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Electromagnetic and thermal studies of microwave processing of foods

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ELECTROMAGNETIC AND THERMAL STUDIES OF
MICROWAVE PROCESSING OF FOODS
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Hua Zhang
January 2000
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(c) Hua Zhang 2000
ALL RIGHTS RESERVED
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BIO G RAPHICAL SKETCH
The author was bom to an elementary school teacher’s family in September of 1962,
in central China. Growing up in a remote countryside village where people can see
green mountains and a clear river, he was nurtured by th e great nature more than
his education from schools when kids were taught very little m ath and science during
a turmoil period in the Chinese history. Fortunately, th e author has made up for
what was missed in his childhood by obtaining several degrees from both Chinese
and American u n iversities for his postgraduate education. He obtained a Master of
Science degree from Florida A&M University in Manufacturing Engineering prior
to arriving a t Cornell University for his Ph.D. degree in A gricultural and Biological
Engineering in May of 1996. Before coming to USA as a visiting scientist at National
High Magnetic Field Laboratory in Tallahassee, Florida, th e author had worked as
an electric engineer and engineering educator for many years since he obtained his
Bachelor and Master of Science degrees in Electric Engineering from HuaZhong
University of Science and Technology in 1982, and The Chinese Academy of Science
in 1987 respectively.
iii
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To Ping, Hao-hao a n d m y p aren ts for th e ir su p p o rts, patience, an d .
iv
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ACKNOW LEDGEM ENTS
The author would like to express his sincere gratitude to Professor Ashim K. D atta
for his guidance and patience as major advisor through his Ph.D. program. Profes­
sor D atta’s professionalism and strict scholarly attitu d e benefit the author’s career
for the years to come. My special thanks go to Professors Charles E. Seyler, Ken­
neth E. Torrance and Craig Saltiel for serving as my thesis committee members.
Their suggestions and critical reviews of my research work are very valuable, most
importantly their encouragement is indispensable to the completion of this thesis.
The author would like to thank the US Department of Agriculture and the US
Army for p rovid in g financial support during this study. Assistances in experimen­
tation from Dr. Irwin A. Taub and Dr. Chris Doona of the US Army Laboratory
at Natick were essential in completing the portion on microwave sterilization.
The author also would like to th ank his lab-mates - Xiaolan Shi, Haitao Ni,
Montip Chamchong, Jifeng Zhang for their supports and understanding. In partic­
ular, the help the author received from Mr. Hurf Sheldon and Mr. James Throop
for computer support and measurement setup are greatly appreciated.
v
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Table of C ontents
1 INTRODUCTION AND OBJECTIVES
1.1 Microwave Heating of F o o d s ...................................................................
1.2 Interactions Between Microwaves and Dielectric Materials (Foods) .
1.3 Previous w o rk .............................................................................................
1.4 O b jectiv es...................................................................................................
1.5 Thesis O u tlin e.............................................................................................
1.5.1 Coupled electromagnetic and thermal modeling of microwave
oven heating of f o o d s ....................................................................
1.5.2 Electromagnetics, heat transfer, and thermokinetics in mi­
crowave ste riliz a tio n s....................................................................
1.5.3 Microwave power absorption in single an d multiple compo­
nent m a terials.................................................................................
1.5.4 Heating concentration of microwaves in spherical and cylin­
drical foods for plane wave and cavity h e a t i n g .......................
2
Coupled Electromagnetic and Thermal Modeling of Microwave
Oven Heating of Foods
2.1 IN T R O D U C T IO N ...................................................................................
2.2 PROBLEM FO R M U L A T IO N ................................................................
2.2.1 Approximation of the 3D Microwave Cavity for Electromag­
netic C alculations..........................................................................
2.2.2 Governing Equations and Boundary Conditions for Electro­
magnetics in a Multimode C a v ity .............................................
2.2.3 Governing Equation and Boundary Conditions for Heat Transfer
2.2.4 Coupling of Electromagnetics and Heat T ra n s fe r....................
2.2.5 Input P a ra m e te rs ...........................................................................
2.3 M E T H O D O L O G Y ...................................................................................
2.3.1 Numerical S o lu tio n ........................................................................
2.3.2 Coupling Between Temperature and Electromagnetics . . . .
2.3.3 Experimental S e tu p ........................................................................
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1
2
4
5
6
6
7
7
8
9
11
14
14
16
19
20
20
24
24
26
28
2.4
2.5
RESULTS AND DISCUSSION ............................................................
2.4.1 Heating P attern s for Different Placements of Food in the Cav­
ity: Experimental Verification of Numerical M o d el................
2.4.2 Heating P atterns as Affected by Dielectric Properties . . . .
2.4.3 Effect of Power Levels on Heating U n if o rm ity .......................
CONCLUSIONS .....................................................................................
29
30
33
37
39
3 Electromagnetics, Heat Transfer, and Thermokinetics in Microwave
Sterilization
42
3.1 In tro d u ctio n ...............................................................................................
45
3.2 Mathematical F o rm u la tio n s ..................................................................
49
3.2.1 Governing Maxwell’s equations and boundary conditions . .
49
3.2.2 Governing heat conduction equation and boundary conditions 51
3.2.3 Kinetics of marker formation and s te riliz a tio n .......................
52
54
3.2.4 Input p a r a m e te rs ..........................................................................
3.3 Numerical S o lu tio n ..................................................................................
57
3.3.1 The microwave s y s t e m ................................................................
57
3.3.2 Coupling m e th o d o lo g y .................................................................
59
3.3.3 Convergence s t u d y .......................................................................
59
3.3.4 Comparison with experimental marker y ie ld s ..........................
60
3.3.5 Calculation of therm al time and its d is tr ib u tio n ....................
61
3.4 Experimental P r o c e d u r e ........................................................................
63
3.4.1 Sample p re p a ra tio n .......................................................................
63
3.4.2 Temperature control during microwave h e a tin g .......................
63
3.4.3 Measurements of marker yields .................................................
64
3.5 Results and Discussions ........................................................................
64
65
3.5.1 Electric field d is trib u tio n s ..........................................................
3.5.2 Tem perature history during microwave h e a tin g .......................
66
3.5.3 Spatial distributions of ste riliz atio n ..........................................
67
3.5.4 Effect of property on the uniformities of tem perature and ster­
ilization ............................................................................
68
3.5.5 Need for coupled so lu tio n s..........................................................
73
3.5.6 Changes in sterilization during come-up arid holding times .
75
3.5.7 Industrial Implementation of Microwave Sterilization . . . .
77
3.6 C o n clu sio n s...........................................................
79
4 Microwave Power Absorptions in Single and Multi-Component Ma­
terials
81
4.1 In tro d u ctio n ............................................................................................
83
4.1.1 Microwave Power Absorption in F o o d s ....................................
84
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4.2
4.3
4.4
4.5
4.6
4.7
4.1.2 Factors affecting power absorption .........................................
86
4.1.3 O b jectiv es................................................................................
87
Problem F o rm u latio n ...............................................................................
87
4.2.1 System co n fig u ratio n ............................................................
87
4.2.2 Governing equations and boundary c o n d itio n s ................
89
89
4.2.3 Current e x c i t a t i o n ................................................................
Numerical Solutions..................................................................................
92
4.3.1 Input param eters for m o d e l ................................................
93
Experimental S e tu p ..................................................................................
98
4.4.1 Heating of a single m a te r ia l................................................
98
4.4.2 Simultaneous beating of two materials ...................................
98
Results and D iscu ssio n ...........................................................................
98
4.5.1 Efficiency of power absorption as related to dielectric proper­
ties and v o lu m e ....................................................................... 100
4.5.2 Total power absorption as related to dielectric properties and
volum e....................................................................................... 103
4.5.3 Power absorption as related to surface a r e a ....................... 105
4.5.4 Power absorption as a function of aspect r a t i o ................ 106
4.5.5 Differential power absorption in heating multiple materials . 106
C onclusions.............................................................................................. 112
Im plications.............................................................................................. 113
5 H e a tin g C o n c e n tra tio n s o f M icrow aves in S p h e ric a l F oods. P a r t
O ne: I n P la n e W aves
114
5.1 In troduction.............................................................................................. 117
5.1.1 Distributions of microwave h e a t i n g ........................................ 117
5.1.2 O b jectiv es................................................................................ 121
5.2 Mathematical Formulations forLossy Dielectric Spheres Under Mi­
crowave H eating
121
5.2.1 Distributions of electric fields under plane w av e s............. 122
5.2.2 Distributions of electric fields under standing waves in a cavity 123
5.3 Parameters in describing heating concentration.................................. 125
5.3.1 Wavelength as an important param eter in characterizing fo­
cusing
126
5.3.2 Dielectric property based grouping of fo o d s............................ 127
5.4 Heating Concentrations in Plane Waves:Effect of Dielectric Proper­
ties and S iz e
....................................................... 128
5.4.1 Small electromagnetic s i z e ................................................... 129
5.4.2 Large electromagnetic rad iu s................................................ 130
5.4.3 Intermediate electromagnetic ra d iu s .................................... 131
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5.4.4 Prediction of focusing effect in the literature experimental data 134
5.4.5 Radius for maximum focusing in spherical f o o d s ................... 134
5.5 Effect of frequency on heating c o n c e n tra tio n ........................................ 135
5.6 C o n clu sio n s................................................................................................. 135
6 Heating Concentrations of Microwaves In Spherical and Cylindrical
Foods. Part Two: In a Cavity
138
6.1 In tro d u ctio n ................................................................................................. 139
6.2 Numerical Solution of Dielectric Spheres and Cylinders in a Cavity
140
6.2.1 Heating concentration (HC) i n d e x .......................................... 142
6.2.2 Input p a ra m e te rs .......................................................................... 143
6.3 Experimental D e ta ils ................................................................................. 143
6.3.1 Materials ...................................................................................... 143
6.3.2 P ro c e d u re ...................................................................................... 146
6.4 Results and Discussions .......................................................................... 146
6.4.1 Spatial distributions of heating potentials as functions of food
geometry and p ro p e rtie s ...................................
146
6.4.2 Intensity of focusing as a function of food geometry and di­
electric p ro p e rtie s.......................................................................... 154
6.4.3 Effect of placement in the oven on distributions of heating
p o te n tia l.......................................................................................... 158
6.4.4 Focusing in cavity heating as compared w ith plane wave heatingl61
6.4.5 Comparison with experimentally measured fo c u sin g ............. 167
6.4.6 Comparisons with published experimental d a t a ................... 168
6.5 C o n clu sio n s................................................................................................. 171
6.6 Implications in Microwave Food Process and Product Development
173
A The Definitions of Penetration Depth and Wavelength
174
B Sensitivities of a and (3 With Respect To e' and e"
175
Bibliography
177
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List of Tables
2.1
2.2
Parameter values used in the c a lc u la tio n s ............................................
Placements of foodstuff in the oven for experimental studies . . . .
3.1
3.2
3.3
Parameter values used in the c a lc u la tio n s ...........................................
55
Properties of teflon and plexiglass used in electromagnetic simulation 56
Stages of microwave heating to control the sample tem perature . .
67
4.1
4.2
4.3
4.4
94
Dielectric properties of materials used in this s t u d y ...........................
Parameter values used in the numerical sim u latio n s...........................
95
96
Dimensions of rectangular blocks in sim u latio n s..................................
Dimensions of rectangular blocks for two compartment simulations
and e x p e rim e n ts.......................................................................................
97
The dimensions of the rectangular blocks with same volume (260 cm3) 108
4.5
5.1
5.2
6.1
6.2
6.3
Literature d ata on diameters th a t lead to heating concentration in
cavity heating compared with plane wave model prediction (Q O I >
1 predicts the focusing) ..........................................................................
Largest radii for maximum heating occurring at surface at different
frequencies for different m a te ria ls ......................
Dielectric properties of various food materials used in this study . .
Parameter values used in the numerical simulations. The coordinates
for placement of the food are measured from the lower left comer of
the cavity to the center of the f o o d ...................
Properties of tylose at room temperature used in this study . . . .
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29
120
136
144
145
147
List o f Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
Schematic of the microwave o v e n ...........................................................
Schematic of the coupling of electromagnetic and therm al calculations
Relative changes in dielectric and thermal properties in the temper­
ature range used in this study for high loss f o o d s .............................
The total electromagnetic energy (normalized) calculated from the
models with different mesh sizes . ...................................................
Calculated (left) and experimental (right) tem perature profiles after
15 seconds of heating in top view (a), bottom view (b), and front
view (c)........................................................................................................
The thermal image of the bottom view at location 2 with calculations
on the left and experimental profile on the r i g h t ................................
The thermal image of the bottom view at location 3 with calculations
on the left and experimental profile on the r ig h t................................
Range of temperature (Tmax — Tmin) calculated using the coupled
model compared w ith th a t from uncoupled model for low loss foods.
Dielectric loss factors vs temperature for salt contents at 0.5%, 1%,
and 4 % ................................ ‘ .............................
Non-uniformity of tem peratures (calculated using Equation 2.24) vs
heating time for low loss foods................................................................
Range of temperature (Tmax — Tmin) calculated using the coupled
model compared w ith th a t from uncoupled model for moderate loss
foods .........................................................................................................
Range of temperature (Tmax — Tmi-„) calculated using the coupled
model compared w ith th a t from uncoupled model for high loss foods.
Temperature profiles (in four vertical slices along the axis of the
cylinder) for high loss foods after 5 seconds (left) and 45 seconds
(right) of heating. For the left figure, tem perature ranges from 1.5
to 20.9 °C, as for the right figure, tem perature ranges from 5.7 to
98.5 ° C ..........................................................
xi
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22
23
26
31
32
32
34
34
35
36
36
38
2.14 Non-uniformity of temperatures (calculated using Equations 2.24)
during various levels of energy absorption during heating for uncou­
pled (a) and coupled (b) calculations...................................................
3.1
3.2
Schematic of the CEM microwave oven and pressure v essel............
The dielectric properties of ham with 0.7% and 3.5% salt contents
(data from Romaine and Barringer [7 5 ]).............................................
3.3 Schematic of the pressure vessel, plexiglass layer, and food sample
for heating in the microwave oven.........................................................
3.4 Schematic of the coupling of electromagnetic and therm al calculations
3.5 Convergence of the electromagnetic calcualtions as the mesh size is
r e d u c e d .....................................................................................................
3.6 Positioning of tem perature sensor and sectioning of the cylindrical
shaped food for marker analysis............................................................
3.7 Electric fields in food sample, plexiglass, and teflon vessel at stage
one in heating 0.7% salt ham. For dimensions, see Figure 3.3 . . .
3.8 Measured temperature history (see Figure 3.6 for location) following
the stages of heating in Table 3.3 for ham with different salt levels
3.9
Marker yields at the end of heating following the stages of heating in
Table 3.3 by experimental measurement (a) and numerical prediction
(b) for whey protein g e l ...................................• ....................................
3.10 Marker yields at the end of heating following the stages of heating in
Table 3.3 by experimental measurement (a) and numerical prediction
(b) for ham with 0.7% salt ..................................................................
3.11 Marker yields at the end of heating following the stages of heating in
Table 3.3 by experimental measurement (a) and numerical prediction
(b) for ham with 3.5% salt ..................................................................
3.12 Effect of composition on uniformity of temperatures at the end of
heating following the stages of heating in Table 3.3. D ata is for ham
with different salt le v e ls .........................................................................
3.13 The density functions for volume fraction for ham with 0.7% (a),
1.6% (b), and 3.5% (c) salt contents vs tem perature after 500 seconds
microwave tre a tm e n t...............................................................................
3.14 The volume fraction for ham with 0.7%, 1.6%, and 3.5% salt contents
for thermal time Fq after 500 seconds microwave treatm ent . . . .
3.15 Numerically predicted increase in marker yields (vertical axis) during
sterilization. D ata for ham with 0.7% salt content following the
stages of heating in Table 3 . 3 ...............................................................
3.16 Numerically predicted change in heating potential from initial to 130
. seconds for ham with 0.7% s a l t ...........................................................
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50
56
58
60
61
62
65
66
69
70
71
72
74
75
76
77
3.17 Numerically predicted marker yields at the end of heating using
uncoupled (a) and coupled (b) electromagnetics and heat transfer.
D ata is for ham with 0.7% salt heated following th e stages of heating
in Table 3 .3 .................................................................................................
78
4.1
Schematic of the microwave oven system for simultaneously heating
of two compartments. For single compartment heating, the material
is placed along the central a x i s .............................................................
88
4.2 Schematic of current excitation model used to characterize the power
coupling into the microwave c a v ity .......................................................
91
4.3 The relative power absorption cross section (RACS) for (a) spheres
of 2.2 and 524 ml volume, (b) cylinders of 21.2 and 250 ml volume,
and (c) rectangular blocks of 10 and 540 ml v o lu m e.......................... 101
4.4 Efficiency of power absorption using plane wave analysis [97] . . . .
102
4.5 The efficiency of power absorption (RACS) in spherical foods for
different radii and p r o p e r t i e s ................................................................. 102
4.6 Power absorbed in different volumes of water in a cylindrical con­
tainer. Numerical calculations use experimental d ata at 250 cm3
volume for c a lib r a tio n .......................................................... : ............... 104
4.7 Power absorption as a function of volume for rectangular foods . . 104
4.8 Power absorption as a function of volume for low loss materials
(frozen vegetables) in different s h a p e s ................................................. 105
4.9 Numerical calculations (a) of surface flux vs surface area compared
w ith experimental d a ta (b) from the work of Ni et al. [56] for cylin­
drical containers (dimensions listed in Table 4.2) ............................. 107
4.10 The effect of aspect ratio on power absorption for rectangular blocks
of 260 ml v o lu m e....................................................................................... 108
4.11 Ratio of power absorption in water to th a t in com oil placed in
rectangular c o n ta in e rs ............................................................................. 110
4.12 Relative power absorption in simultaneous heating of two different
materials as rectangular blocks, (a) water and frozen vegetables, (b)
ham and vegetables, (c) water and h a m ............................................. I l l
5.1
5.2
5.3
5.4
Microwave refracton inside a sphere .....................................................
Scatter plot for the dielectric constants and dielectric losses . . . .
Scatter plots showing the expected focusing region for radii equal to
(a) 0.5 cm .and (b) 1.0 c m .......................................................................
Scatter plots showing the expected focusing region for radii equal to
(a) 6 cm and (b) 13.5 c m .......................................................................
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128
132
133
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
Schematic of th e microwave oven system with food used in this study 141
Distribution of electric field in a spherical potato of radius 4 cm
placed in location 1 (Table 6 . 2 ) ............................................................. 149
Electric fields in spheres (potato) for radii of (a) 4 cm and (b) 5 cm,
showing more focusing for the small rad iu s.......................................... 150
Heating potential distributions along radial direction for different
spherical foods of radii (a) 0.8 cm, (b) 2.5 cm, and (c) 5.0 cm . . . 152
Electric fields a t the XY plane through the center for a sphere of 0.8
cm radius showing the shifting of the maximum value for different
materials: (a) frozen vegetables and (b) m e a t ................................... 153
Electric fields a t the XZ plane through the center along axial direc­
tion in cylinders of 0.8 cm radius for different materials: (a) frozen
vegetables and (b) p o ta to e s ................................................................... 155
Heating concentration index (HC) as a function of the radius of
spheres for different food m a te ria ls...................................................... 156
Heating concentration index (HC) as a function of the radius of
cylinders for different food m a te r ia ls ................................................... 158
Electric fields a t the YZ plane through the center of a sphere (water)
of 0.5 cm radius at (a) placement 2 and (b) placement 3 (listed in
Table 6 .2 ) ................................................................................................... 160
Heating potentials inside spheres (properties of potato) of 4cm radius
placed at different locations (Table 6.2)
161
The electric fields at the YZ plane through the center for spherical
potatoes of 4.0 cm radius for (a) placement 2 and (b) placement 3
(Table 6.2)
...............................................
162
Heating concentration index as a function of wavelength for spherical
foods of 0.8 cm r a d i u s ............................................................................ 163
Electric field of cross-section of a sphere of 0.8 cm radius for different
food materials: (a) ham and (b) p o t a t o ............................................. 164
Electric field a t th e XZ plane through the center of a sphere of water
for different radii: (a) 0.8 cm (b) 0.5 c m ............................................. 165
Heating concentration index as a function of Q O I for spherical foods
of 5 cm radius .....................................................
167
Numerically calculated (left) and experimentally measured (right)
temperature distributions for tylose spheres of 2.5 cm radius for (a)
0% , (b) 2% and (c) 4% salt contents placed at th e center and 4 cm
above the floor in the oven (Figure 6.1)
169
The experimental tem perature distributions (a) by Prosetya et al. [70]
and the predicted heating potentials (b) for a cylindrical water load 170
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6.18 The experimental moisture distributions in the spheres (potato) by
Lu [44] for different diameters: (a) 3.2 cm and (b) 6 c m ................
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Chapter 1
INTRO DUCTION A N D
OBJECTIVES
1.1
Microwave Heating of Foods
As microwave energy is quick, efficient, and convenient, now more than 90% of
American households own a microwave oven, and the production of microwaveable
foods is a sizable industry. Microwave food processes, such as microwave cooking,
pasteurization, sterilization, blanching, dehydration, freeze-drying, and thawing, are
either emerging technologies or already in use in the food industry. Recently, mi­
crowave applications in many areas are growing as the preferred process method [79],
such as ceramics and polymer processing, enhancing catalytic reaction, treating haz­
ardous waste, etc..
In food processing applications, there are many advantages th a t need to be
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exploited over conventional methods, for example, microwave heating is a short
time, high tem perature process, thus it can preserve most of the nutrient and colors.
Unfortunately, most industrial applications are still in lab scale, and the wider uses
are yet to come [51, 80]. Microwaveable products, such as TV-dinners, often fall
short of quality expectations.
Because of the cross-disciplinary nature of microwave processing and the compli­
cated patterns of electric fields in a multiple mode cavity, it is easy to understand
why many microwave food developers and processors are doing their jobs based
mostly on guesswork. The m ajor problems are the lack of understanding of non­
uniformity of heating and the behaviors of microwaves in a cavity. Addressing these
problems requires in-depth studies that are possible only w ith the recent advances
in computational resources.
