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MICROWAVE MODELING AND VALIDATION IN
FOOD THAWING APPLICATIONS
Tim Tilford1*, Ed Baginski2, Jasper Kelder2, Kevin Parrott1 and
Koulis Pericleous1
1
University of Greenwich, Park Row, Greenwich, London, United Kingdom
2
Unilever Research and Development, Vlaardingen, The Netherlands
*
[email protected]
Developing temperature fields in frozen cheese sauce undergoing microwave heating were
simulated and measured. Two scenarios were investigated: a centric and offset placement on the
rotating turntable. Numerical modeling was performed using a dedicated electromagnetic Finite
Difference Time Domain (FDTD) module that was two-way coupled to the PHYSICA multiphysics
package. Two meshes were used: the food material and container were meshed for the heat transfer
and the microwave oven cavity and waveguide were meshed for the microwave field. Power densities
obtained on the structured FDTD mesh were mapped onto the unstructured finite volume method
mesh for each time-step/turntable position. On heating for each specified time-step the temperature
field was mapped back onto the FDTD mesh and the electromagnetic properties were updated
accordingly. Changes in thermal/electric properties associated with the phase transition were fully
accounted for as well as heat losses from product to cavity. Detailed comparisons were carried
out for the centric and offset placements, comparing experimental temperature profiles during
microwave thawing with those obtained by numerical simulation.
Submission Date: 18 December 2006
Acceptance Date: 27 September 2007
Publication Date: 11 January 2008
INTRODUCTION
Microwave heating of a typical food load is
a complex process. The electromagnetic field
distribution within the oven cavity is highly
sensitive to the food load it is required to
heat. The changes in temperature of the food
load lead to variation in electromagnetic and
thermophysical properties (see, e.g., [Metaxas
and Meredith, 1983]). These changes in turn
alter the field distribution and intensity within
the cavity and cannot be ignored. Food properties
are especially sensitive to phase change, which
Keywords: Heat transfer, Maxwell’s equations,
microwave heating, moving meshes, numerical simulation,
rotation, thawing
41-4-30
can have a highly localized impact. Additionally
the food load may rotate during the cooking
process, smoothing what would otherwise be a
poor quality heating process. Thus, despite the
very different thermal and electromagnetic timescales, there is a requirement to strongly couple
the electromagnetic and thermal computational
analysis.
This contribution focuses on coupled
electromagnetic-thermophysical simulation of
microwave heating of frozen food products. The
solution approach utilizes a Finite Difference
Time Domain (FDTD) electromagnetic analysis
solver linked with a Finite Volume (FV) method
multiphysics package. Using separate meshes
for each analysis tool, an otherwise complex
Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
meshing problem can be solved through general
cross-mappings between two separate meshes.
Using a thermal analysis mesh confined to the
food material (with appropriate surface heattransfer boundary models) allows this mesh to
follow the food material as it rotates. This is then
automatically mapped to an appropriately refined
region within the electromagnetic mesh.
There is extensive literature concerned with
the computational analysis of microwave heating
of food material. The Lambert law approximation
can be used to avoid numerical calculation of the
field distribution and is accurate in a number of
cases (see, e.g., [Zhang and Datta, 2005] and
[Liu et al., 2005]). However, the approximation
requires the magnitude of the electric field at the
surface of the dielectric to be specified and so is
unable to deal with surface field variation. Early
examples of a coupled approach to microwave
heating include, e.g., [Zhang and Datta, 2000;
Ratanadecho et al., 2000; Ayappa et al., 2002].
The latter paper predicted temperature profiles
in multi-layer slabs by simultaneously solving
temperature and Maxwell’s equations using
a Galerkin Finite Element Method. Zhang
and Datta [2000] described one way coupling
between two separate finite element packages
(one electromagnetic (EM), one thermophysical).
The analysis linked dielectric properties to
temperature. The model was applied to heating of
foods and to microwave processing of polymers.
