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COUPLED ELECTROMAGNETIC-THERMODYNAMIC
SIMULATIONS OF MICROWAVE HEATING
PROBLEMS USING THE FDTD ALGORITHM
Paweł Kopyt* and Małgorzata Celuch
Institute of Radioelectronics, Warsaw University of Technology, Warsaw, Poland
*
[email protected]
A practical implementation of a hybrid simulation system capable of modeling coupled
electromagnetic-thermodynamic problems typical in microwave heating is described. The paper
presents two approaches to modeling such problems. Both are based on an FDTD-based commercial
electromagnetic solver coupled to an external thermodynamic analysis tool required for calculations
of heat diffusion. The first approach utilizes a simple FDTD-based thermal solver while in the second
it is replaced by a universal commercial CFD solver. The accuracy of the two modeling systems
is verified against the original experimental data as well as the measurement results available in
literature.
Submission Date: 18 December 2006
Acceptance Date: 23 August 2007
Publication Date: 19 December 2007
INTRODUCTION
It is hard to overestimate the importance
of numerical models in solving coupled
electromagnetic-thermodynamic problems often
encountered in the microwave power industry,
where dissipation of microwave power induces
other physical effects, such as heat transfer and
evaporation in heated foods [Bengtsson and
Ohlsson, 1974], mechanical stress in mineral
processing of rocks [Bradshaw et al., 2005],
or phase changes during thawing [Basak and
Ayappa, 1997]. These effects, in turn, may
cause rapid changes of material parameters
[Kenkre, 1991], resulting in a fully coupled,
highly non-linear physical system. Microwave
heating [Püschner, 1964] is an example of such
a coupled effect.
An approach often employed by the
Keywords: Microwave heating, coupled numerical
modeling, FDTD algorithm, multiphysics
41-4-18
researchers in modeling of coupled multiphysics
problems that involve microwaves entails
splitting the problem into constitutive parts and
applying appropriate modeling methods to each
of them. In the literature one can find a number
of publications that address the issue of modeling
the microwave heating effect using either Finite
Element Method (FEM) or Finite Differences
in Time Domain (FDTD) algorithm to handle
the partial differential equations that govern
the electromagnetic (EM) and thermodynamic
phenomena. In [Ohlsson and Bengtsson,
1971], [Ma et al., 1995], [Torres and Jacko,
1997], [Celuch et al., 2001a], [Ratanadecho et
al., 2002] and [Al-Rizzo et al., 2007] FDTDbased simulations were employed to predict
temperature profiles within lossy media heated
by microwaves. In [Pangrle et al., 1991],
[Sekkak et al., 1994], [Zhang and Datta, 2000]
and [Akarapu et al., 2004] a FEM analysis of
the coupled problem of microwave heating was
described.
Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
(a)
(b)
Figure 1. Physical phenomena, their interaction in microwave heating effect and two approaches
to solving the coupled problem: approach with inhomogeneous heat transfer equation
(a) and approach with homogeneous heat transfer equation (b).
The goal of this paper is three-fold:
(1) to illustrate two approaches to building
a multiphysics EM-thermal modeling tool,
whose thermal solver utilizes either a simple
FDTD-based tool of limited functionality, or
a fully-fledged universal Computational Fluid
Dynamics (CFD) software;
(2) to cross-validate these two
implementations by comparing the simulations
results; and
(3) to experimentally validate accuracy
of an alternative coupling method that we have
reported earlier in [Kopyt and Celuch, 2006],
as no such measurement-based verification has
been done so far.
GOVERNING EQUATIONS OF THE
MICROWAVE HEATING EFFECT
In this paper, it is assumed that out of multiple
thermodynamic phenomena only the heat
conduction in solids will be accounted for, as it
seems to be dominant in the majority of typical
problems of microwave heating. The graphical
representation of the microwave heating
phenomena is shown in Figure 1(a). The wave
propagation through the medium depends on
International Microwave Power Institute
media properties, among other factors. In
this paper, we assume that they change with
temperature, which results in continuous changes
of the EM field distribution in the domain as
the temperature rises (or falls). The coupling
also works in the opposite direction with the
heat generation rate strongly depending on the
electric field amplitude. Thus, the changes in the
EM field distribution can increase or decrease
the rate of temperature changes.
The mathematical model that is used to
describe a thermal part of the coupled problem
is based on the heat transfer equation with
appropriate boundary conditions (BCs) and
constant initial condition. The equations are
presented below:
c pρ
du
− ∇ k ∇u = q(u )c p ρ
dt
(
)
(1)
where the medium properties, thermal
conductivity k, specific heat cp and density ρ,
are functions of temperature u:
k = k (u ), c p = c p (u ) and ρ = ρ (u )
(2)
Several kinds of BCs are used with
41-4-19
equation (1) to model various physical scenarios
[McOwen, 1996]. The Dirichlet condition
requires that temperature at the boundary be
declared explicitly. The Neumann condition
assumes that the heat flow through a boundary
is declared, while the Robin condition, known
also as the convective BC, is a combination of
these two.
The EM part is described by Maxwell’s
equations:
∂H
∇ × E = −µ
(3)
∂t
∇ × H = ε0 (ε ′r − jε r′′ )
∂E
∂t
∇⋅D= ρ e and D = ε0 (ε ′r − jε r′′ )E
∇ ⋅ B = 0 and B = µH
(4)
(5)
(6)
where μ is permeability, ε0 is the permitivitty
