Finite element modelling of fish cooking by microwave - 11th ...

Finite Element Modelling of Fish Cooking by Microwave
Shixiong Liu, Mika Fukuoka, Noboru Sakai*
Department of Food Science and Technology
Tokyo University of Marine Science and Technology
4-5-7 Konan, Minato-ku, Tokyo, Japan, 108-8477
sakai@kaiyodai.ac.jp (N. Sakai).
ABSTRACT
Three-dimensional finite element models of solid food in microwave oven has been built to predict food
temperature distribution during microwave cooking by using commercial software FEMAP (Siemens Co.
Ltd.), which is a kind of pre-processor. The temperature distribution of food was simulated, based on heat
transfer equation including the internal heat generation owing to microwave heating. To obtain the internal
heat generation, the distribution of microwave electric field in the oven was estimated by using PHOTO-
Wavejω (Photon Co. Ltd.).
Mashed potato with 1wt % sodium chloride, 1wt % salted water were used as solid material and liquid
material for experimental validation. The input microwave power of 600W was used for heating. Results
showed good agreement between the experimental data and predicted values of temperatures. Finally, a
model for fish cooking in salted water had been built for predicting temperature distribution during
microwave heating.
Keywords: Microwave cooking; Dielectric properties; Finite element analysis; Electromagnetic field analysis;
Temperature distribution
INTRODUCTION
For its rapid heating and convenience, microwave heating is widely used in our daily life. However,
disadvantages also exist, such as non uniform temperatures rising under microwave irradiation which depend
on various factors. It demonstrates the necessity of producing an accurate model to predict the temperature
distribution in the materials.
Many researchers have been focused on temperature analysis of microwave heating which has been the
biggest problem for modelling due to the non uniform temperature distribution. Models for temperature
simulating for microwave heating had been built since 30 years ago [1]. With the rapid development of
computer technology, now most models were built based on commercial computer software with analytical
and numerical techniques [2], and complex mathematic analytical methods were brought in to use. Some
researchers simulated models by analyzing heat and mass transfer and used Lambert’s law for assuming a
source term with exponential decay [3]. Others analyzed models by coupling the heat transfer and
electromagnetic equations (Maxwell’s equation) such as FDTD (finite deference time domain) [4] and FEM
(finite element method) [5] analytical method.
As conditions are intricate, few studies could be found on simulating microwave cooking. Although lots of
models had been built to predict the microwave heating in the past decades, conditions of microwave heating
such as the shape and the size of microwave oven cavity and existence of turntable in cavity had always been
selectively neglected in these studies. More importantly, dielectric properties of materials actually were
changed with the temperatures during cooking. However, they were all regarded as the constant in most of
the studies due to the great difficulties of simulating heating including all of these factors together in a 3D
model. As few research has been found on predicting heating the mixture of solid and liquid by microwave
simultaneously, a more considerable and more accurate 3D model needs to be built for simulating this kind of
microwave heating.
Marine fishes which are good to human’s health have a large number of stable consumers in the world,
especially in Japan, while Japanese has the custom of eating fish that cooked in salted decoction. However,
few studies could be found on investigating or modelling cooking fish by using microwave. In this study, we
investigated the cooking of fish, using salmon (Atlantic salmon) as the sample which is one of the most
popular marine fishes for its delicious taste and health benefits.
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Objective of this study is to build a 3D finite element model for simulating fish cooking under microwave
irradiation. Real experimental environment for microwave heating would be considered, mixture of solid
(fish) and liquid (water) would be heated and simulated in this simulation. Temperature distribution of the
sample have been investigated, based on heat transfer equation including the internal heat generation which
was obtained by analyzing of electromagnetic field.
MATERIALS & METHODS
For experimental validation, mashed potato with 1wt % sodium chloride and 1wt % salted water were used as
solid material and liquid material, respectively. Microwave of 600W output at the frequency of 2450MHz
was used for heating, Initial temperature of the environment is 15˚ C, microwave transmitted in a waveguide
that designed to transmit maximum possible power. In this kind of waveguide, the dominant mode of the
transverse electric wave is the TE 10 mode [6]. Finally, a model for fish cooking in salted water had been built
for predicting temperature distribution during microwave heating.
