The microwave heating of three-dimensional blocks: semi-analytical solutions

The microwave heating of three-dimensional blocks, by the transversemagnetic waveguide mode TM₁₁, is considered in a long rectangularwaveguide. The governing equations are the forced heat equationand a steady-state version of Maxwell's equations, while theboundary conditions take into account both convective and radiativeheat loss. Semi-analytical solutions, valid for small thermalabsorptivity, are found using the Galerkin method. The electricalconductivity and the thermal absorptivity are assumed to betemperature dependent, while both the electrical permittivityand magnetic permeability are taken to be constant. Both a quadraticrelation and an Arrhenius-type law are used for the temperaturedependency. As the Arrhenius-type law is not amenable analytically,it is approximated by a rational–cubic function. A multivaluedsteady-state temperature versus power relationship is foundto be possible for both types of temperature dependency. Atthe critical power level thermal runaway occurs when the temperaturejumps from the lower (cool) temperature branch to the upper(hot) temperature branch of the solution. The semi-analyticalsolutions are compared with numerical solutions of the governingequations for various special cases such as the limits of smalland large heat loss at the edges of the block. An excellentcomparison is obtained between the semi-analytical and numericalsolutions, on both temperature branches for the Arrhenius-typelaw. For the quadratic temperature dependency the comparisonis excellent on the low branch but the semi-analytical theorysignificantly underpredicts the temperature on the upper solutionbranch.     

Simple exact solutions applicable to microwave heating

Summary There is presently considerable interest in the utilization of microwave heating to novel industrial applications. Mathematically such problems involve Maxwell's equations coupled with the heat equation and for which all thermal, electrical and magnetic properties of the material are nonlinearly dependent upon temperature. Accordingly such problems are highly complex and very little theoretical work has been undertaken. The purpose of this paper is to obtain simple exact solutions applicable to microwave heating in the simplest situation, involving only one spatial dimension and assuming that all thermal, electrical and magnetic properties exhibit a power law dependence on temperature. Similarity solutions and other special solutions are examined. These generally result in highly nonlinear coupled systems of ordinary differential equations and although some new closed results are obtained in special cases, in general, such complex systems of ordinary differential equations need to be solved numerically. Roughly speaking, we show that stretching similarity solutions exist only if the power law electrical and thermal conductivities and magnetic permeability with indicesl, m andn respectively are such thatl+m+n=0. Similar constraints on the indices apply for the existence of other simple solutions.     

Application of the two-dimensional differential transform method to heat conduction problem for heat transfer in longitudinal rectangular and convex parabolic fins

In this article, approximate analytical (series) solutions for the temperature distribution in a longitudinal rectangular and convex parabolic fins with temperature dependent thermal conductivity and heat transfer coefficient are derived. The transient heat conduction problem is solved for the first time using the two-dimensional differential transform method (2D DTM). The effects of some physical parameters such as the thermo-geometric parameter, exponent and thermal conductivity gradient on temperature distribution are studied. Furthermore, we study the temperature profile at the fin tip.     

Thermal runaway in microwave heating: a mathematical analysis

A study is made of the solution of a differential equation modelling the heating of a layer of material specimen by microwave radiation. Depending on the microwave power bistable steady-state temperatures may be expected. When changing the power, a switch from one stable branch to another one may arise. The sudden increase of temperature, known as thermal runaway, is studied from the differential equation using asymptotic methods. Such analysis reveals distinct stages in the process of thermal runaway. At the moment the solution leaves a branch, and becomes unstable a particular type of behaviour is observed (onset of runaway). The most specific element at this stage is a time shift delaying the rapid change in temperature. For this shift a simple expression in terms of the parameters of the system is given. Next it is shown that the rapid transition from one branch to the other can be put in a mathematical formula that smoothly matches the two steady state solutions.     

An application of a boundary element method to natural convection

The direct boundary element method is applied to the numerical modelling of thermal fluid flow in a transient state. The Navier-Stokes equations are considered under the Boussinesq approximation and the viscous thermal flow equations are expressed in terms of stream function, vorticity, and temperature in two dimensions. Boundary integral equations are derived using logarithmic potential and time-dependent heat potential as fundamental solutions. Boundary unknowns are discretized by linear boundary elements and flow domains are divided into a series of triangular cells. Charged points are translated upstream in the numerical evaluation of convective terms. Unknown stream function, vorticity, and temperature are staggered in the computational scheme.

