ion cell impedance data. We provide a grouped-parameter SPMe and analyse the features in the impedance related to each parameter. Using the open-source software PyBOP, we estimate 18 grouped parameters both from simulated impedance data and from measured impedance data from a LG M50LT lithium-ion battery. The parameters that directly affect the response of the SPMe can be accurately determined and assigned to the correct electrode. Crucially, parameter fitting must be done simultaneously to measurements across a wide range of states-of-charge. Overall, this work presents a practical way to find the parameters of physics-based models.

A porous material that has been contaminated with a hazardous chemical agent is typically decontaminated by applying a cleanser solution to the surface and allowing the cleanser to react into the porous material, neutralising the agent. The agent and cleanser are often immiscible fluids and so, if the porous material is initially saturated with agent, a reaction front develops with the decontamination reaction occurring at this interface between the fluids. We investigate the effect of different initial agent configurations within the pore space on the decontamination process. Specifically, we compare the decontamination of a material initially saturated by the agent with the situation when, initially, the agent only coats the walls of the pores (referred to as the ‘agent-on-walls’ case). In previous work (Luckins et al., European Journal of Applied Mathematics, 31(5):782–805, 2020), we derived homogenised models for both of these decontamination scenarios, and in this paper we explore the solutions of these two models. We find that, for an identical initial volume of agent, the decontamination time is generally much faster for the agent-on-walls case compared with the initially saturated case, since the surface area on which the reaction can occur is greater. However for sufficiently deep spills of contaminant, or sufficiently slow reaction rates, decontamination in the agent-on-walls scenario can be slower. We also show that, in the limit of a dilute cleanser with a deep initial agent spill, the agent-on-walls model exhibits behaviour akin to a Stefan problem of the same form as that arising in the initially saturated model. The decontamination time is shown to decrease with both the applied cleanser concentration and the rate of the chemical reaction. However, increasing the cleanser concentration is also shown to result in lower decontamination efficiency, with an increase in the amount of cleanser chemical that is wasted.

We consider a liquid containing impurities saturating a porous material; when the liquid evaporates, the impurities are deposited within the material. Applications include filtration and waterproof textiles. We present a mathematical model incorporating coupling between evaporation, accumulation and transport of the impurities, and the impact of the deposited impurities on the transport of both the suspended impurities and the liquid vapour. By simulating our model numerically, we investigate the role of temperature and repeated drying cycles on the location of the deposited impurities. Higher temperatures increase the evaporation rate so that impurities are transported further into porous material before depositing than for lower temperatures. We quantify two distinct parameter regimes in which the material clogs: i) the dry-clogging (high-temperature) regime, in which impurities are pushed far into the material before clogging, and ii) the wet-clogging (high-impurity) regime, in which liquid becomes trapped by the clogging. Clogging restricts the extent to which drying time can be reduced by increasing the temperature.

This report describes work performed during SWI 2023 at the University of Groningen in relation with Problem 1 posed by the company ASMPT. They have detailed simulation software of a machine and they compare the results of this with physical experimental results. There is a significant difference between the simulated and measured data, and it is the goal of this work to study how to estimate the parameters in the simulation model using the ex-perimentally measured frequency response. First, two toy models are studied to understand the challenges of pa- rameter estimation in the frequency domain. Later, optimization methods are applied. Several different approaches of reducing the dimensionality of the parameter space are explored, including determining the parameter sensitivity. A suggestion for increasing the detail of the model, specifically related to the machine base, is also outlined. In the summary, we supply a discussion of the key insights we gained.

The WITT gearbox is a device that harvests energy from its environment by responding to sufficiently high frequency and amplitude harmonic acceleration in any plane of motion. It is a simple apparatus, consisting of pendula, gears, bearings, clutches, and power conversion electronics, that can be immersed in a sealed way into a harsh, turbulent environment. It then generates uninterrupted power by turning vibration to rotation, always in the same direction, regardless of the direction of the external vibration. In this work, we expand the analysis of the mechanical properties of the WITT device, making more realistic assumptions compared to previous studies. Moreover, we investigate the response of the device to various external configurations and parameter values (e.g. attaching the device to pair of pendula, or to a pole, etc.), exploring how this could enable it to respond to lower-frequency excitation.

