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Heating characteristics of simulated solid foods in a microwave oven

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H eating characteristics of sim ulated solid foods in a microwave
oven
Lin, Yah-Hwa Eva, Ph.D.
The Pennsylvania State University, 1991
UMI
300N.ZeebRd.
Ann Aibor, MI 48106
The Pennsylvania Stace University
The Graduate School
Department of Food Science
HEATING CHARACTERISTICS OF SIMULATED SOLID FOODS
IN A MICROWAVE OVEN
A Thesis in
Food Science
by
Yah-Hwa Eva Lin
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 1991
We approve the thesis of Yah-Hwa Eva Lin.
Date of Signature
Ramaswamy C. Anantheswaran
Assistant Professor of Food Science
Thesis Advisor
Chair of Committee
ST.(jf> \ X i___
Rbbert B. Beelman
Professor of Food Science
c
Virendra M. Puri
Associate Professor of
Agricultural Engineering
sjix-h/
•
&>\ n f t l
Food Science
C / z z / 1? /
Carlos ... Zuritz
Assistant Professor of
Agricultural Engineering
rofessor of Food Science
Head of the Department of
Food Science
Ill
ABSTRACT
The power distribution and power-load relationship in a Tappan 500
microwave oven were characterized.
Sinusoidal power distributions were
found inside the oven at different heights during microwave heating.
The power absorption by a load heated in the oven correlated with the
logarithm of the size of load.
Effects of salt content, geometrical configuration, and electrical
shielding on product surfaces on the rates of temperature increase (RTR)
at different locations in the heated samples were studied using sodium
alginate gel.
Center heating was observed in cylindrical samples of
small diameters.
As diameter increased, surface heating predominated.
Corner heating and surface heating occurred in large rectangular slabs.
When salt concentration was increased, microwave power absorption was
higher at the sample surface.
Shielding of microwave power at the
surface of the sample using aluminum foil allowed the investigation of
microwave heating from different directions (radial or axial).
It was
found that reflected waves from the oven floor were not as uniform as
primary waves incident from the top of the oven.
A finite element package, TWODEPEP, was used to solve the
transient heat transfer equation with absorbed microwave power density
as heat source.
Microwave power absorbed at any location in a sample
during heating was derived as a function of dielectric properties and
sample geometry.
Convective and evaporative heat losses at the surface
of the foods were incorporated as boundary conditions in addition to the
initial temperature condition.
The model predictions were compared to
the experimental results at the 0.01 significance level.
The
experimental data were significantly different from the predictions
mostly in the center regions of the cylindrical samples.
The
differences between the experimental results and the model predictions
were not significant in slabs.
Sensitivity analyses of the model on
selected parameters were performed.
Changes in material properties
(thermal diffusivity and dielectric properties) and microwave power
output resulted in a significant change on the predicted temperature.
V
TABLE OF CONTENTS
Eage
LIST OF T A B L E S .................................................
ix
LIST OF FIGURES .................................................
xi
LIST OF SYMBOLS ...............................................
ACKNOWLEDGEMENTS
.............................................
xvii
xxi
Chapter 1
INTRODUCTION .........................................
1
Chapter 2
LITERATURE REVIEW
...................................
5
Electromagnetic Waves.. ...............................
7
2.1
2.1.1
2.1.2
2.1.3
2.1.4
2.1.5
2.1.6
2.2
Wave Equations...............................
Standing W a v e s ...............................
Penetration Depth ...........................
Transmission, Reflection andRefraction . . . .
Power I n t e n s i t y .............................
Field Distribution ...........................
7
10
11
12
13
14
Microwave Cavity OvenSystem ..........................
17
2.2.1
2.2.2
2.2.3
2.3
Magnetron and Power Supply ...................
Power Distribution System ...................
Oven C a v i t y .................................
17
19
20
Product Properties ...................................
21
2.3.1
Permittivity and Conductivityof Dielectrics . .
2.3.1.1
2.3.1.2
2.3.1.3
Effect of Composition ...............
Temperature Effect ...................
Effect of Multiphase Mixture .........
26
27
28
Transmission Properties .....................
Thermal and Physical Properties .............
Geometrical Configurations ofthe Product . . .
Loading E f f e c t ...............................
P a c k a g i n g ...................................
29
32
32
34
35
Modeling of Microwave HeatT r a n s f e r ..................
36
2.4.1
2.4.2
36
37
2.3.2
2.3.3
2.3.4
2.3.5
2.3.6
2.4
21
Heat Transfer M o d e l s .........................
Heat and Mass Transfer M o d e l s ...............
Vi
TABLE OF CONTENTS (continued)
Pape
2.4.3
2.5
Chapter 3
3.1
Numerical Techniques .........................
38
S u m m a r y .............................................
39
...............................
44
MATERIALS AND METHODS
Characterization of the Microwave O v e n ................
3.1.1
Empirical measurement of the Power
Distribution ...........................
3.1.1.1
3.1.1.2
3.1.2
3.2
3.3
44
45
Microwave Power Measurement ..........
Measurement of Power Distribution . . .
45
46
Model Development for the Power Distribution . .
48
Determination of Microwave Power in Relationship to
Load i n g .............................................
49
Derivation of Heat Transfer Model with Microwave Heat
G e n e r a t i o n .........................................
49
Model Development ...........................
Equation for Microwave Power Source (#). . . .
50
51
3.3.1
3.3.2
3.3.2.1
3.3.2.2
3.3.3
3.3.4
Cylindrical Coordinate ................
Rectangular Coordinate ................
Boundary Conditions
Material Properties
51
54
.........................
.........................
56
58
3.4
Finite Element Method and TWODEPEP ...................
59
3.5
Experimental Design
64
3.5.1
3.5.2
3.5.3
3.6
Effects of Size and S h a p e ................
Effect of Shielding of P o w e r ..............
Effect of Salt Content....................
Temperature Measurement
3.6.1
3.6.2
3.7
3.8
3.9
.................................
....
66
66
69
.....................
Temperature Measurement by Fluoroptic Probes . .
Temperature Measurement by ThermocoupleProbes .
Data A n a l y s i s .......................................
Model Verification ...................................
Sensitivity Analysis of the Model Prediction .........
69
70
70
74
75
75
vii
TABLE OF CONTENTS (continued)
Page
Chapter 4
RESULTS AND DISCUSSION ...............................
77
4.1
Experimental Results .................................
77
4.1.1
Mapping the Power Distribution................
4.1.1.1
4.1.1.2
4.1.2
4.1.3
4.1.4
Three-Dimensional Presentation of Power
Distribution ......................
Multiple Regression Model ............
Effect of Loading on Energy Coupling
E f f i c i e n c y .............................
Effect of Salt Concentration..................
Effect of Geometrical Configuration on
Temperature Distribution ................
4.1.4.1
4.1.4.2
Effect of Shape ......................
Effect of S i z e ........................
4.1.5 Effect of Shielding the Surface Microwave Power
4.1.5.1
4.1.5.2
4.2
Axial Shielding ....................
Radial Shielding ....................
Mathematical Modeling
.............................
4.2.1 Microwave Power absorption
4.2.1.1
4.2.1.2
4.2.2
4.2.3
4.2.4
Effect of Geometrical Configuration on
Microwave Power Absorption . . . .
Effect of Sample Temperature on
Microwave Power Absorption . . . .
Effective Microwave Power Gain ..............
Sensitivity Analysis .......................
Model Verification .........................
4.2.4.1
4.2.4.2
4.2.4.3
CHAPTER 5
..................
CONCLUSIONS
Temperature Distribution in Slabs and
Cylinders with Thermal Insulation .
Temperature Distribution in Cylinders
and Slabs with Heat Loss from the
B o u n d a r i e s ......................
Summary ...........................
78
78
79
85
86
91
91
97
107
108
112
113
115
115
118
121
122
129
129
133
154
155
vtii
TABLE OF CONTENTS (continued)
Page
B I B L I O G R A P H Y .................................................
Appendix A
Appendix B
Appendix C
160
FORTRAN PROGRAMS USED IN TWODEPEP FOR PREDICTION OF
TEMPERATURE DISTRIBUTION DURING MICROWAVE
HEAT I N G .....................................
165
THERMOCOUPLE TEMPERATURE MEASUREMENT USING 21X
MICROLOGGER AND AM32 M U L T I P L E X E R ............
172
SUMMARY OF THE STATISTICAL ANALYSIS ON THE RESULTS
FROM EXPERIMENTAL MEASUREMENTS AND MODEL
PREDICTIONS.................................
174
ix
LIST OF TABLES
Page
3.1
Element Sizes and Time Steps Used In TWODEPEP Programs for
Cylinders and Slabs of Different Sizes ....................
62
Experimental Design for the Effect of Size and Shape of the
Model Food Systems during Microwave Heating ................
64
4.1
Multiple Regression Model for Field Distribution............
82
4.2
Variations of the characteristic parameters of the microwave
heating model for sensitivity analysis ..................
124
4.3
Relative Sensitivity Calculated from Sensitivity Analysis .
128
4.4
The Means and Standard Deviations of RTR at Different
Locations in Cylindrical Gels of Various Sizes Measured by
Thermocouples and Fluoroptic Probes and Compared to the
Model Predicted RTR
...................................
143
Sample Program Listings for Rectangular Geometry Used in
T W O D E P E P ...............................................
165
Sample Program Listings for Cylindrical Geometry Used in
T W O D E P E P ...............................................
168
A-3
Specifications for the Inputs in the TWODEPEP Programs
171
B-2
Program Instructions for Differential Thermocouple
Measurement Using 21X with A M 3 2 .........................
173
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in 2 cm Radius Cylindrical Gels and
Compared at 0.01 Significance Level ......................
