Download Effect of radiation losses on hotspot formation and propagation in

IMA Journal of Applied Mathematics (1996) 57, 165-179
Effect of radiation losses on hotspot formation and propagation in
microwave heating
C. GARCIA REIMBERT AND A. A.
MINZONI
Department of Mathematics and Mechanics, IIMAS, National Autonomous
University of Mexico, Apdo. 20-726, Del. Alvaro Obregon, 0100 Mexico DF,
Mexico
N. F. SMYTH
Department of Mathematics and Statistics, The King's Buildings, University of
Edinburgh, Edinburgh, Scotland, EH9 3JZ, UK
[Received 14 July 1995 and in revised form 9 April 1996]
The use of microwaves for the rapid heating of materials has found widespread
industrial use. However, a number of potential problems are inherent in this rapid
heating, including the hotspot phenomenon. A hotspot is a type of thermal
instability which arises because of the nonlinear dependence of the electromagnetic and thermal properties of the material on temperature. The evolution of a
hotspot in a cylindrical material is studied. The propagation of the microwaves is
treated in the Wentzel-Kramers-Brillouin (WKB) limit (geometric optics), and
the thermal behaviour is studied in the limit of small thermal diffusivity. The
resulting temperature distributions are in excellent agreement with full numerical
solutions of the governing equations, even outside the strict range of the
asymptotic validity of the WKB approximation.
1. Introduction
Recently, microwaves have found widespread application in the heating of
materials in industrial processes, ranging from the smelting of metals, food
processing, sintering of ceramics, to the drying of wood and wool (see Metaxas &
Meredith, 1983; Kriegsmann, 1992). The reason for this wide range of industrial
applications is the inherent advantage of microwaves: the rapid rate at which
heating can occur because microwaves propagate through the material so that
heating occurs in the bulk of the material, and not just at the surface as in
conventional heating. However, this rapid rate of heating causes a number of
serious problems. One of these is the so-called hotspot, which has been observed
in a number of industrial applications. A hotspot is a thermal instability due to
the nonlinear dependence on temperature of the thermal and electromagnetic
properties of the material being heated. If the rate at which microwave energy is
absorbed by the material increases faster than linearly with temperature, then
heating does not take place uniformly, and regions of very high temperature can
form, the hotspots (Roussy et ai, 1987; Hill & Smyth, 1990; Coleman, 1991;
Kriegsmann, 1991a,b, 1992; Kriegsmann & Varatharajah, 1993). The temperature
of the hotspots can be high enough to melt the material, which is obviously
undesirable in most applications.
165
O Orford University Preu 1996
166
C GARCIA REIMBERT ET AL.
The microwave heating of a dielectric material is described by a coupled system
consisting of a damped wave equation, which describes the propagation of the
microwave radiation, and a forced heat equation, which describes the resulting
heating. The forcing in the heat equation is proportional to the square of the
amplitude of the electric field of the microwaves (see Pincombe & Smyth, 1991).
Since the electromagnetic and thermal parameters are nonlinearly dependent on
temperature (von Hippel, 1954; Metaxas & Meredith, 1983), this system is highly
nonlinearly coupled and hence difficult to solve analytically. Therefore, all
analytical work to date has used approximations to reduce the coupled system to
a manageable form. These approximations fall into three broad classes.
First, it is assumed that the electric field is known (usually taken to be constant)
or that the electromagnetic properties are constant so that the electrical and
thermal equations decouple and the electric field can be calculated (Roussy et at,
1987; Hill & Smyth, 1990; Coleman, 1991; Kreigsmann et al, 1990; Kreigsmann,
1991a,b, 1992; Booty & Kreigsmann, 1994).
Secondly, the electric field is calculated for temperature-dependent electromagnetic parameters in the Wentzel-Kramers-Brillouin (WKB) limit (geometric
optics) or in the limit of small electrical conductivity (Smyth, 1990; Pincombe &
Smyth, 1991, 1994).