When a contaminated liquid evaporates from within a porous material, the impurities or dirt accumulate and deposit within the pore space. This occurs during the cleaning of filters and fouling of textiles, and is related to the ‘coffee-ring’ problem. To investigate how and where dirt is deposited in the pore space, we present a model for the motion of an evaporation front through a porous material, and the related accumulation, transport, and deposition of dirt, assuming that the liquid remains stationary. For physically relevant parameters, vapour transport out of the porous material is quasi-steady and we derive a single ordinary differential equation describing the motion of the evaporation front in time. Model solutions exhibit spatially non-uniform profiles of the deposited dirt-layer thickness through the porous material. The dirt accumulation and evaporation problems are coupled: deposited dirt hinders vapour transport through the porous material, slowing the evaporation. We identify two scenarios in which the porous material becomes clogged with dirt. Accumulation of suspended dirt at the evaporating interface along with slow dirt diffusion results in the deposited dirt layers clogging the pores at the evaporating interface, halting the drying and trapping liquid in the porous material. Alternatively, slow dirt deposition results in the suspended dirt being pushed far into the porous material by the evaporation, eventually leaving only dirt (with no liquid) in the pore space. We investigate the dynamics of both clogging scenarios, characterising the parameter regimes for which each occurs. Both clogging scenarios must be avoided in practice since they may be detrimental to future filter efficacy or textile breathability.

Mechanical deformations induced by expansion within an elastic material which is spirally-wound in layers with a thin inextensible reinforcing material are considered. The motivation is to understand behaviour of spirally-wound batteries where both the active material and the metal current collectors expand due to changes in lithiation and/or temperature. This paper considers a spiral made from a single reinforcing layer with a matrix layer of linear elastic material, whose properties may vary through the layer. The layers undergo prescribed isotropic expansion, where the matrix expansion may depend on the macroscopic radial coordinate. Asymptotic homogenisation, exploiting the small scale of the layer thickness relative to the large scale of the overall spiral structure, reveals the bulk of the spiral has an unexpected simple behaviour while there are boundary layers in a surface region near the inner and outer windings. There are further finer-structure boundary layers at the very beginning and very end of the spiral. In all these regions analytical solutions are found providing simple expressions for the deformations and in particular the tension in the inextensible layer. Comparisons are shown between these expressions and detailed finite-element solutions of the problem. These reduced-order models provide a simple way of accounting for stresses induced by expansion of the spiral structure.

It has been observed that pollutants mainly cause pollution in the Narmada river and the estuary, which is driven there during high and low tides. The modelling presented here concentrates on phenomena affecting the distribution of pollutants in the Gulf of Khambhat and the Estuaries. The model used for the tidal water movement in the Gulf is the shallow water equations with turbulent friction. A simplified linearised version of these equations is used in two idealised geometries to explore why the tidal amplitude and velocity can increase so significantly up the Gulf, and both resonance effects and the rapid narrowing of the Gulf are identified as being contributors. The wave elevation and velocity are described by a coupled system of partial differential equations that are spatially discretized and solved using an ODE solver.

Traditional refining of silicon generates carbon dioxide emissions that are released into the atmosphere. An alternative process under experimental exploration uses quartz particles coated in a layer of porous carbon as the raw material, instead of lumps of quartz and carbon. The quartz–carbon pellets are processed at a lower temperature than in an industrial furnace, and hence, different chemical reactions are dominant, reducing the greenhouse gas emissions. The quartz core shrinks as it is consumed, and the carbon is converted to silicon carbide, which can subsequently be processed into silicon. We develop a model for chemical and transfer processes within a single quartz–carbon pellet. We derive governing equations for the concentration of silicon monoxide, carbon monoxide, and carbon dioxide, and conservation equations on the moving quartz interface. Furthermore, we then focus on a reduced model in an industrially relevant distinguished limit, and solve numerically the resulting leading-order system. We show examples of reaction-limited behaviour as well as diffusion-limited behaviour. Both regimes are physically admissible due to the large potential range of the parameters. Finally, we sweep through the parameter space, and characterize the dynamics based on the utilization of the carbon and the silicon yield. We find that the diffusion-limited regime is best for carbon utilization and silicon yield, as the silicon monoxide reacts with the carbon before it is transported out of the pellet.

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.

In this work we analyse the local nonlinear electrochemical impedance spectroscopy (NLEIS) response of a lithium-ion battery and estimate model parameters from measured NLEIS data. The analysis assumes a single-particle model including nonlinear diffusion of lithium within the electrode particles and asymmetric charge transfer kinetics at their surface. Based on this model and assuming a moderately-small excitation amplitude, we systematically derive analytical formulae for the impedances up to the second harmonic response, allowing the meaningful interpretation of each contribution in terms of physical processes and nonlinearities in the model. The implications of this for parameterization are explored, including structural identifiability analysis and parameter estimation using maximum likelihood, with both synthetic and experimentally measured impedance data. Accurate fits to impedance data are possible, however inconsistencies in the fitted diffusion timescales suggest that a nonlinear diffusion model may not be appropriate for the cells considered. Model validation is also demonstrated by predicting time-domain voltage response using the parameterized model and this is shown to have excellent agreement with measured voltage time-series data (11.1 mV RMSE).