174
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in 3 cm Radius Cylindrical Gels and
Compared at 0.01 Significance Level ......................
175
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in 4 cm Radius Cylindrical Gels and
Compared at 0.01 Significance Level ......................
176
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in 6 cm Radius Cylindrical Gels and
Compared at 0.01 Significance L e v e l ......................
177
3.2
A-l
A-2
C-1
C-2
C-3
C-4
. .
X
LIST 07 TABLES (continued)
Page
C-5
C-6
C-7
C-8
C-9
C-10
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in 4 cm Radius by 2.5 cm Height
Cylindrical Gels with 1% Total Salt and Compared at 0.01
Significance Level .....................................
179
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in 4 cm Radius by 2.5 cm Height
Cylindrical Gels with Axial Shielding and Compared at 0.01
Significance Level .....................................
180
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in 4 cm Radius by 2.5 cm Height
Cylindrical Gels with Shielding on the Top Surface and
Compared at 0.01 Significance Level ......................
181
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in 4 cm Radius by 2.5 cm Height
Cylindrical Gels with Shielding on the Bottom Surface and
Compared at 0.01 Significance Level ......................
182
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in 4 cm Radius by 2.5 cm Height
Cylindrical Gels with Radial Shielding and Compared at 0.01
Significance Level .....................................
183
Means and Standard Deviations of Measured and Predicted RTR
at Different Locations in Rectangular Gels of Various Sizes
Under Two-Dimensional Microwave Heating and Compared at 0.01
Significance Level .....................................
184
xi
LIST OF FIGURES
Page
2.1
Dielectric properties as functions of frequency (von Hlppel,
1 9 5 4 ) .....................................................
24
Discretization of the power measurement points for microwave
power distribution in a Tappan 500 microwave o v e n ..........
47
3.2
Shell balance within a cylindrical model
53
3.3
Initial triangulations for cylinders of various sizes . . . .
61
3.4
Geometrical configurations of cylindrical and rectangular
gel samples ...............................................
67
Arrangement of microwave shielding using aluminum foil
(shaded areas represent surfaces shielded with aluminum
f o i l ) .....................................................
68
Gel sample temperature measurements using thermocouple
assemblies and fluoroptic probes with data acquisition from
a P C .....................................................
71
Positioning of thermocouple probes on templates for
temperature measurements in (a) cylindrical gels, and (b)
rectangular gels of different sizes ........................
72
Position and the orientation of sample with respect to
microwave oven during heating ..............................
73
Measured power distribution in a Tappan 500 microwave oven
at three levels (a) z - 2.5 cm, (b) z - 0, and (c) z - -2.5
c m .......................................................
81
Time-temperature profiles of cylindrical gels of sizes (a)
3 cm radius by 5 cm height, and (b) 4 cm radius by 2.5 cm
height measured by fluoroptic probes at different distances
from the center at z - 0 l e v e l ............................
83
Power distribution in a Tappan 500 microwave oven determined
by regression model at (a) z - 2.5 cm, (b) z - 0, and (c)
z - -2.5 cm l e v e l s .......................................
84
3.1
3.5
3.6
3.7
3.8
4.1
4.2
4.3
4.4
...................
Effect of loading on the microwave power absorption by
cylindrical and rectangular shaped loads ..........
86
xil
LIST OF FIGURES (continued)
P a pe
4.5
RTR values In 4 cm radius by 2,5 cm height cylindrical gels
at different salt concentrations versus radial distance from
the center at z - 0 level ...............................
88
Effect of NaCl concentration and temperature on the
attenuation factors of aqueous Ionic solutions (Mudgett,
1986) ...................................................
88
The means and standard deviations of RTR In a 4 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z - 0, and z - -1.25 cm three levels (thermally insulated on
the top and bottom surface) .............................
90
Mean RTR in a 3 cm radius by 5 cm height cylindrical gel at
different locations from the center at z - 0 and z - -2.5 cm
levels ...................................................
92
Mean RTR in
z - -2.5 cm
and y axes,
to the axes
a 6 cm x 6 cm x 5 cm rectangular gel at z - 0,
(both measured in the directions parallel to x
and z - -2.5 cm measured in diagonal directions
...............................................
93
4.10 Mean RTR in a 4 cm radius by 5 cm height cylindrical gel at
different locations from the center at z - 0 and z - -2.5 cm
levels ...................................................
95
4.11 Mean RTR in
z - -2.5 cm
and y axes,
to the axes
a 8 cm x 8 cm x 5 cm rectangular gel at z - 0,
(both measured in the directions parallel to x
and z - -2.5 cm measured in diagonal directions
...............................................
96
The means and standard deviations of RTR in a 2 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z - 0, and z - -1.25 cm three levels .......................
98
Themeans and standard deviations of RTR in a 3 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z - 0, and z - -1.25 cm three levels .......................
99
Themeans and standard deviations of RTR in a 4 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z - 0, and z - -1.25 cm three levels ....................
100
4.6
4.7
4.8
4.9
4.12
4.13
4.14
x iit
LIST OF FIGURES (continued)
Page
4.15
4.16
4.17
4.19
4.20
4.21
4.22
101
Themeans and standard deviations of RTR in a 2 cm radius by
5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 2.5 cm,
z - 0, and z - -2.5 cm three l e v e l s ......................
102
Themeans and standard deviations of RTR in a 3 cm radius by
5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 2.5 cm,
z - 0, and z - -2.5 cm three l e v e l s ......................
103
The means and standard deviations of RTR in a 4 cm radius by
5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 2.5 cm,
z - 0, and z - -2.5 cm three l e v e l s ..................
104
Themeans and standard deviations of RTR in a 6 cm radius by
5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 2.5 cm,
z - 0, and z - -2.5 cm three l e v e l s ......................
105
Themeans and standard deviations of RTRmeasured at
different orientations and distance from the center at z 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm
radius by 2.5 cm height cylindrical gel shielded with
aluminum foil on the z - ±1.25 cm s u r f a c e s ..............
109
The means and standard deviations of RTR measured at
different orientations and distance from the center at z 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm
radius by 2.5 cm height cylindrical gel shielded with
aluminum foil on the z - -1.25 cm surface................
110
The means and standard deviations of RTR measured at
different orientations and distance from the center at z 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm
radius by 2.5 cm height cylindrical gel shielded with
aluminum foil on the z -1.25 cm s u r f a c e .................
Ill
.
4.18
Themeans and standard deviations of RTR in a 6 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z - 0, and z - -1.25 cm three l e v e l s ....................
xlv
list
OF FIGURES (continued)
Page
4.23
4.24
4.25
4.26
4.27
4.28
4.29
4.30
4.31
4.32
4.33
4.34
The means and standard deviations of RTR measured at
different orientations and distance from the center at z 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm
radius by 2.5 cm height cylindrical gel shielded with
aluminum foil on the r - 4 cm surface....................
114
Two-dimensional microwave power absorbed at z - 0 in a 6 cm
x 6 cm x 5 cm slab gel at 3 0 ° C .........................
117
Two-dimensional microwave power absorbed in a 3 cm radius by
5 cm height cylindrical gel at 3 0 ° C .....................
118
Power absorption in a 3 cm radius by 2.5 cm height
cylindrical gel along the (a) z direction, and (b) r
direction at different temperatures .....................
120
Effect of attenuation factor on the radial distance of
concentrated heating from the center in cylinders of various
r a d i i ...................................................
122
Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5
cm height by varying microwave power output ± 5% and ± 10%
125
Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5
cm height by varying the..... attenuation factor ± 30% ....
126
Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5
cm height by varying the thermal diffusivity ± 10% and ±
20%
126
Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5
cm height by varying the heat transfer coefficient ± 20% and
± 4 0 % ...................................................
127
Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5
cm height by varying the difference of relative humidity
between sample and ambient air ± 30% and ± 6 0 % ..........
127
(a) Predicted RTR and (b) experimental RTR in a 4 cm radius
by 2.5 cm height cylinder with no heat loss from the
b o u n d a r i e s .............................................
130
(a) Predicted RTR and (b) experimental RTR in an 8 cm x 8 cm
x 5 cm slab at z - 0 with no heat loss from the boundaries
132
XV
LIST OF FIGURES (continued)
Page
4.35
4.36
4.37
4.38
4.39
4.40
4.41
4.42
4.43
The
RTR
a 2
cm,
means and standard deviations of predicted and measured
at various radial distances from the center (r/R - 0) in
cm radius by 2.5 cm height cylindrical gel at z - 1.25
z - 0, and z - -1.25 cm levels .....................
134
The
RTR
a 3
cm,
means and standard deviations of predicted and measured
at various radial distances from the center (r/R - 0) in
cm radius by 2.5 cm height cylindrical gel at z - 1.25
z - 0, and z - -1.25 cm levels .....................
135
The
RTR
a 4
cm,
means and standard deviations of predicted and measured
at various radial distances from the center (r/R - 0) in
cm radius by 2.5 cm height cylindrical gel at z - 1.25
z - 0, and z - -1.25 cm levels .....................
136
The
RTR
a 6
cm,
means and standard deviations of predicted and measured
at various radial distances from the center (r/R - 0) in
cm radius by 2.5 cm height cylindrical gel at z - 1.25
z - 0, and z - -1.25 cm levels .....................
137
The
RTR
a 2
z -
means and standard deviations of predicted and measured
at various radial distances from the center (r/R - 0) in
cm radius by 5 cm height cylindrical gel atz - 2.5 cm,
0, and z - -2.5 cm levels ...........................
138
The
RTR
a 3
z -
means and standard deviations of predicted and measured
at various radial distances from the center (r/R - 0) in
cm radius by 5 cm height cylindrical gel at z - 2,5 cm,
0, and z - -2.5 cm levels ...........................