These two approximations reduce the coupled damped wave equation and
forced heat equation to a simpler system which can be solved for certain
parameter dependencies on temperature. The WKB and small electrical conductivity limits are equivalent, and they lead to the same set of equations (see
Pincombe & Smyth, 1994). Both of these two types of approximations show the
formation of a hotspot as a localized thermal instability.
Thirdly, the smallness of the Biot number is used as the basis of the
approximation. Kriegsmann (1991a, b, 1992, 1993, 1995) has developed an
asymptotic approximation as the Biot number tends to zero. In this limit the heat
loss at the boundary prevents the formation of a localized hotspot in the
material. Instead, the temperature settles to a steady-state value given by an
S-shaped-response diagram in which there are three branches. When the input
power of the microwave radiation is greater than a threshold value, the
upper-branch steady state is the only one available. Thus a slight change in the
input power can lead to a large change in the steady-state temperature. The
temperature of the steady state on the upper branch can be large enough to melt
the material. It is this large change in the steady-state temperature which leads to
the observed phenomenon of a hotspot.
The present work considers the evolution and propagation of a hotspot in a
cylindrical rod heated inside a semi-infinite cylindrical waveguide. The input of
energy is from the microwave radiation, and it is assumed that energy can be lost
due to radiation leakage from the boundary of the rod (see Kriegsmann &
Varatharajah, 1993). The bistable nature of the heating allows a hotspot to
propagate as a travelling temperature front. Like Smyth and Pincombe (Smyth,
1990; Pincombe & Smyth, 1994), we treat the microwave radiation in the WKB
HOTSPOT FORMATION AND PROPAGATION IN MICROWAVE HEATING
167
approximation, which is valid in the limit when the ratio of the electrical
conductivity to the microwave frequency is small. The propagation of the
microwave radiation is now modified by the presence of the temperature front. It
will be assumed that the thermal diffusivity is small, so the hotspot is treated as a
moving boundary layer (see Fife, 1988). Since the propagation of microwave
radiation is temperature dependent (see von Hippel, 1954; Metaxas & Meredith,
1983), the equation for the envelope of the electric field is found to be coupled
with the ordinary differential equation (ODE) for the motion of the boundary
layer. This coupling between the electrical and thermal effects then leads to a
closed first-order nonlinear ODE for the motion of the boundary layer.
The thermal boundary condition used in the present work is that of a thermally
insulated boundary at the beginning of the rod at x = 0. Due to this boundary
condition the maximum value of the temperature is at the boundary and so the
hotspot first forms at the boundary. Subsequent to its formation, the hotspot
propagates into the material and stops when the heat input from the microwave
radiation balances the heat output due to radiation loss. Expressions are derived
in closed form for the position of the temperature layer and for the distributions
of temperature and the electric field. These expressions are compared with
numerical solutions of the full initial-boundary-value problems; excellent agreement is found, even outside the strict range of the asymptotic validity of the
WKB approximation.
2. Formulation
Consider the microwave heating of a semi-infinite cylindrical rod placed on the
axis of a cylindrical waveguide. A similar problem was considered by Kiegsmann
& Varatharajah (1993) in which the electric field inside the rod was taken as
given. However, in the present work the electric field inside the rod will be
calculated along with the temperature. To describe the geometry let us take the
axis of the waveguide to be the .x-axis for x s* 0 and introduce polar coordinates
with the radial coordinate measured from the axis of the waveguide. The rod then
lies with its axis along the axis of the cylinder, occupying the region O^r^sa,
x 5= 0, while the wall of the waveguide is r = b.
The equations governing the propagation of the microwave radiation are
Maxwell's equations together with appropriate boundary conditions at the
boundary of the waveguide and at the boundary of the rod. In the present work
the electrical conductivity of the rod is small, so that the electric and magnetic
fields of the microwave radiation decouple (see Pincombe & Smyth, 1991). Since
a rod inside a waveguide is being considered, it is assumed that the electrical
permittivity and magnetic permeability take their free-space values e0 and /AO,
respectively, outside the rod and that they take constant values e and ii inside
the rod, while the electrical conductivity is assumed to be zero outside the rod
and to take some value a inside the rod. These assumptions mean that the
temperature dependencies of the electromagnetic parameters e and ft are being
neglected. In the case of small conductivity, it is assumed that the temperature
derivatives of e and n are smaller than a/ew, where w is the frequency of the
168
C GARCIA REIMBERT ET AL.