A very general coupled approach to microwave
drying of timber is described by Zhao and Turner
[1999]. The method used an unstructured Finite
Volume (FV) time domain approach to obtain
an EM solution and an unstructured FV heat
transfer solver. The model was able to accurately
determine the temperature distribution within the
load. Numerical models of microwave heating
of a phantom gel load were developed by Ma
et al. [1995], in which numerical results were
contrasted against experimental measurements.
The numerical model utilized the FDTD
method to solve Maxwell’s equations in order
to obtain an EM solution and used an explicit
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FV scheme to obtain a thermophysical solution.
Liu et al. [1994] simulated microwave heating
of a polymer material inside a ridge waveguide.
Again, an FDTD scheme was employed to solve
Maxwell’s equations and a finite difference
scheme was used to solve the heat transfer
equations. Kopyt and Celuch [2003] described
an approach to coupling FDTD computations to
thermal computations based on a single thermal
mesh and multiple FDTD meshes for a rotating
load. This approach takes the average of the
power distributions computed at each load
position as input to the thermal computation. The
FDTD meshing was conformal and preserved
the meshing of the food load during rotation. A
general approach to microwave heating without
rotation is described in [Kopyt and Celuch,
2005] that couples existing computational fluid
dynamics (CFD) and FDTD software using a
common mesh. This work successfully linked
the two computations dynamically for a variety
of CFD discretization.
This paper extends the earlier work of Dincov
et al. [2004], which described the coupling of a
Yee FDTD scheme to the PHOENICS CFD code
[PHOENICS, 1974-2007]. The food material
was modeled as a porous medium with dielectric
properties dependent upon both temperature and
moisture content. The coupling required the colocation of the cells of the two domains, i.e the
cells, in the two domains had identical vertex and
centroid coordinates allowing a direct cell-tocell relationship between the solution domains.
This was restricted to non-rotating loads and
could not allow for automatic mesh refinement.
The approach described in this paper couples a
cartesian mesh FDTD solver to an unstructured
mesh FV multiphysics solver using a crossmapping algorithm that allows fully independent
meshing for each solver. The approach is able to
simulate complex product shapes rotating within
the oven cavity, with non-linear phase-changes
occuring within the rotating load. Another
advantage of independent meshing is that the
EM mesh can be varied as heating progresses,
41-4-31
e.g., in response to load wavelength changes, to
reduce the computational cost.
For feasible simulations supporting actual
product development, a number of requirements
have to be met. First of all, successful interaction
with drawing packages is needed to analyze
complex product shapes. Secondly, both spatial
and temporal accuracy need to be sufficient to
capture heating and the phase transition over full
heating time, which is on the order of minutes.
Finally, problem set-up, computation and results
interpretation should take on the order of days.
This contribution describes a model which meets
these requirements.
ELECTROMAGNETIC ANALYSIS
The electromagnetic analysis is carried out in
a predominantly rectangular domain consisting
of a domestic microwave oven cavity excited
via a standard waveguide. This allows for the
use of a classical Yee FDTD scheme [Yee, 1966]
with tensor product meshes [Monk and Suli, 1994],
where Maxwell’s equations, in the form shown below,
are solved in the time domain with harmonic
excitation.
∂H
µ
= −∇ × E
(1)
∂t
ε
∂E
= ∇×H−J
∂t
(2)
∇ ⋅ (ε E) = 0
(3)
∇ ⋅ (µ H ) = 0
(4)
where E is the electric field strength, H is
the magnetic field strength, μ is the magnetic
permeability, ε is the electric permittivity and J
is the current density:
J = σ eff E
41-4-32
(5)
A conventional effective conductivity σeff in (5)
generates the microwave power dissipation. In
the most general case, σeff can vary both with
temperature T and moisture content. In this case,
the values of σeff and ε are mapped directly at
each thermal time-step from the FV mesh to the
FDTD mesh, i.e., the load geometry is captured
implicitly as a spatial distribution of material
properties via the mapping process. The implicit
construction of the load through the use of the
cross-mapping process simplifies the simulation
of materials that are of non-rectangular shape or
are moved within the applicator. The key benefit
of the approach is that the FDTD mesh is not
dependent upon the geometry or positioning
of the load. One single FDTD mesh could be
created and used throughout the heating process.