1.2
Interactions Between Microwaves and Dielec­
tric Materials (Foods)
A description of the interactions between microwaves and dielectric materials is
briefly described to understand the complexity of this physical phenomenon.
Foods composed of water, starch, fiber, fat, protein etc., are essentially dielectric
materials. Basically, th e dielectric properties are functions of frequency (details can
be found in [49, 78]), tem perature, and moisture contents [53, 88]. Generally, the
ability of coupling electromagnetic energy for dielectric materials, as well as the
distributions of heating potentials inside the materials, depends on their dielectric
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3
properties, geometry, and size. In a multiple mode cavity, more variables, such, as
the size of the cavity, the locations of the power entry p o rt and the food, are also
to be considered.
Microwave processing of foods has three main features-one is volumetric, the
other two are non-uniformity and complexity. These features can be simply ex­
plained by the characteristics of th e interactions between microwaves and foods,
e.g., reflection, transmission, and attenuation of the waves.
W hen a dielectric object is p ut in electromagnetic waves, the surfaces have
electric charges. Therefore, the dielectric object becomes an antenna to irradiate
electromagnetic waves due to the surface electric charges, e.g., reflection. Similarly,
these surface charges generate a field inside the material. After superimposed to
the incoming field, the result is th e transm itted field.
The reflected and transm itted waves depend on the dielectric properties, the
geometry and size of the object. Physically, the amount of electric charges on the
surfaces, and the geometry and size of the object, determ ine the electromagnetic
field inside. If it is a metallic object (equivalent to an infinitely large dielectric loss
factor), the surface charges are sufficient to generate fields th a t cancel all incoming
waves inside the object since m etal materials can easily assemble a large amount of
free electric charges. In this case, the incoming waves are all reflected. Very lossy
dielectric materials, and materials with high dielectric constants, such as water or
meat juices, can generate large am ount of electric charges on the surfaces, though
significantly fewer than th at of metals, but they still cancel most of the field inside.
As a result, majority of the waves are reflected. When materials have small dielectric
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4
constants and loss factors, such, as frozen foods, the surface electric charges axe much
fewer than th a t in previous cases. Therefore, inside fields are stronger and less wave
energy is reflected.
Not only the amount of the electric charges but also the distributions of them,
which reside on the surfaces of the object, axe im portant factors in determining the
inside fields. For cylinders and spheres, the inside fields axe likely concentrated due
to their curved surface on which electric charges reside.
For a large size (larger than wavelength) object, the transm itted waves can
propagate inside. If the materials are lossy, waves axe attenuated as the energy
is converted into heat. Usually, this decaying in the magnitude of the waves is
exponential with relation to the loss factor in a slab.
In short, the transm itted waves no longer have their strengths, phases, and
directions inside a food material. All of these changes are functions of the dielectric
properties and the distributions of the surface electric charges. Worse yet, these
incom in g waves are also functions of the microwave system and the locations of the
food material in a multiple mode cavity. All of these factors make the microwave
processing of food uneven and hard to predict.
1.3
Previous work
Though detailed literature reviews are given in each of the following chapters, some
of the main points are summarized here. Microwave heating is a well established
science based on Maxwell’s equations. Due to the complexity of the equations, the
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5
general solutions are not obtainable. Therefore, a large volume of publications on
the microwave heating of foods are still empirical and experimental. Numerical
solutions of Maxwell’s equations in recent years started to appeax in the literature,
but the coupled electromagnetic and therm al model is not common in the context
of heating foods. Using numerical methods to characterize microwave heating, in
terms of uniformity and total power absorption, is not found in the literature.
1.4
Objectives
A comprehensive study of microwave heating of foods coupling the elctromagnetics
and heat transfer is proposed here. T he goal is to provide a better understanding
and engineering fundamentals of microwave process, thus, to improve the heating
performance. The specific objectives are to:
1. Develop an electromagnetic model th at includes cavity, waveguide, and cur­
rent excitation to predict the electromagnetic fields in foods for microwave
heating. Use thermal images from an infrared camera to validate the numer­
ical predictions
2. Develop a coupled electromagnetics and thermal model for strongly varying
properties, and investigate the criteria in which coupled solutions should be
sought after.
3. Develop a model to predict therm al-tim e distributions for the application of
microwave sterilization in pressurized vessels. Verify the model using the
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6
measurements of m arker formations and further obtain guidelines for more
uniform cook values for cylindrical shaped solid foods.
4. To correlate the magnitude of power absorption to the volume, shape and
properties of foods in microwave heating using numerical simulations, and
characterize the relative power absorption by different food blocks th a t are
heated simultaneously.
5. Study the sizes and properties th at enable the focusing of microwaves in spher­
ical and cylindrical foods. Using a quantified definition of focus proposed in
this study, develop the engineering fundamentals in characterization of this
most im portant feature of microwave heating.
1.5
Thesis Outline
The thesis consists of five papers and each paper is presented as a chapter. An
overview of each individual chapter is given as follows.
1.5.1
Coupled electromagnetic and thermal modeling of mi­
crowave oven heating of foods
In the first paper, e.g., C hapter two, the numerical model th a t couples electromag­
netics and heat transfer are developed with the experimental verifications using
infrared images. Then, th e criterion is established for a numerical solution th at
needs the coupled model, and the heating uniformity, as affected by power levels, is
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7
also discussed.
1.5.2
Electromagnetics, heat transfer, and thermokinetics
in microwave sterilizations
Chapter three describes the studies of microwave sterilization using the coupled elec­
tromagnetics and heat transfer model as developed in C hapter two in conjunction
w ith first order kinetics of biochemical changes. Using sophisticated technology,
chemical marker formations were measured and compared to numerical predictions
at different locations in the cylindrical samples, thus the model for predicting ster­
ilization is verified. Unlike conventional sterilization, heating patterns can change
qualitatively with geometry and properties (composition) of the food material, and
optim al heating is possible w ith suitable combinations. Combined with marker yield
measurements, the numerical model can give comprehensive descriptions of the spa­
tial time-temperature history, and thus can be used to verify the sterilization.
1.5.3
Microwave power absorption in single and multiple
component materials
Chapter four is the third paper th a t characterizes the microwave power absorptions
as related to the size, geometry, and dielectric properties of foods. It is found that
foods with small loss factors m ay not absorb less power if th e size is sufficiently large,
and high loss foods usually absorb more power at small sizes. Using numerical
simulations and experimental measurements, heating of two compartment meals
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8
is also studied for the relative power absorption of individual compartment with
dissimilar properties.
1.5.4
Heating concentration of microwaves in spherical and
cylindrical foods for plane wave and cavity heating
Chapter five and six are two companion papers. Chapter five first presents the
mathematical formulations for spherical foods heated by plane waves and standing
waves. Then it gives simplified criteria for focusing as affected by the food size and
dielectric properties. Chapter six further studies the focusing of microwaves in a
cavity using numerical simulations to give the intensity information of the focusing
with respect to the radii, for different food materials in spherical and cylindrical
shapes. The numerical and experimental studies of focusing in the cavity heating
reveals th at some focusing characteristics of plane wave heating can be used to
interpret the cavity heating.
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Chapter 2
Coupled Electrom agnetic and
Therm al M odeling of Microwave
Oven H eating of Foods
Abstract
Temperature distributions from heating in a domestic microwave oven is studied by
considering the coupling between the electromagnetics and heat transfer through
changes in dielectric property during heating. The Maxwell’s equations for elec­
tromagnetics and the thermal energy equations are solved numerically using two
separate finite-element software. The coupling between th e software was devel­
oped by writing special modules th a t interfaced these software at the system level.
Experimentally measured tem perature profiles were compared w ith the numerical
predictions. The importance of coupling was evident when th e properties change
9
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10
significantly with temperature, as for low and high dielectric loss materials, more
so for the high loss materials. For m oderate loss materials, when th e properties do
not change as much with tem perature, coupled solution leads to results very close
to those for uncoupled solution.
List of Symbols
Cp
specific heat, J/k g K
E
electric field, V /m
/
frequency, GHz
H
Magnetic field, A /m
h
convective heat transfer coefficient, W /m 2K
I
current, A
kt
therm al conductivity, W /m K
k
wave number
P
power density, W /m 3
T
tem perature, °C
Ti
initial temperature, °C
Tnu non-uniformity of tem peratures, °C
t
time, s
vol volume, m3
Vf
volume of an element, m3
x
position
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11
e
permittivity, F /m
€0
permittivity in air, F /m
e*
complex permittivity, F /m
e'
dielectric constant
e"
dielectric loss
< ff
effective dielectric loss
P
apparent density of the food, kg/m ;
P
magnetic permeability, H /m
U)
angular frequency, ra d /s
cr
electric conductivity, S /m
2.1
INTRODUCTION
The prediction of electric fields and temperature distributions varying with heating
time is im portant in improving the design of the microwave oven systems (microwave
cavity, location of waveguide, etc.)
and microwaveable foods (packaging), food
composition, and etc., that ultim ately leads to more efficient and more uniform
microwave food processing.
In m any microwave heating instances, an exponential decay of power absorp­
tion from the surface of a heated material to its interior (Lamberts’ law) is as­
sumed [44, 66, 98]. Such distributions of power absorption can be proven only
for a plane wave penetrating into a sem i-infinite slab. This is an over simplified
model because the microwaves in a resonant cavity are not plane waves and the
food block has finite dimensions. The interaction between foods and microwaves
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12
r
is quite complicated as microwaves propagating in different media have different
behavior. The microwave medium, in general, can be characterized by its geometry
and dielectric properties. During microwave processing, the properties are functions
of temperatures and always change during heating, sometimes dramatically. These
changes redefine the microwave heating patterns. As a result, the microwave power
absorbed in the food is changed not only in term of overall energy dissipated but
also its spatial distributions. Consequently, the rate and spatial variation of tem­
perature are changed during microwave heating of materials th a t are temperature
sensitive. Thus, ideally a coupled electromagnetic and therm al model in 3D is re­
quired for accurate prediction of tem perature in microwave heating in a cavity such
as the domestic microwave oven.
Solutions for electric fields and heating rates in a multimode cavity such as
the domestic microwave oven will require the solution of Maxwell’s equations in
3D. This is computationally quite challenging. Analytical methods to describe the
electric field intensity distribution in microwave ovens were restricted to stratified
loads th a t have a particular thickness and cover the entire floor of the cavity [84, 94].
Such loads are not representative of food heating situations as the food block would
not typically extend to the walls. In the past, various numerical approaches have
been tried on simplified special cases [19, 65, 95]. Detailed 3D numerical solution of
Maxwell’s equations for a loaded multimode cavity has started to appear relatively
recently [9, 7, 25, 46, 62, 67, 92, 100], perhaps due to extensive computational
resources required for these studies. Outside of food heating, 3D electromagnetic
modeling has also been used by several authors [30, 33, 41].
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13
The change in dielectric properties during heating couples the electromagnetics
with heat transfer. Coupled electromagnetics and heat transfer studies have been
performed in much simpler systems such as ID plane waves [85], ID and 2D plane
waves [3], and 2D cavity [11, 86]. In the case of microwave cavity, 1D/2D studies
are not enough because most of the modes vary in three directions. Moreover,
these studies are only able to give qualitative results th a t are not sufficient for the
applications of microwave processing of foods. Relatively few researchers [9, 21, 46,
92, 100] have performed 3D electromagnetic studies on cavity heating. Although
energy equation was included in some of the studies [9, 46, 92], only Ma et al. [46]
have discussed some coupling issues in their studies. The work of Burfoot et al. [9]
possibly included coupling, but they did not elaborate on this. They also reported
sign ifican t differences between predicted and measured tem peratures which they
attributed to insufficient mesh resolution. The works by Ma et al. [46] and Torres
and Jecho [92] do not consider strong variation in dielectric properties that can lead
to extensive variation in heating patterns. They also did not discuss the criterion
for which coupling should be considered. Consideration of coupling for a 3D cavity
with load requires extensive computing resources and development time, and is not
explicitly included in most of the commercial codes of today. Thus, it is of major
benefit to even find the criterion under which coupled studies are a necessity. The 3D
studies in the literature [9, 46, 92, 100] use Finite Difference Time Domain (FDTD)
method th a t is typically difficult to apply to complex geometry in microwave food
processing, and can require relatively long computational time.
Thus, there is a need to develop a coupled 3D electromagnetics and heat transfer
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14
model for foods with, various shapes heated in a microwave oven w ith temperature
dependent properties and provide information on the extent of such coupling and the
criterion under which such coupling should be considered- The specific objectives
of this study are:
1. To develop a coupled 3D electromagnetics and therm al solution for heating in
a microwave oven with tem perature dependent properties.
2. To verify th e predictions by the coupled model using experimentally measured
temperatures.
3. To develop criterion for food processing situations in which coupling effects
are im portant and should be considered.
2.2
PROBLEM FORMULATION
2.2.1
Approximation of the 3D Microwave Cavity for Elec­
tromagnetic Calculations
A domestic microwave oven (Model No. R-5H06, Sharp Inc., NJ) is modeled here,
as shown in Figure 2.1. To compare w ith experimental results on the same oven, the
dim ensions, location of waveguide, etc., in the model need to be as close as possible
to the actual oven. The cavity dimensions are 40 x 40 x 23.5 cm. The waveguide is
on the top of th e cavity and 4 cm away from center.
To take into consideration th e reflected microwave power, the antenna and
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15
waveguide
excitation
cavity
40 cm
Figure 2.1: Schematic of the microwave oven
waveguide are to be included in the model. During heating, the dielectric prop­
erties of food vary due to tem perature, composition, etc., thus changing the pattern
of microwave scattering and propagation in the cavity and the food. The absorbed
power and its spatial distributions are perturbed by a varying load. Using an
antenna as a microwave source can account for the changes of load without consid­
ering the magnetron, which is th e most complex element in a microwave system.
The antenna which is modeled as a piece of metal carries current as a stimulus to
the microwave system. The am plitude of input power corresponds to the amount
of this excitation current. The power delivered at a certain time is a function of
the system, including waveguide, cavity and the properties of load (food) at that
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16
instance.
As in the real system, the antenna is one quarter of a wavelength away from
the end wall to maximize the excited fields. In practice, waveguides are designed
to propagate only the H iq mode. However, the length of the waveguide is usually
short enough to have plenty of evanescent modes which usually resist analytical
approaches. In the numerical model the waveguide is also short enough (9 cm
long) which is close to the real one. This will not pose any problem for numerical
modeling as the solution is the electric field distribution which includes all the
modes, propagating and evanescent.
2.2.2
Governing Equations and Boundary Conditions for
Electromagnetics in a Multimode Cavity
The electromagnetic fields inside the microwave oven (Figure 2.1) th a t is responsible
for the heating of the food material, are described by the Maxwell’s Equations
V x E = ju n H
(2.1)
V x H = —iu/e0e*E
(2.2)
V • (eE) = 0
(2.3)
V -H = 0
(2.4)
where E = E e_Ju;t and H = He~3Ut are the harmonic electric and magnetic fields,
respectively. The complex perm ittivity e* is given by
e* = e '+ j( e " + — )
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(2.5)
17
where th e real part, c7, is called the dielectric constant and represents the material’s
ability to store electromagnetic energy. The imaginary part, e77+ ^
is the effective
dielectric loss of materials representing the energy dissipation. The perm ittivity e
is defined as
e = e -h je"
(2.6)
Note th a t although e is different from the effective complex permittivity, e*. in gen­
eral, they have the same values in this paper since the conductivity a for microwave
processing of foods is usually considered zero. In equation 4.5, e can be a function
of locations in the food due to tem perature variations. Thus curl E can be written
as:
V -E =
-E
(2.7)
Equation 4.5 indicates th at the non-uniform distribution of perm ittivity causes ad­
ditional electric fields, equivalent to free charges.
After mathem atical transformations, the governingequation for electric field is
V ( f - E ) 4- V 2E + fc2E =
0
(2.8)
where
k 2 = u 2fj,Q€oe*
(2.9)
The wave number, k, is complex and can be expressed as:
k = a + j(3
(2.10)
where th e real and imaginary parts can be derived from Equation 2.9 and Equa­
tion 2.10 as:
a
=
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( 2 11}
18
2 irf / e " (\/l + tan2 6 — 1)
2
(2.12)
where
tan. S = —
(2.13)
The term /3 is associated with the decay of microwave fields. Conventionally, (3 1 is
known as th e characteristic penetration depth, while 2ttcx~1 is the wavelength.
Approximating the cavity walls as perfect conductors in microwave frequency,
the boundary conditions for the microwave system can be stated at the cavity wall
surface as follows
(2.14)
'tangential
The excitation of the microwave power is modeled as:
(2.15)
I = I qsin(u/£)
It takes a sinusoidal form with amplitude I0 at
= 2irf.
Equation 2.8 is not a simple wave equation. It contains an extra term with
first order derivative in non-linear form. The implication is obvious. In the simpLe
case of u n iform dielectric medium, the fields are defined by wave vector k and
boundary values. While for non-uniform medium, which is common in microwave
heating, th e fields are also affected non-linearly by the gradient of permittivity as
in tem perature sensitive materials. This is why coupled solution has to be sought
after for tem perature sensitive materials.
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19
2.2.3
Governing Equation and Boundary Conditions for Heat
Transfer
The electric fields described above lead to volumetric heat generation and is included
in the therm al problem as a source term. The non-uniformity of electric fields inside
the food lead to a distribution of temperature and conduction of energy. Thus, the
transient heat equation for the problem is given by:
= v ' { k t V T ) + p<-x ' T )
( 2 ' 16)
where P (x , T ) is the microwave power and is related to the electric fields by:
P{x, T) = i u - e o ^ E 2
(2.17)
The initial temperature is assumed to be constant:
T = Ti
(2.18)
Inside the microwave oven, air flows at a low velocity over the food, leading to the
boundary condition
- htV T = h ( T - T air)
(2.19)
where h is the surface convective heat transfer coefficient. It includes the effect of
free convection as the food warms up and a small am ount of air flow due to a fan.
The primary purpose of the fan (that is mounted on th e outside of the cavity of
a microwave oven) is to cool the magnetron, the source of microwave energy. It
does not blow directly over the food, but causes some air flow through small cavity
openings. To consider this, the value of h is set to 10 W /m 2K for the temperature
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20
calculations in this paper. In the literature, heat transfer coefficient for microwave
heating without forced convection ranges from 5 [43], 10 [2, 9, 91], and 39 W /m 2K
[89]. In [98], it was suggested th a t a zero h be accurate enough in the modeling of
the microwave thawing process. A sensitivity analysis by Chamchong [10] indicates
that the variations in h have little impact on the predictions of tem peratures since
the heating rate by microwave is much higher than the rate of heat loss due to free
surface convection.
2.2.4
Coupling of Electromagnetics and Heat Transfer
As the material heats up non-uniformly, its properties change. Changes in dielectric
properties, d(T(x, t)) and e"(T(pc,t)) in Equation 2.8, cause the electric fields inside
food to change, thus changing the energy deposition (Equation 4.1) and tempera­
ture profiles (Equation 3.9). This interaction is shown schematically in Figure 2.2.
Implementation of this coupling process is discussed under methodology.
2.2.5
Input Parameters
Table 2.1 fists the values used for the parameters in the simulations. To study the
effects of sample placement, a cylindrical potato is used as the food sample with
dimensions and other parameters shown in Table 2.1. For the tem perature range
from 20 to 80 °C, the properties do not change significan tly for potato and constant
properties are assumed.
In the study of the dielectric property effects, it is expected th a t heating patterns
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Parameter
potato
meat
Radius of load, cm
1.9
2.5
Height of load, cm
4
4
i
51.5
Equation 2.20
[17]
e"
16.3
Equation 2.21
[17]
€0, F /m
8.854 x lO"12
8.854 x 10"12
Ho, H /m
47r x 10-7
47r x 10—7
/ , GHz
2.45
2.45
io, A
1.2
1.2
This study
0
o
Table 2.1: Param eter values used in the calculations
source
20
1
This study
Tair, °C
23
23
This study
p, kg/m 3
1130
1390
[71]
cp, J/kgK
3515
3520
[71]
kt, W /m °C
0.56
0.476
[71]
h, W /m 2oC
10
10
This study
Geometry of the oven
40 x 40 x 23.5 cm
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22
Calculate heat
source from
electric field
Power Loss
module
Solve energy
equadon to get
temperature
profiles
Solve electro-magnetics to
get electric field
Calculate material
properties at new
temperatures
Dielectric Properties
module
Figure 2.2: Schematic of the coupling of electromagnetic and therm al calculations
will be significantly affected due to tem perature increase. To cover the range of
typical food properties, low, medium and high loss foods are included. Dielectric
properties as function of tem perature and salt content are used from Sun et al.
[88]. Their equations for dielectric constant and loss for one group of food materials
(meats and meat juices) are given by
e' = m water(1.0707 - 0.0018485T) + m asft(4.7947) + 8.5452
(2.20)
and
e" =
m^ater (3.4472 —0.01868T -I- 0.000025T’2)
+Tnash.(0.23109T - 57.903) - 3.5985
(2.21)
To compare with the coupled solutions, uncoupled solutions were obtained where
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23
3 .5 -1
Dielectric lo ss
3 .0 2 .5 2.0 -
Dielectric
constant
1 .5 -
Thermal conductivity
Specific h e a t \
1.0
0 .5
20
40
60
80
Figure 2.3: Relative changes in dielectric and thermal properties in the temperature
range used in this study for high loss foods
the electric field pattern and the heating potential calculated from solving the elec­
tromagnetics were kept constant in the thermal analysis.
Figure 2.3 shows th a t dielectric properties change significantly more with tem­
perature than therm al properties for this group of food products. Thus, thermal
properties are kept constant in the temperature calculations due to their relatively
insignificant changes w ith respect to the temperature in the range from 5 to 80° C.