139
The
RTR
a 4
z -
means and standard deviations o . predicted and measured
at various radial distances from the center (r/R - 0) in
cm radius by 5 cm height cylindrical gel at z - 2.5 cm,
0, and z - -2.5 cm levels ...........................
140
The
RTR
a 6
z -
means and standard deviations of predicted and measured
at various radial distances from the center (r/R - 0) in
cm radius by 5 cm height cylindrical gel at z - 2.5 cm,
0, and z - -2.5 cm levels ...........................
141
Model predicted RTR and measured RTR at z - 0 in slabs of
(a) and (d) 4 cm x 4 cm x 5 cm, (b) and (e) 6 cm x 6 cm x 5
cm, and (c) and (f) 8 cm x 8 cm x 5 cm, respectively . . .
145
xvi
LIST OF FIGURES (continued)
Page
4.44
Effectof 1% total salt on the predicted and measured RTR In
a 4 cm radius by 2.5 cm height cylindrical gel at different
locations at (a) z - 1.25 cm, (b) z - 0, and (c) z - -1.25
cm levels ...............................................
148
4.45
Effect
of radial microwave heating on predicted and measured
RTR In a 4 cm radius by 2.5 cm height cylindrical gel at
different locations at (a) z - 1.25 cm, (b) z - 0, and (c) z
149
- -1.25 cm l e v e l s ............. .........................
4.46
Effect of microwave heating with the top surface shielded on
predicted and measured RTR in a 4 cm radius by 2.5 cm height
cylindrical gel at different locations at (a) z - 1.25 cm,
(b) z - 0, and (c) z - -1.25 cm levels ..................
151
Effect of microwave heating with the bottom surface shielded
on predicted and measured RTR in a 4 cm radius by 2.5 cm
height cylindrical gel at different locations at (a) z 1.25 cm, (b) z - 0, and (c) z - -1.25 cm levels .........
152
Effect of axial microwave heating on the predicted and
measured RTR at different locations in a 4 cm radius by 2.5
cm height cylindrical gel at (a) z - 1.25 cm, (b) z - 0, and
(c) z - -1.25 cm levels .................................
153
Circuitry of 21X micrologger with AM32 multiplexer
172
4.47
4.48
B-l
....
LIST OF SYMBOLS
empirical coupling coefficients
element size or surface area
surface area at r
outside surface area at r - R
magnetic flux density in [weber/m2]
a material property parameter
material property
electric flux density in [coulomb/m2]
electric field intensity in [volt/mj
electric field
incident electric field
reflected electric field
transmitted electric field
incident wave
reflected wave
frequency
element forcing function
conductance
magnetic field in [ampere/m]
convective heat transfer coefficient, [W/m2*
latent heat of vaporization, [kJ/kg*C]
current density in [ampere/m2]
xviii
LIST OF SYMBOLS (continued)
Jt0t8l
sum of conduction-current density (oe) and
displacement-current density (oee)in time-phase
domain
k
thermal conductivity in [W/m°C]
empirical coupling constant
k
mass transfer coefficient of water vapor in air in
[kg/s*m2,kPa]
L
inductance
my
mass flow rate of water vapor in [kg/s]
OXX
material property * dU/dx
OXY
material property * dU/dy
P
value of the parameter chosen
Pm
maximum power generated by the frequency generator
P0
surface power coupled by unmatched load [W]
Py
water vapor pressure in [kPa]
Pr"
power per unit area at a distance r
the r direction in [W/cm2]
PR"
power per unit surface area at r » E
Pz"
power per unit area at a distance z
the z direction in [W/cm2]
P“
volumetric microwave power absorption in [w/m3]
P"c
absorbed power density at r and z in a cylinder in
[W/cm3]
P * B h e U -r
from the origin in
from the
center in
absorbed power density of the shell 2irArAz in the r
direction in [W/cm3]
P"ghell.z
absorbed power density of the shell 2trArAz in the z
direction in [W/cm3]
RH
equilibrium relative humidity
xix
LIST OF SYMBOLS (continued)
ARH
difference of relative humidity between sample and the
ambient air
RTR
rate of temperature rise In [Deg C/s]
5
Poyntlng vector; a vector in the direction of wave
propagation
At
duration of microwave heating time in [s]
AT
temperature difference between the initial temperature
and the final temperature
Ta
ambient temperature in [Deg C]
U
unknown variable in question
v
velocity of the wave in [m/s]
Vg
volume fraction of suspended phase in the mixture
W#
electric energy density
Wra
magnetic energy density
X
fraction of phase
Y
model response
Greek Symbols
a
attenuation constant in [cm*1]
at
thermal diffusivity in [m2/s]
/3
phase shift constant
Y2
propagation constant
6
angle of a phase difference between the field
intensity E and the current density
e
permittivity of the material in [farad/m]
ec
complex permittivity of continuous phase
LIST OF SYMBOLS (continued)
effective permittivity
complex permittivity of a mixture
permittivity at free space
complex permittivity of suspended phase
dielectric constant
dielectric loss
relative dielectric loss
a time-shape function, which equals to 0.5
critical angle - sin*1 /e2/«i
time coefficient
permeability of the material in [henry/m]
relative permeability
density in [kg/m3]
conductivity of the material in [siemens/m]
effective conductivity
irreversible rate of energy change due to viscous
dissipation
xxi
ACKNOWLEDGEMENTS
First, I want to thank Dr. Swamy Anantheswaran for his guidance,
wisdom, and consideration throughout the course of my Ph.D. program.
I
am very thankful to Dr. Virendra M. Puri and Dr. Carlos A. Zuritz for
initiating the part on mathematical modeling.
I would like to extend my
appreciation to Dr. Puri for assisting me with the use of the finite
element method and TWODEPEP program.
Dr. Robert B. Beelman and Dr. Gregory R. Ziegler are acknowledged
for their valuable inputs and the critical reviews of the manuscript.
1
am also thankful to the Departments of Food Science at Penn State and
University of Nebraska for granting me the graduate assistantship which
made this research possible.
1 want to thank Mr. Soumya Roy in assisting me in this research in
many ways.
I am also very thankful to all my friends in State College,
especially the McClellans, in providing support during the writing of
the manuscript.
Last but not least, I am grateful to my parents, Mr. Shen, Yun-Jen
and Mrs. Shen, Dai Shio-jen, and my husband, Chu-Neng John Lin, for
their encouragement and sustaining help that pressed me toward my goal.
1
Chapter 1
INTRODUCTION
One of the revolutions In the American kitchen during the
twentieth century has been microwave cooking.
Due to rapid changes In
life styles, 56% of American women are working and 54% of households are
dual-lncome families (Messenger, 1987).
Today, microwave ovens have
become an essential appliance in homes.
Microwave ovens are also making
their way into work places and fast food restaurants, and are even found
beside vending machines.
The ownership of microwave ovens In the United
States has reached a level of 70%, and the sales of microwave ovens are
still increasing (Anonymous, 1989).
Oven manufacturers are now producing smaller economy size ovens
for use by low income families or as a second microwave oven in a
household.
the market.
This has further stimulated the sales of microwave ovens in
In pace with such a strong increase in popularity of
microwave ovens, consumers' demands for microwaveable food products are
constantly growing.
This has created opportunities for food processors
and manufacturers to develop food products and packages specifically for
microwave heating.
However, regardless of the success and the acceptance by the
consumers, there are many problems associated with microwave cooking.
They can be summarized as follows: 1) unsatisfactory product quality-nonuniform temperature distribution, rubbery or soggy texture in the end
products and unacceptable flavor development; 2) concerns about
Insufficient microbial destruction due to uneven cooking; and 3) safety
hazards such as over-heating of the center in infant formula bottles or
the volcano heating effect due to localized superheating in casseroles
or high salt/fat sauce products.
The uneven microwave radiation within a microwave oven and
different rates of microwave power absorptions by various food
components cause non-uniform heating and temperature distribution In the
food product.
This is often recognized as the major shortcoming in
microwave food processing.
Methods such as stirring, standing time and
rotation of the container may help to equalize the temperature
distribution, but they actually detract the user from the convenience of
microwave cooking.
Organoleptic properties, such as good texture and
appearance, browning of roasted meats and baked products, and crispness
of bread crust, pizza crust and potato chips, are demanded by the
customer but often cannot be achieved satisfactorily by microwave
heating alone.
Knowledge of the temperature distribution or temperature profile
during microwave heating is important for the calculation of lethality
or time of inactivation of enzymes during pasteurization/blanching or
sterilization.
Also, a better understanding of the interaction between
the food components, packaging system and the microwave during microwave
heating will be useful for the design of microwave processes, packaging
and product formulation to combat the problems associated with microwave
heating of foods.
Microwaves propagate as electromagnetic waves.
In order to solve
problems of nonuniform heating, a thorough understanding of the physical
characteristics of the oven design, mechanisms of electromagnetic wave
transmission, the effects of geometry, chemical composition and physical
structure of the food and container, and the mechanisms by which food
products absorb microwave energy and dissipate it as heat is required.
In the microwave oven, the microwave field distribution is affected by
the transmission properties of waves in the oven cavity.
On the other
hand, microwave absorption and dissipation by the food products is
governed by the dielectric and thermal properties of foods during
microwave heating.
Based on the complete knowledge of the microwave oven system and
the products, the microwave heating of foods can be modeled using the
fundamental theories of electromagnetic wave propagation and heat and
mass transfer.
The solutions to the mathematical models can simulate
the time-temperature distribution of a food load being heated in a
microwave oven.
The best possible improvement in microwave heating can
then be obtained by the optimization of all of the variables in the
model.
If the model can successfully simulate the heating behavior of a
specific food product in a microwave oven, product development
scientists will then be able to test the effects of different variables,
including packaging and food formulation, without having to go through
expensive experimental work.
