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IMA Journal of Applied Mathematics (1994) 52, 141-176 Microwave heating of materials with power law temperature dependencies A . H . PlNCOMBE Department of Mathematics, Levels Campus, University of South Australia, Main North Road, Pooraka, South Australia 5095, Australia N. F. SMYTH Department of Mathematics and Statistics, The King's Buildings, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, Scotland [Received 15 May 1992 and in revised form 25 March 1993] The time-dependent microwave heating of a semi-infinite material, which is governed by a coupled system of Maxwell's equations and the forced heat equation, is considered. The dielectric properties of the material are assumed to be temperature-dependent, with this dependence being of a power law form. Asymptotic solutions of the governing equations are found in the high-frequency geometric-optics limit. These solutions are compared with numerical solutions of the governing equations, and good agreement is found. 1. Introduction In recent years there has been a great deal of interest in the development of industrial applications of microwave heating. This interest is driven by the significant advantages which microwave heating holds over conventional heating for many heating and drying applications. But the widespread application of microwave heating has also uncovered some problems, one of which is the phenomenon of a 'hotspot', which is a small region which has a significantly elevated temperature. In some cases, the temperature at the hotspot is sufficient to melt the material, which is desirable for applications such as smelting, but is undesirable for applications such as sintering (see Araneta et al., 1984). Because of these problems, there has been an upsurge of interest in the theory of microwave heating, most of which is based on the fact that the electromagnetic properties of materials, i.e. the magnetic permeability, the electric permittivity, and the loss factor, are all dependent on the temperature, and thus thermal runaway can occur if the absorption rate increases sufficiently rapidly as the temperature rises. Experimental measurements (von Hippel, 1954) show that the absorption rate rises with temperature for most materials. The classical analysis of microwave heating is based on the assumption that the absorption rate and the permittivity and permeability are all constant, so that Maxwell's equations reduce to the telegrapher's equation, which can be solved without reference to the temperature. The temperature variation is then calculated by solving a forced 141 © Oxford University Preu IW4

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IMA Journal of Applied Mathematics (1994) 52, 141-176

Microwave heating of materials with power lawtemperature dependencies

A. H. PlNCOMBE

Department of Mathematics, Levels Campus, University of South Australia,Main North Road, Pooraka, South Australia 5095, Australia

N. F. SMYTH

Department of Mathematics and Statistics, The King's Buildings,University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, Scotland

[Received 15 May 1992 and in revised form 25 March 1993]

The time-dependent microwave heating of a semi-infinite material, which isgoverned by a coupled system of Maxwell's equations and the forced heatequation, is considered. The dielectric properties of the material are assumed tobe temperature-dependent, with this dependence being of a power law form.Asymptotic solutions of the governing equations are found in the high-frequencygeometric-optics limit. These solutions are compared with numerical solutions ofthe governing equations, and good agreement is found.

1. Introduction

In recent years there has been a great deal of interest in the development ofindustrial applications of microwave heating. This interest is driven by thesignificant advantages which microwave heating holds over conventional heatingfor many heating and drying applications. But the widespread application ofmicrowave heating has also uncovered some problems, one of which is thephenomenon of a 'hotspot', which is a small region which has a significantlyelevated temperature. In some cases, the temperature at the hotspot is sufficientto melt the material, which is desirable for applications such as smelting, but isundesirable for applications such as sintering (see Araneta et al., 1984). Becauseof these problems, there has been an upsurge of interest in the theory ofmicrowave heating, most of which is based on the fact that the electromagneticproperties of materials, i.e. the magnetic permeability, the electric permittivity,and the loss factor, are all dependent on the temperature, and thus thermalrunaway can occur if the absorption rate increases sufficiently rapidly as thetemperature rises. Experimental measurements (von Hippel, 1954) show that theabsorption rate rises with temperature for most materials. The classical analysis ofmicrowave heating is based on the assumption that the absorption rate and thepermittivity and permeability are all constant, so that Maxwell's equations reduceto the telegrapher's equation, which can be solved without reference to thetemperature. The temperature variation is then calculated by solving a forced

141© Oxford University Preu IW4

142 MICROWAVE HEATING OF MATERIALS

heat equation with the forcing term dependent on the square of the amplitude ofthe electric field (see e.g. Metaxas & Meredith, 1983). When the electromagneticproperties are temperature-dependent, the forced heat equation and Maxwell'sequations become a highly nonlinear coupled system, which is difficult to solve ingeneral.

Most of the theoretical work to date (Roussy et al., 1987; Hill & Smyth, 1990;Brodwin et al., 1993; Coleman, 1991) has used the simplifying assumption that theamplitude of the electric field is constant. This assumption can be justified for thinmaterials or for hotspots where the analysis centres on a small region of thematerial, and reduces the problem to one of solving the forced heat equation

for the temperature T, where the forcing term y(T) represents the temperature-dependent absorption of energy from the microwave radiation. The function y(T)is not known theoretically, but it can be deduced from experimental measure-ments for any particular material. Various representations for y(T) have beenused in the literature. Hill & Smyth (1990) used the form

y = yPeVlT, (1.2)

and, with a boundary condition T = 7, = constant, they identified a condition forthe formation of hotspots. The exponential dependence (1.2) has been found tobe valid for various ceramics (Barker et al., 1976; Kingery et ai, 1976; Wo, 1986).

Coleman (1991) used the power law

and assumed that heat diffusion could be neglected at a hotspot because it wasmuch smaller than the rate of heat absorption. He showed that a hotspot willoccur if

y 2 ( y 3 - i ) > 0 . (1.4)

Roussy et al. (1987) numerically solved equation (1.1) for a cylindrical body, withy depending quadratically on temperature, and, using a heat loss boundarycondition, found an approximate condition for a hotspot to form. Brodwin et al.(1992) found steady-state solutions of equation (1.1) in a thin slab, usingconvective and radiative heat loss at the boundary. They found solutionscorresponding to hotspots for certain types of dependencies of y on temperature.Hill & Pincombe (1992) used a nonlinear thermal diffusivity and a space-dependent energy absorption rate,

7>[v(7m + -KVr. (1-5)

and found that the formation of hotspots dependended on both the energyabsorption rate y(T) and the heat diffusion coefficient v(7).

The criterion for the formation of a hotspot quite clearly depends on thefunctional dependency of the absorption rate y on the temperature, but it is true

A. H. PINCOMBE AND N. F. SMYTH 143

that, when the heat diffusion coefficient v is small, hotspots will form if theabsorption rate y(T) is an increasing function of temperature, and d2y/dr2 >0.

When the size of the body cannot be considered small, the full system ofMaxwell"s equations and the forced heat equation must be solved. Owing to thenonlinear coupling of this system, little work has been done on solving theseequations. Kriegsmann et al. (1990) found the steady-state solution for asemi-infinite, one-dimensional body with temperature-dependent conductivity,but with constant permittivity and permeability. The boundary condition used wasconvective and radiative cooling, and the nondimensionalized conductivity wasassumed to be small which enabled a solution to be found as a perturbationseries. Kriegsmann (1991a. b, 1992) considered the microwave heating of aone-dimensional slab of arbitrary thickness in the small Biot number limit inwhich the slab is essentially insulated. Steady-state solutions were found byperturbing on the small Biot number and these solutions were found to havemultiple steady states for the temperature. The stable steady state of highertemperature was found to correspond to a hotspot in that it tended to have amuch higher temperature than the stable steady state of lower temperature.Smyth (1990) explored the high-frequency geometric-optics limit and assumedthat the thermal diffusivity was small and that the conductivity, permittivity, andpermeability depended linearly on temperature, and found the full time-dependent solution of the coupled system. Pincombe & Smyth (1991) consideredthe time-dependent heating of a semi-infinite material with low conductivity andused power laws of the form

to represent the electromagnetic properties and the energy absorption rate. Forthe case where the thermal diffusivity v was small, they used a multiple-scalesanalysis to produce approximate solutions for particular cases, some of whichdemonstrated the existence of hotspots.

