Timothy R. Marchant

Modelling microwave heating

Abstract Although microwave radiation is best known for heating food in the kitchen, in recent years it has found new applications in many industrial processes, such as those involving melting, smelting, sintering, drying, and joining. Heating by microwave radiation constitutes a highly coupled nonlinear problem giving rise to new and unexpected physical behavior, the best known of which is the appearance of “hot spots.” That is, in many industrial applications of microwave heating it has been observed that heating does not take place uniformly but rather regions of very high temperature tend to form. In order to predict the occurrence of such phenomena it is necessary to develop simplified mathematical models from which insight might be gleaned into an inherently complex physical process. The purpose of this paper is to review some of the recent developments in the mathematical modelling of microwave heating, including models that consider in isolation the heat equation with a nonlinear source term, in which case the electric-field amplitude is assumed constant, models involving the coupling between the electric-field amplitude and temperature including both steady-state solutions and the initial heating, and also models that control the process of thermal runaway. Numerical modelling of the microwave heating process is also briefly reviewed.

The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography

The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.

A DRBEM model for microwave heating problems

Abstract Due to the speed of processing, the microwave heating of materials is becoming industrially important. One of the major difficulties in using this technology is the occurrence of “hot spots,” which are localized areas of high temperature that develop as the material is being irradiated. It is of great practical significance to predict the condition under which hot spots arise so that their occurrence can be avoided or utilized. For example, hot spots are desirable in smelting since they quicken the process and undesirable in sintering ceramics since they can lead to damage.

The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane

The initial boundary–value problem for the Korteweg–de Vries (KdV) equation on the negative quarter–plane, x < 0 and t > 0, is considered. The formulation of this problem is different to the usual initial boundary–value problem on the positive quarter–plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter–plane problem, in contrast to the one boundary condition needed at x = 0 for the positive quarter–plane problem. Solutions of the KdV equation on the infinite line, such as the soliton, cnoidal wave, mean height variation and undular bore solution, are used to find approximate solutions to the negative quarter–plane problem. Five qualitatively different types of solution are found, depending on the relation between the initial and boundary values. Excellent comparisons are obtained between these solutions and full numerical solutions of the KdV equation.

Cubic autocatalytic reaction–diffusion equations: semi–analytical solutions

The Gray–Scott model of cubic–autocatalysis with linear decay is coupled with diffusion and considered in a one–dimensional reactor (a reaction–diffusion cell). The boundaries of the reactor are permeable, so diffusion occurs from external reservoirs that contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations. The ordinary differential equations are then analysed to obtain semi–analytical results for the reaction–diffusion cell. Steady–state concentration profiles and bifurcation diagrams are obtained both explicitly, for the one–term method, and as the solution to a pair of transcendental equations, for the two–term method. Singularity theory is used to determine the regions of parameter space in which the four main types of bifurcation diagram occur. Also, in the semi–analytical model, a fifth bifurcation diagram occurs in an extremely small parameter region; its size being O(10–10). The region of parameter space, in which Hopf bifurcations can occur, is found by a local stability analysis of the semi–analytical model. An example of a stable limitcycle is also considered in detail. The usefulness and accuracy of the semi–analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations.

High-Order Interaction of Solitary Waves on Shallow Water

The interaction of solitary waves on shallow water is examined to fourth order. At first order the interaction is governed by the Korteweg-de Vries (KdV) equation, and it is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at third order. To resolve this, a mass conserving system of KdV equations, involving both right- and left-moving waves, is derived to third order. A fourth-order interaction term, in which the right- and left-moving waves are coupled, is also derived as this term is crucial in determining the fourth-order change in solitary wave amplitude. The form of the unidirectional KdV equation is also discussed with nonlocal terms derived at fourth order. The solitary wave interaction is examined using the inverse scattering method for perturbed KdV equations. Central to the analysis at fourth order is the left-moving wave, for which the solution, in integral form, is derived. A symmetry property for the left-moving wave is found, which is used to show that no change in solitary wave amplitude occurs to fourth order. Hence it is concluded that, for surface waves on shallow water, the change in solitary wave amplitude is of fifth order.

Solitary wave interaction for the extended BBM equation

Solitary wave interaction is examined, for the case of surface waves on shallow water, by using an extended Benjamin–Bona–Mahony (eBBM) equation. This equation includes higher–order nonlinear and dispersive effects, and hence is asymptotically equivalent to the extended Korteweg–de Vries (eKdV) equation. However, it has certain numerical advantages as it allows the modelling of steeper waves, which, due to numerical instability, is not possible using the eKdV equation. Numerical simulations of a number of collisions of varying nonlinearity are performed. The numerical results show evidence of inelastic behaviour at high order. For waves of small amplitude the evidence of inelastic behaviour is indirect; after collision a dispersive wavetrain of extremely small amplitude is found behind the smaller solitary wave. For steeper waves, however, direct evidence of inelastic behaviour is found; the larger wave is increased and the smaller wave is decreased in amplitude after the collision. Conservation laws for the mass and energy, exact and asymptotic, respectively, are derived for the eBBM equation, and numerically verified. Data from the collisions, such as the change in solitary wave amplitudes, the higher–order phase shifts and the mass and energy of the dispersive wavetrains are all tabulated. These are used to show that the change in solitary wave amplitude is of O(α4), verifying previously obtained theoretical predictions. A good comparison is also obtained between the numerically obtained phase shifts and existing asymptotic results. Lastly, three different eBBM solitary wave interactions are examined in detail and compared with existing numerical data obtained from alternative weakly nonlinear models and the Euler water–wave equations.

Analytical solution for electrolyte concentration distribution in lithium-ion batteries

In this article, the method of separation of variables (SOV), as illustrated by Subramanian and White (J Power Sources 96:385, 2001), is applied to determine the concentration variations at any point within a three region simplified lithium-ion cell sandwich, undergoing constant current discharge. The primary objective is to obtain an analytical solution that accounts for transient diffusion inside the cell sandwich. The present work involves the application of the SOV method to each region (cathode, separator, and anode) of the lithium-ion cell. This approach can be used as the basis for developing analytical solutions for battery models of greater complexity. This is illustrated here for a case in which non-linear diffusion is considered, but will be extended to full-order nonlinear pseudo-2D models in later work. The analytical expressions are derived in terms of the relevant system parameters. The system considered for this study has LiCoO2–LiC6 battery chemistry.

APPROXIMATE TECHNIQUES FOR DISPERSIVE SHOCK WAVES IN NONLINEAR MEDIA

Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained.

Microwave thawing of slabs

Abstract Microwave thawing of a semi-infinite one-dimensional slab is examined. The system is governed by the forced heat equation and Maxwells equations. Both the electrical conductivity and the thermal absorptivity are assumed to depend on temperature. Convective and radiative heating occurs at the leading edge of the slab, while the Stefan condition governs the position of the moving phase boundary. An approximate analytical model is developed using the Galerkin method. Approximate analytical solutions are found for the temperature and the electric-field amplitude in the slab, which when combined with the Stefan condition allows the position of the moving front to be found. It is shown that the model produces accurate results in the limits of no heat-loss (insulated) and large heat-loss (fixed temperature) at the leading edge of the slab when compared with the full numerical solution for a number of different parameter choices. The approximate model is coupled with a feedback control process in order to examine and minimise slab melting times. A thawing strategy is developed which greatly shortens the thawing time whilst avoiding thermal runaway, hence improving the efficiency of the thawing process.