Simple iteration is found to converge to the quasi steady-state flow. Boundary solutions for two-dimensional examples at a Reynolds number 100 and Grashoff number 10⁷ are obtained.     

A new family of unsteady boundary layers over a stretching surface

In this paper, a new family of unsteady boundary layers over a stretching flat surface was proposed and studied. This new class of unsteady boundary layers involves the flows over a constant speed stretching surface from a slot, and the slot is moving at a certain speed. Depending on the slot moving parameter, the flow can be treated as a stretching sheet problem or a shrinking sheet problem. Both the momentum and thermal boundary layers were studied. Under special conditions, the solutions reduce to the unsteady Rayleigh problem and the steady Sakiadis stretching sheet problem. Solutions only exist for a certain range of the slot moving parameter, α. Two solutions are found for −53.55° < α < −45°. There are also two solution branches for the thermal boundary layers at any given Prandtl number in this range. Compared with the upper solution branch, the lower solution branch leads to simultaneous reduction in wall drag and heat transfer rate. The results also show that the motion of the slot greatly affects the wall drag and heat transfer characteristics near the wall and the temperature and velocity distributions in the fluids.     

Magnetohydrodnamics nanofluid flow of shaped nanoparticles over a porous stretching wall and slip effect

In this study, we analyze the magnetohydrodynamic flow of magnetite-engine oil nanofluid in the presence of nonidentical shaped nanoparticles subject to the porous medium and velocity slip effect. Energy analysis is carried out with the Ohmic heating and thermal radiation impacts. The system of partial differential equations are transformed into the system of ordinary differential equations using similarity variable. The Hamilton–Crosser model is used. The exact solutions for the momentum and heat transport analysis are found. The impact of various emerging parameters on the velocity and temperature profiles are analyzed by graphs. Furthermore, the local skin friction and heat transfer rate are examined graphically. It is examined that the velocity field increases with an increment in the magnitude of ϕ and L. An increase in the value of Hartman number enhancing the temperature profile.     

A Semenov approach to the modelling of thermal runaway of damp combustible material

Semenov theory for the self-heating of a reactive slab is extendedto take account of the presence of water vapour. In this paper,mass changes due to evaporation/condensation are neglected butheat exchange is retained in the energy equation. By doing this,a simple easily solvable set of equations can be set up to representthe thermal behaviour of the slab. No account is taken of possiblewet exothermic reactions in this paper. The aim is simply tounderstand the effects of evaporation/condensation on the overallthermal history. Using a simple model which treats the masschanges within the material as negligible, the competitive effectsof condensation and evaporation are shown to produce a two-timesituation which depends crucially on the surface mass transfer/heattransfer ratio h_m. Either self-heating occurs at a lower ratethan that due to dry oxidation, or else a maximum temperatureis reached before a lower equilibrium steady-state temperatureis achieved. Thus, compared to the dry case, in general terms,evaporation certainly encourages stability. However, the finalstrictly subcritical steady state will not always be achieveddue to the competitive process between recondensation and evaporationloss at the surface at medium timescales. A set of quasi-steadystates is identified which yield plots of a more restrictivecritical value of temperature against the Frank-Kamenetskiiparameter (proportional to the thickness of the slab and itsreactivity). If the value of h_m is such that the maximum temperaturereaches this critical value, then thermal runaway can stilltake place even though the starting value of temperature wasstrictly below the true (damp) final steady-state critical value.     

2‐D solitary waves in thermal media with nonsymmetric boundary conditions

Optical solitary waves and their stability in focusing thermal optical media, such as lead glasses, are studied numerically and theoretically in (2 + 1) dimensions. The optical medium is a square cell and mixed boundary conditions of Newton cooling and fixed temperature on different sides of the cell are used. Nonlinear thermal optical media have a refractive index which depends on temperature, so that heating from the optical beam and heat flow across the boundaries can change the refractive index of the medium. Solitary wave solutions are found numerically using the Newton conjugate‐gradient method, while their stability is studied using a linearized stability analysis and also via numerical simulations. It is found that the position of the solitary wave is dependent on the boundary conditions, with the center of the beam moving toward the warmer boundaries, as the parameters are varied. The stability of the solitary waves depends on the symmetry of the boundary conditions and the amplitude of the solitary waves.     