The load geometry and position are considered
by mapping dielectric properties into the
correct spatial location. A sufficient number
of cells per wavelength should be provided to
maintain numerical accuracy. The automatic
mesh generation algorithm outlined below
optimizes the FDTD mesh spacing to reduce
computational expense while maintaining this
accuracy criterion. The drawback to crossmapping is the smearing of the interface between
dielectric materials. This may effect reflection
and refraction at the interface and adversely
affect accuracy. A possible alternative to crossmapping would be to utilize a conformal FDTD
mesh that remained fixed within the load as it
rotated – a method employed, for example, by
Kopyt and Celuch [2003].
The Yee scheme update of the E field is
dependent upon only the H field and, likewise,
the H field update is dependent upon only the E
field. This greatly simplifies the parallelization
process. The scheme is second order accurate
both temporally and spatially. However, the
accuracy of the scheme is dependent upon the
resolution of the mesh in terms of the number
of mesh cells per wavelength. In the case of
frozen foods, the dielectric properties vary
significantly during the heating process. The
Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
(a)
(b)
local microwave propagation wavelength is
inversely proportional to the loss factor of the
material. Consequently, increases in temperature
and the loss factor due to microwave heating
result in progressive reduction of propagation
wavelength during the heating period.
The user is required to define a criterion for
the number of cells per wavelength needed to
ensure numerical accuracy prior to the simulation
commencing. The implemented scheme utilizes
a tensor-product Cartesian mesh. The locations
of the cell vertices in such a mesh are defined by
a tensor product of X-, Y- and Z-direction cell
distribution vectors. The mesh is re-specified at
each thermal time-step to capture any thermally
induced wavelength changes and the overall
movement of the load by turntable rotation. This
is achieved by an automatic mesh generation
step which first subdivides the domain in the X-,
Y- and Z-directions separately, as illustrated in
Figure 1. The algorithm identifies the location
of walls and the boundary of the food load in
each direction. These locations are then used
to subdivide the X/Y/Z-extent of the domain
into a number of sub-domains. The minimum
wavelength for each sub-domain is determined
by assessing the dielectric properties of each
cell in the sub-domain. With this minimum
wavelength determined, the algorithm is able
to define the mesh spacing in each sub-domain in
a manner meeting the accuracy criterion. Figure
1 shows the sub-divisions of the EM domain
in the XZ-plane with the load in two different
positions; one can see that the sub-domains are
bounded by walls or load locations.
A target of 20 cells per wavelength has
been used to accurately capture the spatial
distribution of electric field. Time-step length
has been determined using a Courant criterion
and is therefore a function of grid dimension and
wave propagation velocity.
The automatic mesh generation process
is rapid and has proven numerically stable.
Generating new FDTD meshes every time-step
incurs minimal additional computational cost
since new mappings from the thermal analysis
mesh to the EM analysis mesh are required in any
case at each time-step to deal with rotation.
The time-domain electromagnetic fields
are integrated to a time harmonic solution for
each thermal analysis time-step. The Fourier
transform of the electric field is used to yield
the power source term for the heating step.
The differences between successive values of
absorbed power at successive Fourier transfer
analyses are used to determine when a converged
time harmonic solution has been obtained.
Figure 1. XZ-plane automesher domain sub-division with load in position 1 (a) and 2 (b).
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THERMAL ANALYSIS
The thermo-physical analysis has been carried
out by incorporating the FDTD code into the
PHYSICA multiphysics package [PHYSICA,
1996-2007] developed at the University of
Greenwich. This software has been created to
solve combined stress, heat and fluid transfer
41-4-33
problems and has a flexible programmable
interface that allows complete modules with
independent meshes to be incorporated. The task
of obtaining a solution to the thermal physical
problem could be carried out using a wide range
of software packages utilizing a number of
solution methods and approaches. The PHYSICA
package was selected for a number of reasons.
Firstly, the package has been co-developed by
the authors and is therefore a practical choice.