of vacuum, ε ′r is relative dielectric constant, ε ′′r
is the relative dielectric loss, ρ e is the electric
charge density, the vectors fields E and H are,
respectively, the intensities of the electric and
magnetic field, and D and B are the electric and
magnetic flux density vectors.
The thermal-to-EM coupling results from
the temperature dependence of the electric
properties of the medium:
ε = ε0 (ε ′r − jε r′′ ) = ε (u )
(7)
The reverse EM-to-thermal coupling stems
from the source term function:
q(u ) =
2
1
ωε0ε r′′ (u ) E(u )
2c p ρ
(8)
where ω is the angular frequency.
The most popular approach to running a
coupled simulation of the microwave heating
effect makes use of the significant difference
between the time-scales of electrodynamic and
41-4-20
thermodynamic phenomena, like in [Ohlsson
and Bengtsson, 1971], [Pangrle et al., 1991],
[Sekkak et al., 1994], [Ma et al., 1995], [Torres
and Jacko, 1997], [Celuch et al., 2001a], [Zhang
and Datta, 2000], [Ratanadecho et al., 2002]. All
the simulation methods based on this feature of
microwave heating perform a sequence of steps
which is repeated until the distribution of all
fields is obtained for the specified heating time.
First, the solution of the EM part of the problem
is found by numerically solving the equations
(3)-(6). Next, with the electric field magnitude
and medium properties known, one can derive
the source term function q given in (8). Assuming
that the freshly obtained source term function q
will stay constant over some time ΔT, the new
temperature field approximation û(t + ΔT) can
be calculated by numerically solving equation
(1). Using this temperature approximation, one
can obtain thermal and EM medium properties
given by equations (2) and (7).
The most common is the approach found in
[Ohlsson and Bengtsson, 1971], [Sekkak et al.,
1994], [Zhang and Datta, 2000], where the heat
transfer equation (1) is solved directly by taking
into account both the initial condition being the
temperature field obtained previously and the
source term function.
In this paper an alternative approach is
employed, illustrated in Figure 1(b). It solves
the heat transfer equation (1) in two stages. First,
the zero thermal diffusivity is assumed reducing
the equation to an ordinary differential equation,
which can be solved when source term function
q is assumed constant:
ˆ � �� ) � �(�)
ˆ �
�(�
ˆ �
� �(�)
1
� ��
���
� ��
����
�
�� �
�
(9)
In the second stage the newly calculated
temperature û(t + ΔT) is fed as the initial
condition into equation (1) with no source
term function assumed (q ≡ 0). The two-stage
Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
approach reduces half the amount of data that are
exchanged between solvers while maintaining
high computational accuracy as long as the
time-step ΔT is sufficiently small. This has
been shown in [Kopyt and Celuch, 2006],
where the performances of the two approaches
are compared using an analytical benchmark
problem.
implementation as well as a high computational
accuracy.
The STM was built so that it can operate
in two modes. In the first mode (called the
linear mode in further sections) the thermal
computations are performed using the discretized
equation (1) in the following form:
u nj +1
ELECTROMAGNETIC AND THERMAL
SOLVER IMPLEMENTATIONS
Because this paper emphasizes coupling
methods and thermal solvers, a ready-made
and already verified EM solver is necessary.
Out of approximately a dozen suitable EM
software packages available on the market and
examined in [Yakovlev, 2006], QuickWave3D [QuickWave-3D, 1997-2007] was selected
because of its fast and accurate conformal
FDTD solver with extra features desirable in
microwave heating applications.
Concerning the thermal solver, another
software review was reported in [Kopyt
and Gwarek, 2004]. This time, 10 out of 20
considered CFD packages met the goals set
as: (i) possibilities of importing mesh, initial
and boundary conditions; (ii) varying media
parameters during the simulation; and (iii) a
robust enthalpy-based model of phase changes.
The Fluent package [Fluent, 1988-2007] was
finally selected. It is based on the Finite Volume
Time Domain (FVTD) method which is similar to
the conformal FDTD algorithm implemented in
the QuickWave-3D package. It facilitates direct
data exchange between the two numerical tools
without the need for error-prone interpolation as
shown in [Kopyt and Celuch, 2005].
In order to properly evaluate the performance
of a CFD-based thermal solver employed in the
multiphysics modeling tool, we have developed
a specific FDTD-based reference tool – Simple
Thermal Module (STM). A standard explicit
formulation of the algorithm by Morton and
Mayers [1994] was selected due to the ease of its
International Microwave Power Institute
= u nj
+
− u nj
P
∆t
∑K
V j ρ j c j i ij
P
∆t
∑K u
V j ρ j c j i ij i
(10)
where Δt is the length of the FDTD thermal
analysis time-step (typically smaller than the
heating time ΔT) and Vj is the volume of the
current cell. The symbols j and i are spaceindices, whereas n is the time-index.
Equation (10) can be derived based on
energy balance of individual cells as seen in
[Szargut, 1992]. Energy change in the ith cell
is a sum of individual heat flows between this
cell and its neighbors:
Vi ρ i ci
dui
dt
P
= ∑ Qij
j
(11)
The heat flow Qij depends on the temperature
gradient through the interface between the cells,
thermal conductivities ki and kj of media that
fill these cells, the area Sij of the interface as
well as liS and lSj – the distances between the
interface and the centres of the ith and jth cells
respectively:
Qij = K ij u j − ui
(
and
)
l
lSj  1
iS