In the present work, we shall consider the time-dependent microwave heatingof a semi-infinite material where the energy absorption rate and the electromag-netic properties are represented by power laws of the form

p=Pl + ap2T"< (a«\) (1.7)

and the loss factor is small. The analysis uses the high-frequency geometric-opticslimit adopted by Smyth (1990), and the complete time-dependent solution of thecoupled system of equations is found as a perturbation series. These solutions areshown to be in good agreement with full numerical solutions of the coupledsystem of equations. The approximate equations governing the microwaveheating of the material in the geometric-optics limit will be found to be the sameas those governing the heating in the limit of small conductivity, found byPincombe & Smyth (1991). This is because the dimensionless perturbationparameter cr,,/(we(,), where w is the microwave frequency and an and e,, aretypcial values of the conductivity and permittivity respectively, is small if either wis large or cr<, is small. Pincombe & Smyth, through the use of power laws of the

144 MICROWAVE HEATING OF MATERIALS

form (1.6), obtained solutions that in the present work are equivalent to takinga = 0(1) in (1.7). The extra assumption of the present work that a « 1 enablesmore solutions to be obtained.

2. Governing equations

We shall now derive the equations governing the microwave heating of ahalf-space, as was done by Pincombe & Smyth (1991). If the material ishomogeneous and isotropic and the current j and the displacement current Dinduced by the microwave radiation are both proportional to the electric field E,and if the magnetic flux density B is proportional to the magnetic field strength //,then Maxwell's equations are

= div(e£') = O, divfl = div (M//) = 0,1curl£= -(fJ.H),, curl H = (eE), + crE. J

The permittivity e, permeability fi, and the conductivity a are, in general,functions of temperature and of the frequency of the microwave radiation (vonHippel, 1954; Metaxas & Meredith, 1983). Microwave heating uses radiation of asingle frequency, but the local frequency of the radiation will vary as it passesthrough a material due to changes in the values of the electric permittivity and themagnetic permeability at any point, brought about by the changing temperature.Thus all changes in the electromagnetic properties can ultimately be attributed tochanges in the temperature. However, the explicit dependence of the dielectricproperties e, n, and a on the frequency of the radiation will be neglected in thepresent work. The dependence of these properties on temperature means that wemust couple Maxwell's equations (2.1) with some form of the forced heatequation. Hill (1989), Smyth (1990), and Kriegsmann et al. (1990) used theequation

|2, (2.2)

which we shall also adopt. The forced heat equation (2.2) was derived fromenergy conservation by Pincombe & Smyth (1991).

For a plane wave propagating in the positive x-direction, we can, without lossof generality, regard the wave as polarized, so that the electric field is in the_y-direction and the magnetic field is in the z-direction. For the one-dimensionalheating of a half-space .v >0, the electric and magnetic fields are functions of xand time / only. Maxwell's equations (2.1) then become

(2.3)

//v = -(e£), -aE, (2.4)

where £ and H are scalar quantities.We can combine equations (2.3) and (2.4) in the usual way to obtain the

damped wave equation

c2Exx = -LH - MH, + PE + QE, + £„, (2.5)

A. H. P1NCOMBE AND N. F. SMYTH 145

where

L = — , M= — , c = (£/xy-, (2.6)IME flE

p = El£1+ojL1+EJ1+_a1^ ( 2 ? )

(JLE llE £

Q = » + *L±*. (2.8)II £

Note that the function Q defined by (2.8) will cause changes in the amplitude ofthe electric field as it passes through the material, but these changes do notrepresent absorption of energy. Rather, they represent either storage of energy(in the case of E,/E and iijfj.) or reflection or scattering of energy (in the case oftr/e). Some of the energy thus imparted to the electrons will be converted intoheat by processes such as stresses within atoms or by collisions. Thus someproportion of the energy given to conduction electrons will be converted into heatand. similarly, some proportion of the energy which distorts atoms or reorientsmolecules will be converted into heat. The atomic and molecular dissipativeprocesses have not been specifically included in the analysis of Maxwell'sequations. They are usually included via an imaginary term in the permittivity (inthe case of dielectrics) or in the permeability (in the case of magnetic materials).This results in an extra term being added to the conductivity term to obtain thetotal attenuation (see e.g. Metaxas & Meredith, 1983). As an illustration, considerthe case where the only loss term is ohmic, where the electric field is sinusoidalwith frequency w, and where the permittivity E is constant. Then equation (2.4)becomes

Hx = —aE — IEWE = —\£l)£*a>E, (2-9)

where

e* = ^-'x7Z' (210)

and £„ is the permittivity of free space. Thus the loss term appears in theimaginary part of the dielectric constant. All other forms of loss (e.g. due topolarization) can also be represented by the imaginary part of the dielectricconstant

e* = e'-\e", e' = e/ett. (2.11)

Here we also adopt that simplifying approach and assume that the atomic andmolecular energy absorption can be represented by an extension of theconductivity term, by replacing the expression a/WE by the losss term e". Thus, inthe case of dielectrics, where the conductivity is low, our loss factor E"(T) mightbe quite large. In the work below, we assume that

e"«\, (2.12)

which allows us to model the microwave heating of a wide range of materials.

146 MICROWAVE HEATING OF MATERIALS

However. (2.12) excludes some highly polar materials, such as water, for whichf"»l.

Smyth (1990) set

E = 4>(x, f)eio>("A-'>, H = .A(.v, /)ei luH<r", (2.13)

and considered the high-frequency (geometric-optics) limit {a I we « 1 in dimen-sionless variables) and was thus able to separate equation (2.5) into equations ofO(a>2), O(w), and 0(1), etc. We use a similar approach here, where we take w tobe the constant frequency of the incident radiation, and obtain

,</>., - w24>9l)= -Lip- M(4>, + \wd,4i)

P(f> + Q(4>, + \u)e,4>) + <(>„ + iu)G,,(f) + 2\w9,(f>t - w24>6l (2.14)

upon substituting (2.14) into (2.5) and keeping terms to 0(1).Pincombe & Smyth (1991) demonstrated that for slowly varying e and /x, and

for small a, the intrinsic impedance relationship between E and H, that is,

/y = (c/M)-£, (2.15)

holds to first order. This result can be extended to the present case where cr is notsmall, but where a/we is small, that is, where w is large, by using thedimensionless variables defined by

t' = w t , x ' = —, E ' = T T > / / ' = — . £ ' = - , M ' = — • c ' = - , ( 2 . 1 6 )co £•() H[) So Md co

with c(|, £,„ Hu, £(„ yu.,,, and w being the values of the wavespeed, the electric fieldstrength, the magnetic field strength, the electrical permittivity, the magneticpermeability, and the frequency, respectively, of the incident radiation. Thus c(, isthe speed of light in a vacuum, and H() and £„ satisfy the intrinsic impedancerelation, that is.

Hit = (e ( )/Mo)- l£ ( ), c,, = (enMo)"*- (2-17)

By substituting (2.16) and (2.17) into equations (2.3) and (2.4). we obtain thedimensionless equations

H\. = -(e'E')r - e"e'E\ (2.18)

£. ;=-(M 'W') , . . (2.19)

where e" = a I we is the dielectric loss factor.From now on. the prime denoting a dimensionless variable will be dropped,

and all variables are understood to be dimensionless. The dielectric loss factor e",which is essentially a dimensionless variable, will continue to be denoted by thedouble prime, since this is the way it is normally denoted in the literature.

A. H. P1NCOMBE AND N. F. SMYTH 147

If we assume that

£, + e"«\ and /x, « 1 . (2.20)

so that E" is small and e and /x are slowly varying, we have

Hx = -eE,, (2.21)

(2-22)

to first order, from which it follows that the intrinsic impedance relation holds tofirst order (see Bleaney & Bleaney, 1976). Hence

, (2.23)

,, (2.24)

to first order. Since from (2.20) e"« 1 and e,,fi,« 1, it follows that

' . (2-25)

to first order, one using (2.13) in (2.23) and (2.24). This can also be shown directlyfrom equations (2.3) and (2.4) by substituting expressions (2.13) and then usingthe normal method (see Bleaney & Bleaney, 1976) that is used when /x and e areconstant, assuming that (2.20) hold.