Microwave power delivered to the load is a function of the microwave system
(Figure 2.1), including size and geometry of the cavity and the antenna, locations
of the wave entry port and the antenna, amplitude of the excitation current (io),
and dielectric properties of the load. All other factors remaining constant, the
power absorbed by th e load is a function of the excitation current, To, as discussed
in detail in [12, 96]. For this study, experimental power absorbed by a water load,
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24
as obtained from its tem perature rise, is used to estim ate Jo-
2.3
METHODOLOGY
2.3.1
Numerical Solution
Previous studies [11,46, 92,100] used the FDTD method to study rectangular blocks
of food materials- The FDTD m ethod typically requires larger memory and longer
CPU times [90]. As it is a march-in-time method, the overall dynamic range may
not reach the maximum perm itted by Adaptive Boundary Conditions (ABC) due
to accumulating numerical dispersion [90]. Finally, the FDTD method has more
difficulty with complex geometries th at axe important to consider in microwave
heating of foods. For example, curved shapes can lead to focusing in the heated
materials [67], and need to be considered routinely. Finite element method is more
efficient in computational model building, as in the case of a cylindrical food load
inside a rectangular microwave cavity.
The finite element-based (FEM) computational electromagnetic program EMAS
is chosen considering the need to handle the geometrical complexities of 3D grid
generation for many different shapes of food inside the rectangular cavity. The
software MSC/ARIES is used for building the 3D geometry as well as generating
the finite element mesh. Equations 4.3-4.6 are in frequency domain. In EMAS,
the three components of vector potential A and the tim e-integrated electrical scalar
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25
potential represent th e four degrees of freedom in a m atrix u of unknowns:
[A/]M + [B]M + [KIM = {£}
(2.22)
where the matrices [M\, [B] and [K] are related to m aterial properties, and m atrix
[L] is related to the excitation. Equation 2.22 is solved using iterative conjugate
gradient solution technique.
Tetrahedron elements w ith 10 nodes are used.
A
quadratic shape function is usually preferred in non-linear problems. The to tal
number of nodes is around 15,000 for a full cavity problem. Though a small volume
compared with th at of cavity, the load has a large fraction of the total elements in
the model as smaller element size is needed in the load due to: a) shorter wavelength
of microwave in load (1.8 to 4.5 cm) than th at in air (12.24 cm); and b) the food
block being the domain of interest. The cpu time normally is about 6 hours on a
Hewlett Packard 9000, model 735 workstation.
The determination of maximum size of elements used in the model is based on
a convergence study. Shown in Figure 2.4, the results of computed total electro­
magnetic energy tends to be stable with very little variations (less than 3 percent)
when the element size (food block) is smaller th an 3 mm.
For thermal analysis, Equation 3.9 is solved using the FEM software, MSC/NASTRAN. In NASTRAN, tem perature is discretized and the unknown temperatures
at each node are assembled into a set of linear equations as:
[C;]{Ta} + [Klc + K ? \{ T ,} = IQ t} + {QVt
(2.23)
where the Cl is specific heat matrix, K lc is surface convection matrix, K lb is con­
duction matrix, Q% is convection surface heat flow, and Q9e is heat generation due
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26
1. 0 0 -1
0M
■o
1
0 .9 5
o
f
a> 0 .9 0 ®
c
<0
smo
0 -8 5 ■
0 .8 0 - i
1--
0 .1 5
— I---------- 1----------[---------- 1--------- 1---------- 1---
I
0 .2 0
0 .2 5
0 .3 0
0 .3 5
0 .4 0
0 .4 5
R ec ip ro c al o f m axim um e le m e n t size(1 /m m )
0 .5 0
Figure 2.4: The total electromagnetic energy (normalized) calculated from the mod­
els with different mesh sizes
to microwaves.
The mesh used for thermal analysis is the same as the one for electromagnetics,
which is necessary for coupling- As it is a 3D transient heat transfer problem,
a backward scheme and adaptive time step are selected to ensure the convergence.
The cpu tim e is about 30 m inute s for thermal analysis as only one variable is solved.
2.3.2
Coupling Between Temperature and Electromagnet­
ics
The two software EMAS and NASTRAN have no built-in ways to be coupled for the
thermal-electromagnetic analysis. In order to obtain coupled solutions, it was de­
cided to link EMAS and NASTRAN at the operating system level. Although both
are FEM software and share the same mesh generator interface (MSC/ARTES),
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27
they axe really independent programs. To develop the coupling, as shown in Fig­
ure 2.2, two modules of ‘Power Loss’ and ‘Dielectric Properties’ were developed.
The coupled solution starts with solving the electromagnetic fields in foods using
EMAS. The power density data, obtained from the electromagnetic calculations, is
converted into power loss d ata using Equation 4.1 and inserted as a module into the
NASTRAN input d ata file for temperature calculation. Temperature distributions
are calculated using NASTRAN. Although this bridges the two software EMAS
and NASTRAN, this coupling is still unidirectional and not capable of providing
feedback to EMAS as the dielectric properties change w ith temperature.
In order to complete the coupling, the dielectric properties of foods have to be
modified, based on the temperature profiles obtained from NASTRAN. This is done
by the ‘Dielectric Properties’ module (shown in Figure 2.2) that reads the temper­
ature data on each node and calculates the mean tem perature for each element by
averaging the tem perature values of all the ten nodes associated with the element.
Simple arithmetic mean of the node temperatures is found reasonable for averaging
when the element size is small (the volume of element is about 0.03 cm3) and the
m axim um ratio of two edges does not exceed 4 for any element in the model. The
‘Dielectric Properties’ module uses this average tem perature for an element to calcu­
late its dielectric properties based on the relationships between dielectric properties
and temperatures of th e specific material (Equations 2.20 and 2.21). This ‘Dielec­
tric Properties’ module updates the dielectric properties in the EMAS input data
file. New electromagnetic fields can now be calculated and the coupling of the heat
transfer and electromagnetics is complete. Temperatures calculated in the previous
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28
cycle are used as initial values to march in time. This is done by developing another
module th at inserts tem perature data for each individual node in the NASTRAN
input file.
For updating the dielectric properties for each element, a UNIX script file is
written th at searches for the specific place in the EMAS input file, deletes the
old value and writes the new value. It also updates th e related pointers in th e d ata
base. S im ilarly, another UNIX script file is w ritten th a t searches and updates initial
temperatures and heat generation for each element in the NASTRAN input data
file.
The time step for marching in each coupling cycle depends on the microwave
heating rate and the sensitivity of materials to tem perature. For all calculations,
other than high loss food, a maximum of 10% change in the dielectric loss factor
in any of the elements was used as the basis for a new cycle of coupling. For high
loss food materials, the dielectric properties are updated when the largest increase
in the arithmetic mean temperature for an element is more than 3°C.
2.3.3
Experimental Setup
The samples are heated in a Sharp domestic microwave oven with cavity dimen­
sions as shown in Figure 2.1. After heating, the samples are quickly withdrawn
(i.e., within a few seconds) and exposed to an infrared camera (Model 782 AGA
Thermovision, Agema infrared system, Secaucus, NJ, USA) and therm al images
are obtained. The microwave heating time is set to 15 seconds for limiting the
highest tem perature in foods well below 100° C to minimize evaporation and related
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29
Table 2.2: Placements of foodstuff in the oven for experimental studies
Distance
Distance
from bottom (cm)
from center (cm)
1
4
0
2
4
4
3
6
0
Location
tem perature changes.
Size, shape and dielectric property of foods, as well as the locations of foodstuff
in the cavity are the m ajor factors contributing to the electric fields inside the food.
Several different locations of th e potato cylinders inside the oven axe considered, as
shown in Table 2.2. T he distances in Table 2.2 are measured at the center points
of the bottom faces for the cylinders relative to th e origin located at the bottom
center of the cavity.
2.4
RESULTS AND DISCUSSION
T he model is verified using experimental results, which is described first. Computed
tem perature profiles considering the coupling between therm al analysis and electro­
magnetics are presented for low, medium and high loss food materials. Finally, the
effect of power level on tem perature uniformity is discussed.
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30
2.4.1
Heating Patterns for Different Placements of Food in
the Cavity: Experimental Verification of Numerical
Model
Verifications in heating patterns with placement of food in the oven is discussed here
by comparing the model predictions with infrared thermographs for three different
locations in the cavity (as provided in Table 2.2). The model predictions in this
section are based on the input parameters for potato as shown in Table 2.1. For
placement 1 in Table 2.2, top, bottom, and front views of calculated and measured
temperature profiles are shown in Figure 2.5. Similarly, bottom views of placements
2 and 3 are shown in Figures 2.6 and 2.7, respectively. Considering the multiple
views and at three placements, the thermal images seem to agree with calculations
reasonably well. The time lapse (and consequent convective and evaporative cool­
ing) between end of heating and imaging would probably explain most of the small
discrepancies between the calculated and observed profiles. The material used was
potato th at is over 80% water. As the temperatures reached 70°C, evaporation
can become sign ifican t in such material. Such cooling may explain Figure 2.5a,
for example, where the hot spot location being slightly away from the edge in the
experimental therm al image as compared to the calculations. In Figure 2.5b, the
slight angular shift of the hot spot in the image is due to sample handling and
positioning error. The temperatures in the therm al images are quite close to the
calculated values, but are slightly lower, perhaps due to the surface cooling effect.
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31
Figure 2.5: Calculated (left) and experimental (right) temperature profiles after 15
seconds of heating in top view (a), bottom view (b), and front view (c).
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32
Figure 2.6: The therm al image of the bottom view at location 2 w ith calculations
on the left and experimental profile on the right
Figure 2.7: The therm al image of the bottom view at location 3 w ith calculations
on the left and experimental profile on the right
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33
2.4.2
Heating Patterns as Affected by Dielectric Properties
Uniformity of heating is often the criterion for successful heating of food- To com­
pare the tem perature distributions from computations of coupled and uncoupled
problems, the uniformity of microwave heating is computed using a) the standard
deviation of tem peratures, Tnu given by
where Vf is the volume fraction of the m aterial at tem perature T, and b) temper­
ature differences between the coldest and hottest spots, Tmax —r min. The model
predictions in this section are based on th e input parameters for meats as shown in
Table 2.1.
Foods Having Low Dielectric Loss
A salt content of 0.5% (m a3h = 0.5 in Equations 2.20 and 2.21) is used to repre­
sent low loss foods. The predicted tem perature ranges (Tmax —Tmin) are shown in
Figure 2.8. T he range of temperatures calculated for the coupled solution is lower
than th a t for the uncoupled solution. The dielectric loss factor in the tem perature
range decreases (Figure 2.9), slightly decreasing the heating rate and tem perature
non-uniformity for the coupled solution (Figure 2.10).
Foods Having Moderate Dielectric Loss
A salt content of 1% {mash = 1 in Equations 2.20 and 2.21) is used to represent
moderate loss foods. Temperature ranges in moderate loss foods are shown in
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34
80-
uncoupled
60-
coupled
i-4 0 -
i20-
0 -I
0
20
10
30
40
Time (s)
Figure 2.8: Range of temperature (Tmax —Tmin) calculated using the coupled model
compared w ith th at from uncoupled model for low loss foods.
80
4% salt
60
40
0 .5 % sa lt
20
20
40
60
80
T e m p e ra tu re (°C)
Figure 2.9: Dielectric loss factors vs tem perature for salt contents a t 0.5%, 1%, and
4%
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35
uncoupled
108
*-
-
coupled
4 -
2O -l
0
10
20
30
40
Time(s)
Figure 2.10: Non-uniformity of tem peratures (calculated using Equation 2.24) vs
heating time for low loss foods.
Figure 2.11. It shows that predictions from the uncoupled solution are quite close to
those from the coupled solution, the latter predicting slightly higher non-uniformity.
As the loss factor is rather flat with the increase of temperature, shown in Figure 2.9,
the updating of dielectric properties does not bring significant changes in coupled
solutions.
Foods Having High Dielectric Loss
A salt content of 4% (mash — 4 in Equations 2.20 and 2.21) is used to represent high
loss foods. As seen in Figure 2.12, the range of temperatures are significantly higher
for the coupled solution as compared to uncoupled. In contrast with Figures 2.8
and 2.11 for low and moderate loss food, th e coupling effects are most significant
for high loss foods such as ham and other salted meats.
For the high loss foods, calculated tem perature profiles at 5 and 45 seconds
are shown in Figure 2.13. The pictures show a quarter of four vertical slices at
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36
coupled
80
uncoupled
C
C
H* 4 0
’i
i—
20
0
10
20
30
40
Time (s)
Figure 2.11: Range of temperature (T^ax—Tmtn) calculated using the coupled model
compared with th a t from uncoupled model for moderate loss foods
100
Oo
coupled
80
e
60
.i
40
uncoupled
20
0
10
20
30
40
Time (s)
Figure 2.12: Range of temperature (Tmax—Tmin) calculated using th e coupled model
compared w ith th a t from uncoupled model for high loss foods.
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37
heights of 0.5, 1.5, 2.5, and 3.5 cm, respectively. W hen the tem perature is low (at
5 seconds) some hot spots are close to the center. The dielectric loss is relatively
low at these tem peratures and microwave energy is able to penetrate more easily.
In this case, few visible maxima (hot spots) of the standing waves remain near the
center. As tem perature near the surface increases at a faster rate w ith time, the
dielectric loss near the surface also increases rapidly. Consequently, a shield develops
making penetration of microwave power difficult and most of the power is deposited
near the surface. Thus, at a later time, 45 seconds, hot spots from near the axis
have disappeared and an exponential decay of tem perature along radial direction is
obvious. This trend of surface heating is expected to continue or be enhanced at later
times. Thus, changes in dielectric properties seem to be the m ajor factor deciding
on the need for a coupled solution. Coupling effects are significant for tem perature
sensitive foods such as high salt content meats, and these effects diminish as foods
with lower dielectric loss values are considered.
2.4.3
Effect of Power Levels on Heating Uniformity
Microwave oven power levels affect heating rates. Thus, it is useful to consider the
importance of coupling effects at various power levels. Changes in power level are
implemented in the calculations by reducing the excitation. Non-uniformities, cal­
culated using Eqn. 2.24 for high loss foods, are shown in Figure 2.14 as increasing
amount of energy is absorbed during heating. T he energy absorption is computed
from calculated temperatures by integrating over volume and heating time. To com­
pare the non-uniformities for different power levels, same values of energy absorption
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38
h o t s p o t s in s id e
tim e = 5 s
tim e = 4 5 s
Figure 2.13: Temperature profiles (in four vertical slices along the axis of the cylin­
der) for high loss foods after 5 seconds (left) and 45 seconds (right) of heating.
For the left figure, temperature ranges from 1.5 to 20.9 °C, as for the right figure,
temperature ranges from 5.7 to 98.5 °C
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39
that correspond to different heating times, are obtained through interpolation. Fig­
ure 2.14 shows that the non-uniformities axe always underpredicted for uncoupled
calculations. Reducing the power level reduces the non-uniformity, primarily by
allowing more time for diffusion. Effects of power level are more significant for high
loss materials, as the power reflected from the food surface (unabsorbed power)
increases with, temperature for such materials. Thus microwave heating is a com­
promise between the rate of heating and its uniformity.
2.5
CONCLUSIONS
1. A methodology was developed for coupling the elctromagnetics and heat trans­
fer in th e analysis of microwave oven heating. Calculated heating patterns
were verified using infrared therm al images.
2. Results show th a t heating patterns in food change substantially w ith location
inside th e oven.
3. Coupled solutions should be sought when dielectric property changes signifi­
cantly w ith temperature, such as salted meats. For moderately lossy materials,
when dielectric properties are not th a t sensitive to temperature, uncoupled so­
lution provides reasonably good estimates of temperatures.
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40
6-
full power
2 0 power
1/3 power
5-
2
-
0.0
0.5
1.0
Energy absorbed (kj)
1.5
(a)
7 —r
6
fuOpower
2/3 power
t/3 power
-
54-
2-
00.0
0.5
1.0
Energy absorbed (kJ)
1.5
(b)
Figure 2.14: Non-uniformity of tem peratures (calculated using Equations 2.24) dur­
ing various levels of energy absorption during heating for uncoupled (a) and coupled
(b) calculations
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Acknowledgment
This research was supported by grant num ber 94-37500-0586 from the United States
Department of Agriculture under the National Research Initiative Competitive
Grant Program . Supercomputing support from the Cornell Theory Center is also
acknowledged.
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Chapter 3
Electrom agnetics, H eat Transfer,
and Therm okinetics in Microwave
Sterilization
Abstract
Sterilization of solid foods using microwave power is studied using numerical mod­
eling and specialized experimental verification. The Maxwell’s equations and the
heat conduction equation are coupled using two separate finite-element software
through specially w ritten modules. Spatial distributions of thermal-time, repre­
senting sterilization, are calculated from tim e-tem perature history and first order
kinetics. Experimentally, concentrations of marker chemicals formed during heat­
ing are measured and taken as indicators of therm al time. Experimental d ata on
marker formation combined with numerical calculations provide an accurate and
42
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43
comprehensive picture of the sterilization process and is a m ajor step in estab­
lishing microwave sterilization processes. Unlike conventional sterilization, heating
patterns can change qualitatively with geometry and properties (composition) of the
food material and an optimal heating is possible with their suitable combinations.
Combined w ith marker yield measurements, the numerical model can give compre­
hensive descriptions of the spatial tim e-tem perature history, and thus be used to
verify the sterilization.
List of Symbols
Cp
specific heat, J/kgK
c
mass concentration, kg/m 3
E
electric field, V /cm
Ea,b
activation energy for microorganism, kcai/mol
E a>m activation energy for marker, kcai/mol
F0
therm al time at 121°C, minute
H
Magnetic field, A/cm
h
convective heat transfer coefficient, W /m 2K
I
current, A
kt
therm al conductivity, W /m K
k0
frequency factor
kb
rate constant for marker
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km
rate constant for microorganisms
kfj,TR rate constant for marker at reference tem perature
km,TR ra te constant for microorganisms at reference tem perature
M
marker concentration
P
power density, W /m 3
R
universal gas constant, J/km olK
T
tem perature, °C
Tnu
non-uniformity, °C
Tr
reference temperatinre, °C
t
tim e, s
v
volume, m3
x
position
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45
e
perm ittivity F /m
eo
perm ittivity in air F /m
e*
complex perm ittivity F/m
er
dielectric constant
e"
dielectric loss
effective dielectric loss
P
apparent density of the food, kg/m 3
OJ
angular frequency
a
electric conductivity, S/m
P
magnetic permeability, H/m
3.1
Introduction
In sterilization of foods and biomaterials, undesirable changes such as loss of nutri­
ents accompany th e desired destruction of bacteria. However, many of the undesired
changes are less sensitive to temperature as compared to the bacterial destruction.
Thus, a higher tem perature and shorter tim e process has long been recognized as
desired, i.e., one th a t would destroy the microorganisms in a short time while de­
stroying relatively less of the desired quantities, such as nutrients. The difficulty
with a higher temperature-shorter time process is its practical implementation for
reaching a high tem perature quickly. The long come-up time often destroys a sig­
nificant quantity of desired components. Microwaves offer an unique opportunity
to raise the tem perature quickly without th e diffusional limitations, thus leading
to lower therm al destruction during the come-up time. However, the spatial non-
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46
uniformities of energy deposition during microwaves and their changes during the
heating process can be dramatically different from conventional heating. This study
describes a comprehensive understanding of the process using modeling and exper­
imentation.
Microwave sterilization has been a topic for research [31, 52, 82] as well as com­
mercial application [29]. However, its commercialization has been plagued with
several issues [80]. Simply having one or a few temperature readings during the
microwave heating is not enough to provide a complete thermal picture of the pro­
cess because the heating patterns are uneven, difficult to predict, and change dur­
ing heating. Hence, research on microwave sterilization have inconsistent, even
conflicting conclusions about the effectiveness and advantages over conventional
methods [31], due to a lack of comprehensive information on the time-temperature
history.
The ability of microwaves to provide less therm al degradation is not univer­
sally true and it depends on a number of food and oven factors. Food factors,
such as dielectric properties, size and shape, play more important roles as com­
pared to conventional heating because they affect not only the magnitude of heat
generation, but also its spatial distribution. As the food is heated, its dielectric
properties th a t determine the microwave absorption changes. This changes the
heating patterns qualitatively, unlike conventional heating where the qualitative
nature of the heating remains unchanged even as thermal properties change during
heating. Thus microwave sterilization would involve coupled electromagnetic and
thermal studies. A complete picture of the microwave sterilization involving coupled
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47
thermal-electromagnetic studies w ith a comprehensive experimental verification of
the computations has not been performed, which are the intended objectives in this
study.
Coupled solutions of the Maxwell’s equations in a cavity (e.g., the domestic
microwave oven) and the energy equation in 3D, that is fundam entally necessary
for the study of microwave sterilization, is computationally dem anding and only few
studies have been reported. Solutions of Maxwell’s equations for simplified special
cases have been presented [3, 18, 65, 95]. More general solutions of Maxwell’s
equations for 3D cavity have been reported [99, 9, 92, 24, 46, 62, 100]. Only a
few of these have coupled electromagnetics with the energy equation [99, 9, 92, 46]
but did not elaborate (except [99]) on the effect of property changes during heating
on the spatial distribution of the electromagnetic field patterns. These coupled
studies also did not cover high enough temperatures, as is true for sterilization where
temperatures greater th an 121°C are reached starting from room tem perature. Such
a large tem perature change for some materials (such as foods w ith significant water
and ions) can cause dram atic changes in heating (and sterilization) patterns during
the heating process. This is a unique feature of the microwave sterilization process
and can only be studied by strongly coupling the electromagnetics and heat transfer.
Considerable spatial non-uniformity of microwave heating an d its significant
changes during the heating process effectively limits the commercialization of the
microwave sterilization process. Comprehensive description of th is non-uniformity
for a sterilization process has not been made. A powerful descriptor of the integrated
time-temperature effect and its spatial variations is the therm al tim e distribution
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48
[16, 54], which, has not been used in the context of microwave sterilization. A
computational study of coupled heat transfer and electromagnetics would also re­
quire comprehensive experimental verification of the tim e-tem perature effects, for
such sterilization processes to be ever accepted commercially and by the regula­
tory authorities. Thus, computed thermal-time distributions would also need to be
experimentally verified. A direct way to verify would be to spatially sample the
sterilized m aterial and analyze for bacterial concentration— this has major difficul­
ties [69]. An exciting alternative is the use of intrinsic chemical markers th at are
formed during the heating process due to the reaction between protein and ribose or
glucose [37, 69]. Measured marker formations can comprehensively verify the calcu­
lated thermal-time distribution and, together with the com putational results, can
provide the confidence and the insight into the development of successful microwave
sterilization processes.