K ij =
+ 
k

 i k j  Sij
(12)
Equation (10) can be used for simple linear
problems where thermodynamic properties of
41-4-21
media do not depend on temperature. Because
properties of media that undergo a phase change
(such as thawing or freezing) cannot be assumed
constant in temperature, this equation must be
reformulated in simulations of scenarios where
phase change effect is present. The modified
equation based on the enthalpy method
[Furzeland, 1980] that is used in the second
mode of operation of the thermal solver (called
the non-linear mode in further parts of the paper)
is shown below:
H nj +1 = H nj − f −1( H nj )
+
P
∆t
∑K
V j i ij
P
∆t
∑ K f −1( H in )
Vi i ij
(13)
where f is a function defining the dependence of
the scalar enthalpy field H on the temperature:
H = f(u). The enthalpy is given as a function
of temperature and medium properties in the
following way:
 u

 ∫ cs u ρ u du, u ≤ um
uRe f


u

H u =
 ∫ cl u ρ u du + L pc ρ um +
um


um


∫ cs u ρ u du, u > um

uRe f
(14)
() ()
()
() ()
( )
() ()
where uRef is an arbitrarily chosen reference
temperature, cs(u) is the specific heat of the
medium in the solid phase, cl(u) is the specific
heat of the medium in the liquid phase, Lpc is
the latent heat of phase change and um is the
temperature of melting.
The computations performed using
41-4-22
(a)
(b)
Figure 2. The coupled simulation process with
the FDTD-based thermal solver (a) and with
Fluent CFD package (b).
equations (10) and (13) are stable on the
condition that for each cell in the domain the
time-step Δt fulfils the following requirement
[Szargut, 1992]:




V j ρ j c p j 
∆t ≤ min 

P
j


 ∑ K ij 
 i

Journal of Microwave Power & Electromagnetic Energy ONLINE
(15)
Vol. 41, No. 4, 2007
(a)
(b)
Figure 3. An example of a temperature-enthalpy curve (a) and the enthalpy-temperature curve
used by the implemented algorithm (b).
Table 1. Electromagnetic Properties of the Phantom Food Used in the Experiment by
Ma et al. [1995].
Temperature [°C]
12.3
25.0
34.7
39.5
45.5
48.0
ε ′r – j ε ′′r
52.0 – j20.0
43.0 – j14.5
36.0 – j11.5
36.0 – j11.3
28.4 – j7.2
22.4 – j7.0
IMPLEMENTATION OF THE COUPLED
MODEL
We have implemented two multiphysics
simulation systems shown schematically in
Figure 2. Since Fluent was not especially
designed to co-operate with an additional EM
solver, an additional interfacing module is
necessary in order to couple it to the QuickWave3D simulator; no such interface is needed for
the STM.
The geometry of the coupled problem is
discretized in the QuickWave-3D environment.
As shown in [Kopyt and Celuch, 2005], the
EM part of the problem imposes more severe
requirements on cell sizes, thus the EM mesh
can be used as the only discretization throughout
the computations. Although it is much finer than
needed for the accuracy of the thermal solution,
it does not slow down the whole system, since
the largest part of the overall computations is
spent on the Maxwellian part.
Regarding the data sent from one solver to
International Microwave Power Institute
another, it is believed that it should contain the
enthalpy field, as it seems to be the only way to
provide complete history of the heating process.
The temperature field cannot be used in this role
because the phase-change effect may cause the
temperature to temporarily stop changing in the
phase-change area when it reaches levels close
to the melting temperature um as illustrated in
Figure 3(a). After enough heat is provided (or
absorbed) the temperature is no longer locked at
um. However, when coupling is considered, the
consecutive exchanges of the temperature field
between the solvers would interrupt the flow of
heat (or heat generation) into the phase-change
region. Because the temperature field holds no
information on the energy stored in particular
cells, each time the thermal solver is invoked it
would calculate the heat diffusion starting from
the same temperature field. This would result in
temperature field stagnation (i.e., the temperature
rises slower than expected) and erroneous results
of the coupled numerical analysis. In order to
avoid the described effect, the enthalpy field
41-4-23
must be exchanged because, despite the ongoing phase-change, it is constantly growing as
presented in Figure 3(b).
FORMULATION OF THE TEST PROBLEMS
In order to compare the performance of the two
presented multiphysics systems we conducted
the two tests described below.
Test Case I
In the first test the measurement results reported
by Ma et al. [1995] were used. Their scenario
involved a microwave oven employed to heat
a sample of gel with properties close to these
of meat. A microwave cavity of dimensions
350×320×271 mm was used. It contained a load
heated over a period of 180 s by microwaves
of average power 600 W, which were fed by
a rectangular waveguide into the cavity. The
thermodynamic properties of the medium were
assumed to be constant: ρ = 1000 kg m-3, cp =
3600 Jkg-1K-1 and k = 0.55 JK-1m-1s-1. The EM
properties are given in Table 1.
Test Case II
We have designed and performed an original
experiment serving as the second test for the
presented computational tool. The experimental
system is shown in Figure 4. It consists of a
microwave oven by Plazmatronika S.A.1 with
an adjustable power level controlled by a PC
and a signal conditioner with a set of eight fiber
optics thermometers by Fiso2 that can register
temperature changes simultaneously.
The sample of bread with the dimensions
60×60×20 mm was placed centrally on the shelf
within the oven. The sample and the temperature
sensors are shown in Figure 4(b). The power
level was set to 100 W and the heating lasted
approximately 5 minutes. The temperature
(a)
(b)
Figure 4. The measurement system (a) and
the oven cavity with bread sample (b).
distribution was registered at 8 locations in
the middle layer of the sample. The EM and