The 3D Finite element model following the real structure including sample, turntable, and microwave oven
were made by pre-processor software. The distribution of microwave electric field and the internal heat
generation in the oven was estimated by using PHOTO-Wavejω (Photon Co. Ltd.). The electromagnetic field
distribution inside the product is solved according to the theory of Maxwell’s equations. The governing
equations for a propagating electromagnetic wave in different form are written as the followings [6]:
∇ ⋅ E = − jωµ
H
(1)
∇ ⋅ H = j εE
(2)
ωε 0
∇ ⋅ E = 0
(3)
∇ ⋅ H = 0
(4)
In the equations of (1) to (4), E is the electric field intensity; H is the magnetic field intensity. µ is
permeability of free space (4π×10 -7 H/m), ɷ is angular frequency (rad/s), ε 0 is the permittivity of free space
(8.85 ×10 -12 F m -1 ), ε is the complex permittivity, andε′ is the dielectric constant while ε ′ is the dielectric
loss factor with j 2 =-1.
jωt
E( x,
y,
z,
t)
= E0(
x,
y,
z)
e
(5)
jωt
H ( x,
y,
z,
t)
= H
0
( x,
y,
z)
e
(6)
' ''
ε = ε − jε
(7)
The energy of microwave which is transmitted through a rectangular wave guide along the z-direction
distributed in the shape of sine curve. Electric intensity of microwave at different position could be calculated
by using following equations:
Z0
E 2
ab
P
t
max = (8)
λ 2
Z
0
377/ 1−
(
λ c
)
= (9)
Where, a, b are the size of wave guide, P is the microwave power, Z 0 is the wave characteristic impedance,
and λ is the microwave length of 147.9 (mm), which transmitted in the wave guide size of 109.2×54.6 (mm),
λ c is the wave length of microwave in free space. Here, λ c = 2Y = 218.4 (mm). Boundary conditions on the
inner wall surface of the oven cavity could be described in equation (11)
yπ
E x
= Emax
sin( )
(10)
y0
E = 0 = 0
tan
H
n
(11)
Microwave heating involves EM interaction of the lossy material (dielectric material having considerable
power dissipation capability) placed between the two plate electrodes. The absorbed MW power per unit
volume (Q, W m -3 ) in the material is proportional to the square of the electric field strength (E, V m
-1 ) and
directly proportional to the dielectric loss factor (E, Vm -1 ) and the frequency (f, Hz). E rms is the root mean
square value of the electric field which is equal to 1/ times of the E-field amplitude as shown in equation
(12) [7].
2 ''
2
Q( x,
y,
z,
t)
= 2π fErms ε ε = πfε
ε '' E (12)
0
0
2

Dielectric properties of 1wt % salted mashed potato, 1 wt % salted water and Atlantic salmon has been
measured by using dielectric constant equipment (HP85070B; Hewlett-Packard Corp., Santa Rosa, CA, USA)
at varied temperatures in a constant temperature and humid chamber (SH-240; Espec Corp., Osaka). Then,
Penetration depth had been calculated by following equation (13).
0.5
λ ⎛ 2
4
2
'( 1 ( ''/
') 1)
⎟ ⎟ ⎞
d = ⎜
π ⎜
⎝ ε + ε ε − ⎠
(13)
Temperature distributions of food were simulated based on Three-dimensional heat transfer equation
including the internal heat generation owing to microwave heating. Heat conduction equation including the
heat generation was described in Eq. (12). Q em is the internal heat generation source of food, which quantifies
the amount of power that dissipated in the sample by dielectric losses.
∂T
ρ C
p
= ∇(
k∇T
) + Q em
(12)
∂t
Experiment of microwave heating had also been done at the same time. Temperature measuring system
(LUXTRON 712; LumaSense Technologies Inc., Santa Clara, California) with optical fibre sensor was used
to measure and record the temperature histories, while image of temperature distribution was captured by
infrared thermal camera.
RESULTS & DISCUSSION
As both solid and liquid materials were contented in the model for simulating fish cooking, we first used
simple materials of solid and liquid separately for simulation and experiment for validation.
Figure 1. Schematic of microwave heating system and container size
Simulation and experiment on heating solid material under microwave irradiation were carried out by using 1
wt % salted mashed potato. Container full filled with mashed potato that located on the turntable in
microwave oven (270 mm×270 mm×185 mm) was rotated and heated for 190s (The continuous rotation
speed is 0.2 π/s) as shown in Fig. 1. Dielectric properties have been measured and shown in Table 1.
Table 1 Dielectric property of 1 wt % salted mashed potato
Temperature (˚C) 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00
Dielectric constant 54.73 52.67 51.02 49.30 46.86 42.81 33.35 27.28
Dielectric loss 20.26 19.95 19.54 18.79 18.22 18.00 15.35 13.79
Penetration depth (cm) 0.72 0.72 0.72 0.74 0.74 0.72 0.75 0.76
Dielectric properties were used as temperature dependent during heating in simulation 1. Heat generation
result of simulation showed the significant change from top to bottom due to the location of wave guide and
the sample size. Temperature distribution of sample on surfaces of horizontal cross-section had been
observed to validate the calculation data. Predicted results and experimental results of temperature
distribution were compared while good agreements have been shown in Fig. 2.