The microwave processes can then be
utilized more effectively and successfully.
The study of the effect of size and shape of food on the
temperature distribution can provide information about cold spots and
hot spots within foods heated within a container of various shapes.
This information will be useful for microwave food package design.
Based on this information, the intensity of the microwaves incident on
the food can be altered by the insertion of susceptor materials to
facilitate improved microwave absorption or by partially blocking the
waves by aluminum foil (Keefer, 1986, and Huang, 1987).
Thus, the
problem of uneven heating of foods can be eliminated by the design of
appropriate packaging materials.
The study on the effect of physical
structure and chemical constituents on the microwave heating
characteristics will be useful for product formulation and for
developing microwave cooking instructions for microwaveable foods.
For
example, the heating rates of food with different salt concentrations
are drastically different.
Chemical composition influences the
microwave penetration depth and power absorption directly.
The effects
of these factors can be simulated by a mathematical model which can take
into account all of the possible conditions of the food, microwave heat
source and heating environments during microwave heating.
5
Chapter 2
LITERATURE REVIEW
Microwaveable foods are becoming increasingly popular in the
marketplace.
The number of microwave products in the market has been
growing rapidly.
The 1988 sales of microwaveable foods in the U.S. have
been estimated at eight billion dollars, and the sales of microwave
ovens have also been reported as breaking the record of the sales of any
kitchen appliance (Anonymous, 1987).
Nevertheless, there has been only
a limited application of microwave processing in the food industry.
Microwave energy was first used in food processing in the early
1940's (Decareau, 1984).
Since then, attempts have been made to use it
in thawing, cooking, reheating, drying, blanching, pasteurization,
sterilization, puffing, baking, proofing and in many other unit
operations.
Though microwave ovens are popular at home, established
microwave applications at the industrial level are scarce in the food
industry.
Food processors were not able to adapt microwave processing
to commercial applications due to insufficient knowledge and technical
support (Osepechuk, 1984).
Due to the many advantages in microwave
processing, such as improved product quality, increased yield, savings
in operating time and floor space, and cool operation of the plant,
there is a growing need for more applications of microwave energy in the
food industry.
It is necessary to conduct basic research on the effect
of different parameters on the microwave heating of food products to
provide the food industry with valuable information necessary for the
more successful use of microwave processing.
Thermal processing is one of the most important unit operations in
food processing.
Microwave heating can be utilized for thermal
processing, by which food is heated to an end point temperature at a
desired rate of heating.
Most food products prepared from raw
ingredients to the stage of ready-to-eat require some form of heating.
The amount of heat treatment and the method of heating depends on the
final form of the product and the shelf-life requirement.
During conventional thermal processing, heat is transferred from a
heat source to the product by at least one of the principle modes of
heat transfer: conduction, convection and radiation.
In conduction,
heat or energy is transferred from molecule to molecule by collision,
which occurs primarily in solids.
In fluids, energy is transferred by
the movement of molecules possessing high energy from one point to
another which describes the mechanism of convective heat transfer.
Heat
radiation transfers energy in the form of electromagnetic waves between
materials at different temperatures without a need for a medium.
Unlike the three mechanisms of energy transport mentioned above
for conventional heating, microwave heating results from the
interactions of chemical constituents of materials with an
electromagnetic field.
During conventional heating, heat transfer
occurs when there is a temperature gradient, and the rate of heat
transfer is dependent on the temperature difference.
However,
microwaves are absorbed by dielectric food components, such as molecules
with dipole moments like water, and conductive ions.
The electric
energy then dissipated into thermal energy by molecular friction and
transferred into the surroundings.
Thus food is heated up
"instantaneously" along with the absorption of microwave power.
When
microwaves generate heat within the food, components with different
dielectric activities heat up at different rates.
The temperature in
the food is equilibrated by conduction.
Factors influencing microwave heating include: properties of the
material, operating frequency, configurations of the power distribution
system, source of electromagnetic power and field distribution.
These
factors are reviewed in the following sections.
2.1
Electromagnetic Waves
Electromagnetic waves consist of coupled electric and magnetic
waves in the form of accelerated charged particles.
The electric wave
(E) and the magnetic wave (H) propagate perpendicular to each other.
The progress of the wave is accompanied by a flow of energy which is
propagating perpendicular to the plane of electric wave and magnetic
wave (Strother, 1977).
2.1.1
Wave Equations
In 1873 James C. Maxwell, a professor at Cambridge University,
England, published the first unified theory of electricity and magnetism
and founded the science of electromagnetics (Kraus, 1984).
A wave of both E and H transverse to the direction of propagation
is called a "Transverse Electromagnetic (TEM) wave."
Maxwell described
the relationships of electric and magnetic field by the following two
equations (Kraus, 1984):
V x B - J *
|£
(2.1)
V x - . - H
where E is
(2.2)
theelectric field in [volt/m],
D is
theelectric flux density in [coulomb/m2],
H is
themagnetic field in [ampere/m],
B is
themagnetic flux density in [weber/m2], and
J is the current density in [ampere/m2].
The field components are related by material properties as follows:
J - oE,
D - eE, and B - pH
where o
isthe conductivity of
the material in [siemens/m],
e
isthe permittivity of
the material in [farad/m],
fj
isthe permeability of the material in [henry/m].
and
Therefore, Eqs. 2.1 and 2.2 become:
V x JST - oJT + e j ?
<2,3)
V x l . - i . |f
(2 .4 )
E, H, D, B and J are complex vector quantities of rms (root-mean square)
value and e^-t dependency, which are time-harmonic functions.
The time-
harmonic functions can also be expressed as sinusoidal function of time
since e^ut - cos(ot)
V x H
+ j sin(ut). Therefore,Eq. 2.3 becomes
- oE0 sin(ot)+ ueEQcos(ut)
(2.5)
Rewriting Eq. 2.5 In a phasor form,
V x H
- oE + jueE
(2.6)
Similarly, Eq. 2.4 can be written as
V x E
- -ju/iH
(2.7)
For most materials other than ferromagnetic substances, the value
of fj is essentially the same as p0. This holds true for biological
materials and agricultural products (Nelson, 1973).
Substituting for the complex conductivity a with a ' + jo" and
complex permittivity e - e' - Je", Eq. 2.6 becomes
V x H - (o'+oe")E + ju(e'-o"/w)E
(2.8)
or it can be expressed as
V x H - o.E + jue.E
(2.9)
where oe and ee are the effective conductivity and permittivity,
respectively.
Since V x H - Jtotal (total current density), Jtotai is a
sum of a conduction*current density (oa) and a displacement-current
density (oea) in time-phase domain.
The operator j with the
displacement-current density indicates that it leads the conductioncurrent density, which is in phase with the applied field, by 90°.
Eqs. 2.6 and 2.7 are the wave equations which relate the space and
time variation of the electric field intensity.
The solution for the
electric field intensity as functions of space and time as a
trigonometrical expression is shown below (Kraus, 1984).
Ey - E0sin(/3x+ut) + E0sin(/3x-ot)
(2.10)
10
Eq. 2.10 represents two waves, one to the left and one to the right,
which make a complete transverse sine wave.
The trigonometric solution
can also be expressed in exponential form as
Ey - E0 eJ(Bt^ K>
(2.11)
The electric field has an imaginary part and a real part of the
exponential function.
The imaginary part is
Ey - E0 Im eK"*’*> - E0 sin(ot-px)
(2.12)
and the real part is:
Ey - E0 Re
2.1.2
- EQ cos(ut-/3x)
(2.13)
Standing Waves
When a wave propagates from one medium to a second medium, a part
of the wave is transmitted through the boundary of the two media and
continues on in the second medium while the other part of the wave is
reflected from the boundary back to medium one.
travels in the opposite direction.
The reflected wave
At the point of reflection, the
reflected wave leads the incident wave by a phase shift 6 in time-phase
(Kraus, 1984). The reflected wave has the same frequency and the same
sinusoidal form as the incident wave.
When the incident wave is defined
as Ey0 - E0 e ^ <*t+^x) and the reflected wave as Ey1 - E1 e ^ Bt‘^x+4>, then the
total electric field is Ey - Eyo + Ey1 - E0 sin(ot+/3x) + E, sin(ut-0x+6).
In most situations, the amplitude of reflected wave is smaller than the
incident wave, and the general form of the standing wave equation is
Ey - /(E0+E,)2 cos20x + (Eq -E^2 sin2/3x sin(ut+/3x)
(2.14)
11
The two waves add to each other at some points (constructive) and
subtract from each other at other points (destructive).
the resultant wave Is like an envelope.
The shape of
The maximum value of the
envelope corresponds to the sum of the amplitudes of the Incident and
reflected waves (E0 + Ej), while the minimum corresponds to the
difference between the two (EQ - Ej).
2.1.3
Penetration Depth
The differential form of the general wave equation can be
expressed as follows (Kraus, 1984)
— -f
(2.15)
vm13E v
ox2
where y* i-s the propagation constant.
The solution of Eq. 2.15 for a
wave traveling in the positive x direction is Ey - E0e‘*x.
Since y - a +
j/3, the solution becomes
Ey - E0 e-« e*W*
(2.16)
where, a is the attenuation constant and j3 is the phase shift constant
which determines the wavelength in the material.
Eq. 2.16 gives the
variation of Ey in both magnitude and phase as a function of x.
The
electric field attenuates exponentially and is retarded linearly in
phase with increasing x.
When Eq. 2.16 is rewritten as Ey - EQ e'K/l)e'^x/0>, D is the 1/e
depth of penetration, which is equal to 1/i £rr/jo.
The penetration depth
decreases inversely proportional to the square root of the frequency.