thermodynamic properties of bread were taken
from measurements [Risman, 1997]. The results
are presented in Figure 5. It is seen that the bread
undergoes a phase change at the temperature
of approximately 100°C where an increase of
enthalpy is accompanied by a much slower rise
of the temperature field. This feature of bread
makes it possible to use this medium in tests of
the STM operating in the non-linear mode.
http://www.plazmatronika.pl
http://www.fiso.com
1
2
41-4-24
Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
εr´, εr˝
Temperature [˚C]
Temperature [˚C]
(a)
Enthalpy [J/g]
(b)
Figure 5. Properties of bread used in the experiment [Risman, 1997]: temperature dependence
of dielectric loss and dielectric constant (a) and temperature-enthalpy relationship (b).
SIMULATION RESULTS
The model of the microwave oven used in
Test Case I was prepared in the QuickWave3D environment. The medium of the same
properties as the one used in [Ma et al., 1995]
was defined. The simulation was performed
using the QuickWave-3D simulator responsible
for the EM analysis, while the thermal analysis
was done using the STM as well as Fluent. STM
operating in the linear mode was employed
because the thermal properties of the phantom
food assumed in [Ma et al., 1995] remained
International Microwave Power Institute
constant as a function of the temperature. In
simulation the heating period of 180 s was
divided into 36 steps of 5 s. This value is
regarded sufficient due to relatively small
changes of the EM properties with temperature.
The results of the simulation obtained with the
two hybrid systems (QuickWave-3D+STM and
QuickWave-3D+Fluent) are shown in Figure 6.
In order to compare the obtained distributions, the
differences between corresponding temperature
values in the corners were calculated and are:
ΔuNW = 0.0304°C, ΔuNE = 0.1440°C, ΔuSE =
0.0016°C and ΔuSW = 0.2216°C.
41-4-25
A similar approach was assumed with
Test Case II. Again, a numerical model of the
microwave oven used in the experiment was
constructed using the QuickWave-3D software.
Bread properties were defined appropriately.
Due to the missing data on the cooling of the
sample as a result of convection, two simulations
were performed for each hybrid system. One
simulation was done with the Neumann BCs
applied at the walls of the bread sample (no
heat flow through the boundary). The second
numerical experiment involved the Dirichlet
BCs with the temperature explicitly fixed at
20°C (the temperature of the environment).
In Figure 7(a), the results obtained with the
STM module for two BCs are compared with
measurements. Figure 7(b) presents four curves
– two for each multiphysics system. The curves
represent the temperature history recorded in the
center of gravity of the sample.
(a)
DISCUSSION
As shown in Figure 6, the results of
simulations performed for Test Case I using
the two multiphysics systems (QW+STM and
QW+Fluent) are in close agreement with each
other and with the data originally reported in
[Ma et al., 1995], which were also used for
verification purposes by Torres and Jecko
[1997].
The results obtained in simulations of
Test Case II also agree with measurements.
The temperature measured in the center of
the sample lies between the curves illustrating
the temperature history obtained in the same
place with simulations performed for two
different thermal boundary conditions. It is
shown that the measurements are much closer
to the temperature curves resulting from the
Dirichlet boundary conditions imposed on the
sample. This behavior stays in agreement with
expectations. The surrounding air does not act
as an insulator, but rather it absorbs heat from
the gradually heated sample.
41-4-26
(b)
Figure 6. Simulated temperature distribution
within the sample after 180 s of heating
obtained using two hybrid simulation systems:
data obtained with the system employing
STM (a) and data obtained with the system
employing Fluent (b).
The cooling effect becomes more significant
for higher temperatures, which is further
corroborated by close agreement between
the simulated curves obtained for Dirichlet
and Neumann BCs at lower temperatures. In
order to properly account for this phenomenon
the convective boundary condition should
be imposed at the walls of the sample. This