PHYSICA is able to solve partial differential
equations using either an unstructured FV solver
or using a finite element (FE) solver. Solutions to
a number of physical problems (such as structural
mechanics analysis) are most efficiently obtained
utilizing the FE approach. Problems such as
heat transfer and fluid flow are often solved
using the FV approach although FE solution
is a viable alternative. In this contribution the
FV approach has been used to solve the heat
transfer problem because the conservative
form of the finite volume formulation is more
efficient in large non-linear problems than the FE
method. Solution of electromagnetic problems
using the PHYSICA FE implementation is
not viable. As it is suggested by Ehlers et al.
[2001], an edge-element formulation may be
more suitable for accurate EM solutions while
the PHYSICA FE solver utilizes a vertex based
approach. However, FDTD is more accurate and
more efficient for regular geometries (see, e.g.,
[Monk and Parrott, 2001]). The implementation
of the FV method within PHYSICA used an
unstructured co-located (non-staggered) grid
and a pressure correction approach by Rhie
and Chow [1983].
PHYSICA’s multiphysics modules can
obtain solutions to a wide range of physical
problems. In this case, the energy equation, in
a form allowing for phase change and localized
heat sources, has been solved:
(
∂ ρ C pT
41-4-34
∂t
) = ∇ ⋅ ( k ∇T ) + S + Q
(6)
where k is the coefficient of thermal conductivity,
ρ is the density, Cp is the specific heat capacity
and S and Q are the source terms for phase
changes and microwave power dissipation
respectively. The thermal analysis was carried
out on a mesh that was body fitted to the food
material alone. Heat losses to the oven space
were modeled with appropriate boundary
conditions. The energy equation was discretized
on a cell-by-cell basis on this unstructured mesh
using the FV method with cell average values.
These are phase averaged quantities when more
than one phase is present in the cell, where the
phase values of ρ , Cp and k are functions of
temperature; values are calculated from a set of
look-up tables prior to each time-step. Power
dissipation Q is determined by mapping power
values from the FDTD scheme solution:
2
1
Q = σ eff E
2
(7)
An additional source term in the energy
equation is S, the latent heat source/sink due
to phase change. The value of this source is
dependent upon the change in liquid fraction in
each cell. Foods are complex multi-component
materials and so phase change will usually take
place over a finite temperature range. A liquid
fraction is defined for each cell and approximated
as a linear variation of liquid fraction between
solidus (Ts) and liquidus (Tl) temperatures, i.e.
( )
f T = 0 , T < Ts
(8)
f T = 1 , T > Tl
( )
(9)
T − Ts
(10)
( )
f T =
Tl − Ts
, Ts ≤ T ≤ Tl
The heat source term due to phase change is
then:
∂f
S = −ρ L
(11)
∂t
Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
where L is the latent heat and f is the liquid
fraction.
An iterative approach is used to obtain
converged temperature and liquid fraction
values. The food material within a computational
cell typically undergoes phase change over a
number of time-steps. Calculating the liquid
fraction directly from a relationship linked to
temperature can lead to large oscillations in the
solution as a very small change in temperature
can cause an element to change from liquid to
solid resulting in a substantial latent heat energy
release. To eliminate this type of oscillation a
Voller-Prakash correction model is used (see, for
example, [Voller and Prakash, 1987], [Voller et
al., 1989, 1990, 2004] and [Swaminathan and
Voller, 1997], where the accuracy and stability
of the model are discussed). This model takes a
correction approach in which the liquid fraction
can take intermediate values between liquid
and solid allowing gradual phase change over a
number of time-steps.
The loss of heat energy from the food load
into the oven cavity occurs through convective
and radiative heat transfer. Convective and
radiative losses, Qconv and Qrad, respectively, were
represented using a surface heat loss boundary
condition as follows:
Qconv = hc Ts − Tamb
(
)
(12)
(
)
(13)
4
Qrad = ζθ Ts 4 − Tamb
where Ts is the surface temperature, Tamb is
ambient temperature, ζ is the Stefan-Boltzman
constant and θ is relative emissivity. The
convective heat transfer coefficient hc used in
(12) has been chosen empirically based upon a
Rayleigh number approximation.