radiation. Physically, this amounts to neglecting the effect of temperature on the
speed of the waves, but retaining its effect on the attenuation of the waves. Also,
since it is assumed that the electrical conductivity is small, the amount of energy
fed into the material is also small and the results of the present work will be
applicable to relatively weak hotspots. With these assumptions on the electrical
and magnetic parameters, Maxwell's equations reduce to
p.,,
for x s= 0 and 0 =£ r =s b (as in Pincombe & Smyth, 1991). In these equations the
superscript 0 denotes the electric and magnetic fields E and H inside the rod (that
is, for 0 « r « a ) , and the superscript 1 denotes the fields outside the rod (that is,
for a^r^b).
The transverse Laplacian is denoted by VT-. The speed of light c
takes the free-space value (e0fi0)"i outside the rod and the value (efi)~^ inside
the rod.
The initial conditions are that the electric and magnetic fields are switched on
at t = 0. The boundary conditions are the usual continuity equations for the
tangential components of the fields fi01 and H0'1 and for the normal components
of the displacements D 0 1 and B°A at the boundary of the rod r = a. The wall of
the waveguide at r = b is taken to be a perfect conductor. At the end of the
waveguide at x = 0 we prescribe the tangential components E%] and H%x in the
form U^1 = £^J. and H^1 =//?«, corresponding to the microwave radiation input
into the cavity. Hence the initial and boundary conditions are
HT\0,y,
z, t) = H?£(y, z, t),
(£°-£')xn=0
(D°-D])-n
Hol=H1}i=0
l
and (H°-H )*n
at r = (
= 0 onr = a,
•
(2.3)
=0 and ( B ° - B 1 ) - n = 0 on
B1 • n = 0 and £ ' x « = 0 at r = b,
where n is the outward normal from a surface.
As found by Pincombe & Smyth (1991), by Kriegsmann & Varatharajah (1993),
and by Booty & Kriegsmann (1994) the equation for the temperature field is
dT
vvr
y(T)P (x^O.O^r^a),
dt
(24)
where P is the average of \E\2 over a microwave period, with the boundary and
initial conditions
dT
— =-L(T)
at r = a, x > 0,
dn
8T
— = 0 at.x = 0,
aX
T(x, y, z, 0) = 0.
HOTSPOT FORMATION AND PROPAGATION IN MICROWAVE HEATING
169
Here T is the temperature of the rod, v is the thermal diffusivity, y is the rate at
which the material takes up heat energy from the microwave radiation, and n is
the outward normal from the surface of the rod. The nonlinear function of
temperature L(T) represents thermal losses via radiation and convection; it does
not take into account any microwave-energy loss through the boundary of the
rod.
The system (2.1-2.5) is a coupled nonlinear system of partial differential
equations, and, in general, it cannot be solved exactly. To simplify this system, in
the present work the limit of small losses will be considered, as considered by
Kriegsmann & Varatharajah (1993) and by Booty & Kriegsmann (1994). In this
case, Kriegsmann & Varatharajah introduced the Biot number B = ha/K (where
h is the effective heat-transfer coefficient and K is the thermal conductivity of
the rod, which has a typical dimension a). They showed that for B =* (a/I)2, where
/ is the length of the rod, to leading order, the temperature becomes a function
of x and t alone; that is, T = T(x, t). They showed that T(x, t) satisfies the
equation
dT
d2T L(T)
1 tTJ^r,P,2
. Aa
— =v—r
h—jy(T)\
\E\ rdrdO,
dt
dx2
a
JUJ2
rK
Mo
nf,
(26)
V
where the average P is now given by the square of the modulus of the complex
field \E\2.