The specific objectives of this study are:
1. To develop a numerical model th at couples the electromagnetics, heat transfer
and first order kinetics to predict tem peratures and therm al-tim e distributions
during a sterilization process.
2. To comprehensively verify the model predictions of the therm al-tim e distri­
butions during sterilization using the experimental measurement of intrinsic
chemical marker formation during the heating process.
3. Develop insight into the heating and sterilization patterns as they unfold dur­
ing th e heating process as a function of material (food) property.
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49
3.2
Mathematical Formulations
3.2.1
Governing Maxwell’s equations and boundary condi­
tions
A schematic of the microwave heating system with the cavity and a cylindrical food
is shown in Figure 3.1. The food is placed inside a pressure vessel to be able to reach
high tem peratures needed for sterilization. Since high pressure developed in the
vessel stops further evaporation, changes in water content of the sample is considered
small and is neglected in this study. Microwave reflection and transmission inside
the cavity and the foods are described by the Maxwell’s equations:
(3.1)
V
X
H = —jcd€Q€'E
(3.2)
V • (eE) = 0
(3.3)
V -H = 0
(3.4)
where E = E,e~jujt and H = He~Jwt are th e harmonic electric and magnetic fields,
respectively. The complex permittivity, e*, is given by:
(3.5)
The cavity an d waveguide walls are metalhc and can be considered as perfect electric
conductor (PEC), where the following boundary condition applies:
(3.6)
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50
waveguide
excitation
cavity
28.5 cm
pressure
vessel
33.5 cm
39 cm
Figure 3.1: Schematic of the CEM microwave oven and pressure vessel
The excitation of the microwave power is treated as sinusoidal at 2.45 GHz (domestic
oven frequency)- given by:
I = I0 sin (art)
(3.7)
The microwave power absorption in food, P (x ,T ), is related to the electric fields
by:
P (x ,T ) = i We0 < / / E 2
This absorbed energy raises the food temperature.
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(3.8)
51
3.2.2
Governing heat conduction equation and boundary
conditions
The uneven volumetric heat generated by microwaves, P (x ,T ), diffuses inside the
food and is convected into surrounding air th at is typically colder. This is described
by the transient heat equation:
QT
pCp^ = V - (kV T ) + P (x , T )
(3.9)
Inside the microwave oven, air flows at a low velocity over the pressure vessel, leading
to the boundary condition
- k V T = h { T - T air)
(3.10)
where h is mainly due to free convection on the pressure vessel surfaces as the sample
warms up. There is small air flow inside the cavity due to a small fan outside the
cavity, leading to a small h. In this study, h is assumed to be 5 W /m 2K (see analysis
in [99]). The initial tem perature of the food is assumed uniform
T = Ti
(3.11)
For most bio-materials including foods, the heat generation is a non-linear func­
tion of tem perature due to the varying dielectric properties. As the m aterial is
heated, its dielectric properties change, which changes the electric field causing the
heat generation. Thus, a solution has to couple th e Maxwell’sequations
(Equa­
tions 4.3 to 4.6) w ith the heat conduction equation (Equation 3.9).W hen E in
Equation 4.1 changes during the process, it is obvious th a t the set of non-linearly
coupled partial differential equations need to be solved numerically.
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52
3.2.3
Kinetics of marker formation and sterilization
The spatial tem perature variations during microwave heating described in the pre­
vious sections lead to spatial variation of time-temperature history and thus the
thermal destruction of microorganisms, nutrients and any other biochemical reac­
tions, which are now discussed.
Sterilization
During sterilization, our concern is the destruction of microorganisms from a safety
standpoint and destruction of nutrients from a quality standpoint. These destruc­
tion rates are usually described as first order reactions, and can be w ritten for
bacterial destruction as
~
at
= he
(3-12)
where c is the concentration of any component at time t, and kb is the reaction rate
constant. The rate constant is a function of tem perature T, generally assumed to
follow Arrhenius law
kb = koe s r
(3.13)
where kQis called the frequency factor, E a,b is the activation energy for th e bacterial
destruction, and R is the universal gas constant. The solution to Equation 3.13 is
given by
] n - = k 0 T exp{ - E a b/R T )d t
c
Jo
(3.14)
Instead of referring to a final concentration c, the food literature uses an equivalent
heating time Fq th a t provides the same final concentration when th e tem perature
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53
T is constant at a reference temperature T r (usually chosen as 121°C). Using this
definition, equivalent heating time Fq can be derived as
*-^
^ & (g '-?))"
( 3 ' 1 5 )
where F a.6 is the activition energy for C-botulinum [45]. The lowest value of Fq is
of interest from food microbiological safety considerations.
Marker Formation
Direct experimental verification of the spatial variation of bacterial destruction
(Equation 3.15) poses challenges. Instead, the intrinsic marker chemicals whose
extent of formation is a function of the tim e-tem perature history has been used to
verify such effects[37]. For example, the reaction between protein precursors and
ribose or glucose yield some chemical compounds. Measured spatial distribution of
these marker chemicals can be compared w ith the numerically calculated thermal­
time (Equation 3.15) obtained from electromagnetic and thermal modeling.
Formation of intrinsic chemical markers axe described as first order reactions
given the assumption of an infinite source of protein precursors in food materials [50,
76]
^
= M M x, - M)
(3.16)
where M is the concentration of intrinsic chemical marker at time t, Moo is the
final amount of marker formation, and km is the rate constant. The solution to this
equation is
M
= 1 - exp (-fcn .fT )
M.
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(3.17)
54
where F™ is given by Equation 3.15 w ith E a^ for bacterial destruction is replaced
by E a,m for marker formation
(3-i8>
Since the marker formation defined by Equation 3.17 and the bacterial destruc­
tion defined by Equation 3.15 depend on the time-temperature history in the same
manner, verification of the calculated marker formation using experimental mea­
surements is analogous to verification of bacterial destruction in sterilization.
3.2.4
Input parameters
Table 3.1 fists the numerical values for the parameters used in th e simulations.
Dielectric properties of the food materials as functions of tem perature are given in
Figure 3.2 for one food material with two different compositions (ham with 0.7%
and 3.5 % salt, d ata for 1.6% salt has intermediate values and is om itted for clarity).
The thermal conductivity and specific heat of the two materials, whey protein and
ham, used in the experiments increase slightly with temperature [71]. Their values
are extrapolated beyond 100° C in this study due to lack of availability of such
data. Normally, at high temperature (up to 130°C) food therm al properties follow
the trend of th a t under 100° C [28]. T he properties of pressure vessel materials
(plexiglass and teflon) are also fisted in Table 3.2.
The dielectric properties of the experimental material (ham) are shown in Fig­
ure 3.2 as a function of temperature from the work of Romaine and Barringer [75].
Temperature variation in dielectric loss factor is stronger at higher salt content
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55
Table 3.1: Parameter values used in the calculations
Parameter
Whey protein gel
Ham
Radius, cm
1.45
1.45
Height, cm
6
6
46.3
€q., r m F /m
8.854 x 10-12
8.854 x 10-12
Ho H /m
47r x 10~7
4?r x 10-7
/G H z
2.45
2.45
0
o
Figure 3.2
S + je "
20
20
this study
Taxy, °C
23
23
this study
0.595
0.595
this study
tr,° c
121
121
[45]
p, kg/m 3
1100
1390
[71]
Cp, J/k g K
4.032 + 1.46 x 10_3r
0.541 + 1.35 x 10~3T
[71]
kt, W /m °C
3.49 + 1.46 x 10~3r
0.449 + 1.35 x 10~3T
[71]
h, W /m 2K
5
5
this study
Ea.,m, kcai/m ol
13.0
13.0
[76]
Eajb, kcai/m ol
69.3
69.3
[76]
fcm, marker-2, /m inute
11.4
11.4
[45]
kb, C-Bot, /m inute
10.96
10.96
[45]
ib, A
/17.3
Source
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[75]
56
140120 -
80-
i o s s f a c to r (3 .5 % )
c o n s t a n t (0 .7 % )
c o n s t a n t (3 .5 % )
6040fo ss f a c to r (0.7% )
20-
20
60
40
80
100
120
T e m p e r a tu r e (°C)
Figure 3.2: T he dielectric properties of ham. with 0.7% and 3.5% salt contents (data
from Romaine and Barringer [75])
Table 3.2: Properties of teflon and plexiglass used in electromagnetic simulation
Teflon
Plexiglass
Source
2.1 + jO.00063
3.45 + /0.138
[15]
kt, W /m °C
0.024
0.052
[15]
p, kg/m 3
1427
1121
[15]
Cp, J/kgK
1372
1078
[15]
Param eter
e'+je*
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57
where ionic effects dominate.
Microwave power absorbed by the load is a function of the microwave system
(Figure 3.1), including size and geometry of the cavity and the power source (as
modelled by the excitation probe), locations of the wave entry port, amplitude of
the excitation current (To), and the dielectric properties of th e load (food). All
other factors remaining constant, the power absorbed by the load is a function of
the excitation current, IQ, as discussed in detail in [12, 96]. For this study, I q is
estimated from the experimental power absorbed by a water load as obtained from
its tem perature rise.
3.3
Numerical Solution
3.3.1
The microwave system
Figure 3.1 shows the schematic of the microwave cavity, waveguide, food and pres­
sure vessel. The microwave oven used in this research is MDS-2000 Microwave
Digestion System (CEM Corp. Matthews, NC). Using the programmable control
system of the oven with a tem perature sensor (Figure 3.3), a pressure vessel con­
taining the food is set to maintain a sterilization temperature at 121°C.
The pressure vessel is made of teflon having 5 mm wall thickness. Between the
food sample and pressure vessel, there is a plexiglass layer as shown in Figure 3.3.
The dielectric loss factor of teflon and plexiglass are small (Table 3.1) so th at the
pressure vessel and plexiglass do not absorb a significant amount of power. However,
they affect th e heating pattern of food sample inside due to their large dielectric
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58
temperature probe
%
food sample
plexiglass
teflon vessel
12 cm
%
2 cm
5 cm (dta.)
Figure 3.3: Schematic of the pressure vessel, plexiglass layer, and food sample for
heating in th e microwave oven
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59
constants (Table 3.1).
T he dimensions shown in Figure 3.1 are used in the numerical simulations. The
microwave source is modeled as an excitation probe th at carries alternating current
at a frequency of 2.45 GHz. Along with the waveguide, the excitation can account
for the reflected power from the cavity (for details, see [99]). Thus the model can
be used to quantify the relative power absorptions by different loads.
3.3.2
Coupling methodology
The governing equations were solved using finite element methods. Two commercial
FEM software packages were used— EMAS (Ansoft Corporation, Pittsburgh, PA)
for electromagnetic fields and NASTRAN (MacNeal-Schwendler Corporation, Los
Angeles, CA) for temperature distributions. Coupling is not built-in these software
and system level codes were •written in C language to develop the two way coupling
as shown in Figure 3.4. Additional details of the coupling process are not included
here and the reader is referred to [99]. The computational tim e was about 6 cpu
hours for each run of electromagnetics and about 0.5 hour for each run of heat
transfer on a HP 9000, model 735 workstation.
3.3.3
Convergence study
Figure 3.5 shows how one of the solution variables (total energy aborbed) changes as
mesh size is reduced. W ithout losing much accuracy, a size of 3 mm between nodes
are selected for th e meshes generated in this study. This is necessarily a compromise,
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60
Power Loss
Calculate heat
source from
electric field _
Solve electro-magnetics to
get electric field
Yes
No
Solve energy
equation to get
temperature
profiles
Start
Calculate material
properties at new
temperatures
Dielectric Properties
Marker
Yield
Figure 3.4: Schematic of the coupling of electromagnetic and therm al calculations
otherwise the computation time can be significantly longer. A n acceptable criterion
for finite element solution of the Maxwell’s equations for plane wave has been shown
to require six grids per wavelength [47]. In cavity, however, the wavelengths are
not unique as multiple modes are resonated. Nevertheless, 3 mm node size was
considered a compromise between accuracy and excessive computing time.
3.3.4
Comparison with experimental marker yields
As explained earlier, measured marker formation th a t is a first order reaction is
treated as representative of sterilization. Marker formation is calculated using Equa­
tion 3.17 for every finite element in the sample. The com putational results are com­
pared with experimental marker measurements, in which samples are divided into
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61
I.O O -i
co 0 . 9 5 E
w
O
f 0.90g>
cfflCD
0.85-
0.800.15
0.20
0.30
0.35
0.40
0 .5 0
R e c ip ro c a l o f m axim um e le m e n t size(1 /m m )
Figure 3.5: Convergence of the electromagnetic calcualtions as the mesh size is
reduced
six sections as shown in Figure 3.6. An average value of computed marker formation
in each of the six sections are calculated as
M = — T M iV i
voUt
(3.19)
where vq is the total volume of th e section, Vi is the volume of zth element, and n
is the to tal number of elements in the section.
3.3.5
Calculation of thermal time and its distribution
The numerical model calculates the thermal time Fq using Equation 3.15 using
average tem perature for each element. The small volume of a typical element (0.03
cm3) should lead to sufficiently accurate spatial distribution of the thermal time
values. Since thermal tim e is a continuous variable, it is presented as a density
function for volume fraction following the procedure developed by D atta and Liu
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62
Tip of
temperature
sen so r
Top
Center
Bottom
Outer ring
‘Core
Figure 3.6: Positioning of tem perature sensor and sectioning of the cylindrical
shaped food for marker analysis
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63
[16].
3.4
3.4.1
Experimental Procedure
Sample preparation
A solution containing 20% whey protein concentrate (Alacen 878, from New Zealand
Milk Product, Santa Rosa, CA), 1% ribose and water was heated in a water bath a t
80°C for 45 min. The resulting gel was cut into cylinders of diameter 1.45 cm and
height 6 cm. The ham material was specially prepared to control the uniformity and
salt content as well as the moisture level. Two days before the experiments, ham
was injected w ith 1% ribose, which is to enhance the M-2 marker yields (see [37] for
details about M-2 marker). It is assumed th a t during this time period the ribose
will diffuse to a uniform concentration within the sample. The ham was also cut
into the same shape and size as the whey protein gel.
3.4.2
Temperature control during microwave heating
Samples are heated in the microwave oven starting from room temperature. The
microwave power level is varied from 10% to 40% of th e full power (630 W). To
reduce the temperature overshooting and increase the heating uniformity, the heat­
ing duration is divided into three stages, as detailed in Table 3.3. After stage 3,
the probe temperature reaches approximately the target tem perature of 121°C (the
time for reaching this tem perature is considered come-up time) and further tem ­
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64
perature control at ±2°C around the target tem perature is achieved by turning the
microwaves on and off. Heating is continued for another four minutes or so, which
is considered holding period.
Following the microwave treatment, the pressure inside the vessel was released
by unscrewing the lock nut, which reduces the tem perature immediately to less than
100°C. The sample was removed and immediately (within ten seconds) sectioned
for marker analysis.
3.4.3
Measurements of marker yields
About one gram of material from a randomly selected location within a section
(Figure 3.6), was homogenized with four times as much volume of water using a
Polytron (Brinkman Instruments, Westbury, NY), centrifuged, and filtered through
a 0.45 micron membrane filter. The amount of marker in this aqueous extract was
measured by anion exclusion chromatographic (AEC) separation and photodiode
array (PDA) detection. The AEC-PDA system comprises of a Wescan (Deerfield,
IL) anion exclusion column and a Waters 990 PDA detector (Milford, MA). The
eluant used was a 10 mM sulfuric acid solution and the flow rate was 1 mL/m in.
The marker concentration data are recorded in a computer.
3.5
Results and Discussions
Temperature and marker yields, computed numerically and measured experimen­
tally, are presented in this section for microwave heating of whey protein gel and
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65
V/cm
2.09E+1
9.10E +0
food sample
5.08E+0
3.06E+0
plexiglass
teflon vessel
Figure 3.7: Electric fields in food sample, plexiglass, an d teflon vessel at stage one
in heating 0.7% salt ham. For dimensions, see Figure 3.3
ham samples. Effect of property changes on the sterilization process is described
using thermal-time distributions.
3.5.1
Electric field distributions
An example of computed field distributions in the pressure vessel and food assembly
is shown in Figure 3.7. As expected, electric fields in th e food sample are generally
lower because of higher dielectric constant. Even though high electric fields exist in
the vessel, there is very little energy loss due to the sm all loss factor of teflon.
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66
1.6% salt
100 -
80-
60-
40-
20-
O
100
200
300
400
500
Time (seconds)
Figure 3.8: Measured tem perature history (see Figure 3.6 for location) following
the stages of heating in Table 3.3 for ham with different salt levels
3.5.2
Temperature history during microwave heating
Temperature history for the probe location in Figure 3.6 is shown in Figure 3.8 for
two materials with different lossiness (ham with two different salt contents). During
the initial stages of heating, microwave focusing effect in a less lossy material (ham
with 0.7% salt content) leads to faster tem perature rise near the center. The higher
salt content (3.5%) ham prevents much of the microwave energy from reaching the
center (no focussing effect here), making the tem perature at the center increase
slower. As heat gradually diffuses from outside warmer locations to colder inside,
probe temperature rises to 127°C even after microwaves are turned off (following
Table 3.3) at 121°C. Therefore, the probe tem perature is not representative or
informative unless th e overall heating pattern is clear due to the significant spatial
variation (discussed later).
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67
Table 3.3: Stages of microwave heating to control the sample tem perature
stage
3.5.3
power level
target
holding
whey protein gel
ham
tem perature (°C)
time (s)
1
40%
30%
60
30
2
30%
20%
90
90
3
20%
10%
121
240
Spatial distributions of sterilization
Spatial distribution of the numerically predicted and experimentally measured marker
formation are shown in Figures 3.9, 3.10 and 3.11 for three different materials (whey
protein gel, ham with 0.7%, and 3.5% salt contents, respectively). The higher con­
centrations of markers in the center compared w ith the outer region in these figures
indicate focussing of th e microwaves for lower loss materials (whey protein gel and
0.7% salted ham). This effect is not seen in Figure 3.2 in the heating of high loss
material (3.5% salted ham ). The model predictions agree with experimental data
very well (Figures 3.9, 3.10, and 3.11). The experimental and numerical d ata are
also in agreement in predicting overheating in the upper p art of the cylinders, an­
other feature of microwave heating. Consistent higher heating at the top of the
samples indicates th a t the cavity standing wave p attern is not significantly changed
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68
by the loads with different material properties. It can be concluded th a t th e over­
heating at the top is mainly associated with th e sample locations in this specific
cavity (Figure 3.1), as the same location is used for the three cases.
There are, however, some discrepancies between model predictions and experi­
mental d ata on marker formation. For example, th e measured values at the bottom
of whey protein cylinder are higher th an the predicted values (Figure 3.9), and the
measured value at the bottom outer location for 0.7% salted ham is much lower than
predicted (Figure 3.10). The non-uniform distribution of added ribose and glucose
in the sample may contribute to the low yield at the bottom outer region in salted
ham, while water released in whey protein gel a t high temperature and migrating
from top to bottom might elevate the level of marker yields at the bottom.
3.5.4
Effect of property on the uniformities of temperature
and sterilization
One way to quantify heating non-uniformity is to use the standard deviation, Tnu
defined by
Till
where Vf is the volume fraction of the material at temperature T. Figure 3.12 shows
this non-uniformity for three materials with different lossiness, after six minutes
of heating. The n o n -uniform ity is higher for low loss material (ham at th e low
salt concentration) due to focusing effect and for high loss material (ham with
the highest salt content) it is also higher due to the increased surface heating.
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69
1.0 -i
■o
0 .9
-
center
®0.8 CM
0 .7
-
outer
0.6 H
Middle
Bottom
Top
(a)
1 .0 - i
center
0.8
outer
0 .4
-
Middle
Bottom
Top
(b)
Figure 3.9: Marker yields at tlie end of heating following the stages of heating in
Table 3.3 by experimental measurement (a) and numerical prediction (b) for whey
protein gel
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70
1.0 -I
cen ter
0.8
*o
0.6
>* ^"O
w
<D ©
-
CO
■£ —
CO " 5
2
-
0.4 -
E
2I
2
o .o
outer
-
Bottom
Top
Middle
(a)
1 .0 - i
^
04
0.8
-
0 .6
—
center
outer
c
0 .2
-
Middle
Bottom
Top
(b)
Figure 3.10: Marker yields at the end of heating following the stages of heating in
Table 3.3 by experimental measurement (a) and numerical prediction (b) for ham
with 0.7% salt
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71
1.0 - i
0 .8
-
"5 0.6
-
outer
0 .4 -
CM
0.2
center
-
Bottom
Top
Middle
(a)
1 .0 - i
co
0 .8
-
0 .6
-
.®
>»
©
outer
INI
2 e
5
5 £
cm
0 .4 -
0.2
-
center
Middle
Bottom
Top
(b)
Figure 3.11: Marker yields at the end of heating following the stages of heating in
Table 3.3 by experimental measurement (a) and numerical prediction (b) for ham
w ith 3.5% salt
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72
30-1
25-
c 20-
152
3
S a lt c o n te n t (% )
Figure 3.12: Effect of composition on uniformity of temperatures at the end of
heating following the stages of heating in Table 3.3. Data is for ham w ith different
salt levels
Thus, improvement in uniformity can be achieved by manipulating the dielectric
properties by changing composition for a compromise between focusing in the center
and intense surface heating. However, the heating differences between the upper
and the lower parts of the samples (Figures 3.10 and 3.11) due to the cavity heating
pattern are not alleviated by the changes of dielectric properties (salt contents).
The density function for temperature for ham with different brine levels are
plotted in Figures 3.13. Over the tem perature range from 88 to 122°C, the density
function for volume fraction is above 0.2 for 1.6% salt ham, while this range is
from 75 to 125°C for 0.7% ham and from 86 to 126°C for 3.5% ham for the same
volume fraction. To sum up, these curves lead to a conclusion th a t an appropriate
salt content, such as 1.6% for the size and shape of ham in this study, exists with
which the sterilized food material has less volume of overheated and underheated.