As there’s almost no change in penetration depths of this material at different temperatures showed in Fig. 1,
heat generation distribution were considered as the same at any temperature. We assumed that there would be
same temperature distribution results inside sample which simulated by using the dielectric properties of any
one of the certain temperatures as constant. Dielectric property at 20 ˚C was then used as the constant for
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testing (simulation 2). No significant difference could be observed between simulation 1 and 2 as showed in
Fig. 2, which means, simulation could be simplified by regarding dielectric properties as constant if
penetration depth of this material doesn’t change with temperature.
Figure 2. Comparison of simulating and measured temperature distribution on middle surface
From Fig. 3, penetration depths of all the three different materials were observed that there is nearly no
change at different temperatures. Therefore, we would use the dielectric properties of 20 ˚C (Table. 2) of all
the three materials as the constant value in the following simulating.
Table 2. Dielectric properties of 1% salted mashed potato and water
Dielectric properties Mashed potato Water
Atlantic salmon
(20˚C) (1wt % salted) (1wt % salted)
ε ’ 54.73 74.44 43.46
ε ’’ 20.25 23.97 13.63
Figure 3. Penetration depth of the three materials at different temperature
1wt % salted water was then used as the material of liquid for simulating and experiment validation. During
heating, convection inside water can easily occur which has to be considered in our simulation. If the real
value of thermal properties of water was used in simulating temperature distribution of water under
microwave irradiation as the usual way, hot spot inside the water would be founded in the results of
simulation, which means non-uniform temperature inside the water (Fig. 4, simulation p1’ and simulation
P2’). And, temperature difference will become bigger as time steps pasted. However, from the real
experiment, temperatures inside water were observed being uniform as convection occurred during heating.
Solution has been proposed by assuming water as a material of much more bigger thermal conductivity.
Equals to the effect of convection, the heat would transfer very fast during heating while temperature would
be averaged quickly in a result.
As the model that we used before for simulating mashed potato had been confirmed by experiment, it would
be used again by changing the material property into 1 wt % water for simulating while experiment would be
done at the same time to find out the suitable thermal conductivity setting for water in simulation. And, as the
convection occurs inside water that averaged the temperature, rotation would not be considered here in both
experiment and simulating. Good results of uniformed temperature showed in simulation demonstrated that
thermal conductivity of water of 1000 times’ bigger was suitable for our simulation. By comparing with
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experiment data, temperature results inside water (1 wt % salted) of prediction were found to meet with it
very well (300s) (Figure 4) in the condition of setting thermal conductivity of water into 1000 times’ bigger.
Figure 4. Temperature comparing of assigned point between simulation and experiment
Finally, a model for simulating cooking fish in 1 wt % salted water had been built (Figure 5). Atlantic salmon
in rectangular shape was located inside the water in the container for microwave (600 W) heating for 300s.
Rotation would not considered in this model due to the including material of water and the shape of container.
Figure 5. Schematic of fish cooking model (half)
Table 3. Thermal properties of different materials
Thermal conductivity Specific heat
Material
(W/m/K) (J/K/Kg)
Density
(Kg/m 3 )
Container 0.2 1800 905
Salmon 0.47 3589 1050
Water (1 wt % Salted) 600 (Real : 0.6) 4200 1000
Turn table 1.05 840 2230
Air 0.024 1000 1.3
Figure 6. Heat generation of Atlantic salmon and water
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Figure 7. Temperature distribution of Atlantic salmon and NaCl-water solution in longitudinal section (190s)
Table 3 shows the parameter of thermal properties of different materials we used for simulating. Heat
generation and temperature has been simulated, temperature distribution inside sample could be easily
showed and observed in colour figures (Figure 6 and Figure 7).
CONCLUSION
Models for predicting temperature of solid and liquid materials under microwave irradiation had been
validated separately by using mashed potato (1 wt % salted) and water (1 wt % salted) with experimental
measurements. For solid material, good agreement has been found between simulation and experimental
results. For liquid material, as the convection occurred inside the water during heating, 1000 bigger value of
thermal conductivity of water has been set in the simulation. Equals to the effect of the convection,
simulation results meets the experiment histories very well. Consequently, methods of this study for
modeling microwave heating could be used to predict temperature distributions of sample accurately in
which penetration depth of microwave do not change significantly during heating.
Finally, a three-dimensional finite element model for modeling fish cooking inside salted water under
microwave irradiation has been developed to predict heat generation distribution and temperature distribution
inside fish and water during microwave heating. Internal heat generation was simulated by analyzing the
electric field distribution in the microwave oven. Temperature distribution of food was simulated based on
heat transfer equation including the internal heat generation owing to microwave heating. Temperature
distribution inside sample could be easily showed and observed in colour figures.
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