A significant factor in the heating of food
loads is the effect of evaporation. In the later
stages of the cooking process a substantial
amount of the absorbed microwave energy is
used to overcome the latent heat of vaporization
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and evaporate water from the food. In order to
account for this, an evaporative volume sink
term has been implemented in the source term
of energy equation (6). The approximation is
a simplification in that it assumes evaporation
only occurs once the food temperature reaches
the boiling point of water. That is, the effects
of evaporation below the boiling point are not
considered. The sink term effectively caps the
temperature in a FV cell to a pre-specified
boiling temperature. The energy used in the
evaporation process is accounted for in a change
in water content. The sink term is switched off
if no further water remains in the cell.
The PTFE bowl containing the food material
has not been considered in the dielectric or
thermal simulations. The plastic material
occupies very little volume and has low
dielectric constant and specific heat capacity.
Consequently, the authors have assumed the
magnitude of the errors generated by this
approximation to be small in comparison with
system global approximations; this matter will
be addressed in their further work.
CROSS-MAPPING ALGORITHM
The electromagnetic analysis is carried out on
a domain consisting of the oven (containing
product) and exciting waveguide, with
permittivity and conductivity values mapped
directly from the thermal analysis FV mesh of
the food load at every thermal time-step. This
mapping is re-constructed each thermal timestep allowing the food to rotate either placed
centrally on the turntable or offset by a distance
from the turntable center. Restricting the thermal
analysis domain to the food material permits the
entire thermal mesh to be moved with the load
as it rotates. A second mapping is required to
transfer the predicted power distribution from
the electromagnetic FDTD mesh directly to the
unstructured FV thermal analysis mesh.
Both the electromagnetic and thermal
analyses use piecewise constant material
41-4-35
properties, thus the mapping is based on
an elementary quadrature approach. The
mapped cell average value is an integral of the
originating values over the intersection of that
cell with the originating mesh. In Figure 2, the
rectangular area represents an FDTD cell and
the three triangular areas represent cells from
the CFD domain. The FDTD cell is recursively
subdivided into sixteen sub-volumes, each with
a sample point located at its center. The pointin-cell algorithm determines which of the CFD
cells contains the sample point. This is repeated
for each of the sample points. The cell value
of the mapped variables (e.g. loss factor or
dielectric constant) is then the average of the
values determined at each sample point. In
practice, around 100 quadrature points per cell
are used in both mapping directions using a FV
rule based on recursive volumetric subdivision
of each cell. The approach is similar in some
respects to the use of overset meshes in CFD
[Benek et al., 1983] .
Generating an efficient set of quadrature
points is straightforward for quadrilateral
elements, but more complex for tetrahedral
elements, using recursive sub-division for the
sample points. An efficient point-in-cell search
algorithm (see, e.g., [O’Rourke, 1998]) is used to
create fast and conservative mappings between
the meshes.
SOLUTION PROCEDURE
The material properties are initialized based
upon the initial temperatures. The initial location
of the load is determined and the automesher
generates an FDTD mesh. The electromagnetic
properties of the load are mapped from the FV
food material mesh onto the FDTD mesh, and
the FDTD scheme is run until a converged
EM solution has been obtained. The power
distribution calculated on the FDTD mesh
is then mapped back onto the FV mesh. The
thermophysical solution is marched forward
in time by a predefined time-step. The mapped
41-4-36
Φ FDTD
CELL
=
5
7
4
Φ1 + Φ2 + Φ3
16
16
16
Figure 2. Illustration of cross-mapping
algorithm with 16-point rule.
power density is used as a source term in solution
of temperature and liquid fraction. After a
thermophysical solution has been obtained, the
electromagnetic and thermophysical properties
are updated using the new temperature
distribution. The simulation progresses to the
following thermophysical time-step. The load
location is recalculated and the EM solver is
called again. This process is repeated until the
thermophysical solution has reached the userdefined simulation end time.