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73
However, It shall be noticed th at the differences due to different salt contents are
not overwhelming since the cavity heating patterns and edge overheating effect are
not altered by the content of salt.
Another way to quantify and provide improved understanding of th e steriliza­
tion process is using the concept of thermal-time distribution, as proposed by Nau­
ruan [54] and used, for example in [16]. Calculated thermal-time distributions are
shown in Figure 3.14 for three different dielectric properties. The therm al time
(Equation. 3.15) distribution is analogous to residence time distribution, but ac­
counts for the tem perature history. As shown in Figure 3.14, the volume fraction
for a certain therm al time represents the portion of material below th a t therm al
time. For the low loss material (0.7% salt ham samples), large volume fraction of
material falls in the low thermal time range, indicating that most of the m aterial is
underheated. For a high loss material (ham w ith high salt content), the portion of
material th at exceeds 2000 second thermal tim e is over 40%, representing significant
overheating in the outer ring. Intermediate lossy material gives a better therm al
time distribution th a t has the smallest range of therm al time, eventhough the range,
from 0.1 to 1200 seconds, is still quite large due to the nature of heating in a cavity.
3.5.5
Need for coupled solutions
As the dielectric properties are tem perature sensitive, the distribution of electric
fields, defined as heating potential, change w ith th e heating time. The initial distri­
bution of heating potential, as shown in Figure 3.16a, changes as heating progresses
(Figure 3.16b). The distribution of initial heating potential is different from th a t
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74
0. 8 -
S 0.6C 0 .4 -
*
ao
80
60
TOO
Temperature (°C)
120
(a)
£ 0.8 0.6 -
5 0^0.0
80
70
90
100
110
120
130
Temperature (°Q
(b)
1. 0 -
2 0.6 I
0.4 -
0.0
120
80
140
(c)
Figure 3.13: T he density functions for volume fraction for ham w ith 0.7% (a),
1.6% (b), and 3.5% (c) salt contents vs tem perature after 500 seconds microwave
treatment
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75
1 .0 - 1
0. 8 0.7%
0.6 3.5%
o
0 .4 -
0.2 1. 6%
0.0
10'-t
.0
10'
,1
2
10 '
10
Thermal time F0 (seconds)
10".3
10*
Figure 3.14: The volume fraction for ham with 0.7%, 1.6%, and 3.5% salt contents
for therm al tim e F q after 500 seconds microwave treatm ent
during the holding period. The focusing effect, which is initially presented, is not
significant after the material reaches th e sterilizing temperature. This qualitative
and significant change in heating p attern has to be taken into consideration in the
calculations of marker yields by coupling the electromagnetics w ith energy trans­
fer, as was done in this study. This is shown in Figure 3.17, where the uncoupled
solution predicts increased marker yields due to focusing.
3.5.6
Changes in sterilization during come-up and holding
times
Come-up tim e is the time needed to reach near sterilizing tem perature from an initial
temperature. In therm al sterilization of food utilizing conventional (non-microwave)
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76
c e n te r
o u te r
tim e= 440 s
0.1
tim e= 2 4 0 s
2-
0.01
-
8;
642-
0.001
-
0.0
0.5
1 .0
1.5
2.0
Figure 3.15: Numerically predicted increase in marker yields (vertical axis) during
sterilization. D ata for ham with 0.7% salt content following the stages of heating
in Table 3.3
heating, come-up time contributes signific a n tly to sterilization (and loss of nutrients)
since the heating process is slow. In contrast, rapid microwave heating has a short
come-up time and accumulates little sterilization during th at time. As shown in
Figure 3.15, during the first 65 seconds of heating, probe tem perature reaches only
about 62 °C) and the marker yields are small due to the low temperatures. After a
certain h old ing tim e (t — 440s) the marker yields reach high levels. The distribution
of marker yields at later times is also more uniform due to the saturation in marker
formation [37].
The distribution of heating potenials change between the come-up and holding
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Figure 3.16: Numerically predicted change in heating potential from initial to 130
seconds for ham with 0.7% salt
times (Figure 3.16). This change, however, has limited impact on the distributions
of marker yields because less amount of microwave power is delivered during the
holding period (power is off for most th e time). The pattern of marker formation
shown in Figure 3.10 does not change significantly during the holding period.
3.5.T
Industrial Implementation of Microwave Sterilization
Information on the least thermal tim e an d its location are of prim ary importance
for the microbial safety of sterilized food. Successful application of microwaves
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78
1.0 - i
0.8
-
0. 2
-
c e n te r
outer
o.o
Top
Middle
Bottom
(a)
1 .0 - i
0 .8
center
-
outer
CVI
0.2
-
Bottom
Middle
Top
(b)
Figure 3.17: Numerically predicted marker yields at tlie end of heating using un­
coupled (a) and coupled (b) electromagnetics and heat transfer. Data is for ham
with 0.7% salt heated following the stages of heating in Table 3.3
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79
in sterilization requires a comprehensive and verifiable picture of the spatial time
tem perature history. This study shows th at the numerical simulations and marker
formation analysis together can provide such comprehensive information.
3.6
Conclusions
1. A coupled thermal-electromagnetic model was combined with kinetics to de­
scribe microwave sterilization in a comprehensive way. Coupling of heat trans­
fer and electromagnetics was necessary to account for significant changes in
dielectric properties during heating. The model predictions were verified by
obtaining experimental data involving chemical marker formations th at are
functions of time-temperature history in the material.
2. The tim e-tem perature history and thus the sterilization varies spatially in a
very significant way. Additionally, the tim e-tem perature history changes qual­
itatively during heating, changing the relative spatial variation of sterilization.
The spatial non-uniformity of sterilization an d its transient changes can be
improved significantly by changing the m aterial dielectric property th at is a
function of its composition. Effect of salt content was found to be particularly
pronounced in using food materials.
3. Since heating pattern can change dramatically compared to conventional heat­
ing for different food materials, placement in the oven, and oven design,
and the patterns can also change during heating, a combination of a cou­
pled thermal-electromagnetic model complemented with experimental marker
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80
formations is needed for comprehensive verification of microwave sterilization.
Acknowledgment
We acknowledge the financial support by the US Army Laboratory at Natick, Mas­
sachusetts. This research was also partly supported by the USDA NRI program
with grant number 94-37500-0586. We appreciate the high temperature dielectric
property data provided by Prof. S. Barringer of The Ohio S tate University.
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Chapter 4
M icrowave Power A bsorptions in
Single and M ulti-C om ponent
M aterials
Abstract
The amplitude of microwave power absorbed by foods in cavity heating are stud­
ied using numerical and experimental methods for single and two compartment
arrangements. Maxwell’s equations are solved numerically for foods in different
shapes, sizes and having different dielectric properties heated a domestic oven. Re­
lationships are developed to characterize the power absorption of foods as affected
by their sizes and properties. Interestingly, foods w ith small loss factor may not
absorb less power if the size is sufficiently large, while high loss foods usually absorb
more power at small sizes. This is also true for the heating of multiple component
81
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82
materials which, is proven both numerically and experimentally in this study.
List of Symbols
a
radius, m
c
speed of light, m /s
E
electric field, V /m
/
frequency, GHz
h
height, m
H
Magnetic field, A /m
I
current, A
k
wave number
ko
wave number in the air
I
length, m
L
penetration depth, m
P (x ,T )
power density, W /m 3
P
power, W
T
temperature, °C
w
width, m
x
position
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83
e
permittivity, F /m
eo
perm ittivity in air, F /m
e*
complex permittivity, F /m
e'
dielectric constant
e"
dielectric loss
< ff
effective dielectric loss
P
apparent density of the food, kg/m 3
X
wavelength, m
P
magnetic permeability, H /m
Ul
angular frequency, rad/s
Subscript
T
transverse direction
2
normal direction
ex
experimental
nu
numerical
n
mode number
4.1
Introduction
In many applications of microwave processing of foods, such as frozen foods, com
oil, meats and vegetables, the total power absorption as functions of the dielectric
properties, sizes, and shapes are of interests. W hen heating two dissimilar compo­
nents simultaneously in an oven, the amount of power th a t one absorbs compared
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84
with another is a critical factor to the quality of multi-compartment frozen dinners.
Such engineering fundamentals can help food process developers and microwaveable food designers to better understand this physical phenomenon and reduce the
number of experiments, therefore lead to better quality of microwave foods.
4.1.1
Microwave Power Absorption in Foods
In microwave heating, the power absorbed by foods is generally expressed as [49]:
P (z ,T ) =
(4.1)
As stated by Osepchuk [63], Equation 4.1 is not of much direct practical value even
though it gives the basic concept of how microwave power is converted into heat.
In microwave process and product development, very little is told by Equation 4.1
in terms of the total power absorption by a food as single component or one part
of a multiple component setting, such as a TV-dinner in a multiple mode cavity.
From Equation 4.1, one can not generalize th a t foods with small loss factor are
less capable of absorbing power because its small dielectric constant may result
in a large electric field, e.g., E . It is specially true in plane waves irradiation as
explained by Fresnel Formula [13], which is for perpendicular incident plane wave
to a dielectric material.
Efppd _
E a ir
2e0____
^
2
)
e 0 " b ^ m a te ria l
where catena/ = e7 -+- je". Obviously, if the dielectric constant is small, then the
electric field in the m aterial is high. In reality, the distributions and magnitudes of
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85
electric fields inside foods, which, axe also affected by th e sizes, shapes, and dielectric
loss factors, makes it impossible to estimate the to ta l power absorption.
Foods with high dielectric constants and low loss factors, such as water, usually
absorb less energy according to Equations 4.1 and 4.2 for flat surfaces. It may not
be true for some specific cases as the impedance of th e materials at certain sizes
and shapes may be very small for specific modes [13], thus most of the microwave
energy can be delivered to the material.
As shown in Equation 4.1, the loss factor is an im portant factor and directly
related to power absorption. However, it is generally not true th at a large loss
factor results in a high efficiency in absorbing microwave power. The reason is
th a t microwaves penetrate very short distances and th e electric fields inside are
relatively low for high loss foods [67]. If the size of a food is many times larger
th an the penetration depth, a majority part of th e food will not absorb energy.
Therefore, the average absorbed microwave energy p er unit volume will be less.
Proportional to tem perature, as in meat juices and salty hams, the loss factor
increases dramatically during heating [10], as in the thawing of frozen foods. This
increased loss factor may not increase the power absorption as presumed in the
therm al runaway theory [77, 78], because the reflection of microwaves increases when
the wave number \k\ (oc (•v/e'2 -F e,,2)i) is large. Therefore, without the detailed
information of the changes in electric fields inside foods due to the increases of
tem perature, the therm al runaway theory in the literature ([77, 78, 93, 22]) becomes
less convincing.
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86
4.1.2
Factors affecting power absorption
Microwave power absorption in foods is related to many factors, such, as shape, size,
dielectric properties, location of the food, and the oven system. Usually, the oven
system and the location of the food are hard to generalize. Thus, most the studies
in the literature are mainly focused on the volume, properties and frequency.
The output of microwave power of an oven in heating of foods is generally not
the power th a t is claimed by the oven manufacturer. Usually, a testing method
recommended by IEC is employed [8] to check the output power of the oven. Before
the IEC standard, there were many methods [26] th a t gave different results. Ac­
cording to Gerling [26], the Japanese method typically gave 10 to 15 % more output
power than the American method because they used 2000 grams of water instead of
1000 grams. This indicates that the power absorption increases as the load volume
increases.
Barringer et al. [4] experimentally studied the power absorption of com oil and
water as affected by their volume in a regular domestic oven and special modified
oven system, and reported the significant effects of volumes and properties. The
shape also has the effect on the power absorption. It is reported th a t sphere is the
best shape and cylinder follows for microwave heating [27].
Frequency of microwave is another im portant factor for the coupling of mi­
crowave energy in the dielectric materials. For spherical materials, such as biolog­
ical tissues of hum an bodies, the total power absorption was theoretically calcu­
lated in terms of Relative Absorption Cross Section (RACS) [1] varying along with
[40, 63, 97]. According to Weil [97], the value of the RACS monotonically in­
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87
creases for ^
less th an 0.3. Between 0.3 and 1, the RACS has its maximum value,
but fluctuates due to resonances. For large value of
it deceases down to zero
for an infinite radius or zero frequency.
Previous works, also including [40, 83, 97], mainly focuses on the hazard exposure
of a human body to plane waves. Nevertheless, the studies of power absorption of
foods as related to the size, shapes and properties for cavity heating, are hardly
extensive and comprehensive.
4.1.3
Objectives
• To correlate the efficiency of power absorption to the volume, shape and prop­
erties of foods in microwave heating using numerical simulations.
• To compare the characteristics of power absorption by multiple food blocks
when heated simultaneously.
4.2
Problem Formulation
4.2.1
System configuration
A regular domestic microwave oven (GE Inc., Louisvile, KY) is modeled in this
research. In the simulations of single component, the food is placed in the center
of the oven as shown in Figure 4.1. The model also includes the waveguide and
current excitation. For simulating the two compartment food heated in the same
oven, the two food blocks are placed symmetrically as shown in Figure 4.1.
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88
cavity
- waveguide
food two
20 cm
food o n e
/
/
/
/
„excitation
rfV
/
29 cm
Figure 4.1: Schematic of the microwave oven system for simultaneous heating of
two compartments. For single compartment heating, the material is placed along
the central axis
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89
4.2.2
Governing equations and boundary conditions
Microwave heating in a cavity is governed by Maxwell’s equations:
(4.3)
V x H = —juito€*E
(4.4)
V - (eE) = 0
(4.5)
V •H = 0
(4.6)
where E = Ee~Jurt and H = He~Jujt are the harmonic electric and magnetic fields,
respectively. T he complex permittivity, e*, is given by:
(4.7)
The cavity and waveguide walls are metallic and can be considered as perfect electric
conductors (PEC), where the following boundary condition applies:
(4.8)
The microwave power absorption, P (x ,T ), is related to the electric fields as ex­
pressed in Equation 4.1.
4.2.3
Current excitation
The excitation of the microwave power is treated as sinusoidal form at 2.45 GHz
(domestic ovens) given by:
I — I qsin(u/£)
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(4.9)
90
The current carried by a piece of metal in the waveguide is served as the power source
of the microwave system th a t are numerically modeled in this study. The excitation
current, Jo, is estim ated based on experimental measurements of tem perature rises
in this study for the oven modeled.
The inclusion of excitation in the model (details in [99]) allows the characteriza­
tions of the to tal power coupling into the cavity from a magnetron. This is critical
in this paper as th e power reflected from the cavity is related to the power absorp­
tions of foods inside the cavity. In short, different foods in the same oven have
different power reflection, thus different power absorptions. This effect of foods can
be accounted for by modeling excitation current.
Theory of current excitation model
Considering a waveguide, as shown in Figure 4.2, the electromagnetic fields can be
expressed for z > A/4 (the location of current excitation is about z > A/4 away
from the end wall of th e waveguide):
E = £ { A [ B r n + SI £ „ ] e - fc>I + B „[£ r„ - o,B In]efc*1}
(4.10)
n
H =
+ a j r j e - * - * + B „ [-H Tn + a . f f j e 1*1)
n
(4.11)
where n is the mode number, and An and B n are the coefficients to be determined.
Using mode orthogonality relations (for details see [12, 13]), the coefficients are
found as:
a
^
- C 1 + K ) Jo T(y) * ^T n d y
2 (l-,ltfit)I I .
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,
,
t4'12)
91
end wall
to cavity
a/ 4
current excitation
z
Figure 4.2: Schematic of current excitation model used to characterize the power
coupling into the microwave cavity
where
Tin = f ^ E m x H m ) • azdS
(4.13)
R £ is the reflection coefficient from the cavity, which is a function of the sizes of
cavity and the waveguide, as well as the load inside the cavity, while R~ is the
reflection coefficient from the end wall. For TEoi mode,
is 1 if the antenna is
one quarter of wavelength from th e end wall.
The calibration of current excitation for the model
The total power delivered to the load can be calculated by a close surface (plane
S and the surfaces of cavity and waveguide) integration of the Poynting vector.
Assuming the load, cavity, waveguide, and antenna are not changed in a short
heating period (15-20s), the input impedance seen by the current excitation is fixed.
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92
According to Collin [13] (Equation 4.10, 4.11), the power absorbed by the load can
be written as:
Ptotal oc %
(4.14)
Though the measuring tem perature rises for the w ater loads, the experimental
power absorption, Pex, is calculated and used to estim ate the excitation current.
The exact oven system and the locations of the water containers is modeled and
solved numerically by an arbitrary assumed excitation current, I'Q, and the total
power absorbed, Pnu, is calculated. Using Equation 4.14, th e corresponding I q for
the microwave system is simply given by:
Once calibrated, the excitation current is a property of th e specifics of the oven
system, which represents the input power source for th e oven.
4.3
Numerical Solutions
In the engineering of microwave processing of foods, the comprehensive information
of heating potential is needed, which is only available after the electric fields are
solved. As the theoretical solution of Maxwell’s equations in 3D cavity is neither
possible nor efficient, Finite Element Method (FEM) is used to obtain the spatial
electric field distributions. Details on numerical modeling of a 3D cavity problem
can be found in [99]. As the numerical method is efficient and easy to use, it is
applied here to study the engineering fundamentals of th e energy absorption in
microwave heating, as related to volume, geometry, and properties.
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93
Maxwell’s equations are solved using FEM. Tetrahedron elements with 10 grids
are used. A quadratic shape function is used in all the numerical calculations in
this paper. The total number of nodes is around 24,000 for problems with large
size (a = 5.5 cm) of food materials. T he calculations for a model with the largest
number of nodes (usually in a two compartment setting) take more than 12 hours
cpu time, while for small models, they only take about one hour. The rest of the
jobs have interm ediate computational loads.
4.3.1
Input parameters for model
The dimensions and the location of the power entry port for computations are from
a domestic oven (GE Inc., Louisvile, KY) in order to compare with experimental
data. The cavity dimensions are 29 x 28.5 x 20 cm.
Several food materials, in spherical, cylindrical, and rectangular shapes, were
selected to study the power absorptions in a cavity. They are com oils, frozen
meats, frozen vegetables, ham, raw potato, and water. As the appropriate param e­
ters for characterizing focusing are wavelength and penetration depth of microwave
propagating inside a material (discussed in next chapter), the selected materials
represent th e broader ranges of A and L (defined in Appendix A) th at cover most of
the food materials. As shown in Table 4 .1 , the wavelengths of selected materials are
from 1.38 to 8 .6 5 cm, and the penetration depths are between 0.48 and 3 6 .7 8 cm,
which are typical ranges of food materials at 2.45 GHz. The dielectric constants
and loss factors are also listed as input parameters in the numerical simulations.
O ther input param eters are listed in Table 4 .2 , including the sizes of the spheres
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94
Table 4.1: Dielectric properties of materials used in this study
Material
c '+ ie *
A
L
(cm)
(cm)
\k\
e"/\k\
Source
Com oil
2.0+y0.15
8.65
36.8
72.6
0.0021
[59]
Frozen Vegetables
5 .1 + /0 .9
5.4
4.9
116.7 0.0077
[59]
ham
38 4- j 30
1.86
0.48
356.9
0.0841
[75]
potato
51.5+ j 16.3
1.68
0.87
377.1
0.0432
[55]
water
78+jlO
1.38
1.72
454.9
0.0220
[35]
and cylinders in th e numerical simulations- Table 4.3 and 4.4 detail the input data
for the rectangular blocks considered in the numerical simulations for single and
multiple-compartment foods.
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T able 4.2: P aram eter values used in th e num erical sim ulatio ns
Parameter
spheres
cylinders
Radius, cm
0.5, 0.8, 1.6,
0.5, 0.8, 1.6, 2.5,
2.5, 4.0, 5.0
3.5, 4.5, 5.5
3, 4.0, 4.0, 5.0,
Height, cm
6.5, 9.0, 10.0
Geometry of oven
29 x 20 x 28.5 cm
io, A
0.595
0.595
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T ab le 4.3: D im ensions o f re c ta n g u la r blocks in sim u latio n s
width.
length.
height
volume
cm
cm
cm
cm3
1
3.2
2.6
1.2
10
2
5.0
4.0
2.5
50
3
7.0
5.7
2.5
100
4
9.0
7.2
4.0
259
5
13.0
10.4
4.0
540
No.
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97
Table 4.4: Dimensions of rectangular blocks for two compartment simulations and
experiments
width.
length
height
volume
cm
cm
cm
cm3
1
4.4
3.75
2.2
36
2
6.0
7.0
2.4
100
3
9.5
8.1
3.25
250
4
10.0
8.0
5.0
400
No.
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98
4.4
4.4.1
Experimental Setup
Heating of a single material
Water loads in beakers (cylindrical containers) were heated in a domestic microwave
oven (GE Inc., Louisvile, KY), as shown in Figure 4.1 for a short time (10 to 20
seconds) to minimize the effect of evaporation and the tem perature rises in the
containers. The temperature rises of water are measured after a couple second
stirring using a therm al couple. The power absorption was calculated based on the
temperature rises.
4.4.2
Simultaneous heating of two materials
Water and com oil (Table 4.1) in two compartment configuration, as shown in
Figure 4.1 were heated for a short time (10 to 20 seconds) with plastics covers to
reduce the effect of evaporation. Temperature rises in water and oil were measured
after quick stirring. Four different sizes (Table 4.4) of rectangular containers were
used, volume ranging from 25 ml to 400 ml.
4.5
Results and Discussion
Power absorption is calculated for the foods with different sizes, shapes, and prop­
erties as listed in Table 4.1. In all calculations, the microwave system is kept the
same for heating different foods so th a t the power source is not changed, and the
power absorptions can be compared only for factors such as volume and properties.
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99
In this Chapter, the location effects are not discussed, so th a t the placement of
foods is at lower center of the oven in all calculations. In two compartment sim­
ulations, the foods are placed symmetrically at the same vertical position as that
of single compartments, with a 3 cm gap in between. Experimental results for two
compartment foods are also presented an d compared with the numerical ones.