It was convenient in the above (see (2.16)) to nondimensionalize by usingvalues of c, E, and H associated with the incident radiation. However, if weconsider the radiation to be incident on the surface of the maerial, then we musttake reflection into account. In the situation we are considering, where the lossfactor, the permittivity, and the permeability are all functions of temperature, thisreflection is also temperature-dependent. Since the temperature of the materialand its surface are determined by the solution of the governing equations, thereflection of the microwave radiation at the surface of the material and themicrowave heating of the material are coupled. When the radiation is normallyincident on the surface of the material, the ratio of the amplitude of the electricfield in the transmitted beam to the amplitude of the electric field in the incidentbeam is given by

r = 2/(n + \), (2.26)

where n is the refractive index of the material (see Bleaney & Bleaney. 1976).In general, the refractive index is complex and is given by

/i = [0i + iM")(e + ie")]-. (2-27)

where fj." and e" are the magnetic and electric loss factors, respectively. Thismakes the ratio r in (2.26) complex, which indicates a phase change at theboundary. The ratio r is a function of the temperature through its dependence onthe magnetic and electric parameters, and is thus a function of time. That is,

r = r(T(0, t)) = R(t)ei(il). (2.28)

Hence the amplitude of the electric field at x = 0 is given by R{t). A similar result

148 MICROWAVE HEATING OF MATERIALS

can be obtained for the magnetic field H, and so we can nondimensionalize withrespect to the characteristics of the incident radiation: the incident wavespeed c0,frequency w, and the amplitudes E{) and //„.

If we apply the nondimensionalizations (2.16) to equations (2.6)-(2.8), andreplace a by wee" as discussed above, we obtain

( 2 . 2 9 )

, : i ( )fie fi e e

Q = (oQ', Q'=^ + — + e". (2.31)fi e

We also apply the nondimensionalizations (2.16) to the damped wave equation(2.14), and, after using (2.25), (2.29), (2.30), and (2.31), we obtain

c2(w2tf>rr + i«o3exx<t> + 2\a>*ex4>, - m^e2) =-u>2 — <t>- — («24>, + \w34>e,)fi fl

+ a)2P'4> + Q'(co2(f), + iw30,tf>) + w2<j>,, + \w*6,,<j> + 2io)3e,<f>, - co4<f>82. (2.32)

As we are considering the high-frequency (geometric-optics) limit in which(o » 1, equation (2.32) can be separated into equations of order O(co"), the firsttwo equations being

O(w4): 92 = c 2 ^, (2.33)

O(co*): c2dxr<}> + 2c2ex(t>r = - — dl<j> + Q'ei<t> + 6ll4> + 2dl<1>l. (2.34)ft

The first of these equations, (2.33), is the eikonal equation and the second, (2.34),is the transport equation of geometric optics. We choose the solution

9, = -cdx (2.35)

of the eikonal equation (2.33), representing a wave travelling in the direction ofincreasing x. By substituting (2.35) into the transport equation (2.34), we obtain

(2.36)

The amplitude equation (2.36) is the same as that found by Pincombe & Smyth(1991) under the assumption that the conductivity a is small. This is to beexpected, as the perturbation parameter in the present work is a-Jcoe0, which issmall if either w is large or a0 is small.

A. H. P1NCOMBE AND N. F. SMYTH 149

Thus the microwave heating of the half-space x > 0 will be governed byequations (2.2), (2.35), and (2.36) if it is assumed that e " « l and that theelectromagnetic properties are slowly varying. For plane waves normally incidenton the surface x = 0, the boundary conditions are, from (2.13) and (2.28),

4,(0, t) = R{t), 6(0, t) = -t + f(f). (2.37)

We note that R(t) and £(/) are not determined at this stage since both R(t) and£(f) depend on fi, e, /i", and e", which are functions of the yet to be determinedtemperature. Since we are solving a system where Maxwell's equations (2.1) arecoupled to the forced heat equation (2.2) and we have nondimensionalizedMaxwell's equations, we also need to nondimensionalize the forced heat equation(2.2). Let

T' = T/Tn, (2.38)

where 7J, is some suitable reference temperature (e.g. the absolute temperature ofthe surroundings at time t = 0), so that equation (2.2) transforms to

:x g()^:ct>2. (2.39)c(, a)/,,

The coupled system (2.36) and (2.39) is difficult to solve in general, so thefurther assumption that the dimensionless heat diffusivity vw/co is small will bemade. For the source term to dominate over the heat diffusion term in (2.39), werequire

vw2lcl«EllT{), (2.40)

and, when this condition is satisfied, equation (2.39) becomes, to first order,

T,=g(J)3(f)2 = GiT,)4)2 ( 2 4 1 )

(uTt)

We now drop the prime which indicates the dimensionless temperature. Fromnow on, all references to temperature are understood to refer to the dimension-less temperature unless otherwise stated.

The variation of conductivity, permittivity, and permeability with temperaturecan, for many materials, be represented by a power law (von Hippel, 1954). Weuse the forms

(2.42)

(2.43)

(2.44)

where, for the assumption (2.20) to hold, we require a « l . Here, for con-venience, we consider materials for which the magnetic loss factor is zero, so thatwe are considering dielectric materials. We note that the functional forms(2.42)-(2.44) are not invariant under translational transformations of the tem-perature, and we cannot therefore arbitrarily rescale the ambient temperature tozero. However, the form of the expressions is invariant under stretching

150 MICROWAVE HEATING OF MATERIALS

transformations, and so we are able to use the dimensionless temperature T. Thematerial being heated will have an initial temperature given by

T(x, 0) = 7Xv), (2.45)

and in the important case where the material is initially at equilibrium with itssurroundings, the initial temperature will be constant (i.e. T,(x) = T, = constant).If we are given a relationship of the form of (2.42), we cannot arbitrarily rescalethe temperature so that the initial value is zero. However, it is possible, in manycases, to find an alternative power law, with three different parameters, which canadequately represent e" as a function of a temperature which is scaled to have aninitial value of zero. In this case there is one important restriction. When 7] = 0,to ensure that the derivatives of the electromagnetic properties are finite at t = 0,we require a3,e3,fi3^ 1.

In the forced heat equation (2.2), the source term includes the function g(T),where the temperature has not been nondimensionalized, which represents thetemperature-dependent absorption rate. The heat gain must be related to the lossterm. For example, Metaxas & Meredith (1983) use a relationship where the rateof heat gain is proportional to the product of the frequency and the effective lossterm. In our terminology, this is equivalent to

g{T) = kwe", (2.46)

where k is a constant of proportionality and e" in this work refers to all possibleloss mechanisms. In general, not all losses will be converted into heat and, foreach loss term, a different proportion of the energy lost might be converted intoheat. Because of these complications, we adopt a source term in equation (2.41)(and equation (2.39)) of the form

G(T) = y]+ay2T?\ (2.47)

while noting that (2.46) must be approximately true and that the value of G(T) isinfluenced by the amplitude £{1 of the incoming radiation. All of the othervariables in the dimensionless equations can be expected to have initial values of

Thus, if (2.46) holds, we can define the parameters in (2.47) by

l ka2ElT T r.i = a3. (2.48)

In the current work, we consider the heating of a semi-infinite slab of materialwith an insulated boundary at x = 0, giving

7;r = 0 atJC = O, (2.49)

and a uniform initial temperature of

7 = 7; aU = 0. (2.50)

The thermally insulated boundary condition is a good approximation for dielectricmaterials, for which the Biot number, measuring the ratio of heat convection or

A. H. PINCOMBE AND N. F. SMYTH 151

radiation at the boundary to heat conduction, tends to be small: for example, theBiot number is approximately 0-0001 for ceramics (Kingery et ai. 1974:Kriegsmann. 1991a. b). Expressions (2.49) and (2.50) together with (2.37) are theinitial and boundary conditions used in the present work.

