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Available at: http://www.ictp.it/~pub_off IC/2005/104

United Nations Educational Scientific and Cultural Organization

and

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THERORETICAL PHYSICS

SOLVING MICROWAVE HEATING MODEL USING HERMITE-PADAPPROXIMATION TECHNIQUE

O.D. Makinde1

Applied Mathematics Department, University of Limpopo,

Private Bag X1106, Sovenga 0727, South Africa

and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

We employ the Hermite-Pad approximation method to explicitly construct the

approximate solution of steady state reaction-diffusion equations with source term that

arises in modeling microwave heating in an infinite slab with isothermal walls. In

particular, we consider the case where the source term decreases spatially and increases

with temperature. The important properties of the temperature fields including bifurcations

and thermal criticality are discussed.

MIRAMARE TRIESTE

November 2005

1Group Junior Associate of ICTP. makindeo@ul.ac.za

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1. Introduction

Microwaves are a form of electromagnetic radiation; that is, they are waves of electrical and

magnetic energy moving together through space. Electromagnetic radiation ranges from the

energetic x-rays to the less energetic radio frequency waves used in broadcasting.

Microwaves fall into the radio frequency band of electromagnetic radiation. Microwaves

should not be confused with x-rays, which are more powerful. Microwaves have three

characteristics that allow them to be used in cooking: they are reflected by metal; they pass

through glass, paper, plastic, and similar materials; and they are absorbed by foods, (Hill

and Marchant, 1996). This technology has found new applications in many industrial

processes, such as those involving melting, smelting, sintering, drying, and joining,

(Kriegsmann, 1992). Heating by microwave radiation constitutes a reaction diffusion

problem with radiative heat source term and the long-time behaviour of the solutions in

space may lead to appearance of hotspots in the system i.e. isolated regions of excessive

heating (Coleman, 1991). In order to predict the occurrence of such phenomena it is

necessary to analyze a simplified mathematical model from which insight might be gleaned

into an inherently complex physical process.

The theory of reaction diffusion equations is quite elaborate and their solution in

rectangular, cylindrical and spherical coordinate remains an extremely important problem

of practical relevance in the engineering sciences. Several numerical approaches have

developed in the last few decades, e.g. finite differences, spectral method, shooting method,

etc. to tackle this problem. More recently, ideas on classical analytical methods have

experienced a revival, in connection with the proposition of novel hybrid numerical-

analytical schemes for nonlinear differential equations. One such trend is related to the

Hermite-Pad approximation approach, (Guttamann 1989; Makinde 1999; Tourigny and

Drazin 2000). This approach, over the last few years, proved itself as a powerful

benchmarking tool and a potential alternative to traditional numerical techniques in various

applications in sciences and engineering. This semi-numerical approach is also extremely

useful in the validation of purely numerical scheme.

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In this paper, we intend to construct an approximate solution for a steady state

reaction diffusion equation that models microwave heating in an infinite slab with

isothermal walls using perturbation technique together with a special type of the Hermite-

Pad approximants. The mathematical formulation of the problem is established and solved

in sections two and three. In section four we introduce and apply some rudiments of

Hermite-Pad approximation technique. Both numerical and graphical results are presented

and discussed quantitatively with respect to various parameters embedded in the system in

section five.

2. Mathematical Formulation

Consider a microwave heating in an infinite slab with isothermal walls as shown in figure

(1). It is assumed that energy dissipation has a negligible effect on the electromagnetic field

and so the temperature distribution can be investigated in isolation.

T=T0 y = a

y

Microwave heating x

T=T0 y= -a

Figure 1: Geometry of the problem

For the simplest microwave heating model, the steady state nonlinear reaction diffusion

equation with source term that describes the thermal behaviour can be written in the form

(Hill and Pincombe, 1991);

02

2

=+ nyTEedy

Td , (1)

with the following boundary conditions:

0)(,0)0( TaT

dy

dT== , (2)

where Tis the temperature, T0 the wall temperature, n the thermal absorptivity index, the

constant thermal conductivity of the material, E the amplitude of the incident radiation,

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electric field amplitude decay rate, (x, y) distances measured in the axial and normal

direction respectively and a the slab half width. Furthermore, it is important to note that the

value of n depends upon; the material under consideration, the temperature range under

investigation and the frequency of the microwave radiation. At higher frequencies (say 1010

Hz), materials such as fused glass are very accurately approximated by linear and quadratic

power (n=1, 2). At lower frequencies, higher values of thermal absorptivity index n > 2 will

be required (Hill and Marchant, 1996).