A new boundary element method for the solution of plane steady-state thermoelastic fracture mechanics problems

An efficient numerical technique is developed for plane, homogeneous, isotropic, steady-state thermoelasticity problems involving arbitrary internal smooth and/or kinkedcracks. The thermal stress intensity factors and relative crack surface displacements due to steady-state temperature distributions are determined and compared to available solutions obtained by other methods. In these analyses the thermal boundary conditions across the crack surface are assumed to be insulated. The present approach involves coupling the direct boundary integral equations to newly developed crack integral equations.     

Thermal instability of a layered viscoelastic rectangular prism under high-frequency shear loading

Within the framework of a coupled problem of thermoviscoelasticity, using the method of numerical modeling, we have studied the thermal instability of a rectangular prism, composed of copper and polyethylene or polymethylmethacrylate layers, in the course of its dissipative heating. The prism is subjected to high-frequency force or kinematic shear. We have established that, in the case of polyethylene, thermal instability occurs under conditions of force loading and is absent under a kinematic one. However, for polymethylmethacrylate, thermal instability takes place for both cases of loading. This effect is attributable to the existence of temperature intervals where the shear and volume loss compliances for each of the polymers increases with the temperature. We have also revealed that the critical values of thermal instability for a prism with metal layers are substantially higher than those for a homogeneous prism. In addition to thermal instability, thermal resonance instability is also possible. It is caused by the jump of the thermal state from the low- to high-temperature branch of the soft-type resonance characteristic.     

Analysis of two stationary magnetohydrodynamics systems of equations including Joule heating

We study the coupling of the equations of steady-state magnetohydrodynamics (MHD) with the heat equation when the buoyancy effects due to temperature differences in the flow as well as Joule effect and viscous heating are (all) taken into account. Two models for the gravity force are considered: the first one is the well-known Boussinesq approximation; in the second one density is assumed to be constant except in the gravity force, where it is assumed to be a non-increasing function of the temperature. The equations are posed in a bounded three-dimensional domain. We give existence results of weak solutions to both models under certain conditions on the data. We also give some uniqueness results.     

Free convective boundary-layer flow in a heat-generating porous medium: similarity solutions

The free convection boundary-layer flow on a vertical surfacein a porous medium with local heat generation proportional to(T – T)^p, where T is the local temperature and T is theambient temperature, is considered when there are power-lawvariations in either the wall temperature or the wall heat fluxwhich enables the equations to be reduced to similarity form.When the wall temperature is prescribed, solutions are foundfor p 2 and p p_c (p_c = 10.673) with a saddle-node bifurcationat p = p_c and two solution branches for p > p_c. When thewall heat flux is prescribed, solutions are found only for p< 2. The special case p = 2 is considered and the limitingforms as p 2 and p are obtained and compared with the solutionsobtained from solving the similarity equations numerically     

Stability of a microwave heated fluid Layer

When a time harmonic electromagnetic wave impinges on a slaba certain portion of the wave creates heat within the slab throughdipolar and ohmic heating. The electrical and thermal propertiesof the material dictate the dynamical nature of the heatingprocess, as well as the steady-state temperature profile. Thematerial considered here is a slab of fluid. We consider thecase where the fluid is bounded by thin rigid layers of transparentmaterial. The steady-state heating profile governs the typesof convective motions that can occur and also affects the stabilitycharacteristics of temperature, pressure and velocity perturbationsintroduced in the slab. The main objective here is to examinesuch stability characteristics, initially in the linear regime.Both rigid-rigid and rigid-free configurations are considered.     

Quasilinear parabolic and elliptic equations with nonlinear boundary conditions

This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation.     

Analytical description of the radiative-conductive heat transfer in a gray medium contained between two diffuse parallel plates

Explicit analytical solutions for the temperature and heat flux in a gray medium contained between two diffuse parallel plates are derived for both pure thermal radiation and coupled conduction-radiation heat transfer. This is achieved by combining the integral equations for the heat flux and temperature predicted by the radiative transfer equation with the corresponding predictions of the discrete ordinates method. The algebraic formulation of this well-known method is used to derive analytical results that agree with their corresponding numerical ones with an accuracy greater than 99.9%, for a large interval of optical thicknesses and conduction-to-radiation factors. The explicit and original solutions, for both pure radiation and radiative-conductive heat transfer, therefore solve the problem of one dimensional steady-state heat transfer in gray cavities.     