The fundamental observation made by Kriegsmann & Varatharajah (1993) and
by Booty & Kriegsmann (1994) was that for constant |JB|2, or for a prescribed
value of the mode strength, the nonlinear function
p
(11)
has bistable behaviour; that is, Q(T, \E\) = 0 has, for values of \E\ below a certain
threshold, three roots. The smallest and largest of these roots are stable steady
states of the forced heat equation (2.6), and the intermediate root is an unstable
steady state. When \E\ exceeds this threshold, only the largest root is available.
The smallest root is interpreted as the normal steady state for microwave heating,
and the largest root is interpreted as a hotspot because it corresponds to heating at a high temperature. The function Q(T, \E\) used by Kriegsmann &
Varatharajah (1993) had a complicated form due to the realistic form of their
electrical-conductivity model. In the present work, in order to obtain a simple
model for the bistable behaviour, we replace the function Q used by Kriegsmann
& Varatharajah by the canonical cubic
Q(T, \E\) = A(r0 - T)(T2 - 2T0T + y \E\).
(28)
This simple cubic will give the same qualitative behaviour for the temperature
evolution as the more complicated function used by Kriegsmann & Varatharajah.
Observe that if y \E\ is sufficiently large the only steady state is the hotspot at the
temperature To, but if y \E\ is small enough there are three steady states.
170
C. GARCIA REIMBERT ET AL,
We now consider the simplest possible approximate solution for the electric
and magnetic fields. Since the electrical conductivity is small, we consider the case
a = 0 first. Due to the cylindrical symmetry of the waveguide, the solutions
decouple into transverse-electric (TE) and transverse-magnetic (TM) modes
where the electric and magnetic fields satisfy uncoupled wave equations. For
simplicity, we will only consider the TE modes, in which case the field E is
transverse to the direction of propagation. Moreover, it can be represented as
£ ° ' = (0, <p°\ -<p°zA) where <p0>1 is the longitudinal (x-component) of the
magnetic field B 0 1 . From this representation if follows that the electric field is
purely tangential to cylindrical surfaces whose axes are in the x-direction. Hence
the boundary conditions on the normal component of E at r ~ a and r = b are
automatically satisfied. The continuity of the tangential components of E°A and
H°-j at r = a and r = b give the continuity of <p°^ and <pai at r = a and the
vanishing of <Pa at r = b. The remaining boundary conditions on the normal
component of iff01 are satisfied due to the consistency of Maxwell's equations.
This situation suggests that the eigenvalue problem
V24>°=-X2if>0 (0«r«a),
VV = - A V
I/»° = I/M and ip° = ipl, forr = a,
(a*r*b),
^ = 0 for r = b
(2.9)
(2.10)
should be considered. The modes £ ^ ' have eigenvalues -A^ generated by the
eigenfunctions ipn.
Let us now assume that the transverse dependence of the incident wave
coincides at x = 0 with one of the modes just described, that is, that
Einc = E(fi^lEon\y,z).
(2.11)
The frequency w is a scale frequency for the frequency of propagation of the
mode E%\ This assumed form is an idealization since the incident field in the real
problem has to be determined as part of the solution because the input field is
fed from another waveguide at the place in which the microwave radiation is
generated. Clearly, due to the nonlinear dependence on temperature of the
conductivity <r in the damped wave equation (2.1), the modes will mix and the
solution will in general be a superposition of all the modes. However, since the
electrical conductivity is assumed to be small, and since the frequency ai is
chosen so that it is close to the frequency of the mode £^', the contributions from
the other modes will be of lower order, that is, O(a/(e)). Hence the electric field
will take the approximate form
E = E(x,t)E°n\y,z).
(2.12)
Substituting this separate form for E into the damped wave equation (2.1), using
boundary conditions (2.3), and projecting onto the corresponding mode gives the
scalar equation for the field E(x, t) and its corresponding boundary conditions as
0 - c § j | + A5£ + ~Kr)£] = O (^0,^0),
oi
6X
6 at
(2.13)
HOTSPOT FORMATION AND PROPAGATION IN MICROWAVE HEATING
171
with
£(0, 0 = Eoe"™1,
E(x, 0) = E,(x, 0) = 0 for* 5*0.