Therefore, a high-frequency field penetrates a shorter distance than a
12
low-frequency field In a given medium.
This phenomenon Is known as the
"skin effect."
2.1.4
Transmission, Reflection and Refraction
In the case of a linearly polarized plane wave obliquely Incident
on a boundary between two media, the plane wave can be resolved into
perpendicular (the E field perpendicular to the plane of incident) and
parallel (the E field parallel to the plane of incidence) components.
These waves are perpendicular polarized and parallel polarized,
respectively (Kraus, 1984).
For the perpendicular polarized wave, Snell's law of reflection
says that the angle of reflection is equal to the angle of incidence.
Also, Snell's law indicates sin(6{) - [q1/q2] sin C©f), where 6( is the
angle of refraction and 8( is the angle of incident.
For a dielectric
medium the index of refraction rj is equal to //irep and Snell's law
becomes
sin(8t) *
J
sin(0i)
(2.17)
>|
The critical angle is 6fc - sin'1 V e2/e1• When a wave is incident from
the more dense medium onto the less dense medium (e, > e2) at an angle
greater than the critical angle, the incident wave is totally internally
reflected back into the more dense medium.
In the case of parallel polarization, an incident angle exists
when the wave is totally transmitted into the second medium (Kraus,
1984).
This angle is called the Brewster angle, where 0{B - tan*1
13
/ e2/ei• T*1® Brewster angle Is also sometimes called the polarizing
angle since a wave composed of both perpendicular and parallel
components and incident at the Brewster angle produces a reflected wave
with only a perpendicular component.
2.1.5
Power Intensity
Consider a planewave traveling with averticalelectric
and a horizontal magneticfield H.
field E
The energy density W# at a point in
an electric field is
W„ - h eE2
(2.18)
The energy density Wn at a point in a magnetic field is
WB - h pH2
(2.19)
The instantaneous electric and magnetic energy densities expressed as a
rate of change of energy in the wave is
|f - -V • X
xH
(2.20)
The vector product of the electric and magnetic field is called the
Poynting vector (Baden Fuller, 1979).
8 ■ K x B
The
Poyntingvector (S) is avector in the
and
isperpendicular to theplane of
(2.21)
direction of
E and Hvectors.
wave propagation
Since
the
intrinsic impedance is equal to E/H - /p/e, by rearranging Eq. 2.18 and
2.19 we can arrive at W( - H pH2 - *1 cE2 - tfe. Thus the electric and
magnetic energy densities in a plane traveling wave are equal, and the
14
total energy density is the sum of the electric energy density and the
magnetic energy density, and therefore W - eE2 - juH2. Although equal
amounts of energy are associated with the electric and magnetic fields,
in practice most of the measurable effects of electromagnetic waves on a
material are due to the electric field energy (Strother, 1977).
Considering two plane waves traveling in opposite directions, the
instantaneous value of Ey resulting from the two waves is
(2 .22 )
Ey - E0 sin (ut + J3x) + E1 sin (ut - /Jx)
In the situation of a pure standing wave, E0 - - Ej, the net Poynting
vector is zero, and hence no power is transmitted.
The electric energy
density is We - 2eE02 cos2(ut) sin2(/3x) and the magnetic energy density
is Wm - 2/iHQ2 sin2(ot) cos2(/3x). The electric energy density is at
maximum when the magnetic density is zero, and vice versa.
Furthermore,
the points where they are maximum are a quarter wavelength apart.
The
energy oscillates back and forth from the electric form to the magnetic.
The Poynting vector of a pure standing wave is expressed as
Sx » ~4EJi0 cos(ut)sin(wt)cos (Px)sin(Px)
■ -4
E% cos(u£)sin(cot)cos(px)sin(Px)
and the peak value of the Poynting vector is V e/fj E02.
(2.23)
Eq. 2.23 shows
clearly that Sx is maximum at ut-ir/4.
2.1.6
Field Distribution
In the microwave oven, the cavity dimensions are of the same order
of magnitude as the wavelength of the microwaves.
Thus a number of
standing waves will exist inside the microwave oven.
Each of these
15
possible standing wave patterns Is referred to as a mode.
The mode
pattern field Is unique to every microwave oven and depends on the
geometrical dimensions and the microwave distribution system.
It is
also affected by the geometry and composition of the food present In the
oven.
Calculation of the three-dimensional field distribution inside a
microwave oven requires a good knowledge of all the modes present in the
microwave oven.
The transverse electromagnetic waves (TEH) in the transmission
lines are waves with electric and magnetic
to the direction of propagation.
fields entirely transverse
The mode of wave transmission can be
identified by the subscripts m and n for TE (transverse electric) and TH
(transverse magnetic) waves in a waveguide.
The value of m or n
indicates the number of half-cycle variations of each field component
with respect to z (broad side) and y directions, respectively.
Each
combination of m and n values represents a different field configuration
(Kraus, 1984).
To obtain the field configuration and complete information
concerning waves in a transmission line, one has to solve the wave
equation subject to the appropriate boundary conditions.
equation can be developed from Maxwell's equations.
A wave
The boundary
condition applied to the wave equation is that the tangential component
of the resultant E must vanish at perfectly conducting walls.
The
restrictions are that the field components vary harmonically both with
time and distance and also attenuate with distance.
Then one has to
choose the type of higher-order mode of transmission to be analyzed.
transverse electric (TE) wave has Ex - 0 while a transverse magnetic
A
16
(TM) wave Hx - 0,
Next a
component can be obtained
solution of a scaler-wave equation In one
that fits the boundary conditions.
This
solution is substituted back into the equations for other field
components.
Thus a set of equations can be obtained which give the
variation of each field component with respect to space and time (Kraus,
1984).
The particular mode
or modes that are actually present
the dimension of the system, the method
the discontinuity of the structure.
dependon
of microwave distribution and
Each mode of transmission has a
particular cutoff wavelength, velocity and impedance.
When the
frequency is high enough to permit transmission in more than one mode,
the resultant field is equal to the sum of the fields of all the modes
present.
In a waveguide, there is no attenuation of the wave if the guide
walls are perfect conducting and the medium is a lossless dielectric.
However, if there are any losses incurred by the wave due to imperfect
conducting walls or a lossy medium, there is attenuation.
Therefore,
the propagation factor, y . composes of a real (a) and an imaginary (/3)
component.
A finite tangential component Et is developed from the
electric field due to the finite wall conductivity, and a is not zero
due to the lossy dielectric medium.
The field decays exponentially with
distance in the direction of propagation as given by (Kraus, 1984)
P - P0 e'2tfX
a is the attenuation factor which can be defined by Eq. 2.25.
(2.24)
17
dp
~ 2 Pdx
■ 1 p o w e r lost p a x unit d i stance
2
p o w e r transmitted
(2.25)
The attenuation in this case is due to an actual power loss into Joule
heating.
2.2
Microwave Cavity Oven System
The performance of microwave ovens varies between different
manufacturers and even between models of the same manufacturer.
It is
helpful to know the construction and operation of microwave cavity ovens
so that the variability in the microwave heating process can be better
understood and controlled.
2.2.1
Magnetron and Power Supply
The Federal Communications Commission allocates the radio
frequencies that can be used for heating as ISM (industrial, scientific,
and medical) frequencies (Decareau, 1985).
At present, 915 MHz and 2450
MHz are the frequencies used most in food industry and commercial ovens,
respectively, in the United States.
According to Raytheon, high
frequency (2450 MHz) permits better coupling of microwave energy to
small loads and allows for the presence of a greater number of modes in
the oven, which eliminates hot and cold spots due to the more random
heating patterns, while General Electric has claimed that use of 915 MHz
provides greater penetration and less thermal runaway in defrosting
(Osepchuk, 1984).
18
Most consumer microwave ovens operate at a frequency of 2450 MHz.
The magnetron designed for consumer microwave ovens Is different from
the magnetrons In the Institutional types of ovens and ovens operating
at 915 MHz frequency.
There is a variation in the microwave power come-
up time in different models of microwave ovens.
It takes from 2 s to
4.5 s, depending on the oven manufacturers, for the magnetron in a
consumer oven to operate after the power is turned on.
This is because,
for a magnetron in a consumer oven to generate microwave power, the
filament must come to the operating temperature.
Normally, it takes
about 1.5 s for the filament to come to the operating temperature in a
new magnetron (Gerling, 1987).
Variations in the lag time for a
magnetron to operate after the oven power is on make a great difference
in the final temperature distribution within a small product for the
same time and power setting.
Operation of a magnetron is partly determined by the amount of
power reflected back to the magnetron and the phase of the reflected
power.
This can be measured in relationship to the delivered power as
voltage standing wave ratio (VSWR), which is dependant on the exact
nature of the impedance in the feed system and the load (Gerling, 1987).
The power generated from a magnetron is affected by three factors:
line voltage, heat, and load.
According to Gerling (1987), a change of
±5% on line voltage can lead to ±5 to ±20 % difference in power
supplied.
Overheating of the magnetron can cause a reduction of power
to the oven by as much as 10%.
The size of the load in the cavity
determines the amount of power reflected back to magnetron which in turn
affects the performance of the magnetron.
19
Because of the reduction of power generation due to the reflected
power from the cavity, oven manufacturers have Installed glass or
ceramic shelves to absorb some power and reduce the power reflection.
The efficiency of microwave power absorption by a product is affected by
the load size in the cavity for the same reason.
The larger the load
size, the lower the power reduction by reflection and cavity loss and,
hence, the higher the efficiency of microwave power absorption.
2.2.2
Power Distribution System
Waveguide: Transmission lines, which can convey electromagnetic
waves only in higher-order modes at high frequencies, are usually called
waveguides.