Simulations of both the centric and offset
cases consisted of 72 thermophysical time-steps
each of 5 seconds duration giving a total heating
time of 360 seconds. Each thermophysical
time-step was comprised of 200 Jacobi preconditioned conjugate-gradient method solver
iterations. The food domain mesh was generated
by creating a CAD model of the bowl and
food material using PTC Pro/Engineer [PTC,
2007]. This CAD model was meshed using the
GAMBIT package [GAMBIT, 1988-2007].
Finally, a custom-made filter, developed by the
authors, was used to transfer the mesh into a
format compatible with the PHYSICA package.
The final mesh consisted of 125,000 tetrahedral
cells. Figure 3 illustrates the discretized finite
volume mesh used by PHYSICA to describe the
bowl geometry.
The dimensions of the oven used are shown
in Figure 4 while load material properties,
Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
determined experimentally at the Unilever
Research Center, are given in Table 1. The
temperature-dependent material properties are
determined by linearly interpolating between
listed data points. The EM domain is comprised
of two sections – a WR340 waveguide and the
applicator cavity. The domain is excited by an
incident TE10 field, using a total-scattered field
method, with an incident field excitation plane
located 1⁄4 of the distance along the waveguide
in the Z-direction. The excitation plane emits
waves in both directions. In order to prevent
reflections from the closed end of the waveguide,
a Mur type absorbing boundary condition [Mur,
1981] has been implemented.
The FDTD mesh was generated automatically
for each FDTD solution, using up to 768,000
cells. All simulations were run on a Compaq
Alpha ES45 system with 4 processors operating
at 1.0 GHz and 4 GB RAM. The solution was
obtained after a runtime of approximately 20
hours. The process was accelerated through
the use of Open Multi-Processing (Open-MP)
parallelization of sections of the FDTD solver.
Open-MP is a set of compiler directives enabling
parallelization through multithreading. These
directives have been implemented in a manner
allowing the field updates within the FDTD
Figure 3. Typical mesh for the food geometry
in the bowl.
solver to be processed by a number of CPUs
(or CPU cores) simultaneously. Use of Open-MP
resulted in a reduction in FDTD solver runtime
of approximately 45%. This performance relates
to an update rate of 15.36 million cells per
second on an obsolete machine. The use of the
automatic mesh generation algorithm ensures
optimal mesh sizing through assessment of local
wave propagation speed and system geometry,
which in turn ensures optimal FDTD time-step
length reducing the number of FDTD time-steps
required to reach steady state.
Figure 4. 3D representation of oven and load.
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41-4-37
Table 1. Material Property Data.
Thermal conductivity, W/mK
Specific heat capacity, J/kg
Density,
kg/m3
Solidus
temp,
K
Liquidus
temp,
K
T=223 K
T=265 K
T=271 K
T=363 K
T=219 K
T=265 K
T=280 K
T=308 K
991.44
271.3
271.3
211.0
84.5
86.0
43.0
1389.0
3813.0
3800.0
3551.0
EXPERIMENTAL STUDIES
Temperature measurements were carried out on
samples of a commercially made frozen cheese
sauce that has a composition of 73.7% water,
16.8% fat, 4.0% protein and 3.5% carbohydrate.
1% sodium alginate (Maugel DMB from
International Speciality Products) was added to
prevent convection during heating. The sauce
was filled into a commercial bowl of 550 cm3,
having a diameter of 150 mm and a height of
50 mm.
The dielectric properties of the sauce
were measured over a temperature range of
approximately -20°C to 100°C, using an HP872C Network Analyser with a 14 mm open ended
coaxial probe. Specific heat was measured using
a Holometrix QTA Adiabatic Calorimeter and
thermal conductivity with a Holometrix COM800 instrument.