(2.14)
Note that we are neglecting the reflected wave formed as the incident wave
impinges on the surface of the rod at x = 0. Also, the magnetic fields are
approximated by this expression with an error of O(a/ecj), which does not
influence the solution for the electric field to the order considered. To summarize,
we are considering a signalling problem which neglects the effects of finite cavity
length since no energy reflected back. Also, the proposed solution is valid in the
limit of small electrical conductivity. Moreover this model neglects detuning
effects in which the changing electrical conductivity of the material affects the
strength of the electric field in the cavity (see Kriegsmann, 1995).
In the forced heat equation (2.6) the temperature also depends on the modal
shape E%\ Therefore, in this equation the quantity \E\2 is replaced by
\E(x,t)\2—2\
[\E°S\2rdrdd«Dn\E\2,
(2.15)
na Jo Jo
and so the nonlinear term in the forced heat equation is replaced by
Q = A(T0 - T)(T2 - 2T0T + yDn \E\).
(2.16)
Thus the equations used in the present work to study the microwave heating of
a cylindrical rod are the damped wave equation
0
or
ox
(2.17)
e oi
and the forced heat equation
—dT
d2T
2
dx
Finally, for the electrical conductivity a we shall take a simple linear dependence
on temperature with the form
~fT\
(2.19)
Since the time scale for the electromagnetic radiation is much shorter than that
for heat diffusion, equation (2.17) may be approximated by
This equation, together with (2.18), forms the approximate system used in the
present work to describe the mcirowave heating of the rod.
172
C. GARCIA RE1MBERT ET AL.
3. Asymptotic solutions
The governing equations (2.20) and (2.18) with the boundary and initial
conditions given by (2.3) and (2.5), are nondimensionalized by using the changes
of variables 7 = wt, jp = (DX/CQ, E = E0E, 7 = T0T, yE0 = y , <T0 = <r0, ax = Tocru
v' = VCD/CO, P = ATQJCO, and A = A n /w. On dropping the tildes, these equations
and boundary and initial conditions become
52c
E,
—;2
at
with
-i2j7
o 11/
r2+
dx
/,»
-
_l_ _
T\
11 /T
/ (TQ ~T~ tj\ 1 \ OtL
A£ + l
\
£(0,0 = ew',
o)
— = 0 for* 5=0, t>0,
I at
E(x, 0) = E,(x, 0) = 0 forx>0,
(3.1)
(3.2)
and
— =v'^+o(T2-T
3?
5x
+ y\E\)(l-T) forx^O, t^0,
(3.3)
with
£ ( 0 , 0 = 0,
T(x, 0) = 0 forx^O.
(3.4)
It is important to observe that the nondimensional equations (3.1) and (3.3) still
contain different time scales since electromagnetic radiation travels at a vastly
larger speed than heat diffuses. For these time scales to be of the same order, the
volume of the body being heated would have to be very small, as shown by
Kriegsmann (1993). The relevant time scales used in the present analysis are the
(nondimensional) order-one time scale of the electromagnetic radiation, the slow
diffusion time (since v ' « 1 ) , and the slow time imposed by the small electrical
conductivity of the rod.
To find an asymptotic solution of (3.1-3.4), we consider the solution of the
forced heat equation (3.3) assuming that the electric field £ is known. When the
electric field £ has a constant amplitude (for o-0 = cr, = 0) the temperature
equation (33) has an explicit front solution joining the state 7 = 1 with the state
T = i[l - (1 - 4y |£|)i]. The velocity of the front (which moves from left to
right) is given by
(3.5)
and the temperature profile takes the explicit form
exp [- (p/2v')*(x - Ct))\
l + exp[-(p/2V)l(x-O)]
)
(see Fife, 1988). When v' is small and the variation of |£| in x and t is slow, it is
known (Fife, 1988) that the solution for the temperature is given to leading order
by
T(x,t) = F(x-m),
(3-7)
where the position g(t) of the layer satisfies the ordinary differential equation
I = i(Jv'p)l{l - 3[1 - 4y |£(£(0, Oil*}-
(3-8)
HOTSPOT FORMATION AND PROPAGATION IN MICROWAVE HEATING
173
Thus, if E, is known the motion of the front is determined by integration of this
equation. Observe that the time scale for the temperature evolution, which is
proportional to £, is slow since v' is assumed to be small. This provides a check
on the consistency of the approximation introduced in (2.17) to obtain (2.20).