To compare power absorptions, th e quantity Relative Absorption Cross-Section
(RACS) [97] is used:
RACS =
(4.i6)
o
It normalizes the total power absorption to an effective shadow area (S') that is
normal to the wave propagation direction, and is vra2 for spheres, 2a x h for cylinders
and w x l for rectangular blocks. RACS used in this paper characterizes the efficiency
of microwave power absorption for different types, sizes, and shapes. Even though
RACS is defined [97] for plane wave irradiation, it is applicable in cavity heating,
as the cavity heating modes can be considered as superposed by plane waves [34].
In this study, the RACS is plotted against a parameter expressed as e"/|fc|, in
which e" is proportional to the absorbed power, as shown in Equation 4.1, and
the wave number \k\ (oc
+ e"2)^) is also related to the power reflected from
the food, as shown in Equation 4.2, which indicates a lower electric field inside a
material if its dielectric constant and loss factor are large. Thus, parameter,
characterizes the ability to absorb microwave power. Values of
f\k\,
/\k\ from some
food materials are fisted in Table 4.1, e.g., com oil has the smallest value of
ham has the largest.
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and
100
4.5.1
Efficiency of power absorption as related to dielectric
properties and volume
Efficiency of power absorption for various shapes are plotted in Figure 4.3 as a
function of the param eter e " /|k\. For small volume d a ta for all three shapes in
Figure 4.3a, b and c, power loss increased with
f\k \. To observe the effect of
changing volume, the efficiency of power absorption is plotted as a function of
size for spherical foods in Figure 4.5. It shows th at efficiency of power absorption
increases with volume first, and then decreases. Microwave power absorbed in small
volume is always small, as has been reported previously for plane wave analysis [97].
More interestingly, these curves also show that increasing volume may not increase
the efficiency of power absorption. This volume effect on the efficiency of power
absorption can perhaps be understood using similar results for plane waves, as
shown in Figure 4.4. It shows th a t the efficiency of power absorption increases
for very small values of radius, and decreases asymptotically for large values, with
resonance complicating th e relationship for intermediate values of radius. Volume
effect of power absorption was also experimentally observed by [4]. They showed
th a t for small volume (50g), a high loss material (water) is heated several times
faster than a low loss m aterial (com oil), while for a large volume, low loss material
heats faster than the high loss material. This can be explained using Figure 4.5.
For small volume, relative power absorption is much higher in water. However, for
large volume, the relative power absorptions are quite close in oil and water, and,
therefore, the rate of heating will be higher for com oil due to its lower specific heat.
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101
6-
ham
w ate r
frozen
. vegetable
.com oil
o-
s
a
2
(a)
2 1 .2 m l
w ate r
20-
ham
co
a
<r
frozen
.vegetables
.com oil
2
6
4
8
(b)
10 ml
ham
w a te r
6-
5-
frozen
.v eg eta b les
Q 4540 ml
32-
»m oil
2
*
z
eT/k x10
6
8
(c)
Figure 4.3: The relative power absorption cross section (RACS) for (a) spheres of
2.2 and 524 ml volume, (b) cylinders of 21.2 and 250 ml volume, and (c) rectangular
blocks of 10 and 540 ml volume
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102
1.5
1.0
RACS
0.5
Figure 4.4: Efficiency of power absorption using plane wave analysis [97]
w ater
10
ham
potato
frozen vegetables
i
0. 11
2
3
5
Figure 4.5: The efficiency of power absorption (RACS) in spherical foods for differ­
ent radii and properties
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103
4.5.2
Total power absorption as related to dielectric prop­
erties and volume
Total power absorption, calculated numerically and obtained experimentally, are
plotted in Figure 4.6. The numerical calculations used the same excitation cur­
rent, J0, for all the volumes, and its value was obtained by equating the calculated
power to the experimentally observed value at 250 cm3. As shown in Figure 4.6,
the experimental data agrees quite well with the numerical calculations for other
volumes. The figure shows th a t the absorbed power increases asymptotically with
volume. This trend has been reported from experimental observations in the past,
but this study showed for th e first time the trend using comprehensive numerical
computations for a cavity.
At small volumes, however, there can be additional complications following the
discussion in the previous section in relation to Figure 4.4. This is shown in Fig­
ure ?? for rectangular shaped foods. Although for larger volumes, the absorbed
power increases with volume, at smaller volumes there are local decreases in power
absorption for some materials. Such behavior at small volume is related to many
factors, including properties and shape of the material.
The effect of different shapes of materials on the to tal absorbed power, as a
function of volume, is shown in Figure 4.8 for low loss materials. For all three
shapes, the overall trend shows an increase in absorbed power w ith volume, with
some local variations at small volumes consistent with previous comments. From
this figure, relative power absorption for the three shapes are observed to be different
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104
400
S(0<59
5
w
©
5o
Q.
■o
€
a
<
300
numerical calculation
experim ental m e a su re m e n t
200
100
400
200
300
Volume of w ater in cylindrical co n tain er (cm 3)
100
500
Figure 4.6: Power absorbed in different volumes of w ater in a cylindrical container.
Numerical calculations use experimental d ata at 250 cm3 volume for calibration
CL 300
o
w ater
200
frozen vegetables
Volume (cm )
Figure 4.7: Power absorption as a function of volume for rectangular foods
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105
500-
cyfindrical
spherical
400-
300-
rectangular
'O
200
-
100-
0
100
200
300
400
500
Volume (cm3)
Figure 4.8: Power absorption as a function of volume for low loss materials (frozen
vegetables) in different shapes
as volume changes, and no definite conclusion can be made over the entire volume
range. Therefore, it is difficult to comment on any best shape for total power
absorption, as has been claimed for sphere [27].
4.5.3
Power absorption as related to surface area
Average power flux, defined as the total power absorbed by the material, divided
by the total surface area of the material, is shown in Figure 4.9a from numerical
computations for cavity heating.
As a general trend, the power flux decreases
with increase in surface area. This trend is attributed to the increase in power
absorption (Figure 4.8) being slower than the increase in volume and surface area.
Local variations for small volume, as discussed in the previous section, may occur
depending on other factors, such as shape and properties. Experimental d ata [56]
in Figure 4.9b for heating of soup (a high loss material) for variation of surface
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106
flux with, area show a very similar trend. For a low loss material, the decrease in
power flux may not be as steep, since Figure 4.5 predicts th at efficiencies of power
absorption for low loss foods (com oil and frozen vegetables) do not drop sharply
for the largest volumes.
4.5.4
Power absorption as a function of aspect ratio
In the computations on rectangular blocks presented earlier, the aspect ratio, defined
as the ratio of its w idth to height [10], is kept the same to consider only the volume
and property effects. For a given volume, absorbed power may change with aspect
ratio. Computed power absorptions for three different aspect ratios (Table 4.5) are
shown in Figure 4.10 for a high loss food (ham). Power absorbed increases w ith the
aspect ratio, almost linearly for this particular situation. This significant effect of
aspect ratio is mainly due to the surface heating in ham th a t has a small penetration
depth. The increase in power absorption with aspect ratio may not be true for low
loss materials [10].
4.5.5
Differential power absorption in heating multiple ma­
terials
Experimental verification and simultaneous heating of oil and water
Computational and experimental results for simultaneous heating of two materials
with widely varying dielectric properties (water and com oil) are shown in Fig­
ure 4.11. The two liquids were placed in rectangular containers as shown in Fig-
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Total surface area (cm2)
(a)
4 .0 -
o
Experiment
Curve fitting
2 .0 -
1 .5 100
200
300
400
500
2,
600
700
Total surface area (cm”)
(b)
Figure 4.9: Numerical calculations (a) of surface flux vs surface area compared
with, experimental d a ta (b) from the work of Ni et al. [56] for cylindrical containers
(dimensions listed in Table 4.2)
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108
5
500 -
■g 400-
300 150
250
___
200
300
S u rfa c e a r e a (cm )
Figure 4.10: The effect of aspect ratio on power absorption for rectangular blocks
of 260 ml volume
Table 4.5: The d im e n sio n s of the rectangular blocks w ith same volume (260 cm3)
w idth
length
height
surface area/volume
cm
cm
cm
cm2/cm 3
1
7.62
6.09
5.60
0.48
2
9.00
7.20
4.00
1.00
3
12.40
10.00
2.10
1.32
No.
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109
ure 4.1. The relative power absorption, by water and com oh, shown in Figure 4.11,
agree well except a t a small volume. This discrepancy at small volume is likely due
to reasons mentioned earlier. The experimental setup involving the size and shape
of the compartments can not be controlled as precisely as the computations. A
slight difference in geometry may lead to a resonance in simulation.
As in a single compartment, water absorbs significantly more power than com
oil at small volume. As volume increases, the relative absorption of power for water
reduces considerably, until a t 400 cc, water absorbs only about twice as much power
as oil. It is interesting to note that a t this higher volume, the rate of tem perature
rise in oil was higher than water due to its lower specific heat and density. It was not
possible to compute the trend in relative power absorption for still higher volumes
due to limitations in memory and other computing resources.
Simultaneous heating of various material combinations
Power absorption during simultaneous heating of two different foods in a cavity, as
arranged in Figure 4.1, are computed and shown in Figure 4.12. Three different
foods, frozen vegetables, ham, and water are considered, covering a large range of
dielectric properties. Two materials of same volume are chosen at a tim e for heating,
covering a volume range from 50 to 500 ml. To eliminate the location effect [99],
two computations were made for each combination of materials by exchanging their
locations and averaging for the material.
As shown in Figures 4.12a and 4.12b for combination of low loss (frozen vegeta­
bles) and high loss (water and ham) materials, low loss material absorbs less power
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110
Pwater^oil
18
experiment
model
14
10
■
0
100
200
300
400
500
Volume (ml)
Figure 4.11: Ratio of power absorption in water to th a t in com oil placed in rect­
angular containers
in small volume, but more power in large volume th a n the high loss materials. This
is consistent w ith the predictions for heating of a single compartment, implying that
the relative power absorption for simultaneous heating of two materials follows the
same mechanism as for a single material. In other words, if a material of certain ge­
ometry and size absorbs less microwave power th an another when heated separately,
the same will be true when they are heated simultaneously. Between two materials
that are both lossy, as shown in Figure 4.12c, the relative absorption changes with
volume; however, there is insufficient information to comment on any general trend.
In comparison with single compartment heating, interference between the two
compartments can impact power absorption. Thus, th e above results obtained for
a certain gap (3 cm) may not hold if the two materials are placed very close or
touching, due to the field interference and shielding- Although, at gap distances
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I ll
water
frozen vegetables
0.8
0
too
200
300
400
500
600
500
600
Volums (mi)
(a)
ham
frozen vogatablM
0
100
200
300
400
Volume (mi)
(b)
e
2
1.2
0.8
0.6
0.4
w ater
ham
0
100
200
300
400
500
600
Volume (ml)
(c)
Figure 4.12: Relative power absorption in simultaneous heating of two different
materials as rectangular blocks, (a) water and frozen vegetables, (b) ham and veg­
etables, (c) water and ham
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112
wider than. in. this study, the results presented here are expected to be valid.
4.6
Conclusions
1. Power absorptions for single and multiple com partment heating were numer­
ically computed and verified using experimental d ata from this study and
published literature.
2. Efficiency of power absorption, as measured by power absorbed per unit pro­
jected area (RACS), increases w ith volume first, and then decreases. For small
volumes, efficiency of power absorption is higher for high loss foods. For large
volumes, there is no significant difference in the efficiency of power absorption
between low and high loss foods.
3. T he total power absorption generally increases w ith volume for different shapes
as well as properties, b ut in the range of small volumes, there can be local
breaks in this trend.
4. Power absorption per unit of total surface area decreases with increase in
surface area. This trend can also be different in the region of very small
volumes. Increase in surface area, as occurs when th e aspect ratio is decreased,
causes increased power absorption.
5. Relative power absorptions in simultaneous heating of two different foods fol­
low the same relationship as when heated individually.
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113
4.7
Implications
The efficiency of microwave power absorptions in foods are related to their volume
and properties in single and multiple compartment arrangements, which are very
important in food applications- For example, when heating high loss foods (e.g.,
meats and fresh vegetables) in a single compartment, small volume is preferred.
Results in this study can also explain the commonly observed phenomena of
serious localized overheating for microwave heating of frozen foods. When a frozen
food is microwaved, some small regions thaw first due to spatial non-uniformities
in heating. As heating continues, these thawed high loss regions absorb much more
power than the rest of the frozen low loss regions due to the high efficiency of power
absorption at small volumes for high loss materials (see Figure 4.5). This leads to
thermal runaway. Note th at this volume effect in therm al runaway is in addition to
the effect due to the tem perature coefficient of dielectric loss that is typically used
to explain this phenomena.
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Chapter 5
H eating Concentrations of
Microwaves in Spherical Foods.
Part One: In Plane W aves
Abstract
A comprehensive and systematic study of microwave heating concentration, as in
spherical and cylindrical materials, is presented based on theoretical analysis and
numerical investigations for both plane waves (part one) and cavity standing waves
(part two). In part one, along with penetration depth, the wavelength is introduced
for characterization microwave heating of foods. Based on theoretical analysis, the
distributions of power density in radial direction are studied for various groups of
foods with a radii range from 0.5 cm to 13.5 cm. A t 2.45 GHz, the focusing areas
are pronounced in terms of dielectric properties and radii then presented as heating
114
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115
diagrams, which, are useful and informative. For example, the smallest radius th a t
can lead to heating concentration of microwaves is 0.5 cm for any food materials,
and the upper limit of radius that incurs center heating for unfrozen foods is about
13.5 cm.
List of Symbols
a
radius, m
A Inm coefficient
An
coefficient
Bn
coefficient
E
electric field, V /m
f
frequency, GHz
H
Magnetic field, A /m
i
I
current, A
k
wave number
k0
wave number in the air
kt
therm al conductivity, W /m K
I
length, m
L
penetration depth
P
power density, W /m 3
QOI
quasi-optic index
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R
position vector
r
cylindrical or spherical coordinator
T
tem perature, °C
t
time, s
x
coordinator
y
coordinator
2
coordinator
e
permittivity, F /m
€q perm ittivity in air, F /m
d
dielectric constant
d' dielectric loss
A wavelength, m
y
magnetic permeability, H/m
u> angular frequency, rad/s
Q
cylindrical or spherical coordinator
<fj
spherical coordinator
Subscripts
i
incident wave
m maximum
r
reflected wave
t
transm itted wave
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117
5.1
Introduction
One of the recommendations from The National Materials Advisory Board for mi­
crowave processing of materials in 1994 states as follows:
Develop empirically simplified models and ‘microwave heating diagrams’ based, on
measurements and on the extensive data collected from results o f numerical simula­
tion to make numerical techniques more accessible to processors [5].
An effort th at follows this recommendation is made in this chapter (Part One) and
following chapter (P art Two) for microwave processing of foods.
5.1.1
Distributions of microwave heating
The efficiency and applicability of using microwave energy for industrial and domes­
tic food processing depend on its uniformity and predictability. As the wavelengths
of microwaves for most food materials axe the same order of magnitude of their
sizes at 2.45 GHz, the heating is usually uneven and hard to predict in a standing
wave applicator (microwave oven). Therefore, understanding the characteristics of
heating non-uniformity and accurately predicting the distribution of energy depo­
sition are critical to microwave food product and process development. This needs
thorough studies of th e interactions between microwaves and dielectric materials
(foods). Generally, they are functions of shape, size and properties of foods, as
well as the microwave system. Complicated though, some of the features can be
summarized in term s of heating concentration.
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118
Focusing o f microwaves
Focusing or internal heating concentration [14] is one of the most significant fea­
tures of microwave heating compared with the conventional heating methods. Con­
sequently, this issue was studied by many researchers in food applications.
In the literature, it is known that focusing is related to radius relative to the pen­
etration depth. Given by Buffler [8, 20], criterion for focusing is th at the diameters
are about 1.5 times of th e penetration depths. Based on experimental observations,
Prosetya et al. [70] reported that water in a cylindrical container has core heating
at radius about 5.5 cm.
Focusing is also found in microwave heating of cylindrical agar gels w ith dif­
ferent sugar contents (0%, 40%, and 60%) as reported by Padua [64]. Based on
experimental results and numerical simulations he concluded th at the radius (3 to
3.5 cm) and composition (dielectric properties) affect the power absorptions and
the distributions of temperatures in radial direction.
Heating concentration of microwaves in spherical and cylindrical foods, mainly
fresh vegetables, are also studied by Van Rem men et al. [72] using simplified electro­
magnetic models. T he problem with these models is the singularity of the electric
field at the geometric center [44], which is not true physically.
Table 5.1 lists the observations on the concentration of heating inside cylindrical
and spherical foods by Ohlssion and Risman, Swami, and Nykvist and Decareau [58,
74, 89, 57]. Most of th e observations are for specific foods. Xh°ugh. these d ata are
useful, a generalized relationship between the dielectric properties and the focusing
is not obvious as the diameters for heating concentration listed in Table 5.1 seem
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119
irrelevant to the dielectric constants or th e loss factors. Moreover, these sporadic
results without comprehensive explanations and systematic presentations are hardly
useful in microwave food process and product development. The most valuable
information is, perhaps, the intensity of microwave concentrations as related to the
dielectric properties, which is actually not available in the literature. It is obvious
th a t the focusing of microwaves in foods needs more comprehensive and systematic
studies.
Other uneven heating in foods
Foods with certain size and geometry can be dielectric resonantors (DR) if their
dielectric constants are high, such as water or water rich vegetables. In microwave
com m unica tio n circuits, DR is a widely used element [36, 68]. For low loss foods
in cylindrical or rectangular shapes, it may have resonance with exact dimensions
and properties. T he concentration of energy at edges and comers is evident when
microwave heating solid foods [74]. Generally, at the edges and comers the surface
electric charges are accumulated more th an other places due to the geometric effect,
thus, the fields are stronger and more heating can be observed. In a cavity, only a
small portion of the edges and some of th e comers of the food block have obvious
overheating as a result of s tand ing waves. Whereas, the most common uneven
heating of microwave is probably caused by the maxima of standing waves in foods
when large size (greater than the wavelength) foods are considered, especially for
food with a low dielectric loss factor and a high dielectric constant.
In this research, the heating concentrations inside th e foods due to curved surface
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120
Table 5.1: Literature d ata on diameters th a t lead to heating concentration in cav­
ity heating compared with plane wave model prediction (Q O I > 1 predicts the
focusing)
Reported
M aterial
Phantom food
e '+ J t"
52 + j l 4
A
L
focusing dia.
(cm)
(cm)
(cm)
1.68
1.01
1.8 - 3.5
QOI
Source
2.96 - 4.83
[61]
Agar gel
[89]
0.015M NaCl
70+ JT 5
1.45
1.09
!0
3.04
Meat
41.6 + j'11.3
1.87
0.98
6
4.12
Agar gel
[57]
[64]
0% sugar
7 5 + y i3
1.4
1.3
3.5
10.3
6% sugar
35 + j 18.5
2.0
0.64
None
0.07
Agar gel and
78 + jf'10
1.38
1.72
5.5
2.5
[70]
51.5 -h j 16
1.68
0.87
2.5
3.04
[44]
water (cylinder)
Potato (sphere)
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121
axe the main focus of this study.
5.1.2
Objectives
The objectives of this study are:
1. To chaxacterize the heating concentration in spherical foods under plane waves
based on the theoretical analysis.
2. To enhance the understanding of the relationships and differences of microwave
heating in plane and standing waves.
3. To create microwave heating concentration diagrams th at axe useful in the
food process and product development.
5.2
Mathematical Formulations for Lossy Dielec­
tric Spheres Under Microwave Heating
Plane wave irradiation is discussed in this section. As will be shown later, plane wave
analysis can be the basis for better understanding of microwave food processing even
though heating in the latter is in standing waves in a multimode cavity. Microwave
heating of spherical and other curved dielectric objects in plane wave was studied
in the context of harmful exposure to the human head [23, 39, 38, 40, 42, 83]. In
these extensive studies, frequency is a variable while the material geometry and size
are fixed (such as the head of a man).
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122
5.2.1
Distributions of electric fields under plane waves
A plane wave incident on a dielectric sphere is partly transm itted into th e material
and partly scattered. At the surface of the sphere the boundary condition is:
E i+ E r = Et
(5.1)
In the spherical coordinate system, plane wave solution of Maxwell’s equations,
Eoelkz can be expressed as [87]:
E0e*‘
= E0 j r (a .m 'g , + bnn$„)
(5.2)
n=0
where z is the propagating direction. The terms m ^ n and
are given by
r a £ I = -r—EJn (fcr) + P i (cos Q) cos <f>6 - j n (hr) dP^ ° s d ) sin ^
sin \)
ott
(5.3)
— 3ri(kr) + P i (cos 9) cos (fyr
n ii» =
1 d ,
dPi(cos9)
~
^ d ? [rjn{kr)] - dr - 005(1)6
^
^-[rjn(kr)]Pi(cos9)sm9<f>
hr sin# d r
(5.4)
where j n(kr) is Bessel function in spherical coordinate system, P i (cos 6) is Legendre
function, and the wave number k and ko are given by
k = uJy/JPe
(5.5)
ko — —
c
(5.6)
The fieldsinside a sphere under plane waveirradiation can
on Equation 5.1(a detailed analysis
be deduced based
can be found in [87]), which take the following
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123
form:
(5.7)
where the coefficients can be expressed as:
1
ka
hn (ka) [k0a jn (A70a)]' —j n (k0a) [kaf^tP (ka)]'
(5.8)
hn(ka)[koajn(koa)]'hn(ka) — (!f ) 2j n(k0a )[ka h ^\ka )Y
Therefore, th e electric fields inside a spherical food under plane wave irradiation
can be calculated using Equation 5.7. To obtain the convergent solutions for Equa­
tion 5.7, only a small number of term s (2 |^ ~ |) are needed [38].
5.2.2
Distributions of electric fields under standing waves
in a cavity
To understand the standing waves in a cavity, plane waves are the basis. Mathe­
matically, a plane wave (e^fcx-a;t)) is different from a standing wave (Re{el(-hx~ ^ } ) .