Microwaves can be considered to travel down the waveguide
in a zigzag fashion with waves being reflected repeatedly between the
metal walls of the guide.
The waves of higher-order modes have
components of E or H in the direction of propagation.
The cutoff
frequency (or cutoff wavelength), below which the transmission is
impossible in the waveguide, occurs when the wavelength is of the same
order of magnitude as the dimension of the line (Kraus, 1984). The
longest wavelength which can be transmitted in a higher-order mode is
twice the width of the spacing of the waveguide.
Mode Stirrer: The mode stirrer in a microwave oven rotates when
the power is turned on and causes the magnetron frequencies to shift
slightly as it rotates, changing the field conditions.
It enables
different mode patterns to be alternated in the oven during the heating
20
cycle.
A combination of a number of different patterns of modes during
heating helps to ensure uniform heating.
With the mode stirrer,
unfavorable coupling between modes is avoided by the time separation of
modes.
Such coupled modes are considered as degenerative and generally
give a diagonal power density pattern as opposed to the regular single
mode pattern which parallels the vertical cavity walls (Risman et al.,
1987).
The mode patterns excited at different times and from different
locations are less structured and may be non-symmetrical.
This creates
an effective field which has maxima and minima at different locations.
Rotating food on a turntable in reality is analogous to a mode stirrer.
During microwave heating of food, field pattern will change with the
heating cycle since dielectric properties of food are temperature
dependent.
Moving food in the microwave oven will affect the mode
patterns and allow food to pass through areas of high and low power
density and, thus, improve the heating uniformity (Ohlsson, 1989).
2.2.3
Oven Cavity
The dimensions of the oven cavity are generally several times
greater than the wavelength at 2450 MHz in free space.
Multiple
reflections of microwaves in the microwave oven result in a number of
possible three-dimensional standing wave patterns.
According Watanabe
et al. (1978), the power density distribution in a domestic small sized
microwave oven contained over ten propagating modes and many non­
propagating modes.
In general, the field in a typical microwave oven
21
can be described as a mixture of about 20 to 30 modes.
Thus, the
microwave oven is often referred to as a multimode cavity; the number of
modes Increases rapidly as the cavity size Increases (Lorenson, 1990).
The field distribution will change if the microwave generation and
distribution system is changed.
Also, putting a container of food into
the oven changes the reflections and will, therefore, modify the mixture
of modes in the oven.
2.3
Product Properties
The mechanisms by which the constitutive parameters govern the
conversion of electromagnetic energy to heat are the flow of free charge
carriers and the polarization effects.
These constitutive properties
are, in general, functions of geometry, frequency, and temperature.
2.3.1
Permittivity and Conductivity of Dielectrics
For a dielectric material, the electrons are bound at the
equilibrium positions and cannot be detached by the application of
electric fields; and there is no migration of charges in a dielectric
under the electric field.
Under normal conditions, the negatively
charged electron cloud surrounds the nucleus symmetrically.
When a
dielectric is placed in an electric field, the electron cloud becomes
slightly displaced or asymmetrical with respect to the nuclei.
field is removed, the atoms and molecules return to their normal
unpolarized state and the dipoles disappear.
Once the
22
field is removed, the atoms and molecules return to their normal
unpolarized state and the dipoles disappear.
The major component of most foods is water.
Hater is a dipole
molecule which will become polarized under an applied field.
If the
electric field oscillates along the length of the water dipole, it sets
up a vibration along the bond.
When the electric field is perpendicular
to the bond length, there will be a torque exerted on the molecule which
will set it into rotation.
electric dipole moment.
The polarization of a dielectric produces an
For the energy to dissipate as heat, it must be
transferred into molecular vibration against its surroundings rather
than remain as absorbed internal vibration (Curnutte, 1980).
The constitutive parameters, permittivity, permeability, and
conductivity are important parameters in describing the interaction of
electromagnetic waves with the material.
the capacitance C per unit length.
The permittivity e is equal to
The permittivity
e
is a complex
quantity of great importance in the microwave heating of foods.
The
complex form of permittivity e - e' - je" has real and imaginary parts,
both of which are frequency dependent.
The real part of the complex
permittivity e' is the dielectric constant.
the dielectric loss.
The imaginary part en is
The permeability p is equal to the Inductance L
per unit length; and, the conductivity a is equal to the conductance G
per unit length.
According to Eq. 2.6, the space rate of change of H equals the sum
of the conduction-current and displacement-current densities.
will be three possible conditions: (1) oc »
o.
There
o, (2) oe » o, or (3) oc «
When the displacement current is much greater than the conduction
current, as in the case of condition (1), the medium behaves like a
dielectric.
If o - 0, the medium is a perfect, lossless dielectric.
On
the other hand, when the conduction current is much greater than the
displacement current, as in condition (3), the medium may be classified
as a conductor.
In conductors, current results from movement of free
electrons because of conductivity o.
In dielectrics, the bound charges
are predominant; therefore, the current is normally considered from the
displacement jueE.
Since both conductivity and permittivity are
functions of frequency, frequency is an important factor in determining
whether a medium acts like a dielectric or like a conductor.
Frequency also determines how microwave power is absorbed by a
dielectric and dissipated as heat, since the dielectric properties are
functions of frequency.
As frequencies increase from low values, the
polar molecules can follow the change in the direction of the electric
field up to a critical point.
At that point, the dipole motion can no
longer keep up with the changing field; as a result, the dielectric
constant drops while the dielectric loss increases with increasing
frequencies in this region.
The energy is absorbed asa result of
phase lag between the dipole rotation and the field.
the
These dispersion
and relaxation behaviors, in relation tothe frequency change, were
modeled by von Hippel (1954) as shown inFig. 2.1.
The energy lost from the wave, as a result of attenuation, is
dissipated in the propagating medium as heat.
The real part of Eq. 2.8,
also referred to as effective conductivity oe - (o'+ue")E, represents a
dissipative current in phase with E; and (ue'-o")E, or the effective
permittivity ee, represents a reactive current out of phase with E
24
(Nelson, 1973).
The effective conductivity o0 and the effective
permittivity e# can also be measured experimentally as dielectric loss
and dielectric constant, respectively.
These empirical constants imply
the capacity of a material to interact with the field (dielectric
constant) and the ability of a material to convert the absorbed energy
to thermal energy (dielectric loss) (Curnutte, 1980); a" generally turns
out to be negligible and, in this situation, ee is still referred to as
the dielectric constant, which decreases with increasing frequency as
seen in Fig. 2.1.
Dielectric constant is associated with the stored-
energy density e#E2 - e'E2, while dielectric loss oe (includes both the
o' and the e" components) is considered to be the total loss current
associated with the energy dissipated as heat, oeE2 - (o'+ue")E2
(Vermeulen and Chute, 1987).
ar'T
Km-K*
to
r
100
CUT
Fig. 2.1
Dielectric properties as functions of frequency
(von Hippel, 1954)
25
The ability of a material to absorb microwave energy and dissipate
as heat can be expressed as dissipation factor, which is normally known
as the loss tangent (Nelson, 1973).
The loss tangent is defined as
tan 6 - a ^/na '.
6 is the angle of a phase difference which exists between the field
intensity E and the current density; it is called the dielectric loss
angle.
For a nonconducting dielectric material the conductivity a is
zero, then the loss tangent can be expressed as
tan 6 - «€"/»«' “ «"/«'.
(2.26)
The inductive losses in biological materials are normally
considered zero; i.e., the relative permeability fir is assumed to be
unity.
The dielectric constant of a material is associated with the
capacitive loss which does not affect the heating rate of a material;
whereas the dielectric loss, which is related to the resistive loss,
determines the voltage coupling of the material and power absorption
levels.
The variation of power absorption with dielectric loss can be
shown as (Mudgett and Nash, 1980)
P" - 2jrfe0E2ep"
(2.27)
where P" is the volumetric microwave power absorption [w/m3J,
f
is the frequency,
e0
is the permittivity at free space,
E
is the electric field
er"
is the dielectric loss.
intensity, and
The phases of different intrinsic impedance in a heterogeneous
system may be arranged in series or in parallel analogous to electrical
26
circuits (Mudgett and Nash, 1980).
The phase which intercepts more
power will heat at a faster rate than the phase that absorbs less power.
2.3.1.1
Effect of Composition
The composition of food affects the dielectric properties which,
in turn, affect the heating characteristics of food.
Uater accounts for
much of the primary absorption of microwaves in biological materials.
Dielectric properties of food systems, as functions of frequencies and
temperatures, can be predicted based on the activities of water
molecules, conductive ions and inert solid contents.
The model can be
useful for product formulation of composite food systems to reduce the
phase differential heating effect.
Dielectric properties of biological materials have been studied
extensively by Risman and Bengtsson (1971), Mudgett et al. (1974, 1977,
1979 and 1980), and Ohlsson and Bengtsson (1975).
Most of these works
are related to the measurement and modeling of dielectric properties in
various food products.
It has been found that the dielectric behavior of food components
is mainly affected by moisture (especially unbound water) and ash
content (dissociated salt ions in the aqueous solution).
In general,
the higher the moisture content, the higher the dielectric constant.
Dielectric loss increases with increasing moisture content to a point,
then it decreases with higher moisture contents (Schiffmann, 1986).
ensure that the majority of microwaves are being absorbed by foods,
materials of the mlcrowaveable containers are designed with low
To
27
dielectric loss; therefore, the containers do not absorb much microwave
energy.
Dissolved salts or ions bind water molecules and depress the
dielectric constant to a level lower than that of pure water.
The
dielectric loss of aqueous ionic solutions is elevated above that of
pure water due to conductive and/or electrophoretic migration of free
ions in the applied field.