The fundamental problem is then to determine the electric field E. To do this
we now consider the damped wave equation (3.1). Since the temperature is now
known in terms of £, we can solve the damped wave equation using the WKB
approximation on assuming that aja), ajco = 0 ( a ) « 1. The WKB approximation entails that the electric field is in the form
E = A(X, r) exp {iq[t + e(X)/a]},
(3.9)
where X = ax, T = at. Substituting this WKB expansion for the electric field into
the damped wave equation (3.1) and equating powers of a gives the eikonal and
transport equations
(3.10)
\q/
and
6
uT
M — - H — - T 1/4 = 0,
O^TL
\tL)Qf
(OCX
(3.11)
/
with the boundary and initial conditions
a(0, T) = 1,
A(X, 0) = 0.
(3.12)
The temperature distribution T is now approximated as a slowly moving
discontinuity. For consistency we therefore need to assume that the slow time
scale, O((|v'p)i), is comparable to the slow time scale a of the WKB
approximation. However, as will be seen later in this section, the asymptotic
solution is in excellent agreement with the full numerical solution of the
governing equations (3.1) and (3.3) even if the time scale a is not small.
The solutions of WKB equations (3.10) and (3.11) are easily found. The
solution for the phase 6 for a wave moving to the right is
(3.13)
Using this solution for 8 the transport equation (3.11) becomes
i^4
= 0,
(3.14)
a
where, here and below, a factor of 1/w has been absorbed in a0 and o-j for
brevity, and the temperature is given by 7 = 1 for x =s f (f) (that is, behind the
front) and by T = 0 for x > £(t) (that is, ahead of the front).
The transport equation is integrated along the characteristics starting at X = 0
up to the front x = £(r). At that point there is reflection and transmission of the
microwave radiation, and hence of the electric field. As is usual in geometric
optics, the reflected wave is neglected (the reflected wave having an amplitude of
174
C GARCIA REIMBERT ET AL.
O(a)) and the integration is continued beyond the front at x = £(t) with the same
transport equation, but now with T = 0. At x = £(/) only continuity of the electric
field can be imposed. The solution for the electric field could be extended to
include the reflected wave, but this is not done because the reflected wave is not
needed in the present work. The solution of the transport equation can thus be
found as
A(X,
T)
if £(0 =£*=££',
ifx>pt,
=•
(3.15)
where
(3.16)
It can then be seen that the electric field at the front is
|£(f(0. Ol =
ex
P [-(o"o + 0"i)£(O/20.
(3.17)
Using this expression for the electric field at the front, the differential equation
(3.8) for the position of the front becomes
€ = / « ) = l(iv'p)i(l - 3{1 - 4y exp [-(a* + a-,)£(0/2/8]}*).
(3-18)
Equation (3.18) for the position of the front has a single steady state xd,
Xa=
~~^+vlog(v)-
(119)
Moreover, since / ' ( r d ) < 0 , this steady state is stable. Thus, when when the
hotspot forms at x = 0 it propagates into the material and slows down until it
stops at the position given by (3.19). When the front position reaches its
maximum value, the electric-field amplitude (3.15) reaches the steady state
1
'
rexp[-(<ro + a-,)x/2/3],
lexp(-cr1A:d/2)3)exp(-o-(>x/2/3)
Hx^x6,
if*5*xd.
^ ' "*
(Note that the electric-field front has propagated to infinity when the steady state
has formed.) Further, it can be seen from (3.6) that when the front has reached
its maximum penetration into the material the temperature has the steady profile
1
/l-a-
It can be seen from expression (3.19) for the penetration depth that when the
uptake y of heat energy by the material from the microwave radiation is small
enough (9-y < 2) there is no penetration of the hotspot into the material.