approach, however, requires that the heat transfer
Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
International Microwave Power Institute
Temperature [˚C]
Time [s]
(a)
Temperature [˚C]
coefficient h, given at the boundary, be known.
In order to establish the correct value of h, it is
necessary to perform additional measurements
of the velocity field inside the microwave cavity
as well as the temperature at the boundaries of
the sample. Such measurements are not trivial,
and thus we decided to verify the accuracy of
the multiphysics simulation systems using the
Neumann and the Dirichlet conditions, which
can be treated as specific cases of the Robin
boundary condition: the zero Neumann BC
corresponds to h equal to 0, while the Dirichlet
BC can be described with h fixed at positive
infinity.
In order to investigate possible causes of
minor discrepancy between the measurements
and simulations in the early stage of heating, a
set of additional experiments was performed.
The inertia of the thermometers used in
the measurements – a most probable cause
of the observed differences – was checked
experimentally for bread. The thermometer
was placed first in a sample of bread at room
temperature (25ºC) and then rapidly transferred
to a sample of temperature approximately twice
as high. The output signal was registered and is
shown in Figure 8. For comparison the response
of an ideal sensor is marked with a dashed line. A
substantial contact resistance was observed due
to the air-filled pores of bread. This resistance
results in the rise time reaching 20 s – a period
that is comparable to the delay observed in
Figure 7(a).
It is also interesting to point out that having
an access to the two simulation tools makes it
possible to evaluate the accuracy of the solution.
The explicitly formulated FDTD method and the
FVTD method implemented in Fluent converge
to the exact solutions from opposite directions as
demonstrated in [Kopyt, 2006], which suggests
that for cases where no reference data exist the
accurate solution is expected to lie between the
two approximations.
Time [s]
(b)
Figure 7. Measurements and simulation
results obtained with the simulation system
employing STM (a) and simulation results
obtained with two simulation systems for
various boundary conditions (b).
CONCLUSIONS
This paper presents two hybrid simulation
systems developed in order to model coupled
electromagnetic-thermodynamic problems like
microwave heating. Both systems have been
built with the use of the 3D conformal FDTD EM
simulator QuickWave-3D as a basis. It has been
coupled with an external thermodynamic analysis
tool responsible for heat diffusion calculations.
41-4-27
Temperature [˚C]
Time [s]
Figure 8. Response of the thermometer used in the experiment with bread.
The first of the presented systems employs an
original FDTD-based thermal solver developed
by the authors. In the second case, this module
has been replaced by the universal CFD solver
Fluent. With these two tools major concepts
of coupled electromagnetic-thermodynamic
modeling have been presented. They also
allowed for illustrating two popular approaches
to constructing multiphysics simulation systems
discussed in literature. The first one is based on
using highly-customized solvers (like STM),
while the other one uses commercial tools (like
CFD packages), which can only demonstrate
(and justify) their universality in problems that,
besides the heat transfer and electromagnetic
phenomena, involve effects like free or forced
convection, radiation and more. In such cases
the cost of a custom-made solver would prove
to be prohibitively high.
The computational accuracy of the two
systems has been verified against measurements
obtained in two experiments as well as through
comparing their output data. One of the
experiments has been described in this paper
while the results of the other one have been
taken from literature. Based on the performed
simulations, it seems that for presented test cases
the performance of the two simulation systems is
equally high. The performed tests also allowed
us to verify an alternative coupling method
41-4-28
based on the two-step process of solving the
heat transfer equation.
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