In the aqueous mixtures, however, both the
dielectric constant and dielectric loss are depressed by insoluble and
immiscible constituents such as lipids, proteins, and carbohydrates.
(Mudgett, 1985).
2.3.1.2
Temperature Effect
Vibration of a dipole is not only dependent on the frequency and
intermolecular bindings, but is also temperature dependent.
Foods at
high temperatures absorb less microwave energy than at low temperatures.
At high temperatures, thermal movements are more intense, and energy
necessary for overcoming the inter-molecular bonds is less; therefore,
the dielectric heating decreases with the increasing temperature
(Ohlsson, 1989).
Debye predicted the dielectric behavior of water as
functions of frequency and temperature.
Increasing the temperature
depressed both e ' and e" of free water at fixed frequency and shifted
the region of dispersion to higher critical frequency (Mudgett, 1985).
In aqueous ionic solution, hydrated ions try to move in the
direction of the oscillating electric field.
During their movement, the
ions transfer energy randomly to adjacent water molecules.
Water
28
molecules are more mobile and less tightly bound to the Ions at higher
temperatures.
Thus, Ions absorb and dissipate energy freely, and the
conductive heating Increases with the Increasing temperature (Ohlsson,
1989).
The dipole component of dielectric loss decreases as the
temperature Is Increased, while the conductive dielectric loss increases
as the temperature is Increased.
Thus, the total dielectric loss of a
food initially decreases as the initial temperature Increases, when it
is dominated by the dipole loss effect.
The total dielectric loss then
increases as the temperature increases further, when the conductive
effect dominates (Mudgett, 1985).
2.3.1.3
Effect of multiphase mixture
Mudgett et al. (1974) modeled the effect of structure on the
dielectric
properties of a material by using a combination ofthe
Maxwell and Rayleigh models.
This model was originally proposed for the
prediction of complex conductivity of randomly distributed spheroidal
particles in a continuous phase by Fricke (1955).
Mudgett adapted the
model for the prediction of the complex permittivity for a
non*interactive two-phase mixture:
. _
ecl€t ii+xvt) +eji-v.)x]
e0U+V,)+€#(l-V,)
where «m is complex permittivity of a mixture,
ec is complex permittivity of continuous phase,
eg is complex permittivity of suspended phase,
(2
28)
29
X is fraction of phase,
Vg is volume fraction of suspended phase in the mixture.
Thus, the dielectric properties of a non-interactive, two-phase mixture
may be predicted from the proportion and material properties of each
phase if the particle shape and orientation are known.
The density of a product has an effect on its dielectric constant.
Since the dielectric constant of air is very low, the presence of air
will reduce the material's dielectric constant.
The dielectric constant
of a material increases almost in a linear fashion with density
(Schiffmann, 1986).
The study by Rzepecka and Pereira (1974) on dried
milk and dried whey powder showed that the dielectric permittivity (e)
is linearly correlated with bulk density under constant moisture
content.
Nelson (1980) also found that the dielectric constant of fresh
fruits and vegetables correlated with moisture content and tissue
density.
2.3.2
Transmission Properties
Vhen a wave is propagating in an unbounded medium of permeability
and permittivity e, the velocity of the wave is v - 1/7pe, or the
phase velocity is
- ■“ ■4m ■f
<*•**>
The index of refraction rj is defined as the reciprocal of the relative
phase velocity of the wave in the medium with respect to the velocity of
30
light, or rj - ^/ir«r.
For nonferrous media, fir is nearly unity, so that
The refraction of microwave energy is proportional to the square
root of the dielectric constant.
much greater than that of air.
Most foods have a dielectric constant
The refracted wave in food travels in a
direction close to the surface normal.
Consequently, once the microwave
energy is transmitted into the food, there is very little possibility
for it to travel close to the interface between the food and the air or
escape out of food (Ohlsson, 1990).
The relative intrinsic impedance of a dielectric material, as
defined in Eq. 2.30, determines the level of power transfer from the
field in terms of reflection and transmission at the boundaries.
(2.30)
j*7n
The relationship of power incidence and transmission is expressed by
Snell's law as indicated in Eq. 2.17.
The relative reflected and
transmitted power with respect to the total power incident may be
estimated from two coefficients, the reflectivity coefficient and the
transmissibility coefficient, which are defined as follows:
(2.32)
Et 1* T^CQSflt
T^cosflj
where E{ is the incident electric field,
Ep is the reflected electric field and
E( is the transmitted electric field
(2.33)
31
At the boundary of two media with extremely different dielectric
properties, the reflected wave is nearly as large as the incident wave.
The field is essentially a standing wave.
Whereas at the boundary of
two media with similar dielectric properties,
the reflected wave is
small and the field is a traveling wave.
There are three types of impedance: intrinsic, characteristic, and
wave.
The characteristic impedance is basically related to electricity.
The intrinsic impedance and the wave impedance are field or wave
quantities involving ratios of electric field
Intrinsic impedance refers to the ratio
tomagnetic field.
of
H for a plane (TEH) wave in an unbounded medium.
complex
the phaserfieldsE and
For a medium with
and e, the intrinsic impedance is calculated as
(2.34)
For low conductivity dielectric material, substituting y in Eq. 2.34
with jut/jue, the intrinsic impedance becomes (von Hippel, 1954)
(2.35)
The wave impedance refers to the ratio of an electric field
component to a magnetic field component at the same point for the same
wave.
For a TEM wave the wave impedance is the same as the intrinsic
impedance; but for higher-order modes, there can be as many wave
impedances as there are combinations of electric and magnetic field
components.
The transverse-wave impedance is a function of the
intrinsic impedance of the medium, and is also the dimensions of the
32
waveguide.
As Che dimensions become very large compared to the
wavelength, Che transverse-wave Impedance approaches Che intrinsic
impedance of Che medium (Kraus, 1984).
2.3.3
Thermal and Physical Properties
In solid food materials, the rate of heat conduction is much
slower than the rate of microwave heating.
Uneven microwave absorption
and slow heat conduction result in temperature distributions in food
during microwave heating.
The only remedy is to heat the food product
at a lower power level (Ohlsson, 1983).
The microwave absorption by fats and oils is very low compared to
that of high moisture foods.
their low specific heat.
But fats and oils heat well because of
It takes less time to heat fats and oils to a
certain temperature than water under the same mass and the same heat
flux.
2.3.4
Geometrical Configurations of the Product
Kritikos and Schwan (1975) indicated that the microwave energy
absorbed by a large spherical biological tissue at high frequencies is
similar to that absorbed by an infinite slab.
In this situation,
heating is described as a skin phenomena and microwave energy decays
exponentially into the sphere.
Kritikos and Schwan (1975) also
characterized the maximum microwave heating in the spherical geometry
over a wide spectrum of frequencies in terms of quasi-physical optic
33
region and resonance region.
When the radius of the sphere Is longer
than the wavelength Inside the sphere (high frequency region), the field
converges and Is concentrated In the half wavelength region (quaslphyslcal optic region).
When the wavelength exceeds the radius (low
frequency region), the electric and magnetic modes resonate inside the
sphere and nonuniform heating occurs.
The effective gain of the power
at a certain distance from the center in one dimension can be calculated
by the product of the ratio of the surface area to the inside area and
the attenuation effect at that area as shown in Eq. 2.36 (Kritikos and
Schwan, 1975).
Ap
9Min
where PR"
.
2 * ° --------- .
p"
e-2.(ii-r)
isthe power per unit surface area at r - R,
Pr"
isthe power per unit surface area at r,
Ar
isthe outside surface area at r - R and
Ar
(2.36)
At
is the surface area at r.
If the value of the effective gain in power is greater than one, then a
concentrated heating effect will occur inside the sphere.
Ohlsson and Risman (1978) studied temperature distribution due to
microwave heating in spheres and cylinders of different diameters.
found that the concentrated heating effects are more pronounced for
spheres than for cylinders.
Conductive heat transfer proceeds at a
slower rate than the concentrated heating effect resulting from the
geometrical configuration.
The study of the effect of geometrical
They
34
configuration will help to eliminate the problem of nonuniform heating
by the proper design of the shape and size of a product.
2.3.5
Loading Effect
The efficiency of microwave power absorption is affected by the
load condition in the oven.
efficiency."
This is sometimes referred to as "coupling
Energy coupling efficiency is dependent on the impedance
characteristics of the microwave generator (Mudgett, 1986).
Power
coupling by a loaded microwave cavity is affected by product impedance
characteristics which may vary considerably during heating.
An
acceptable load impedance is one which couples sufficient power from the
magnetron to prevent excessive anode heating, and yet, does not load the
magnetron so heavily that it falls to oscillate at the correct
frequencies, shifting to another mode (Mudgett, 1985).
Furthermore,
when foods with different impedance characteristics are heated together
in a microwave oven, there will be a competition for the microwave power
between them (Ohlsson, 1983).
The coupling efficiency can be estimated
empirically by a correlation based on liquid and solid calorimetry.
The
true power incident at the dielectric surface can be estimated from the
maximum power generated from the magnetron by using either of the two
equations given by Mudgett (1986) as follows:
P0 - PB [l-exp(-aV)]
P ■
0
*
Km + Vb
(2.37)
(2.38)
35
where P0 is the power coupled by unmatched load,
Pm Is the maximum power generated by the frequency generator,
Kg is the empirical coupling constant,
a,b are empirical coupling coefficients.
The coupling coefficients are unique for each heating apparatus.
They can be determined by calorlmetrlc measurement.
The energy coupling
efficiency (Ey) is determined as:
E v - -^2 x 100%
2.3.6
(2.39)
Packaging
Since basic microwave oven performance cannot be changed, food
processors and manufacturers can only manipulate the ingredients and the
packaging system to improve the quality attributes of microwaveable
foods.