Furthermore, the penetration depth increases as q increases (that is, as the
microwave frequency w increases), and if the frequency q is below the cut-off
value of A the hotspot again does not propagate into the material (see 3.13).
HOTSPOT FORMATION AND PROPAGATION IN MICROWAVE HEATING
175
The solution which has been derived illustrates a very simple mechanism
whereby a hotspot can form and propagate into a material until a steady state
forms when heat input balances heat loss. The solutions for the steady
electric-field amplitude (3.20), the steady temperature profile (3.21), and the
penetration depth (3.19) will now be compared with full numerical solutions of
the governing system (3.1) and (3.4).
4. Comparison with numerical solutions
The numerical method used to integrate the damped wave equation (3.1) and the
forced heat equation (3.3) is the same as that used by Pincombe & Smyth (1991),
who give full details of the scheme. The scheme uses central differences to solve
the damped wave equation and a Crank-Nicolson scheme to solve the forced
heat equation. In the Crank-Nicolson scheme, linear extrapolation of the
temperature from the two previous time steps is used to estimate the forcing term
p(T2 - T + y |£|)(1 - T) at the new time step. This scheme is accurate to the
second order and it was found to be stable.
The parameter values in the numerical solutions were taken t o b e p = l,A = l,
a0/co = a-l/w = \, and a wide range of nondimensional frequencies q were
chosen. This choice of a0 and ax provided a severe test on the asymptotic solution
since the WKB solution is expected to be a good approximation only for aolu>,
a,/<o«l. The nondimensional diffusivity was chosen to be v ' = 0-01 for the
comparisons.
Figure 1 displays the evolution of the temperature profile as a function of x and
/ for the full numerical solution. The hotspot can be clearly seen forming at the
10
FIG. 1. T h e temperature evolution as a function of x and / when £ „ = 1-0, 7Jj = 0-0, <TO = 0 - 5 , at = 0 - 5 ,
y = 1 0 , p = 1 0 , <7 = 2O, A = 1 0 , and v' = 0-005.
176
C GARCIA REIMBERT ET AL.
FIG. 2. A comparison of the steady-state solution given by (
) the WKB approximation and (—)
the numerical solution when £ 0 = 1 0, To = 00, a0 = 0-5, a, = 0-5, y = 1 -0, p = 1 0, q = 20, A = 1 -0,
and v' = 0-01, for (a) the steady-state temperature and (b) the steady-state electric field.
origin and propagating into the material until a steady state forms as predicted by
the WKB solution.
Figure 2 compares solutions given by the WKB approximation and the
numerical solution for the steady-state electric field of (3.20) and for the
steady-state temperature profile of (3.21). It can be seen that the agreement is
HOTSPOT FORMATION AND PROPAGATION IN MICROWAVE HEATING
177
3.00
2.95
2.60
(b)
FIG. 3. A comparison of the penetration depth for (
) the WKB approximation and (—) the
numerical solution when £ o = l - 0 , 7"0 = 0-0, ao = 0-5, a ^ O - 5 , p = 1 0 , A = 10, and v' = 001, as a
function of: (a) y for q = 2-0, and (b) q for y = 1 -0.
excellent, especially since this comparison includes parameter values outside the
strict range of the asymptotic validity of the WKB approximation.
Figure 3 compares the penetration depth given by the WKB approximation of
(3.19) with the numerical solution. Since the diffusivity in the numerical solutions
is nonzero, there is no sharp front in the numerical solutions. The numerical
178
C GARCIA REIMBERT ET AL.
value of the penetration depth was calculated by finding the point of inflection of
the numerical solution for the temperature profile. The point at which the
temperature profile changes curvature gives a reasonable value for the penetration depth. In Fig. 3(a) the penetration depth xd is given as a function of the heat
uptake y, and in Fig. 3(b) it is given as a function of the frequency q. It can be
seen that the agreement between the values of the penetration depth given by the
WKB approximation and by the numerical solution is excellent. It can be seen
from Fig. 3(b) that the agreement improves as the frequency q increases, as is
expected from the nature of the WKB approximation. On the other hand, it can
be seen from Fig. 3(a) tha the agreement is uniform over a wide range of values
of the heat uptake y.