Introducing the following dimensionless variables into equations (1) and (2):

0

1

0

2

,,,T

TT

a

yyak

TEa n

====

, (3)

we obtain the dimensionless governing equation together with the corresponding boundary

conditions as (neglecting the bar symbol for clarity);

02

2

=+ nkyTedy

Td , (4)

with

1)1(,0)0( == Tdy

dT, (5)

where and k represents the thermal absorptivity and electric field decay rate parameters

respectively.

3. Method of Solution

For n =1, equation (4) becomes a linear boundary value problem and can easily be solved.

The exact solution is

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,

2222

22

2222

22

)(

2

01

2

01

01

2

01

2

01

01

+

+

+

=

k

eJ

kY

k

eY

kJ

ekJ

kY

k

e

JkYk

e

YkJ

ekY

kJ

yT

kk

ky

kk

ky

(6)

where J0, J1 are Bessel functions of first kind and Y0, Y1 are Bessel functions of second

kind.

For n > 1, the problem becomes nonlinear and it is convenient to take a power series

expansion in the thermal absorptivity parameter , i.e.,

=

=0

)(i

i

iTyT . (7)

Substituting the solution series (7) into equations (4)-(5) and collecting the coefficients of

like powers of , we obtained and solved the equations governing the coefficients of

solution series iteratively. The solution for the temperature and velocity fields are given as

)()484846

34644(4

)(1)(

3)12()1()1()1(2

2

4

2

2

Oekyeeekeekke

ekkyeyekkyk

neeekky

kyT

ykykykkyykkk

kkkk

kky

+++++

++=

(8)

Using the computer symbolic algebra package (MAPLE), we obtained the first 22 terms of

the above solution series (8) as well as the series for the material maximum temperature

)1,,;0(max >== nyTT .

4. Thermal Criticality and Bifurcation Study

The concept of criticality or non-existence of steady state solution of nonlinear reaction

diffusion problems for certain parameter values is extremely important from an application

point of view. This characterizes the thermal stability properties of the materials under

consideration and the onset of thermal runaway phenomenon, Makinde (2004). In order to

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determine the possibility of thermal criticality in the system together with the evolution of

temperature field as the thermal absorptivity parameter increases in value, we employ a

special type of Hermite-Pad approximation technique. The procedure is briefly illustrated

as follows;

Suppose the partial sum

)()()(1

0

1

NN

i

i

iN OUaU +==

= as 0, (9)

is given. We shall make the simplest hypothesis in the contest of nonlinear problems by

assuming the U() is the local representation of an algebraic function of . Therefore, we

seek an expression of the form)3(

3

)2(

2

)1(

101 )()()()(),( UAUAUAAUFd

N

d

N

d

NNNd +++= , (10)

such that

A0N()=1, +

=

=id

j

j

ijiNbA

1

1)( , (11)

and

)(),( 1+= Nd OUF as 0, (12)

where 1d , i =1, 2, 3. The condition (11) normalizes the Fd and ensures that the order of

seriesAiN increases as i and dincrease in value. There are 3(2+d) undetermined coefficients

bij in expression (11). The requirement (12) reduces the problem to a system ofN linear

equations for the unknown coefficients of Fd. Therefore for consistency of the linear

system, we shall take

N=3(2+d). (13)

Equation (12) is a new special type of Hermite-Pad approximants. Both the algebraic and

differential approximants form of equation (12) are considered. For instance, we let

U(1)

=U, U(2)

=U2, U

(3)=U

3, (14)

and obtain a cubic Pad approximant. This enables us to obtain solution branches of the

underlying problem in addition to the one represented by the original series. In the same

manner, we let

U(1)=U, U

(2)=DU, U(3)=D

2U, (15)

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in equation (11), where D is the differential operator given by D=d/d. This leads to a

second order differential approximants. It is an extension of the integral approximants idea

by Hunter and Baker (1979) and enables us to obtain the dominant singularity in the

temperature field i.e. by equating the coefficient A3N() in equation (12) to zero. Hence,

some of the zeroes of )(3d

NA may provide approximations of the singularities of the series U

and we expect that the accuracy of the singularities will ensure the accuracy of the

approximants. Furthermore, it is well known that the dominant behaviour of a solution of

differential equations can often be written as Guttamann (1989),

2,...1,0,for

2,...1,0,for

ln)(

)()(

=

m

m

K

KU

c

m

c

m

c

as c, (16)

where K is some constant and c is the critical point with the exponent m. The critical

exponent m can easily be found by using Newtons polygon algorithm. Meanwhile, for

algebraic equations, the only singularities that are structurally stable are simple turning

points. Hence, in practice, one almost invariably obtains mc = 1/2. If we assume a

singularity of algebraic type as in equation (16), then the exponent may be approximated by

)(

)(1

3

2

CNN

CNN

N DA

Am

= . (17)

For details on the above procedure, interested readers can see Vainberg and Trenogin

(1974), Common (1982), Sergeyev, and Goodson (1998), Makinde (1999), Tourigny and

Drazin (2000), Makinde (2005), etc.