They are, however, closely related. A standing wave can be decomposed into two
propagating waves in opposite directions as:
cos(ktz - cut) = 0.5 * ( e ^ - ^ +
(5.10)
The solutions of Maxwell’s equations for standing waves in an empty cavity can
be grouped into TM and TE modes. For the TM modes, electric fields are expressed
as [34]:
E lm n
=
A lImn
mn
COS (kxx) SUL(kylf) SU1(k2z) COS (cut) X
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124
+ 4 fmn
kyfcz
2 sin(kxx) cos(kyy) sin(k~z) cos(u)t)y
k
f~2 _j_ £2
+i4imn J- -—- sin(fcr r) sin(A^t/) cos(kzz) cos (cut) z
/C
(5.11)
while for the TE modes, the electric fields axe expressed as:
Eimn =
ky
cos(/txxr) sm(kyy) sin(kzz) cos(ast)x
K
k
+4imn sin(Arxr) cos{kyy) sm{kzz) cos(uit)y
k
(5-12)
where
kx = ~
(5.13)
A* = ^
(5.14)
™
(5.15)
and
k = {k£ + k%+ kl) 2 = uj(ey.) 2
(5.16)
In the above equations, Z, m, and n are integers with a condition th at only one of
them can be zero for a cavity mode. These cavity normal modes (given by Equa­
tions 5.11 and 5.12 can be expressed in term of transversely polarized progressive
plane waves as follows [34]:
Elmn(R, 0) = f £ 4 ,e ik' R
(5.17)
8 p- i
where p = 8 and Ep*kp = 0 . Hence, any normal mode in a cavity can be considered
to be a superposition of 8 plane waves.
If a sphere is placed in a multimode cavity, the number of modes it sees depends
on the size of the cavity. Every one of these modes, again, can be expressed as
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125
superposition of the planes waves. Therefore, the total electric field inside the
sphere can be expressed by adding all the solutions (obtained by Equation 5.7) of
the plane waves that compose the cavity modes, if the sphere is sufficiently small
compared with the cavity [34]. Thus, results from plane wave calculations can help
in understanding the heating behavior in a cavity.
5.3
Parameters in describing heating concentra­
tion
Heating concentration inside curved surfaces generally includes the two effects—
focusing and resonance, depending on the wavelength relative to the size of the
material [39]. In focusing, the transm itted waves converge to a smaller volume in
the material and it occurs for size larger than wavelength. Resonance inside the
material can occur when the size is smaller than the wavelength. In both cases, the
maximum heating potential occurs inside the spheres. Heating concentration (H C )
is defined in this study as a binary parameter depending upon whether or not the
largest electric field is on the surface. A more comprehensive description of HC' is
left to the next Chapter.
Although dielectric constants and loss factors are the properties needed to define
the electromagnetic fields, these properties do not provide simple means to have
insights into the heating patterns.
Focusing of microwaves has been related to
penetration depth [67, 49] b u t its relationship to wavelength [39] has not been
elaborated in food applications. They can be used as intermediate parameters for
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126
characterizing the heating patterns. The relationships between wavelength, A, and
penetration depth, L, and dielectric properties are given by:
27T
A= —
a
(5.18)
L = ±
(5-19)
where
y/2Oi = ^ V
c
/v/^ T ^ + f '
^ =
(5-20)
(5.21)
c
5.3.1
Wavelength as an important parameter in character­
izing focusing
When a plane wave is incident on a sphere as shown in Figure 5.1, the angle of
transmission, 9t, is related to th e wavelength, A, by the well known equation [32]
cos 9t = ^1 - ^ sin2 0 ^ '
(5.22)
where
O tt
i
---------------
k = -j-y/e' ~ je "
(5-23)
For a small wavelength (large k), 9t is small. This would lead to focusing since
more energy will tend to be concentrated near the center of a sphere. For a large
wavelength, 9t is higher and energy can leave the sphere, therefore less focusing.
Thus, wavelength is an im portant param eter in focusing.
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127
_
T _
.
Figure 5.1: Microwave refraction inside a sphere
5.3.2
Dielectric property based grouping of foods
For most of the food materials, such as vegetables and meats, L is mostly determined
by e" and A mostly by e7 (see Appendix B for detail). Therefore, the dielectric
properties can be used as key variables in understanding th e interactions between
foods and microwaves through the intermediate parameters, A and L. Foods can be
loosely grouped using these two parameters o n a e '- e " plane as shown in Figure 5.2.
Frozen foods have both low dielectric constants and loss factors. Meats usually have
intermediate dielectric constants and moderate to high loss factors. Their dielectric
loss factors are particularly high when they are salty. Vegetables usually have high
dielectric constants and moderate loss factors. Both dielectric constants and loss
factors are high for meat juices. As will be shown later, this grouping of the food
materials on the e7 — e" plane was found useful in characterizing the focusing of
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128
40->
Meat juice •
at increasing
temperature
1
30
-
20
-
o
ts
CTJ
—
v
Meats and
Vegetables
••-.v ♦
a>
_©
O
%*
;♦y
• mft
♦
•*
♦♦
jk .
~ :♦
♦ »♦ ♦
♦♦
10-
Frozen foods
grain, nuts
“I
10
20
“I
I
30
40
Dielectric constant
T"
50
I
“ I"
60
70
Figure 5.2: Scatter plot for the dielectric constants and dielectric losses
microwaves.
5.4
Heating Concentrations in Plane Waves: Ef­
fect of Dielectric Properties and Size
The extent of heating concentration in a spherical food can be described by com­
paring its radius with wavelength (which is a function of dielectric properties). A
spherical food can be considered electromagnetically small when the radius is much
smaller than the wavelength (a «
A), electromagnetically large when the radius is
much bigger than the wavelength (a »
A) and intermediate electromagnetic sizes.
Heating patterns based on these electromagnetic sizes are now discussed.
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129
5.4.1
Small electromagnetic size
In this region, the maximum electric field occurs at the surface of a sphere (a detailed
analysis can be found in [40]). For small radius, e.g., a < < A, the first term of the
spherical harmonic series is the d o m inant one in Equation 5.7, thus the transm itted
electric field can be expressed approximately as [40]:
E t = E q ^ ° r (cos Ox —sin 8 cos 4>z) -f-
(5.24)
The first term in Equation 5.24 can be considered as due to a magnetic dipole, and
the second term is due to an electric dipole. From Equation 5.24, it is obvious th at
the electric field has its maximum value at r = a.
In microwave food processing, the largest radius (lower limit for which heating
concentration exists) of a food th at permits a maximum heating on the surface is
of interest. Though Equation 5.24 is valid [32] only w hen A »
a, that maximum
heating occurs at the surface may be true for relatively large radius. A more precise
criterion for which maximum heating occurs at the surface was developed by Kritikos
and Schwan [39] from the numerical solutions of Equation 5.7 as
A > 2.73a
(5.25)
This criterion is used to define the region where surface heating is expected. When
their radii decrease, groups of foods start to fall into this region depending on their
dielectric properties (Figure 5.2). Foods with high dielectric constants, such as meat
juices, have the smallest radii th a t still make them stay in this region. A radius
of 0.5 cm for any food will make it the maximum heating on the surface according
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130
to Equation 5.25. Some large size (2-3 cm in radius) frozen foods may still be
considered electromagnetically small due to their long wavelength, and therefore
will have their maximum heating at the surface.
5.4.2
Large electromagnetic radius
In this region, when \ka\ »
1, little energy can reach th e center of the sphere
and heating is mostly on th e surface (no focusing). This will be shown in the next
chapter for a cavity for large size. There are two factors th a t determine how much
wave energy reaches the center [39]. First, the wave attenuation due to the loss
factor is of the order of e~a/L. Secondly, the fraction of energy reaching toward
the center is related to th e wavelength due to refraction, which is of the order of
87
r(a/A )2. A quasi-optic index (QOI) is developed to set a lower radius limit for
this electromagnetically large region:
(5.26)
For Q O I < 1 , the surface heating potential is larger th a n the inner one (geometric
optics region) and there is no focusing. According to Equation 5.26, the criterion
for focusing involves wavelength and penetration depth relative to the size. For
some foods, both wavelengths and penetration depths are small or large, such as
meat juices or frozen foods. Foods can have a small penetration depth and large
wavelength (e.g., salted ham) or large penetration depth and small wavelength (e.g.,
water). Therefore, penetration depth alone can not generally dictate the distribu­
tions of heating potential, except for very large and lossy materials. Thus, the
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131
literature use of Lam bert’s law with, a penetration depth. [8 , 67, 70, 74] to describe
microwave heating is true only for very restricted situations.
Foods with large wavelength and small penetration depths belong to this region
of no focusing at relatively smaller radii according to Equation 5.26. Salty meats
axe the first group (Figure 5.2) th a t loses its ability to focus as radius increases, due
to their small penetration depth. Frozen foods can have relatively large radii before
they lose their ability to focus, due to their large penetration depth.
5.4.3
Intermediate electromagnetic radius
Figures 5.3a and b and 5.4a and b show the expected focusing areas (shadowed
distances given by Equations 5.25 and 5.26) for various sizes (ranging from 0.5
to 13.5 cm in radius) of food materials shown in Figure 5.2. The horizontal axis is
A/ (2.73a), which is proportional to the ratio of wavelength to radius. This param eter
is mainly used to measure a small spherical food’s closeness to the region in which
maximum heating potential is at the surface. The vertical axis is QOI, defined by
Equation 5.26, which takes into consideration both A and L for large radii.
As shown in Figure 5.3, if the radius is less th an 0.5 cm, heating concentration
is not expected for most of the foods. When radius increases to
1
cm, as shown in
Figure 5.3b, most of the meats and vegetables have maximum heating potentials
inside, but frozen foods still have their surface heated more. For large radius (a =
6
cm), as shown in Figure 5.4a, most of the meats and some of the vegetables are no
longer to expected to have any focusing. All of th e m eats and vegetables are not
expected to have any focusing for radii larger th an 13.5 cm (Figure 5.4b). Due to
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132
12 —|
Area focusing
happens
X
a> 10■o
c
Frozen food£
CO 8 -
o
a» 6 o
CO
CO
2
a
4-
For radius = 0.5 cm
Meats and
vegetables
20 -
1
T
t
4
2
1
7
5
w avelength/(2.73* radius)
(a)
3. 0 -
x
2 .5 -1
03
2 .0
03
-o
_c
o
Area focusing
happens
Frozen foods
-
CL
o
1. 5 -
tfl
CO
'
10 ~ *
Meats and
: vegetables
For radius = 1.0 cm
0.5 -
“T0 .5
1.0
~I
1.5
I
—r~
2.0
2.5
~T~
3.0
wavelength/(2.73*radius)
(b)
Figure 5.3: Scatter plots showing the expected focusing region for radii equal to (a)
0.5 cm and (b) 1.0 cm
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133
104 T
For radius = 6.0 cm
03
C3
Q.
O
7(0a
3
cr
^ > "Cleats and
I*
vegetables
Area focusing
happens
Frozen foods
0.1
0.5
0.4
0.3
w avelength/(2.73*radius)
0.2
(a)
10
-
For radius = 13.5 cm
10®
-
X
© . _6
■a 1 0 -
%
f-10*
w
\5 * M e a ts and
v ( vegetables
<0
O 102 -!
Area focusing
happens
Frozen foods
I
0.05
0.15
0.10
0.20
w avelength/(2.73*radius) •
(b)
Figure 5 .4 : Scatter plots showing the expected focusing region for radii equal to (a)
6
cm and (b) 13.5 cm
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134
very small dielectric constants and loss factors, frozen foods can have focusing for
very long radius.
5.4.4
Prediction of focusing effect in the literature experi­
mental data
Table 5.1 shows several results where focusing was observed experimentally when
heated in a microwave cavity. It is interesting to investigate the ability of the Q O I
param eter from plane wave analysis (given by Equation 5.26) in predicting focusing
in a cavity. Thus, the corresponding Q O I param eters are also calculated in the
same table. As shown, the experimentally observed focusing is well predicted by
the corresponding Q O I param eter having high values.
5.4.5
Radius for maximum focusing in spherical foods
Since focusing here is defined by Q O I >
1
(Equation 5.26), the radius corresponding
to maximum focusing for a food material can be obtained from:
dQOI
dr
= 0
(5.27)
r= a m
which leads to:
am — 2L
(5.28)
Thus, a sphere has its maximum focusing if its radius is twice the penetration depth
in it.
Literature data are available supporting Equation 5.28. For example, a spherical
phantom food with
1 .0 1
cm penetration depth has its maximum focusing at
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2
cm
135
radius in the numerically calculated results for plane wave [61]. Experimentally, a
range of radius between 1.8L - 3.8L for focusing was reported in spherical potatoes
and yams [44].
5.5
Effect of frequency on heating concentration
Focusing is also a function of frequency since wavelength and penetration depths are
functions of frequency of the microwaves. The r m for some food materials at different
frequencies are shown in Table 5.2, covering radio frequency (20 to 100 MHz) and
microwaves (915 and 2450 MHz). It is obvious th a t lower frequency heating is more
predictable, as the surfaces are always heated more for the normal range of food
sizes (20 cm wavelength at 915 MHz). This explains why heating a large frozen meat
block (tempering) is one of the most successful industrial applications of microwaves
at 915 MHz [81]
5.6
Conclusions
1. Wavelength is shown to be a major param eter in the characterization of mi­
crowave heating concentration in foods.
2. Heating concentration of spherical foods at 2450 MHz happens within a certain
range of radius depending on the dielectric properties. Focusing in frozen foods
occurs at relative large radii (more th an 13.5 cm). High loss foods, such as
meat juices and salty meats, have focusing a t relatively small radii. Water
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136
Table 5.2: Largest radii for maximum heating occurring at surface at different
frequencies for different materials
frequency
material
MHz
dielectric properties
wavelength
largest radius
e' + je/f
(cm)
(cm)
35
beef, lean
75+7*201
58.52
21.42
915
beef, raw
54+7*22.6
4.29
1.57
2450
beef, raw
52 + j l 7
1 .6 6
0.61
100
water
78 + j'0.39
33.97
12.43
915
water
78.7 + 7*4.2
3.69
1.35
2450
water
7 8 + 7 10
1.38
0.51
100
carrot
72.4+ ./120
25.34
9.28
900
carrot
68.4 + j 20
3.95
1.45
2400
carrot
6 5 + J1 5
1.53
0.56
100
potato
73 + j 208
2 0 .2 1
7.40
915
potato
3.94
1.44
2450
potato
1.67
0.61
6 6
+ j 21
51.5+7*16.3
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137
rich, vegetables and water have focusing at interm ediate radii range (up to
6
cm). For radius less than 0.5 cm, the maximum heating potential is on the
surface for most of the food materials.
3. The maximum focusing of microwaves occurs at am = 2L for spherical foods.
Heating by plane wave is different from standing wave. The question is how much
of the analysis in this part is valid in cavity heating? In the next chapter, numer­
ical simulations and experiments for cavity heating will be presented to verify the
conclusions here, and further detail the heating concentration quantitatively.
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Chapter 6
H eating Concentrations o f
M icrowaves In Spherical and
Cylindrical Foods. Part Two: In a
Cavity
A bstract
A comprehensive quantification for microwave heating of dielectric spherical and
cylindrical materials is presented in this paper based on extensive numerical sim­
ulations of cavity heating. The distributions of power density along radius are
studied for various foods for a radius range from 0.5 cm to 5.5 cm and wavelength
from 1.38 to 5.4 cm. The results from numerical modeling agree with the theoreti­
cal analysis and are also verified by the thermographs an d published experimental
138
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139
data. Moreover, both, experiments and numerical predictions verify the rules that
relate radii and dielectric properties to focusing developed in the previous chapter.
A heating concentration (H C ) index is proposed to quantify th e focusing effect of
microwaves. At 2.45 GHz frequency, some foods, such as w ater rich vegetables, are
found inherently much more likely to focus energy with high H C index, while some
other foods, such as salty hams, are not likely to have significant focusing at any
radius.
6.1
Introduction
This chapter follows the earlier chapter in developing criterion for focusing and
heating concentration. In th e previous chapter, criterion for focusing were developed
for plane waves using analytical solutions. Here, using complete numerical solutions
for a cavity, we will investigate the applicability of these criteria and we will also
develop a quantitative measure for heating concentration in a cavity. In cavity
heating there are s ta n d in g waves instead of propagating plane waves. As discussed
in section 2.2 of Chapter 5, results from plane waves can be applicable to cavity
heating since a normal mode in a cavity can be expressed as superposition of plane
waves [34]. These plane waves have different amplitudes and are traveling in different
directions in general. However, if the heating potentials in a spherical or cylindrical
body from these different plane waves are integrated over its azimuthal direction,
the variation along the radial direction can have qualitatively similar behavior as in
plane waves (as will be shown later in this chapter).
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140
The objectives of this study axe:
1. To numerically simulate the microwave heating in a multiple mode cavity for
foods with different properties, sizes, and shapes (spherical and cylindrical) .
2. Based on the generalizations of the numerical simulation results for standing
waves and theoretical analysis for planes waves, to develop simple quantitative
relationships for heating concentration in spherical and cylindrical foods in
cavities in terms of their sizes and properties.
3. To experimentally verify these simple models for cavity heating.
6.2
Numerical Solution of Dielectric Spheres and
Cylinders in a Cavity
Governing equations and boundary conditions for a lossy dielectric object in a 3D
cavity are presented in Chapter 2. A domestic microwave oven (GE Inc., Louisvile,
KY) is modeled here, as shown in Figure 6.1. The dimensions, geometry and the
location of power entry port for com putation are modeled as the actual oven. The
cavity dimensions are 29 x 28.5 x 20cm.
Details of numerical solution using a finite element method are given in [99].
Maxwell’s equations are solved using EMAS, a FEM software. Tetrahedron elements
with 10 grids are used. A quadratic shape function is used in all the numerical
calculations in this paper. The to tal number of nodes is around 24,000 for problems
with large size (a = 5.5 cm) of food materials. The calculations for a model with the
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141
28.5 cm
waveguide
x excitation
z
20 cm
29 cm
Figure 6.1: Schematic of the microwave oven system w ith food used in this study
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142
largest number of nodes take more than 12 hours cpu tim e on a HP 735 workstation,
while for small models they only take about one hour.
For some of the calculations, heat conduction is also included, as will be noted
at appropriate places. The details of the solution to the heat conduction equation
with free surface convection boundaries can be seen in Chapter 2 .
6.2.1
Heating concentration ( H C ) index
To quantify the focusing of microwaves from the numerically calculated distribution
of potentials, a dimensionless parameter called th e heating concentration {HC)
index, is proposed:
f 1 ?—
L A (r ')dr'
H C = •/o- - g
o—
(X
A ^
(6-1)
d r ')
where r' = r /a and A{r) is defined as:
A s{r) =
J
j p { r , 9, <f)) s in 9d9d<f>
(6.2)
JJ
(6.3)
for spheres and
A c(r) = ^
p{r, 9, z)d6dz
for cylinders. The ps{r,6, <p) and ps(r, 9, z) are normalized power density for spheres
and cylinders, respectively, and are defined as:
p ,(r,M )
=
(6.4)
* m ax
r a \
pc{r,9,z) =
Pc{r,9 ,z)
— -------
,
(6.5)
nr a x
m
Power density (Ps(r, 9 ,4>), Pc{r, 9, z)) and the maximum power (PmQx) are calculated
from numerical solutions.
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143
6.2.2
Input parameters
Five food materials, in spherical and cylindrical shapes, were selected. These are
frozen vegetables, m eat with 30% fat, ham, raw potato, and water. Since the appro­
priate parameters for characterizing focusing are the wavelength (A) and penetration
depth (L) in the material, these food materials are selected to cover the A and L
representing most of the food materials (Table 6.1). O ther input parameters are
listed in Table 6 .2 , including the sizes of the spheres and cylinders considered in the
numerical simulations. The estimation of the excitation current, I0, is discussed in
detail in Chapter 4.
6.3
Experimental Details
Thermographs were used to verify the numerical model by heating the material
tylose, a food analog.
6.3.1
Materials
The material used in the experiments is tylose, a mixture of water (77% by weight)
and methyl cellulose (MH1000, Hoechst Celanese Corp., NC, USA). Tylose is widely
used as an experimental m aterial in food related researches [6 ,
10
] partly because
of its similar properties with meat and its ability to be formed into various shapes.
Another advantage of using tylose is the easy manipulation of its dielectric properties
by adding a different am ount of salt.
Tylose with three levels of salt contents (0, 2, and 4%) were made by mixing
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T able 6.1: D ielectric prop erties o f various fo o d m ateria ls used in th is Study-
Material
d + je "
X
L
(cm)
(cm)
\k\
Source
Frozen vegetables
5.1 -FjO-9
5.4
4.9
116.7
[59]
Meat
17.6 + j'5.7
2 .8 8
1.45
2 2 0 .6
[60]
Ham
3 8 + j3 0
1 .8 6
0.48
356.9
[75]
Potato
51.5 + j'16.3
1 .6 8
0.87
377.1
[55]
W ater
78 + jlO
1.38
1.72
454.9
[35]
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145
Table 6 .2 : Param eter values used in the numerical simulations. The coordinates for
placement of the food are measured from the lower left comer of the cavity to the
center of the food
Param eter
spheres
cylinders
Radius (cm)
0.5, 0.8, 1.6,
0.8, 1.6, 2.5,
2.5, 4.0, 5.0
3.5, 4.5, 5.5
Height (cm)
4.0, 4.0, 5.0,
6.5, 9.0, 10.0
Geometry of oven (cm)
29 x 20 x 28.5
29 x 20 x 28.5
Coordinates for placing
4cm radius potato
1
14.5,10,14.25
2
14.5,7.5,14.25
3
10.5,10,14.25
-
Coordinates for placing
0.5cm radius water
1
14.5,8,14.25
2
7.25,8,14.25
3
14.5,5.5,14.25
Io, A
0.595
0.595
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146
with. 77% water and methyl cellulose (23,
21
, an d 19%). The material was made
into h em isp h ere of different radii using wax molds and joined together to form a
complete sphere before heating. The dielectric and therm al properties of tylose with
different salt contents are listed in Table 6.3.