In the present work it was shown how radiation loss to the surroundings from
the surface of a material can lead to the formation of steady hotspots which can
occupy large regions of the material. A simple ordinary differential equation was
derived for the position of a moving temperature front by combining a WKB
approximation for the electric field with a moving-boundary-layer description of
the heat front. The mechanism proposed for the propagation of the hotspot
strongly suggests that in real three-dimensional problems hotspots are formed
around caustic regions (since the heating is a maximum there). Once the hotspots
are formed they propagate away from the caustics and they stop when the heat
input balances the heat loss. The hotspots thus become steady-state hot regions
surrounding the caustic.
Acknowledgements
A. A. Minzoni would like to thank the Royal Society of London and the National
Academy of Science of Mexico for a travel grant to the University of Edinburgh
that made this work possible. The authors would like to acknowledge the helpful
comments of one of the referees which did much to improve the presentation of
this work.
REFERENCES
BOOTY, M. R., & KRIEOSMANN, G. A., 1994. Microwave heating and joining of ceramic
cylinders: A mathematical model. Methods Appl. Anal. 1, 403-14.
COLEMAN, C. J., 1991. On the microwave hotspot problem. J. Aust. Math. Soc. B 33, 1-8.
FIFE, P., 1988. Dynamics of Internal Layers and Diffusive Interfaces. SLAM Regional
Conference Series. Philadelphia. PA: SIAM.
HILL, J. M., & SMYTH, N. F., 1990. On the mathematical analysis of hotspots arising from
microwave heating. Math. Engng. Indust. 2, 267-78.
KRIEGSMANN, G. A., 1991a. Microwave heating in ceramics. Ceramic Trans. 21, 177-83.
KRIEGSMANN, G. A., 1991b. Microwave heating in ceramics. Ordinary and Partial
Differential Equations, Vol. 3 (B. D. Sleeman & R. J. Jarvis, eds.). Proceedings of the
11th Conference on Differential Equations, Dundee, Scotland.
KRIEGSMANN, G. A., 1992. Thermal runaway in microwave heated ceramics: A one
dimensional model. J. Appl. Phys. 71, 1960-6.
KRIEGSMANN, G. A., 1993. Microwave heating of dispersive media. SIAM J. Appl Math.
53, 665-9.
HOTSPOT FORMATION AND PROPAGATION IN MICROWAVE HEATING
179
KRIEGSMANN, G. A., 1995. Mathematical models of microwave heating bistability and
thermal runaway. Ceramic Trans. 59, 269-77.
KRIEGSMANN, G. A., & VARATHARAJAH, P., 1993. Formation of hot spots in microwave
heated ceramic rods. Ceramic Trans. 36, 221-8.
KRIEGSMANN, G. A., BRODWIN, M. E. & WAITERS, D. G., 1990. Microwave heating of a
ceramic halfspace. SIAM J. Appl. Math. 50, 1088-98.
METAXAS, A. S., & MEREDITH, R. J., 1983. Industrial Microwave Heating. IEE Power
Engineering Series, Vol. 4, (P. Peregrinus, ed.). London: Institution of Electrical
Engineers.
PINCOMBE, A. H., & SMYTH, N. F., 1991. Microwave heating of materials with low
conductivity. Proc. R. Soc. Lond. A 433, 479-98.
PINCOMBE, A. H., & SMYTH, N. F., 1994. Microwave heating of materials with power law
temperature dependencies. IMA J. Appl. Math. 52, 141-76.
ROUSSY, G., BENNANI, A., & THIEBAUT, J., 1987. Temperature runaway of microwave
irradiated materials. /. Appl. Phys. 62, 1167-70.
SMYTH, N. F., 1990. Microwave heating of bodies with temperature dependent properties.
Wave Motion 12, 171-86.
VON HIPPEL, A. R., 1954. Dielectric Materials and Applications. Cambridge, MA: MIT
Press.