5. Results and Discussion

The bifurcation procedure above is applied on the first 22 terms of the solution series and

we obtained the results as shown in tables (1) and (2) below:

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Table 1: Computations for dominant singularity showingthe procedure rapid convergence for k= 1, n = 2.

d N c mcN

1 9 0.798406706029926 0.4978996

2 12 0.798419517502792 0.4999999

3 15 0.798419517935786 0.5000000

4 18 0.798419517941164 0.5000000

5 21 0.798419517941164 0.5000000

Table 2: Computations showing thermal criticality for k= 1.

n 2 3 4 5

c 0.7984195179 0.4713264479 0.3349306829 0.2598772782

mcN 0.5000000 0.5000000 0.5000000 0.5000000

Table 3: Computations showing thermal criticality for n = 2.

k 1 2 3 4

c 0.7984195179 1.0087045320 1.2308810985 1.4611429164

mcN 0.5000000 0.5000000 0.5000000 0.5000000

The result in table (1) shows the rapid convergence of our procedure for the dominant singularity

(i.e. c) together with its corresponding critical exponent c with gradual increase in the number of

series coefficients utilized in the approximants. In table (2), we noticed that the magnitude of

thermal criticality decreases with increase in the value of thermal absorptivity index (n) in the

system. This shows clearly that the higher power of thermal absorptivity will enhance the early

appearance of thermal runaway in some materials due to excessive heating. Table (3) shows that

the magnitude of thermal criticality increases with increased values of electric field decay rate

parameter (k). Hence, increasing the decay rate of electric field in microwave heating will delay the

possibility of thermal runaway or ignition in the system. Figure (2) shows a transverse increase in

the material temperature with maximum temperature along the centerline of the slab. Similar

temperature profile is observed for the case of n = 1 as represented by equation (6). However,

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further increase in temperature is noticed by increasing the value of . In figure (3), we observed

that the material temperature increases with higher values of absorptivity index (n) due to increase

in the intensity microwave radiation absorptivity index and decreases with increase in electric field

decay rate (k). This is in agreement with the results of Hill and Marchant (1996). A slice of the

bifurcation diagram for n >1 is shown in figure (4). In particular, for every k > 0, there is a critical

value c (a turning point) such that, for 0

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6.0

Tmax II

3 .0

c

1.0

I

0 0.5 1.0

Figure (4): A slice of approximate bifurcation diagram in the (, T(0; n>1, k)) plane.

6. Conclusion

The steady state reaction-diffusion equation with source term that arises in modeling

microwave heating in an infinite slab with isothermal walls is studied. Exact solution is

obtained for the linear case while the nonlinear problem is solved using the perturbation

technique coupled with a special type of the Hermite-Pad approximants. The possibility of

thermal runaway phenomenon is observed and the corresponding thermal criticality values

are obtained and clearly demonstrated on the bifurcation diagram.

Acknowledgements

This work was done within the framework of the Associateship Scheme of the Abdus

Salam International Centre for Theoretical Physics, Trieste, Italy. Financial support from

the Swedish International Development Cooperation Agency is acknowledged.

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2. Common, A. K., (1982), Applications of Hermite- Pad approximants to waterwaves and the harmonic oscillator on a lattice,J. Phys. A 15, 3665-3677.

3. Guttamann, A. J., (1989), Asymptotic analysis of power series expansions, PhaseTransitions and Critical Phenomena, C. Domb and J. K. Lebowitz, eds. Academic

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4. Hill, J. M., and Pincombe, A. H., (1991), Some similarity temperature profiles forthe mirowave heating of a half-space. J. Aust. Math. Soc. Ser. B, Vol. 33, 290-320.

5. Hill, J. M., and Marchant, T. R., (1996), Modelling microwave heating. Appl. Math.Modelling, Vol. 20 (3), 3-15.

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