In general, the transmission ratio r, given by (2.28), will be a function of thetemperature T. However, in the case considered here, where /JL" = O. and e, /x.and E" are given by the relations (2.42)-(2.44), it can readily be shown that themodulus R and the phase £ of the transmission ratio are of the form

R = R0+aRl+O(a2), (2.51)

£ = £ , + a£| + 0(a2) , (2.52)

where /?„ and £„ are constant and R, and £,, etc., are functions of thetemperature. The expressions for /?„, /?,, £„, £, in the expansions (2.51) and (2.52)cannot be determined until the relationship between the constant «,. from theexpression (2.42) for the dielectric loss factor E", and the ordering constant a isknown. When a, ~ O(a"), we write

at=a"au, (2.53)

and, by substituting this expression (2.53) into (2.26) and (2.27), we obtain, forn ^ 2-

2

°"l+(M.e, ) - 1 ' (2.54)

where

(M.g.)-Vt2 A (M^i)^ _1 2 [ l + 0 ) ] ' 2 2 e [ l + ( ) i ] 1 3

2£2[1 + (M | £ |)-]

(2.55)

and

=

2 e [ l + ( M £ ) ] ' (2.56)

where

ete2[l+2(/tt,g,)*]o £M2 ete 2 [ l+2( / t t ,g , ) ] =1 2 [ l + ( ) V 2 2 [ l + ( ) 1 ] ' '

and the temperature 7 is evaluated at the surface x = 0.

152 MICROWAVE HEATING OF MATERIALS

3. Perturbation solution for a « e"« 1

The governing equations to be solved are the heat flow equation (2.41) with asource term depending on the square of the amplitude of the electric field, thephase (eikonal) equation (2.35), and the amplitude (transport) equation (2.36).We first consider the case when 0< a «a, « 1. Most materials which are suitedto microwave heating have low conductivities. However, in a lossy material, thee" term is not merely a conductivity term, but also includes the other lossmechanisms (see Metaxas & Meredith, 1983). Thus, if the material is capable ofbeing heated, this term will not be zero. Here we shall assume that this loss factoris small enough to avoid the propagation changes which accompany large lossrates, but that it is larger than the rate at which the loss term, the permeability,and the permitivity are changing. This is not difficult to justify as it is quite likelythat the rate of change of the electromagnetic properties, with respect totemperature, will be smaller than the order of the loss term, and the case wheree"~O(a) has already been treated by Pincombe & Smyth (1991) for differentfunctional forms than (2.42)-(2.44). The case where the rate of change of theelectromagnetic properties is of the same order as the loss term is treated in thenext section. The transport equation (2.36) can be written as

> + O(a2), (3.1)where

_ +£_^ r , - , ( 3 7 ; + C | 7 ) + fl2r<i ( 3 2 )

ac2 + O(a2)], (3.3)and

^ ^ ) (3.4)

The functions /i, and c2 in equations (3.1)—(3.4) are temperature-dependent, andit will be found that they contain secular terms in time. These secular terms arepresent before the perturbation analysis is undertaken and cannot be removed.They are due to the temperature being an increasing function of time due to thecontinual energy input from the microwave radiation. However, it will be shownhere that the perturbation solutions are uniformally valid as long as thetemperature is in the range 0s£r=£7j, where Tx will be defined later in thissection. At the boundary x = 0, we have the condition

<f>(0, t) = 4>t)(0, t) + a<£,(0, 0 + O(a2) = /?„ + a/?, + O(a2). (3.5)

When the material's dielectric properties are constant, so that a2 = 0, e2 = 0,and /x2 = 0 (which is equivalent to taking a = 0), the transport equation (3.1) canbe easily solved to yield

0 = flue-""-"21"', (3.6)

on using the boundary condition (2.51).

A. H. PINCOMBE AND N. F. SMYTH 153

By analogy with this solution for constant dielectric properties, we set

<Kx,t)=f(x,t)e-«'J\ (3.7)

where the function / takes account of the varying electric field strength at .v = 0and g the decay of the electric field in x>0. It is not necessary to assume theform of solution (3.7). However, on solving (3.1), we would find that the solutionis of the form (3.7) with the roles of/and g as stated. On substituting (3.7) intothe amplitude equation (3.1), we obtain

f, + cfx = {g, + cgx - \[ax + ah, + O(a2)]}f. (3.8)

The simplest solution to equation (3.8) is obtained by setting

g, + % = Jfl1 + M . + 0(o2), (3.9)

which leaves

f, + cft = 0. (3.10)

Let us expand / a s

f=ft,+ af1 + O(a2), (3.11)

so that, from (3.10),

flu + cJat = 0 (3.12)

subject to /,(0, t) = /?,,, and

(3-13)

subject to/i(0, /) = Rx(T(0, t)), where the boundary conditions are obtained from(3.5).

Equations (3.12) and (3.13) can be solved by the method of characteristics, toyield

/ , = /?,„ (3.14)

/, = /?,(7(0, r)). (3.15)

where

r = t-x/Cl. (3.16)

We see that, as stated above, / accounts for the variation of the electric fieldstrength at x = 0.

We similary use a regular perturbation expansion for g(x, t) to obtain a solutionof (3.9). We thus set

g(x, t)=go(x, t) + agl(x, t) + O(a2), (3.17)

while at the boundary x = 0 we have gn = g\= ••• = 0 , from (3.5). On substi tutingthe expansion (3.17) into equat ion (3.9), we obtain

at 0(1): gi» + clgOx = kal (3.18)

154 MICROWAVE HEATING OF MATERIALS

subject to go(0, f) = 0, and

atO(a): gu + clgu-c2gae = ±hl (3.19)

subject togi(0, 0 = 0.

Equation (3.18) can be solved, using the method of characteristics, to yield

g,, = ifl,jc/clf (3.20)

and, upon using this result, we can simplify the O(a) equation (3.19) to obtain

g\,+c{gXx = {hx + \c2aJcu (3.21)

From (3.7), (3.14), and the solution (3.20) for g,,, we see that

^(jc,0 = ̂ l)e~<"1-r/2ri) + O(a). (3.22)

Equation (2.41) governing the temperature becomes, upon substitution of (3.7),

T, = (y, +ay2Ty')R2e-2R"-2ai!'—-. (3.23)

If we expand T in terms of the small parameter a,

T= Tn+aT, + a2T2 + •••, (3.24)

we obtain, after substitution of the expansion (3.24) into the reduced heatequation (3.23) and separation of the resulting equation into equations of orderO{a"'\

O(a°): T.)l = yjRffi-2>"< (3.25)

O(a'): Tll = (y2T^ + 2ylRj/Rn-2y]g])Rffi-^», (3.26)

plus equations of order O(a") (n^2). To evaluate the exponent function g,,using equation (3.21), we need to calculate h, and c2, which are both functions oftemperature. However, only terms of first order should be included in (3.21) ashigher-order terms would be taken into account in the equations for g2, gj, etc.Thus we have a simple solution path in which we solve for g(, (equation (3.20)),then use g,, to solve (3.25) for T,,, then use the solutions for gt) and T{) to solve(3.19) for g|, and complete the O(a) solution by using g,,, g,, and T{) to solve(3.26) for Tx.

The initial condition for the temperature,

Hv,0) = 7i (***()), (3.27)

leads us to the following initial values:

T()(x, 0 = 7 ; when / -JC/C, «0 , (3.28)

T, = T2= ••• = 0 w h e n / - j r / c , « 0 , (3.29)

where the condition / xlcx =£0 is a recognition of the fact that heating cannotbegin at any point until the wavefront has reached that point.

A. H. PINCOMBE AND N. F. SMYTH 155

We can integrate (3.25) to obtain the following first-order expression for thetemperature:

f < ^ ) (3.30)

where

T = t-x/c,. (3.31)

The functions h} and c2 can now be evaluated as

+ ^ { T i + yiR»teU"'c'TiRfo<e('"k'\2 - a, T)

+ a2(Tt+JlRfa-('""c<Y (3.32)

and

^ 7 i + yi/??,Te-(-" /c '»r+ ^ - ( 7 ; + y./Jgre-^""-'^'. (3.33)

We can solve (3.21) by using the method of characteristics. On the characteris-tic defined by

we have / -or/c, = x = constant and

T ( 7 ; + y , T / ? J i e )4e,

T (T< + y > T/?ge-(<"*/c"l)r + " ^ l l (71 + yi T/?,2,e('"*A:i))w

2 4/i,c,

(71 + y, r/?^-<"^'»)£'- (3.35)

The first two terms on the right-hand side of (3.35) are directly integrable,regardless of the value of the initial temperature 7j, while the final three termscan be integrated for general a3, /x3, and e, only for the case 7] = 0. For 7! ¥= 0,(3.35) can be integrated numerically. However, an analytical solution can beobtained using Taylor expansions on the final three terms in (3.35), truncated

156 MICROWAVE HEATING OF MATERIALS

when a term in the expansion is reduced to O(a). For clarity, the results of thisexpansion and the corresponding solutions for g, are presented in the Appendix.The case when T, ¥^ 0 is of secondary importance in the present work.