Steady-state solutions in a nonlinear pool boiling model

We consider a relatively simple model for pool boiling processes. This model involves only the temperature distribution within the heater and describes the heat exchange with the boiling fluid via a nonlinear boundary condition imposed on the fluid–heater interface. This results in a standard heat equation with a nonlinear Neumann boundary condition on part of the boundary. In this paper, we analyse the qualitative structure of steady-state solutions of this heat equation. It turns out that the model allows both multiple homogeneous and multiple heterogeneous solutions in certain regimes of the parameter space. The latter solutions originate from bifurcations on a certain branch of homogeneous solutions. We present a bifurcation analysis that reveals the multiple-solution structure in this mathematical model. In the numerical analysis a continuation algorithm is combined with the method of separation-of-variables and a Fourier collocation technique. For both the continuous and discrete problem a fundamental symmetry property is derived that implies multiplicity of heterogeneous solutions. Numerical simulations of this model problem predict phenomena that are consistent with laboratory observations for pool boiling processes.     

Buckling of an axisymmetric vesicle under compression: the effects of resistance to shear

We consider the axisymmetric deformation of an initially spherical,porous vesicle with incompressible membrane having finite resistanceto in-plane shearing, as the vesicle is compressed between parallelplates. We adopt a thin-shell balance-of-forces formulationin which the mechanical properties of the membrane are describedby a single dimensionless parameter, C, which is the ratio ofthe membrane's resistance to shearing to its resistance to bending.This results in a novel free-boundary problem which we solvenumerically to obtain vesicle shapes as a function of plateseparation, h. For small deformations, the vesicle contactseach plate over a small circular area. At a critical value ofplate separation, h_TC, there is a transcritical bifurcationfrom which a new branch of solutions emerges, representing buckledvesicles which contact each plate along a circular curve. Forthe values of C investigated, we find that the transcriticalbifurcation is subcritical and that there is a further saddle-nodebifurcation (fold) along the branch of buckled solutions ath = h_SN (where h_SN > h_TC). The resulting bifurcation structureis commensurate with a hysteresis loop in which a sudden transitionfrom an unbuckled solution to a buckled one occurs as h is decreasedthrough h_TC and a further sudden transition, this time froma buckled solution to an unbuckled one, occurs as h is increasedthrough h_SN. We find that h_SN and h_TC increase with C, thatis, vesicles that resist shear are more prone to buckling.     

Self-heating of plastics during cyclic defformation

Analysis of the phenomenon of self-heating as a result of competition between hysteresis heating and heat losses to the ambient medium shows that for polymers two zones of steady-state heating are possible: a low-temperature zone, corresponding to a high endaurace limit, and a high-temperature zone, corresponding to low endurance. Between these zones is a temperature region in which steady-state heating is impossible. The high-temperature steady-state zone is frequently not realized as a consequence of the sharp drop in strength at high temperatures. The transition from one steady-state zone to the other is discontinuous, the occurrence of one or the other zone being determined by the deformation conditions (stress, cycling speed, size of specimen, heat conductivity, etc.). However, the self-heating temperature at which this transition takes place does not depend on the deformation conditions, but is determined only by the properties of the material. These conclusions have been confirmed experimentally.Mekhanika Polimerov, Vol. 1, No. 3, pp. 93–100, 1965     

A finite element method for heat transfer of power‐law flow in channels with a transverse magnetic field

Heat transfer of a power‐law non‐Newtonian incompressible fluid in channels with porous walls has not been carefully studied using a proper numerical method despite a few constructions of approximate analytic solutions through the similarity transformation and perturbation method for Newtonian fluids (i.e. power‐law index being one). In this paper, we propose a finite element method for the thermal incompressible flow equations. The incompressible condition is treated by a penalty formulation. Numerical solutions are validated by comparing them with an approximate analytic solution of the Navier–Stokes equation in the Newtonian fluid case. Then, the method is used to simulate the heat transfer of various power‐law fluids. Additionally, unlike previous studies, we allow the thermal diffusivity to be a function of temperature gradient. The effect of different values of the parameters on the temperature and velocity is also discussed in this paper. Copyright © 2013 John Wiley & Sons, Ltd.