6.3.2
Procedure
The tylose samples in spherical shape (radii of 0 .8 cm, 1 .6 cm and 2.5cm, respectively)
are heated in a GE domestic microwave oven (GE Inc., Louisvile, KY) w ith cavity
dimensions as shown in Figure 6.1. After heating, the samples are quickly withdrawn
(i.e., within a few seconds), split into hemispheres, and viewed by an infrared camera
(Model 782 AGA Thermovision, Agema infrared system, Secaucus, NJ, USA) th at
obtains therm al images. The microwave heating tim e is set to 15 seconds to limit
the highest tem perature well below 100°C to minimize evaporation. Samples are
placed inside the oven center at a height of 4 cm above the floor of the cavity.
6.4
Results and Discussions
6.4.1
Spatial distributions of heating potentials as functions
of food geometry and properties
Heating concentrations in spherical and cylindrical foods in a microwave cavity are
studied. It is shown th a t for small radius, the distribution of heating potential is
mostly related to th e wavelength, while for large radius, penetration d epth plays an
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1 47
Table 6.3: Properties of tylose at room temperature used in this study
Parameters
0
% salt
2
% salt
4% salt
Source
e7
58
54.8
50.2
[1 0 ]
e"
17.5
29.6
38.4
[1 0 ]
A, cm
1.59
1 .6
1.62
L, cm
0 .8 6
0.504
0.382
\k\
399
404.8
407.8
p, kg/m 3
1060
1060
1060
[48]
cp, J/kgK
3700
3700
3700
[48]
h , W /m°C
0.49
0.49
0.49
[48]
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148
important role, in addition to wavelength.
S p h ere
Electric fields in a sphere placed inside a cavity (Figure 6.1) are computed numeri­
cally for five different properties, as given in Table 6 . 1 , and for three different radii.
An example of interior electric fields is shown in Figure 6.2 where the bottom figure
is a quarter cut off to show the inside distribution. Cross-section of field distri­
butions for two different sizes are shown in Figure 6.3. In these figures, there are
several identifiable waves decaying towards the center-five maxima of electric fields
in Figure 6.3a, and six maxima of electric fields in Figure 6.3b, from the surface
to center along the radial direction. It is clear that the radius is important in the
occurrence of focusing.
S m all R a d iu s
For a small radius (Figure 6.4a) the heating potential along the
radial direction has its m axim um close to the surface for most of the materials,
regardless of penetration depths.
The inward shifting of th e maximum heating
potential can be explained as a function of wavelength. For the longest wavelength
(e.g., frozen vegetables), the m ax im u m heating potential occurs at the very surface.
For the smallest wavelength (e.g., water), the maximum potential happens at the
most interior place. For the intermediate wavelengths, the surface heating potential
decreases and the interior values increase following the order of wavelengths. It is
evident from Figure 6.5a th at for small size, the distributions of heating potential
along radial direction are primarily a function of wavelengths and are not affected
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149
Figure 6.2: Distribution of electric field in a spherical potato of radius 4 cm placed
in location
1
(Table 6.2)
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150
(b)
Figure 6.3: Electric fields in spheres (potato) for radii of (a) 4 cm and (b) 5 cm,
showing more focusing for the small radius
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151
by penetration depths.
This wavelength-only effect on the distribution of heating potential can also be
seen from Figures 6.5a (frozen vegetables) and 6.5b (meat). The field distributions
on the cross-section through th e center of the sphere show a slightly inward shifting
of m axim um heating potentials due to the decrease in wavelength from 5.4 to 2.88
cm.
L arg e R a d iu s
As the radius increases, penetration depths start to play an impor­
tant role, as shown in Figure 6.4b. There is strong focusing at the larger radius for
most of the materials because of their relatively longer penetration depths. For the
smallest penetration depth (ham), there is no focusing and the maximum heating
potential is at the surface as commonly identified in the literature [67, 8 , 78]. As the
radius increases further (Figure 6.4c), for materials with intermediate penetration
depth, focusing begins to diminish as seen for the calculations for potato and meat.
The effect of radius on focusing can also be seen in an obvious way in Figure 6.3
for potatoes of two different diameters. The smaller size shows more focusing than
the larger because of less decay of energy.
C y lin d e r
Electric fields in a cylinder are shown in Figures
. . The input parameters are
6 6
fisted in Table 6.2. Each cylinder has a different height, however the ratio of height
to diameter is kept roughly constant.
For small radius (a = 0.8 cm), focusing is absent when the wavelength is long,
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152
potato
fro zen
02
Ot
02
05
03
os
or
(a)
potato
0.6 -
! . 0. 4 -
2.0
t.O
R adtal tfrta n co (cm)
(b)
potato
€
02-
1
2
3
R adial O ltan c o (cm)
4
5
(c)
Figure 6.4: Heating potential distributions along radial direction for different spher­
ical foods of radii (a) 0.8 cm, (b) 2.5 cm, and (c) 5.0 cm
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153
(b)
Figure 6.5: Electric fields a t the XY plane through th e center for a sphere of 0.8
cm radius showing the shifting of the maximum value for different materials: (a)
frozen vegetables and (b) m eat
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154
as shown in Figure
Figure
. a for frozen vegetables, whereas focusing can be seen in
6 6
. b for a potato th a t has a shorter wavelength. Following the analytical
6 6
calculations for plane waves in the previous chapter and th e numerical calculations
for a sphere in the previous section, Figure
6 .6
implies th a t focusing is only related
to wavelength, but not the penetration depth. More on the radius effect of focusing
is discussed in the next section.
The heating concentration in cylinders is slightly different from th at in spheres
depending upon the radius. For example, a spherical potato (shown later in Fig­
ure 6.13b) has slightly more heating inside (Figure 6 .6 b) th a n a cylindrical one for
the same radius of 0 . 8 cm.
6.4.2
Intensity of focusing as a function of food geometry
and dielectric properties
Comparisons of focusing intensity using the parameter H C for different geometry
and dielectric properties show th a t some foods with a high dielectric constant may
have very strong focusing, while some other foods w ith a high dielectric loss factor
are not likely to have significant focusing.
Sphere
The concentration indices (HC) of heating potentials for spherical foods of different
radii are shown in Figure 6.7. In the previous chapter (Equation 5.26) we saw th at
for plane wave heating, materials w ith a small wavelength will have higher focusing.
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1 55
(b)
Figure 6 .6 : Electric fields at the XZ plane through the center along axial direction
in cylinders of
0 .8
cm radius for different materials: (a) frozen vegetables and (b)
potatoes
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156
x
*c
3.0
-
2. 5
-
water
meat
o 2 .0 -
potato
2
|
1. 5
frozen vegetables
-
o
O
1. 0 -
0. 5
ham
-
1
3
2
4
5
Radius of spheres (cm)
Figure 6.7: Heating concentration index (HC) as a function of the radius of spheres
for different food materials
Figure 6.7 shows th at this is likely to be valid for cavity heating as well, since water,
having the smallest wavelength has a high intensity of focusing over a large range of
radii. Equation 5.26 also predicts that focusing is less intense for short penetration
depths. Figure 6.7 for cavity heating also shows the same trend for ham. As shown
in Figure 5.4 of the previous chapter for plane waves, foods with longer wavelength
and penetration depth, such as frozen vegetables, have a large range of radius where
the maximum heating potential is inside. This is also true for the cavity heating
as seen for frozen vegetables in Figure 6.4. However, as shown in Figure 6.7, the
focusing inside frozen foods is not intensive.
Interestingly, the potato and meat are the two types of foods th a t have approx­
imately the same m axim u m H C , but occurring a t different radii. As potatoes have
a shorter wavelength and penetration depth, th e m a xim um focusing is at a smaller
radius (about 2.0 cm). Due to a longer penetration depth (less decay) and a longer
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157
wavelength, meats have their maximum focusing occurring at a larger radius (about
3.2 cm).
As discussed in the previous chapter for plane waves, a sphere has its maximum
focusing if its radius is twice of the penetration depth. The numerical results for a
cavity confirm this conclusion, as the values of am listed in Table 6.1 axe very close
to that of predicted maximum H C values, as shown in Figure 6.7.
Cylinder
For a cylindrical shape (Figure 6.8), variations in focusing due to dielectric prop­
erties show a similar trend as for spherical foods (Figure 6.7), although th e range
of radii and the occurrence of maxima may vary. Some of the differences between
cylinders and spheres can be attributed to th e cavity standing waves th a t vary
along axial direction. At small radius (a = 0.8 cm), th e H C indices decrease with
wavelength as in a sphere. All food materials, except frozen vegetables, have low
H C indices at large radius. Both the intensity and the radius range of focusing
are less th an th a t for spheres. Theoretically, a cylindrical food has only one curved
surface along its axis, and the wave energy th a t focuses down to the inner surface
is half of th a t in spheres. Thus, the criterion for focusing should be Q O I > 2 for
cylindrical foods with large radii (the criterion for sphere is Q O I > 1 as discussed
in the previous chapter).
Theoretically, the quasi-optics region corresponds to a radius larger th a n wave­
length for a sphere [39]. Therefore, many high loss foods, such as hams, never have
focusing if focusing is understood as geometric optic effect at large radius. For small
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158
2.0 -
meat
w a te r
p otato
frozen vegetables
1
2
ham
3
4
5
Radius of cyffndeitcm)
Figure 6.8: Heating concentration index (HC) as a function of the radius of cylinders
for different food materials
radius (less than the wavelength), the intensity of H C is also low. Thus, high loss
foods, such as ham, never have any significant heating concentration in spherical or
cylindrical shapes, as shown in Figures 6.7 and 6.8.
6.4.3
Effect of placement in the oven on distributions of
heating potential
As mentioned in the previous section, the normalized radial distribution of potential,
averaged over layers of the sphere do not change for heating in a cavity. This
radial distribution of potential is only a function of size and dielectric property of
the sphere. Thus, it is expected that the radial distributions of heating potential
are also invariant for change in placement inside th e cavity. To investigate this,
numerical solutions of heating potentials for several placements in the cavity are
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159
now considered.
For a sphere of small radius (a =0.5 cm) placed in different locations (Table 6.2),
the electric fields in same cross-section (Figures 6.14b, 6.9a and 6.9b) have maximum
heating potentials at surfaces, but with slightly different angular orientations.
For a sphere of larger radius (a = 4 cm), as shown in Figures 6.10, th is in­
variance of heating potential distribution in radial direction is even more evident.
The curves for heating potential along the radial direction for the same material,
but with different placements (Table 6.2), are quite close. Note, however, th e un­
averaged heating potential distributions over th e cross-section are quite different
for the three placements (Figures 6.3a, 6.11a and 6.11b). Since different locations
in the oven are equivalent to different sets of plane waves, the above result can
also be interpreted as the invariance of heating potential distribution for different
sets of plane waves. However, change in oven system is also equivalent to exposing
the material to different sets of plane waves. Thus, the above results for different
locations imply th at the averaged and normalized heating potential distribution is
also independent of oven system.
The focused point in the material varies w ith placements in the oven, as shown
in Figures 6.3a, 6.11a and 6.11b. The focused points are near the center, b u t not at
the center. This discredits the use of Lambert’s law to predict focusing, as is often
used [73]. The reason th a t the focused points are not exactly at the center of the
sphere is th at the angle of transmission (see Figure 5.1 in the previous chapter) may
be small but not zero for the large permitivity of a food material, such as potatoes.
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160
(a)
Figure 6.9: Electric fields at the YZ plane through the center of a sphere (water) of
0.5 cm radius a t (a) placement 2 and (b) placement 3 (listed in Table 6.2)
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161
1 .0 —1
o 0 .8 -
placem ent 1
placem ent 2
0. 6
-
1. 0 .4 os
_c
1ff5l 0 . 2 -
placement 3
1.0
1.5
2 .5
3 .0
2.0
Radial distance (cm)
3 .5
4 .0
Figure 6.10: Heating potentials inside spheres (properties of potato) of 4cm radius
placed at different locations (see Table 6.2)
6.4.4
Focusing in cavity heating as compared with plane
wave heating
Small radius
From the theoretical results [39] for plane waves discussed in the previous chap­
ter, the heating concentration is mainly related to |fca| for small radius. When
\ka\ increases, higher heating concentration shifts from the surface to the inside of
the sphere. This is exactly what the 3D numerical model predicts. As shown in
Figure 6.12, the concentration indices of th e foods (frozen vegetable, meat, ham,
potato, and w ater in decreasing order of th e magnitude of their wavelengths) are
linearly proportional to their \ka\ values a t small radius (a = 0.8 cm).
Moreover, th e numerical model predictions not only qualitatively, b u t also quan­
titatively agree w ith the theoretical results for plane wave heating. According to
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162
(b)
Figure 6.11: The electric fields at the YZ plane through the center for spherical
potatoes of 4.0 cm radius for (a) placement 2 and (b) placement 3 (Table 6.2)
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163
0.78 -I
.water
Radius = 0.8 cm
p o tato
ham
m eat
frozen
v eg etab les
0 .7 2 -
2
3
5
W a v e le n g th (cm )
Figure 6.12: Heating concentration index as a function of wavelength for spherical
foods of 0.8 cm radius
Kritikos and Schwan [39] when \ka\ < 2.3, the maximum heating potential is on the
surface. This can be seen from Figures 6.3a and 6.3b which show the maximum elec­
tric fields occurring at or very close to th e surface for frozen vegetables and meats
whose |A;a|s are 0.934 and 1.765, respectively. For all other m aterials (Table 6.1)
that have the values of \ka\ more th an 2.3, the maximum heating potential occurs
inside the sphere, as shown in Figures 6.13, 6.14a and 6.4a.
The electric field distributions in a -section through the center of the sphere, as
shown in Figures 6.5, 6.13, and 6.14a, show how the hot spot moves away from the
surface of the spheres when the wavelength decreases.
This inward shifting of the hot spot is also a function of radius. At a = 0.5
cm, water (|A;a|=2.27) does not have center heating as shown in Figure 6.14b, in
accordance w ith the criterion (|fca|=2.3) for maximum inside heating. Additionally,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
164
Figure 6.13: Electric field at the XZ plane through the center of a sphere of 0.8 cm
radius for different food materials: (a) ham and (b) potato
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165
Figure 6.14: Electric field at the XZ plane through th e center of a sphere of water
for different radii: (a) 0.8 cm (b) 0.5 cm
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166
it can be concluded th at 0.5 cm is th e lowest value of radius for which focusing can
be expected in food materials, as water has a large |fc[ value (Table 6.1).
Large radius
The theoretical results for plane waves and the numerical results for a cavity also
agree for large radius. In plane wave heating, the focusing of microwaves can be
characterized by Q O I, defined by Equation 5.26 in Chapter 5. If Q O I is larger
than 1.0, the power gained due to the curved surface is more th an the loss due
to the exponential decay for lossy materials which leads to focusing [39]. The
numerical calculations confirm this. As shown in Figure 6.4c, for a = 5 cm, ham
{QOI = 0.0054), and potato {QOI = 0.71) have their maximum heating potential
at the surfaces, while frozen vegetables {Q O I = 9.43), m eat {Q O I = 13.5), and
water {Q O I = 18) still have focusing. Figure 6.15 shows th a t heating concentration
{HC) for a cavity increases with Q O I th a t can be calculated simply (Equation 5.26)
for plane waves, thus demonstrating the usefulness of plane wave results for cavity
heating.
Also, as mentioned in Chapter 5, focusing reported in the experimentally mea­
sured d ata in literature (Table 1 of th a t chapter) is well predicted by the plane wave
focusing param eter QOI. This is yet another validation of the plane wave results
for cavity heating.
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167
O
X
X
D
■cCO
2.5
-
ra d i u s = 5 c m
2.0 -
c
CO 1 . 5 -
c
©
o
§
fro z e n
v e g e ta b le s
p o ta to
o
water
m eat
ham
1.0 -
o
0
2
6
8
10
12
14
16
18
Quasi-optics index (QOI)
Figure 6.15: Heating concentration index as a function of Q O I for spherical foods
of 5 cm radius
6.4.5
Comparison with experimentally measured focusing
Temperature distributions, measured experimentally and com puted numerically, are
shown in Figure 6.16. Numerical computations of coupled heat transfer and elec­
tromagnetics are described in Chapter 2. A convective heat transfer coefficient of
h — 10W /m r K was used in this study. Figure 6.16 shows th e effect of variation in
material properties for a radius of 2.5 cm. Good qualitative agreement between nu­
merical computations and experimental data is obvious. Figure 6.16a shows strong
focusing effect for a low loss m aterial (tylose with no salt), for both numerical pre­
dictions and experimental data. This focusing is also predicted by the criterion
developed for plane waves since Q O I = 3.08 > 1. For more lossy materials (tylose
with 2% and 4% salt), numerical computations also agree well w ith experimental
data, both showing lack of focusing, as seen in Figure 6.16b and Figure 6.16c. Lack
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168
of focusing for both of these materials is also predicted by the plane wave parame­
ter, Q O I, since Q O I = 0.39 < 1 for tylose w ith 2% salt, and Q O I = 0.078 < 1 for
tylose with 4% salt.
6.4.6
Comparisons with published experimental data
Prosetya et al. [70] reported focusing effect in cavity heating of agar gel in a cylinder
of 5.5 cm radius, as shown in Figure 6.17a. To compare w ith th eir results, heating
potentials were computed for the cavity in this study inside a cylindrical agar gel of
the same radius. These calculated heating potentials are averaged along th e height
and shown in Figure 6.17b. The calculated variation of heating potential shows
focusing as in the experimental d ata of [70], and the radial distribution of heating
potential also matches the general shape of radial tem perature profiles reported in
their work. The experimental d ata shows less focusing effect in th e lower region of
the cylinder. This was also qualitatively true for the computed results, but could
not be shown here due to complexities in presenting such data.
Lu et al. [44] reported moisture profiles in the drying of potatoes in a spherical
shape, as shown in Figure 6.18. For th e smaller diameter of 3.2 cm, Figure 6.18a
shows the radial moisture profile to have a much lower value of moisture near the
center. This is well predicted by th e computational results in Figure 6.4b for a
5 cm diam eter potato showing strong focusing in the center area. For a larger
diameter of 6 cm in Figure 6.18b, th e radial distributions of m oisture showed low
moisture contents in both center and surface areas. This is also predicted by the
computational results for a 10 cm diam eter potato in Figures 6.4c, which shows
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169
(c)
Figure 6.16: Numerically calculated (left) and experimentally measured (right) tem­
perature distributions for tylose spheres of 2.5 cm radius for (a) 0% , (b) 2% and
(c) 4% salt contents placed at the center and 4 cm above the floor in the oven
(Figure 6.1)
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170
70*
SO O ttA prG tl
Radius « 5 £ cm
Liquid tfcm ht - 55 cm
SO-
0.0
1.0
R aj!aidiatm m £am m alar <o2>
(a)
1 .0 - 1
§
O
c
0.8 -
5 0.6 -
oo.
ac
§ 0 .4 X
0
1
2
3
Radius (cm)
4
5
(b)
Figure 6.17: The experimental tem perature distributions (a) by Prosetya et al. [70]
and the predicted heating potentials (b) for a cylindrical w ater load
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171
higher heating potentials in the center and surface.
6.5
Conclusions
1. Distributions of heating potentials are computed numerically for spherical and
cylindrical foods in a multiple mode cavity. These computations agree well
with experimental measurements in this study an d in published literature.
2. The numerical computations verified the criteria for focusing developed from
plane wave analysis to be applicable to cavity heating. For a size greater
than the wavelength in the material, Q O I > 1 predicts focusing. Otherwise
\ka\ > 2.3 predicts focusing.
3. The normalized distribution of heating potentials along radial direction for a
spherical food is mainly affected by its size and properties and varies little
with placement inside the oven.
4. The intensity of heating concentration (H C index) was reported for a range
of food materials and axe shown to be significantly different for the materials.
5. Compared with spheres, cylindrical foods have less intensity and lower upper
limit of the radius for focusing.
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0.2
0.4
0.6
OimtntfonlMS mdhis
as
(b)
Figure 6.18: The experimental moisture distributions in th e spheres (potato) by
Lu [44] for different diameters: (a) 3.2 cm and (b) 6 cm
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173
6.6
Implications in Microwave Food Process and
Product Development
A few implications of this study can be stated as guidelines to microwave product
and process development:
1. Frozen foods have very large range of radii for focusing, but the intensity of
focusing is rather low.
2. Microwaves are hardly focused in high loss foods, such as salted ham of any
size and shape.
3. For vegetables and water rich foods, the heating concentration is relatively
strong over a wide range of radii from 1.6 to 11 cm in diameter.
However, precise calculation of heating potentials for a given combination of
oven and food shape, size and properties can only be obtained through detailed
numerical modeling, as being done in this study.
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A ppendix A
The D efinitions of Penetration
D epth and W avelength
Wavelength:
(A.1)
A= —
a
Penetration depth:
(A-2)
where a and
are defined as:
a =
c
yj y/e12 + e"2 + er
(A.3)
+ €"2 ~ r
(A.4)
/3 =
c
174
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ppendix B
Sensitivities o f a and (3 W ith
R espect To e' and en
Taking the derivatives of a and /?, which are defined in Equations A.3 and A.4, w ith
respect to e' and e", we have:
8a
1
“
2 ,/l+ ( f ) »
8a
8^
1
(B .l)
<5e"
(B-2)
+ v/ r + W )
(5/?
j3 ~
1
<5^
2Ve/2 T e"2 e'
£0
1
e'e"
Se'
P ~ 2 yjea + e"2 Ve'2 + e"2 - e' e'
If e' = 51.5 and e" = 16.3, as for potatoes, one has:
— = 0.47—
a
e
175
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(B.3)
(B.4)
176
—
= 0 .0 2 5 6 ^ -
a
e"
(B .5 )
86
St?
= -0.0093—
/3
e
613 = - 2 .9 2 3 ^ 13
6'
(B.6)
To relate to wavelength, and penetration depth, we have:
8X _ _ 8 a
X
a
8L_ _ _ 8 £
L ~
(3
(B.7)
(B.8)
It is clear th at wavelength, A, is more sensitive to the change of dielectric constant,
while penetration depth, L, strongly depends on the loss factor. This trend is held
true for most of the food materials.
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