The microwave oven (Panasonic NNT3
553W) was connected to a stabilized voltage
supply (Claude Lyons, TS-3) to remove any
variation in magnetron output due to changes
in main voltage. The effective power output
from the microwave oven was measured by
monitoring the temperature rise of a 500 g water
load in the bowl. To confirm the power deposition
in the actual product, the average temperature of
the cheese sauce was measured at one minute
intervals using a commercial radiometry system
(Loma Scientific). Heating of the cheese sauce
was temporarily stopped (approximately 10
seconds) while it was placed in the radiometry
chamber for temperature measurement.
During the microwave heating, internal
41-4-38
temperature measurements of the sauce were
made with fibre optic probes (FISO FOT-L)
mounted at specific locations in the frozen sauce.
A jig was used to prevent probe movement. The
probes were connected to the sensor interface
(FISO MWS) which was mounted on the
microwave oven so that the sensor probes corotated with the turntable. Thermal images of
the heated sauce surface were also obtained
(FLIR A40M Researcher camera) at one-minute
intervals during heating.
A total of 8 fiber optic probes were mounted
at specific locations in the frozen sauce. These
locations are summarized in Table 2 giving the
horizontal and vertical displacements of the
probe tips from the bowl center. The general
layout of the probes is illustrated in Figure 5.
RESULTS OF SIMULATION AND
MEASUREMENT
Figures 6-7 show the temperature evolution
as measured and simulated for each of the
probe locations. Figures 6(a) to 6(h) relate
to the centered rotation case while Figures
7(a) to 7(h) refer to the offset rotation case.
Figures 8 and 9 show relative loss factor and
temperature contours at a number of heating
times. The figures show rapidly increasing
temperatures near the edge of the bowl.
Prediction and measurement are in satisfactory
agreement, especially in the view of the spread
in the experimental replicates. Temperatures rise
more slowly in the center of the bowl, due to
attenuation of the electromagnetic fields (and
the resulting reduction in energy deposition)
Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
by surrounding material. The agreement is fair
when the sauce is frozen, but deteriorates for
longer heating times.
Comparing the temperature development
between the same bowl locations in the centered
and offset cases, it is clear that the preferential
edge heating occurs in both cases. In addition,
the heating uniformity seems better in the
centric case. Model and experiment agree best
for short heating times when the sauce is still
frozen throughout.
The analysis of the dielectric properties of
the sauce is displayed in Figure 10. The plots
show the rapid variation in dielectric properties
evident during the phase change. The function
used to govern the phase change rate has therefore
significant influence upon the temperature
traces. The dielectric properties are related
directly to the load temperature. Throughout the
phase change region the load temperature, and
therefore the dielectric properties, are regulated
by the Voller-Prakash liquid fraction function.
In this contribution a significant approximation
has been made in considering the food load to be
a pure material with a single specified melting
point. The cheese sauce considered is in fact
a complex multi-component material which
undergoes phase change in a non-linear manner
over a temperature range.
The effect of dielectric property variation
during heating can further be explored
performing a Q-factor analysis. The Q-factor
relates the energy stored in the system to the
energy dissipated in the system per cycle. The
variation of Q-factor over the duration of the heating
time is shown in Figure 11. This shows that in the
first few seconds of heating the load, which is fully
frozen and with near zero loss factor, it is almost
transparent. After parts of the load undergo phase
change, the Q-factor drops rapidly, eventually
settling to a level approximately 10% of the initial
value. The FDTD solver compares successive values
of the discrete Fourier transform of the electric field
as a convergence criterion. A high Q-factor increases
the number of time-steps required to reach a steady
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Table 2. Monitor Probe Positions
Relative to Center of Bowl.
Probe #
Horizontal (mm)
Vertical (mm)
1
2
3
4
5
6
7
8
-65.0
-52.5
-35.0
-17.5
0.0
17.5
35.0
52.5
-15
-15
-33
-15
-25
-15
-33
-15
Figure 5. Layout of probe locations.
state and therefore increase the simulation time. This
has been evident when assessing the convergence
of the initial EM solutions.