The solution of (3.35) is simplified greatly when 7j = 0, in which case

g^ = kH, + \a,C2|cu (3.36)where

a,

and

which is, of course, the limiting case of (A.7) as 7j—>0. We shall consider thespecial case of T,= 0 from now on, as this initial condition greatly simplifies thesolution for gu

In order for the expansion of the solution for g to be uniformly valid, werequire

I « g . | / M « l . (3.39)

An inspection of (3.20) and (3.36) shows that the condition (3.39) for uniformvalidity can easily be satisfied for large values of x. However, we need to showthat (3.39) is satisfied for small values of x. For x « 1, we have 1 - e~*' = JC, sothat the relation (3.39) reduces to the sum of terms involving powers of (t -x/ct).These terms are either of the form (t — xlcx)

a or of the form (t - jr/c,)""1. Whena 5=2 we have y" > y"'1 for any variable y 5*0, but for 1 s=a « 2 we require y > 1for y" 5=_ya~'. If an expression containing y" is to be small for y>\, thecoefficients need to be small and therefore the expression will necessarily have asmall value when _ys£l. The expression t-x/ct represents the time since thewavefront passed the point x and is thus the heating time, to first order. In orderfor the expansion for g to be uniformly valid, we require

<min {(_**_), (J£^y (2M ( I W ( W ll\aiLp' \aE2E3/ \aa2' \ (J.2 / \ E2 / J

When the above condition (3.40) for uniform validity is satisfied, we can writeour solution as

«>, (3.41)

A. H. PINCOMBE AND N. F. SMYTH 157

where g, is given by (3.36), and R is given by

R = R0 + aR,(T(0, r)) + O(a2). (3.42)

When ag, « 1 , we can write

e-°K> = — + O(a2), (3.43)

and thus we are able to write our solution in the form

D -(oix/2ci)

*(*, 0 = —-f ^ - ^ r , (3.44)l + a(gl -Rt/R0)

which closely parallels the form of the solution obtained by Pincombe & Smyth(1991) for the case where e"~O(a) and for different power law temperaturedependencies for the wave speed c and the loss term e". For example, when thepermittivity and the permeability are constant and 7J = 0, (3.44) becomes

*(*. ' )= " r n , (3-45)1 + aa2l /2aia3

where

H l - e-(a'fl>r/Cl))- (3.46)

Equation (3.26) for the O(a) temperature component 7", is readily integrable.For the case when T, = 0, the solution is

(y,

(-u f?2TP~("<xlc>)^'(y|/?()Te J _fi3 + \

j2 )

2a i L/x,

- ( / ? S y , ) ' - ' ( l - e - ) (e , \ e 3

(RlY'O

f (l_e2C, ^ , M 3 ( M 3 )

*2 ] (3.47)

The above solution is of course unbounded as t—>°°, owing to the thermally

158 MICROWAVE HEATING OF MATERIALS

insulated boundary condition allowing no heat loss, and therefore no mechanism,for a steady state to form. This is not a great drawback, as the major reason whymicrowave heating is used industrially is that it takes place on a rapid timescaleand so the long-time behaviour of the solution is less relevant than forconventional heating. Kriegsmann (1992) used a multiple-scales expansion with aslow timescale based on the low rate of heat loss at the boundaries of a materialwith small Biot number to develop a solution uniformly valid in time. Thesolution was assumed to have only slow time dependence and no fast timedependence. This was valid as the material had a finite width, and hence, becauseof the large value of the speed of light, the transients due to the initiation of theslowly varying temperature profile in the material can be ignored. However, inthe present work, the material being heated is semi-infinite, so that thesetransients cannot be ignored, as there is always a wavefront present in thematerial. The temperature will then depend on the fast timescale as well as a slowtimescale due to the low heat leakage at the boundary. This will make thesolution more complicated, and introducing the slow timescale is not attempted inthe present work.

Solution of the phase equation

The phase equation (2.35) can also be solved using a regular perturbationexpansion. If we take

6{x, t) = eo(x, t) + a0,(jc, t) + O(a2) (3.48)

and substitute into (2.35), we obtain

at 0(1): en + c,d(ix = O, (3.49)

at O(a): dll+clelx-clc26tlx = 0. (3.50)

The O(l) equation (3.49) can be solved to obtain

on using the boundary condition (2.52). Substitution of this result into (3.50) gives

eu + ci0lx = -c2, (3.52)

which has the solution, after applying the boundary condition 6, = f, at x = 0,

(3.53)for Ti = 0.

A. H. PINCOMBE AND N. F. SMYTH 159

It can be seen from solutions (3.51) and (3.53) that, to order O(a), the phase 8has an explicit dependence on the loss factor e" only through the phase £ of thetransmission ratio. However, it is also apparent that 6, is not constant. In thewell-known case where the loss factor, the permittivity, and the permeability areall constant. Maxwell's equations (2.3) and (2.4) transform to the equations oftelegraphy, and these are normally solved by assuming that the frequency of theradiation will remain constant. This assumption can be justified by a strongphysical argument, for example on a conservation of waves basis. In the moregeneral case which we consider here, where the wavespeed varies with thetemperature, the local frequency will not be constant, but will vary from theincident frequency because the wavespeed through the material is changing withtime.

4. Perturbation solution for e"~O(a)

In this section, we shall develop a perturbation solution for the scaling e" ~ O(a).This scaling is more restrictive than that of the previous section, a « e"« 1, butnevertheless, given the huge variation in the behaviour of dielectric propertieswith temperature (see von Hippel, 1954), examples of materials for which thescaling e"~ O(a) holds can be found. Such an example is given in Section 7. Thecompound AlSiMagA-196 has e" = 0-011 + cr7"''825 with a =0-014 (von Hippel,1954; pp. 377-8).

When e" ~ 0(a) , we can set a, = aa,, in (2.42) and write the transport equation(3.1) as

4>, + c4>x = -a(ka,, + \hx)4> + O(a2). (4.1)

We can solve this equation by using the same method as we applied in Section 3.We thus let (f>=fe~l< and we can then show, as before, that f = Ru +a/?i(7"(0, r)) + O(a2), and that, in this case, we have

g, + cg.r = a ( k , + ^ . ) + O(a2). (4-2)

Here we use a multiple-scales perturbation expansion to solve for g:

g(x, t) = g,,(jr. X. t) + agx(x. X, t) + O(a2), (4.3)

where

X = ax. (4.4)

This multiple-scales expansion is necessary to eliminate secular terms which willoccur in the subsequent analysis.

160 MICROWAVE HEATING OF MATERIALS

On substituting the expansion (4.3) into the equation (4.2), we obtain

at 0(1): gai + clgOx = 0, (4.5)

at O(a): gi,+ C|gu-c,c2g,, r + c,g,,A- = ifl,I + |/i,. (4.6)

The 0(1) equation (4.5) can be solved by the method of characteristics to yield

) , (4.7)

where r is the characteristic variable, that is, r can be considered constant along acharacteristic. The characteristic variable r is also the value of the time when thereferenced part of the wave was at the boundary x = 0. At the boundary x = 0, wehave g,, = d(r, 0), but we know that the amplitude of the wave at x = 0 has thevalue R(T). Thus it follows that d{t, 0) = 0 for all values of t, so that

(4.8)

with d(0) = 0. Hence (4.6) simplifies to

. + ^ i . (4.9)

where the prime denotes differentiation with respect to X. The constant term \an

on the right-hand side of this equation, giving the decay of the wave, should be apart of the first-order solution for g, so we set

il'^hajc,. (4.10)

Equation (4.10), when combined with the boundary condition d(0) = 0, has thesolution

(4.11)

Equation (4.9) for g, then reduces to

gu + c,g,.r = i/i,. (4.12)

At this stage we can proceed to an O(a) solution in exactly the same way as inSection 3. We use the expansion (3.24) for the temperature and the subsequentequations for 7J, (3.25) and 7", (3.26), where Tit depends only on g0 which is givenby (4.11) and g, depends only on 7̂ , and g0, while 7, depends on g(l, T{), g,, and theform of the transmission ratio R(T). In this case, the solution of (3.25) is

T^y.Rlre-^'^+T,. (4.13)

In Section 3 we treated the two cases 7] = 0 and 7 ^ 0. Here we deal with only thesimpler case, T, = 0, and demonstrate that the solution is completely analogous tothat in Section 3, thus showing that the solution obtained in Section 3 is correctregardless of the size of a, in (2.42).