We study the endothermic reaction and flow of a granular solid reactant, where energy for the reaction is provided by a counter-current flow of hot gases through the porous reactant bed. Research into reacting flows typically focusses on exothermic combustion processes. However, endothermic processes are common in the metallurgy industry, including the production of cement, silicon and rutile titanium dioxide. Several common features are observed in experimental and numerical studies of these processes, including critical temperatures of the reactant at which the chemical reaction begins, and regions of the reactor with uniform reactant temperature. Motivated specifically by the processes in a silicon furnace, we analyse a model of endothermic, reacting counter-current flow using the method of matched asymptotic expansions. Assuming the Péclet number in the solid is large, we explore the full range of values for the dimensionless inter-phase heat-transfer rate, finding six distinguished limits. In all limits, we find a diffusive boundary layer in which there is a fast chemical reaction rate due to the high temperatures, analogous to exothermic flame fronts. Outside this region, the counter-current flow is crucial to the chemical processes. For intermediate values of the heat-transfer rate, we find the same qualitative properties as those observed across the metallurgy industry, and we quantify the dependence of these properties on the flow rate and heat-transfer rate. In the limit of large heat-transfer coefficient, we derive the single-temperature limit, in which the solution structure is dependent on the direction of net heat flux through the domain.

Modelling the production of silicon in a submerged arc furnace (SAF) requires accounting for the wide range of timescales of the different physical and chemical processes: the electric current which is used to heat the furnace varies over a timescale of around $10^{-2}\,$ s, whereas the flow and chemical consumption of the raw materials in the furnace occurs over several hours. Models for the silicon furnace generally either include only the fast-timescale, or only the slow-timescale processes. In a prior work, we developed a model incorporating effects on both the fast and slow timescales, and used a multiple-timescales analysis to homogenise the fast variations, deriving an averaged model for the slow evolution of the raw materials. For simplicity, in the previous work we focussed on the electrical behaviour around the base of a single electrode, and prescribed the current in this electrode to be sinusoidal, with given amplitude. In this paper, we extend our previous analysis to include the full electrical system, modelled using an equivalent circuit system. In this way, we demonstrate how the two furnace-modelling approaches (on the fast and slow timescales) may be combined in a computationally efficient way. Our previously derived model for the arc resistance is based on the assumption that the dominant heat loss from the arc is by radiation (we will refer to this as the radiation model). Alternative arc models include the empirical Cassie and Mayr models, which are commonly used in the SAF literature. We compare these various arc models, explore the dependence of the solution of our model on the model parameters and compare our solutions with measurements from an operational silicon furnace. In particular, we show that only the radiation arc model has a rising current-voltage characteristic at high currents. Simulations of the model show that there is an upper limit on the length of the furnace arc, above which all the current bypasses the arc and flows through the surrounding material.

This report is a summary of the work performed during 145th European Industrial Study Group on the topic of Simultaneous Transmission and Reception of electromagnetic signals presented by Dstl. We establish that the solution to this problem lies in estimating the reflected signals caused by the outgoing signal. We define the problem mathematically using an integral equation to represent the received signal. From this we derive Cramer-Rao bounds which give lower bounds for the error when estimating the signal based upon error in the calculation of the reflected signal. We perform numerical simulations for a simple example to demonstrate some techniques that can be used in the solving of this problem.

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.

This paper presents the current state of mathematical modelling of the electrochemical behaviour of lithium-ion batteries as they are charged and discharged. It reviews the models developed by Newman and co-workers, both in the cases of dilute and moderately-concentrated electrolytes and indicates the modelling assumptions required for their development. Particular attention is paid to the interface conditions imposed between the electrolyte and the active electrode material; necessary conditions are derived for one of these, the Butler-Volmer relation, in order to ensure physically realistic solutions. Insight into the origin of the differences between various models found in the literature is revealed by considering formulations obtained by using different measures of the electric potential. Materials commonly used for electrodes in lithium ion batteries are considered and the various mathematical models used to describe lithium transport in them discussed. The problem of up-scaling from models of behaviour at the single electrode particle scale to the cell scale is addressed using homogenisation techniques resulting in the pseudo 2D model commonly used to describe charge transport and discharge behaviour in lithium-ion cells. Numerical solution to this model is discussed and illustrative results for a common device are computed.

A mathematical model is proposed for the flow of nutrients in an inflatable hydroponics module being developed by Phytoponics. Simple experiments were performed via the injection of dye into the system enabling a basic understanding of the time and length scales of nutrient flow and mixing. Four different flow regimes are identified. At the scale of a single root, a Stokes-flow approximation may be used. Brinkman flow operates at the individual plant scale which homogenises into a 1D model for macro-scale flow of nutrients. A shear flow model is used to predict the flow in regions dominated by plant roots. Finally, simplified two-phase flow equations are derived for the more turbulent bubble flow during aeration. These are solved within the software COMSOL. The overall conclusion is that both the periodic flow of nutrients and the aeration are required to enable even nutrient spread.