DISCUSSION
The results obtained from the numerical model
broadly match those from the experimental work
and predict the correct trends both in time and
as a function of location. While the agreement is
satisfactory for short heating times, beyond the
melting transition the discrepancy is increasingly
worse. For this, several explanations may be
given. First, the actual cheese sauce is a nonhomogeneous medium with locally varying
properties. Though these variations are small,
in view of the strongly non-linear dielectric
properties around the melting transition they
41-4-39
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 6. Simulated and experimental temperatures at probe position 1 (a), 2 (b), 3 (c), 4 (d),
5 (e), 6 (f), 7 (g) and 8 (h), centric placement.
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Vol. 41, No. 4, 2007
(a)
(b)
(c)
(d)
(e)
(f)
(h)
(g)
Figure 7. Simulated and experimental temperatures at probe position 1 (a), 2 (b), 3 (c), 4 (d),
5 (e), 6 (f), 7 (g) and 8 (h), offset placement.
International Microwave Power Institute
41-4-41
(a)
(a)
(b)
(b)
(c)
(c)
Figure 8. The loss factor distribution contour
plots after 2 (a), 4 (b) and 6 (c) minutes
heating time, centered case.
Figure 9. Temperature distribution contour
plots after 2 (a), 4 (b) and 6 (c) minutes
heating time, centered case.
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Journal of Microwave Power & Electromagnetic Energy ONLINE
Vol. 41, No. 4, 2007
International Microwave Power Institute
Dielectric Constant
Temperature (Cº)
(a)
Loss Factor
may initiate local “hot-spots” that accelerate
the transition into local runaway heating. This
“randomness” in the heating may also explain
part of the spread in the experimental traces.
Secondly, whereas the real sauce has a clear
melting temperature range, in the numerical
model a melting point was set at -2ºC. In view
of the sensitivity of the eventual temperature
distribution to the melting transition, this should
be accounted for in future modeling attempts.
Third, heating was observed to be very nonuniform, with boiling and thawing processes
occurring simultaneously in differing locations
within the bowl after approximately 300
seconds of heating. This suggests that more
detailed attention should be paid to accurately
modeling the thermal boundaries, particularly
the evaporative contribution, as these have
significant influence on temperature distribution
in this case.
Fourth, in the current simulation a heating
time-step of 5 seconds was employed, which may
be decreased to better capture the full effect of
the rotating turntable. For heating times beyond
360 seconds the experimental traces show a clear
periodicity related to the changing distance to
the waveguide.
Fifth, the cross-mapping process smears the
dielectric interface across FDTD cells adjacent
to the interface. This smearing may adversely
affect the ability of the EM solver to consider
focusing effects of the interface and may result
in an inaccurate field distribution within the
load. This would result in an inaccurate heating
solution in the center of the load – a discrepancy
evident in the results presented.
Finally, the experimental uncertainty should
be considered. In addition to the inevitable
scatter in the properties data, probe positional
accuracy is crucial in view of the strong local
property changes.
Temperature (Cº)
(b)
Figure 10. Temperature variation of relative
dielectric constant (a) and the loss factor (b)
of the considered cheese sauce.
CONCLUSION
The results presented demonstrate the feasibility
of fully coupled EM and thermo-physical
simulations of the microwave heating of a
frozen product, and emphasise the importance
of accurately accounting for the melting
transition and evaporative losses. Future work
should include a higher spatial and especially
temporal resolution, combined with an improved
phase change model, both for the melting and
evaporation transition.
The approach presented in this contribution
is applicable to a wide range of applications.
In particular it is suited to problems in which
dielectric properties change significantly during
the heating process and/or in applications in
which the load in moved inside the applicator.
41-4-43
Q-Factor
Time (s)
Figure 11. Variation of the Q-factor during
the heating period.
The work presented exploits only a small portion
of the capabilities of the PHYSICA solver which
is capable of solving a wide range of physical
problems such as Newtonian and viscoelastic
fluid flow, linear/non-linear structural mechanics
and simulation of chemical reactions. The ability
of independent meshing with cross-mapping to
efficiently couple electromagnetic field analysis
with the analysis of a wide range of thermophysical problems means that the approach has
potential to be used in the simulation of many
practical engineering problems.
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