A. H. PINCOMBE AND N. F. SMYTH 161

By substituting (4.13) into expression (3.2) for /?,, we obtain

+ — r?(/?oT)F'"'^<2)e"(fl"fvV/r') + ajiyyRlT)"^-^^1^. (4.14)

We can then integrate (4.12) to yield

g](x, t) = -^-yt'(Rlrr'-1Rl(l - e-2 a M

2a,, e

(4.15)

If we consider the equivalent case from Section 3, set X = ax and a, = aan, anddrop the terms which are relatively of O(a), then solution (3.20) for g() transformsto solution (4.11), and solutions (3.36)-(3.38) transform to solution (4.15), whichshows that the solutions of Section 3 are valid regardless of the size of at.

5. Some exact solutions for power laws of the form p =/?,(l + p2TY}

In Sections 3 and 4 we have developed solutions of the coupled transportequation (2.36) and the temperature equation (2.41) for the case when theelectromagnetic properties (i.e. the loss factor e", the permittivity e, and thepermeability /x) and the heating rate y are all described by power laws of theform (2.42)-(2.44). In a previous paper (Pincombe & Smyth, 1991), power laws ofthe form

were adopted, and approximate solutions were developed for some special cases.In this section, we again use power laws of the form (5.1) and develop somefurther solutions.

When the electrical permittivity e and the magnetic permeability /i are bothconstant, the phasespeed c will also be constant. In this case, the transportequation (2.36) reduces to

4>, + c<t>x = \e"4>, (5.2)

which is analogous to the transport equation, for constant wavespeed c, derivedby Pincombe & Smyth (1991) on the assumption of low conductivity and forproperties which varied on a slow spacescale and timescale (Pincombe & Smyth,1991; eqn (3.8)),

<t>v + c<t>x = 2-e'<t>, (5.3)

where TJ = at, X = ax, and e' = ae". Note that the transmission ratio R is

162 MICROWAVE HEATING OF MATERIALS

constant in this case and can be factored out of the equations as part of thenondimensionalizing process. The temperature equation (2.41) is

T, = y(T)4>\ (5.4)

where, in this section, the heat coefficient y and the loss factor e" are given by

y = 7i(l + y2T)y> and e" = a ,( l + a2T)"\ (5.5a, b)

By making the transformation

0 = (l + y27)'->\ (5.6)

equation (5.4) can be reduced to

y.O^2, (5.7)

for which the transformed heat coefficient has a constant value. For y^< 1, G willgrow with time and the associated temperature will also grow with time, but willalways be finite for finite time. However, for y^>\, 9 will decrease withincreasing heating time and will reach zero in a finite time, with the associatedtemperature approaching infinity as 6 approaches zero. This blow-up of thetemperature in a finite time is associated with the formation of a hotspot (see Hill& Smyth, 1990; Brodwin et al.% 1993; Coleman, 1991; Hill & Pincombe, 1992).

In general, the heating rate y is proportional to the loss factor e", as shown in(2.46). Thus, by combining (2.46) with the power laws (5.5), we expect that

y2 = a2. yj = fl3. (5.8)

and so the expression (5.5b) for the loss factor e" transforms to

£" = a,0w<1-)"). (5.9)

We can combine equations (5.2) and (5.7) to obtain the single equation

0,, + c0tt = -e"0 , (5.10)

On substituting for e" from (5.9), we finally obtain

, (5.11)

where

« = y3 / ( i -y3 ) - (5-12)

Equation (5.11) can be integrated once to yield the first-order hyperbolicequation

^ . (5.13)

A. H. PINCOMBE AND N. F. SMYTH 163

Ahead of the wavefront, 6 = 1 (as T = 0 there by scaling) and 9, = 9X = 0, giving

+ l). (5.14)

In order to maintain the continuity of 0, and hence the temperature T, at thewavefront, we require (5.14) to be valid at the wavefront, giving

In characteristic form, (5.15) becomes

(5.15)

(5.16)dl n + 1 c

From (5.7), the boundary condition is 9 = 1 + -y, y2(l - 73)' at x = 0 since <f> = 1 atJC = O.

Equation (5.16) cannot be solved in general, but exact solutions can beobtained for particular values of n. Solutions are given below for the casesy3 = 0 (n = 0), 73 = kin = 1), y3 = 2 (n = -2) , and y3 = \{n = -3).

Case 1: n = 0. In this case, the heating rate y and the loss factor e" are bothconstant (temperature-independent) and the solution of (5.16) is

ifO*x*cr.l if JC > cr.

where, as before, r = t - x/c is the characteristic variable.

Case 2: n = 1. Here the solution is

\fx>ct, ( 5 1 8 )

where

0C = coth"1 (1 + ^7i72r), r = t-x/c.

Case 3: n = -2. Here 73 = 2 and thus we have a case in which the temperaturecan become infinite in finite time. The solution is given implicity by

(O«jr«cf),

0 = 1 (JC > c/),

where as before r = t-x/c. It can be seen that 0 vanishing, so that T becomesinfinite and a hotspot forms, first occurs at x =0 at the time t = (7i72)"'-

Case 4: n = - 3 . In this case, 73 = £ and 0 ^ 1 , so the appropriate solution is

164 MICROWAVE HEATING OF MATERIALS

given implicitly by

0=1 (x > a),

where as before x = t — x/c. In this case, 9 becomes zero at x = 0, and a hotspotforms there, at time t = 2(yly2)'

]-

6. Numerical solution of the coupled wave and heat equations

In Sections 3 and 4, we developed approximate analytical solutions for the electricfield and the temperature, for the case where the electromagnetic properties varyslowly with temperature. It was shown in Section 3 that the solution for theelectric field can be transformed so that it has a similar form to previously knownsolutions derived by Pincombe & Smyth (1991), which indicates that the solutionsare qualitatively correct. However, the most critical test of the validity, bothqualitatively and quantitatively, of the analytical solutions is obtained bycomparing them with numerical solutions of the damped wave equation (2.5) andthe forced heat equation (1.1), subject to the initial and boundary conditions(2.37), (2.49), and (2.50). The numerical scheme for the coupled damped waveequation and forced heat equation is summarized here and, for full details, thereader is referred to Pincombe & Smyth (1991).

After applying the nondimensionalizing transformations (2.16) to the dampedwave equation (2.5), we obtain

c2E^ = AE + BE, + E,,, (6.1)where

fj.s /x e e fi

B = ^ 3 + £"-^, (6.3)H e /I

and where c, /A, and e are in dimensionless form.When the wavespeed c is either a constant or a decreasing function of the

temperature, equation (6.1) can be solved numerically by using a centreddifference scheme (see Burden et al., 1981), which results in the recurrencerelation

E^=—pEk+ii+ \+P

VEk'+TTpEk-ui~rrpEk'-h (64)

where

A = c Ar/Ax, (6.5)

p = {BM, (6.6)

V = A(At)2, (6.7)

Ekj = E(k AJC, j At), and A and B are evaluated at (xk, tt). Here k ^n,j « m, and

A. H. P1NCOMBE AND N. F. SMYTH 165

At and Ax are the timestep and the spacestep, respectively. The boundaryconditions (2.28) become

£(), = R(j Ar){cos [-j A/ + f(/ At)] + i sin [-/ At + f (; At)]} (6.8)

and

EnJ = 0, (6-9)

where the boundary x = n Ax of the domain of numerical integration is chosen tobe large enough so that E is negligible there, and thus En] = 0 for ;' = 1,..., m.Consideration of computer storage and run time limitations places significantrestrictions on the number of timesteps m which we can use. Initially, the materialis in thermal equilibrium with no microwave radiation present, so that

£*.<• = 0, (6.10)

£*.-.= 0, (6.11)

where we have assumed that E = 0 for t < 0.The numerical scheme defined by (6.2)—(6.11) has second-order accuracy, with

error of order O(At2, Ax2); however, we need to consider its stability. The schemeused here is almost identical to that of Pincombe & Smyth (1991), which wasfound to be stable as long as A =£ 1. If we define the error in our scheme by

A£A, = £(x*,/,)-£*,-, (6.12)

then the difference equation (6.4) holds for AEkj and we can represent the errorby the sum of Fourier components of the form g' exp (i/3& Ax) where /3 is real.Substitution of this form into (6.4) gives a quadratic in g with the solutions

r±{r2-\+P2f-

8= TT (6-13)

1 + p

where

r = 1 - {-q - 2A2 sin21/3 Ax. (6.14)Note that 17 ~ O(At2) and, for At« 1 and At2« Ax, the contribution from the £17term can be ignored, and the stability condition is thus A *£ 1.

When the wavespeed c is an increasing function of temperature, the timestep Atmust be adjusted so that the scheme remains stable. Pincombe & Smyth (1991)derived a four-point scheme, which gave second-order accuracy for variablewavespeed, in order to integrate equation (6.1) with A = 0. In this work, thecoefficient A is nonzero, but the four-point scheme still gives second-orderaccuracy. The recurrence relation is

+Pl)EkJ+l = A,(£ f t + U + £*-,.y) + [1 + <p + X

-2, (6.15)

166 MICROWAVE HEATING OF MATERIALS

where

( 6 1 6 )

c2

A.=

, \( Ar, \2 /Af.2_, —<- + <p + 1

2bx2

(6.18)

(6.19)

(6.20)

/;-i LVA^-i/ J

f y _ ,

A A,,-,/

To this stage, we have treated the coefficients A and B in (6.1) as functionswhich have a known value at the point (xk, tt). However, we need to know thevalues of ^,, ftx, and e, before we can evaluate B, and we need /A,, /xr/, e,, £,„ ande,"before we can evaluate A. Since /A, e, and e" are functions of the temperature7\ evaluation of A and 5 requires the numerical estimation of 7,, Tr, 7

1,.,, and 7,,.The numerical schemes (6.4) and (6.15) result in fluctuations and numerical

dispersion at the wavefront due to the large ^-derivatives there. Pincombe &Smyth (1991) used a moving boundary located at the wavefront to overcome thisand calculated the amplitude of the electric field at the wavefront by using awavefront expansion of the form

E = ap^s), (6.24)

where s = x - c,/ is the distance behind the wavefront, to first order, and a is the

L^

A. H. PINCOMBE AND N. F. SMYTH 167

amplitude of the electric field at the wavefront. We take the same approach here,and find that the amplitude a at the wavefront is given by

[ l + 2 ( f c / f l l ) ( l - e ( " " ) ) ] - ( 6 2 5 )

where

(6.26)

with /i, = fi(T,) and e, = e(7]), in the case of zero diffusivity v = 0. We shall see, inSection 7, that the value of the amplitude a of the electric field at the wavefrontgiven by (6.25) enables us to obtain a smooth result from the numericalintegration of the damped wave equation (6.1) near the moving boundary.

The heat equation (2.2) can be integrated using the Crank-Nicolson scheme(see Burden et al., 1981), except for the fact that the nonlinear source term needsto be evaluated at the (y + l)th timestep. This problem has been satisfactorilydealt with by Pincombe & Smyth (1991), who evaluated the source term at the(; + l)th timestep using linear interpolation, which allowed them to numericallysolve the equation using the Crank-Nicolson scheme. We adopt the same schemehere, which results in the recurrence relation

kj+} \E\2k,J+i

= £Tk + tJ + (\-2OTkJ+£Tk-jj + toyk./\E\lJ, (6.27)

where

C = v Af/A.v2. (6.28)

7. Discussion

In Sections 3 and 4, we derived solutions of the transport equation (3.1). Thesesolutions do not depend on specific values of the parameters, so that in this sensethey are general solutions. However, they do rely on the assumption that theelectromagnetic properties of the material being heated can be adequatelyrepresented by the power laws (2.42)-(2.44) and (2.47). It might well be assumedthat a model of the form

P=Pl+p2T"\ (7.1)

with pi, p2, Py being free parameters and with a free choice of a zero and a scalefor the temperature 7\ will be capable of fitting a wide range of curves for thetemperature dependence of, for example, the real component of the permittivitye' and the loss factor e". This can be easily verified by consulting the tables in vonHippel (1956). The graphs published by von Hippel were not specificallydeveloped for microwave heating, but do contain examples with frequency oforder 0(10'" Hz), which is in the microwave range. Some simple curve fittingshows that a model of the form (7.1), where the zero of temperature can bechosen to help the fit, can represent any curve for which d2p/dT2 does not changesign in the temperature range of interest. The simplest of the solutions derived in

168 MICROWAVE HEATING OF MATERIALS

the previous sections was obtained for the case in which the initial temperature 7jwas zero. To check the applicability of this assumption that we can take T, = 0, wechose some typical workplace temperatures (e.g. 10°C, 25°C) as the zeros of ourscaled temperature and, after some simple curve fitting, find that, using the model(7.1), we can obtain good representations of curves as long as (a) p(T) ismonotonic increasing and (b) d2p/dT2>0.

We can also use the model (7.1) when p(T) is monotonic decreasing andd2p/dT2<0, but this is of little practical interest. As an example, the compoundAlSiMagA-196 (von Hippel, 1954: pp. 377-8) can be represented by

e' = 5-3, (7.2)

e" = 0-011 + 1-4 X 10"27IK2S, (7.3)

where the temperature T is obtained from the Celsius temperature T* by

7 = (7*-25)/500, (7.4)

that is, our scaled temperature is zero at a normal room temperature of 25°C andeach unit of temperature represents 500°C. In terms of (2.42), equation (7.3) gives

a, =0011, a = 0-01, a2=l-4, a-, = 1-825. (7.5)

Thus the evidence shows that the power laws (2.42)-(2.44) and (2.47) arepractical models for the representation of the variation of the electromagneticproperties with temperature, even in the case where the initial temperature isconstrained to be (scaled) zero. However, we still need to check the accuracy ofthe perturbation solutions. We do this by comparing them with the equivalentnumerical solutions generated using the schemes outlined in Section 6.

Because the analytical solutions were derived under the assumption that thetemperature diffusivity v was zero, v will be set to zero for the numericalsolutions unless otherwise stated. In the numerical scheme defined in Section 6,we set the far boundary at a value of the space variable x which will not bereached during the simulation. Use of a semi-infinite block simplifies theanalytical solution, but creates some difficulties numerically. Basically, withlimited computer memory available, we want the simulation to be completedbefore the initial wavefront reaches the limit of storage for the space variable x.This can be arranged by making the coefficient of heat absorption y sufficientlylarge. It turns out, after some numerical trials, that y = e" is large enough. In thedefinition (2.48) of the parameters y,, y2, y> the constant k is very small(k~O(\0']2): Metaxas & Meredith, 1983), so that our choice of y = e" isequivalent to requiring that the amplitude £„ of the incident electric field is verylarge (~O(106) volts/metre), which is equivalent to a power of the order ofO(109) watts. Note that we do not claim that the perturbation model is applicableat such high field strengths, merely that the numerical solution at the high fieldstrength is a valid scaled-up version of what happens at lower field strengths.

We fist consider the case where the electrical permittivity e and the magneticpermeability /x are both constant, but the loss factor e" and the heating coefficienty vary with temperature. In this case, the transmission ratio R(r) is constant and

A. H. PINCOMBE AND N. F. SMYTH 169

can thus be incorporated in the y term. The following temperature variations areused

e" = 01 + a 7 K \ y = 0 1 + a 7 ' \ a =001 , e = ix = l-0. (7.6)

The results, after 600 timesteps (/ = 30), are shown in Fig. 1, with A representingthe numerical solution, B representing the constant-coefficients solution, and Crepresenting the perturbation solution. The constant-coefficients solution refers tothe perturbation solution evaluated for a2 = e2 = M2 = 0 or, equivalently, a = 0.There are some fluctuations in the numerical solution for amplitude function <p(Fig. l(a)) which arise just behind the wavefront and rapidly attenuate. Thesemay be partly caused by numerical errors. The methods outlined in Section 6have been used to minimize these errors, but it has not been possible to eliminatethem completely. Furthermore, the finite size of e" leads to oscillations in theelectric field near the wavefront. This was found by Marchant & Smyth (1992),who solved (2.5) for constant /A, e, and e" using Laplace transforms. Theamplitude of these oscillations was found to go to zero as e" goes to zero. It canbe seen that the perturbation solution is very close to the numerical solution bothfor the amplitude function 4> (Fig. l(a)) and for the temperature T (Fig. l(b)).The greatest error comes, as expected, where T is greatest, at the boundary x = 0.At this time, at x = 0, the perturbation solution predicts a temperature T = 3-65.If we apply the conditions for uniform validity (3.40), we note that a(3-65)' 5 <jt 1,so that we have gone beyond the region of uniform validity. However, theperturbation solution still predicts the temperature at x = 0 with an error of only2-7%, compared with 20% for the constant-coefficients solution. If the perturba-tion solution had been plotted for a time within the range of uniform validity,there would have been no discernible difference from the numerical solution. Theperturbation solution is then accurate even outside its strict range of uniformvalidity. When the same parameter values are used, but the numerical solution isrun for 800 timesteps (t = 40), the perturbation solution is still close to thenumerical solution both for the amplitude function </> (Fig. 2(a)) and for thetemperature T (Fig. 2(b)). In this case, the temperature at .v =0 is T = 5-3 and,since a(5-3)' 's>0-l, we are obviously well outside the region of uniform validity.Yet, at x = 0, where the error is greatest, the error in T is still only 5-1% for theperturbation solution, against 31% for the constant-coefficients solution.

We next consider the case where the loss factor e" and the heating coefficient yare both constant, but the permittivity e and the permeability /x both vary withtemperature. We take

e " = y = 0 l , e = n = \-0 + aT2'\ a = 0 0 1 . (7.7)

There is not a great deal of difference between the three solutions in this case (seeFig. 3), but there is no doubt that the perturbation solution (C) is much closerthan the constant-coefficients solution (B) to the numerical solution (A).

In order to test the solution (A.7) for the case when 7 ^ 0 , we need toseparate the space into two regions: 0 « x < xt and x,«x < *, whereyi/?orexp(-a1jt/c ])> T, in the first region and yl/?flrexp(-fl|.r/cl)=£T, inthe second. A comparison of (A.9)-(A.ll) and (A.12)-(A.14) shows that the

170 MICROWAVE HEATING OF MATERIALS

—|—i—|—i—[—.—|—i—|—i—|—I—|—i—| I—|—1—I—i—r

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

XFIG. 1. Electric field anipniude and temperature for the case a =0-01. e = / i = l-0. o, = y, =0-1,t>2 - 7i - ' 0̂ °;i = 7j = 1 i .ii / = 30. A is the numerical solution, B is the analytical solution to firstorder, and C is the analytical solution to O(a).

A. H. PINCOMBE AND N. F. SMYTH 171

two solution sets are identical when To = Tr The transition point *,, where wechange from one solution form to the other, is the solution of

y,(r-x t/c1)e- ( '" j r i /C l ) = 7;. (7.8)

When the wavespeed c, given by (3.3), is not constant, the evaluation of theelectric field strength at any point is more complicated than it is for constantwavespeed. The amplitude <f> of the electric field, given by (3.7) coupled with(3.11), (3.14), (3.15), (3.17), (3.20), and (3.36)-(3.38), is affected both by theabsorption of energy and by variations in the wavespeed. When the wavespeed cis an increasing function of temperature, it will decrease with an increase of thespace variable x, and the terms containing /j,2, ^3, e2, £3 in (3.37) and (3.38) willcontribute to an increase in the local value of the electric field strength, thiscontribution being caused by compression of the waveform. In the opposite case,when the wavespeed is a decreasing function of the temperature, there is astretching of the waveform and a consequent decrease in the local value of theamplitude of the electric field.

Confirmation of the amplification/reduction effect can be obtained from aphysical argument. If n(x, t) is the photon density, i.e. the number of photons perunit length at (x, t), then we can write the continuity equation for photons in theform

n, + cnx = -(cx + p)n, (7.9)

where /3 is the proportion of photons that are absorbed by the material in unittime and c is the phasespeed, as before. The continuity equation (7.9) can beobtained in the usual way by considering the population of photons in a region ofinfinitesimal width or it can be obtained by considering how the difference in thewavespeed at different points causes the waves to bunch up (cr < 0) or to stretchout (ct > 0). If the energy density is given by

l = hwn, (7.10)

where h is Planck's constant divided by 2n, then it is a simple matter to show that

/, + c/r = ha)(n, + cnx) + hn (to, + cw,). (7.11)

Using the eikonal equation (2.35), we can show that

co, + ccox = cocjc. (712)Thus (7.11) gives

/, + c/, = -(c, + /3-c,/c)/. (7.13)

The energy density / for an elecromagnetic wave is proportional to e<f)2, so that(7.13) leads to

(* * ^ \ (7.14)

where Q is the dimensionless coefficient defined as Q' in (2.31), and we havetaken /3 = e". By rearranging (7.14), we obtain (2.36) after noting that

^ + ^ = - 2 - (7.15)e fj. c

172 MICROWAVE HEATING OF MATERIALS

8 10 12 14 16 18 20 22 24 26 28 30

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

X

FIG. 2. Electric field amplitude and temperature for the case a = 001, e = /* = l-0, a, = y, =0-1,a2=y2= 1-0, a, = r j= 1-5 at r = 40. A is the numerical solution, B is the analytical solution to firstorder, and C is the analytical solution to O(a).

A. H. PINCOMBE AND N. F. SMYTH 173

0.04 6 8 10 12 14 16 18 20 22 24 26 28 30

X

FIG. 3. Electric ff£1d amplitude and temperature for the case a = 0 0 1 , e " = y = 0 1 , £ , = / * , = 1 0 ,£2 = / i 2 = l-0, e 1 = / i j = 2-l at ( = 30. A is the numerical solution, B is the analytical solution to firstorder, and C is the analytical solution to O(a).

174

and

MICROWAVE HEATING OF MATERIALS

-c — = 2cx+c —. (7.16)

Appendix

In this appendix, the solution for the function g, of Section 3 willbe found by using Taylor expansions in the equation (3.35) for g,. Whenyi/??,rexp(-a,x/ci)> 7] we can use the expansion

truncating as soon as

p(p-l)(p-2)-(p-nn\

\a,

, (A.I)

(A.2)

since g, is the O(a) term in the solution for g, while for T,> y,Rlrexp (-fl|jc/c,)we use

"x/c')y = T?

truncating as soon as

p(p-l)-(p-nn\

•• a .

, (A.3)

(A.4)

Thus, when •y,/?f)Texp(-a,x/C|)> 7], we have

q(T, T dp =axp

• ( 1 - .

(A.5)

where (A.5) is truncated in accordance with the criterion (A.2).Similarly, when 7j> y,/?oTexp(-al.t/ci), we have

r dp =

4a,rf-2(y,/?2T)2(l - (A.6)

A. H. PINCOMBE AND N. F. SMYTH 175

where (A.6) is truncated in accordance with (A.4).We thus have from (3.35) two approximate solutions for g,, namely

g, = Z+Wt + W2 + W3. (A.7)

where

+ g 2 ( . 2 ~ f l ' T ) [(7; + y,/?2,rp - (7; + y, r^e"'"^'')' '] (A.8)

and, when y\Rix exp (-a,;t/C|)> 7̂ ,

W, — ( y . ^ H l -e-<""-«"'>)

(y,^Tr- ' ( l - e-""'"'-'^'») + - , (A.9)

W2 = - ^ _4/x,c,

•), (A.ll)/

while, when y^Rirexp (-a]x/cl)< T

4a,

W2 = f 7T'r + — rr'"'r,/?f,r(l - e-""-""')\ a,

3 " 1 } 7T-2(y,/??,r)2(l -e- ( 2"" f c ' » )+ •••),4a, /

4e,c, V a,

4fl,

176 MICROWAVE HEATING OF MATERIALS

The first of these expressions for Wu W2, and W3 is useful for large r and thesecond for small r.

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