Factors such as shape and size of food product, covering
material, and cooking directions can significantly influence the
performance of microwave cooking.
It is very important for food
companies to conduct thorough testing and evaluation to see how the
designed package performs under a variety of conditions in different
microwave ovens.
36
2.4
Modeling of Microwave Heat Transfer
Mathematical modeling can play an Important role in the design and
optimization of the product and process parameters in order to obtain
the best possible improvement in heating.
2.4.1
Heat Transfer Models
The general model takes into account conductive heat transfer,
internal microwave heat generation, and surface cooling due to
convective and evaporative heat losses at the boundaries.
Therefore,
initial temperature, ambient air temperature, and water vapor pressure
at the surface of the product and in the air are needed to describe
initial and boundary conditions.
Ohlsson and Bengtsson (1971) developed microwave heating profiles
by computer simulation using finite difference technique and showed good
agreement with experimental time-temperature data collected in meat
products during microwave heating.
He also showed the influence of
thermal conductivity of the material and the surface heat transfer
coefficient on heat transfer during microwave processing.
Mass transfer
of water vapor from the food surface occurred simultaneously during
microwave heating.
Surface cooling effects were found due to both the
convective heat transfer and evaporative cooling.
Kirk and Holmes (1975) modeled temperature profiles in model foods
being heated in a microwave field.
The model predicted the temperature
profile of high moisture content substances at any given time and
37
thickness of the sample by using the finite difference method.
The
profiles agreed well with the experimentally measured temperature
profiles in gels of 1% ion-agar in water.
Ofoli and Komolprasert (1988) derived a mathematical model in
which the conductive heat transfer was considered negligible in
comparison with the internal heat generation due to absorption of
microwave power.
The temperature increase predicted by this model was
purely a function of electric field intensity and time.
The center
heating or edge and corner heating phenomena which occur normally in
microwave heating was not predicted by this model.
2.4.2
Heat and Mass Transfer Models
Lyons et al. (1972) have done experiments to measure the radial
and axial changes in temperature, moisture, and pressure with respect to
time in porous cotton yarn cylinders during microwave heating.
During
microwave heating, pressure may build up due to the accumulation of
super-heated steam.
Pressure increases significantly as the temperature
rises to the boiling point, and results in explosions and volcano
effects in eggs and sauces cooked in the microwave oven.
The phase
change may cause distortion on the product.
A water-laden sandstone model system was used by Wei and Davis
(1985) to predict heat and mass transfer phenomena in porous materials
during microwave drying.
data.
Their model agreed well with experimental
The model also predicted local moisture concentration, gas
densities, and pressure gradient with respect to time.
During microwave
38
heating, the center of sandstone Is hotter than the surface; air flows
slowly toward the center and water vapor migrates toward the surface as
liquid vaporizes on its way out.
Both heat and mass transfer must be considered when modeling
microwave thawing or dehydration.
Thermal and dielectric properties
vary drastically with the phase change during thawing.
much work reported in these areas.
There is not
Taoukis et al.(1987) used a modified
Isotherm migration method to develop a mathematical model to predict the
thawing time and temperature profiles of cylindrical meat products
during microwave processing.
Finite difference method was used to solve
the heat and mass transfer equations under the moving boundary
condition.
The model and experimental results compared well at low
microwave power at 2450 MHz.
Low frequencies, 300 MHz and 915 MHz, were
found to thaw the frozen meat with more efficient energy consumption as
compared to 2450 MHz.
2.4.3
Numerical Techniques
Numerical models can also be used to predict heat transfer and for
optimization of microwave packaging and food formulation.
The analysis
can be carried out by dividing the food load and container into discrete
elements.
The nonuniform heating effect and the differential heating
effect of multi‘Component meals or layered products can also be studied
by knowing their dielectric, thermal, and physical properties.
So far, the development of computational methods and techniques
for analysis of the multi-mode cavity has been very limited.
A three­
39
dimensional, finite element method makes it possible to numerically
calculate the absorbed power density in every discretized volume
element.
Ohlsson and Risman (1978) developed a finite element program
to investigate the n'onuniform temperature distribution in model foods of
simple geometries.
The microwave power density at a given location in
the sample was modeled for a cylinder and a sphere.
normal plane wave incident at the surface.
The model assumed
TM and TE incident waves in
the x, y, and z axes were considered for sphere, whereas only horizontal
(TE wave in y direction) and vertical (TM wave in z direction) were
considered in the cylindrical model.
Uithin the sizes and shapes of
meat and potato products studied, the model simulated temperature
distribution very well.
Center heating effects were found in spheres of
diameters less than 6 cm and in cylinders of diameters less than 3.5 cm
for the particular material studied.
Swami (1982) developed a finite difference model to describe
microwave heating of high moisture foods in cylindrical and rectangular
shapes.
The volumetric power for each finite shell was derived from the
surface power assuming a 45° or 90° incident angle.
Convective and
evaporative surface heat loss were considered in the boundary
conditions.
Dielectric properties of the gel samples at various sodium
chloride concentrations were predicted by Hasted-Debye equations.
The
model predicted temperature distribution in good agreement with the
experimental measurement for gel samples of high moisture and different
salt concentrations under two-dimensional and three-dimensional
microwave heating.
The finite element method is more versatile and has several
advantages over the finite difference technique.
It can handle
irregular geometries, analyze nonhomogeneous and nonisotropic food
products and is generally more accurate (Segerlind, 1984).
The finite
element method (FEM) is a procedure for solving problems by dividing the
domain of interest into small, basic elements.
The FEM is derived using
either variational methods or Galerkin's method for solving operator
equations.
The starting point for this method is the partial
differential equations describing the process.
The solution is
approximated by writing it as a linear combination of known functions
with unknown coefficients.
This approximation function is then
substituted into the original partial differential equations, and both
sides of the equation are multiplied by the full set of approximation
functions.
Finally, integrating over the solution region results in a
matrix equation, which is solved for the unknown coefficients.
provides the approximate solution.
This
The accuracy of a finite element
analysis depends both on the type of approximation function used in each
element and on the number of elements used.
The sizes of the elements
must be chosen to be a small fraction of the minimum wavelength in the
material.
According to Lorenson (1990), the size of the discretized
element should be less than about 3 mm.
Many FEM software packages are
available for used on mainframes, workstations, and PCs.
They leave the
engineers free to focus on the design issues instead of algorithms
(Cendes, 1989).
Finite element analysis was used to solve the Maxwell equation for
the three dimensional microwave field of a microwave heating furnace
41
system (Tejika et al. 1987).
In this case, the configuration and the
operation of the microwave transmission system and the cavity must be
clearly defined.
The major drawback in using numerical methods for
calculating the power density In a microwave oven is the difficulty in
clearly defining the mode pattern.
Full specification of the mode
pattern may be impossible because of the theoretical complexity of the
microwave propagation within the oven and because of the interference of
the field pattern from various load conditions in the cavity (Risman et
al., 1987).
2.5
Summary
The uneven temperature distribution in food during microwave
heating is affected by the microwave field distribution and by the
characteristics of the food.
The electromagnetic field inside the oven
can be described by Maxwell equations, but are difficult to solve
analytically.
It is necessary to develop a technique to measure the
microwave field distribution.
Thus far, methods of measurement for the
field distribution inside a microwave oven were found in literature to
be mostly descriptive.
Technique to measure microwave oven field
distribution quantitatively has to be developed.
Parameters of the food product to be heated in a microwave oven
are as important as the field distribution in determining the
temperature distribution in a food.
Factors such as geometrical
configuration, chemical composition, and packaging materials may
interact with the electric field and, in turn, influence the microwave
42
power absorption by the product.
The dielectric properties and
transmission properties of a material are functions of microwave
frequency, temperature, and chemical/physical properties.
Therefore,
heating characteristics of food during microwave processing is a complex
phenomenon.
A comprehensive experimental design may help to elucidate
the complex system and identify the most critical factors.
A
mathematical model can incorporate the hypothesis of the mechanisms
behind the microwave heating behavior of a food and be verified by
experiment.
Most of the mathematical models for the microwave heating
process, found in the current literature, were solved by the finite
difference technique.
Though FEM is a fairly new numerical method, with
its flexibility in handling irregular geometrical configurations and
material properties, it would be the best numerical method to solve the
microwave heating process in foods.
Therefore, the objectives of this research were:
1.
to characterize patterns of microwave power field distribution
of the microwave ovens studied;
2.
to study the effect of geometrical configurations of model food
systems on the temperature profiles during microwave heating;
3.
to study the effect of dielectric properties of food on microwave
heating due to the change of composition; and
4.
to develop a heat transfer model and to use a finite element
technique to predict temperature distribution in foods of
cylindrical and rectangular shapes during microwave heating.
This research was designed to fulfill the objectives stated.
Is hoped that more elaborate modeling can be developed In the future
based on this research to simulate the dynamic interaction occurring
during microwave processing.
It
44
Chapter 3
MATERIALS AND METHODS
This research was conducted by two approaches— theoretical and
experimental.
A microwave heat transfer model and a method to measure
the temperature distribution of high moisture foods were developed.
The
experimental results were used to verify the temperature prediction by
the model.
This chapter provides a detailed description of the
derivation of microwave power absorption function at different locations
in the food and the temperature measurement technique used in this
research.
3.1
Characterization of the Microwave Oven
Household microwave ovens vary greatly in the design of microwave
power generation and distribution systems, cavity size, and type.
Variations also can be found In accessories such as turntables and
temperature control features.
The configurations of the oven, together
with the interaction between microwaves and the load, determine the mode
pattern for that oven.
The field distribution can be obtained by
solving the Maxwell equations if the multi-mode pattern can be clearly
defined.
The mixing of the electric field by the mode-stirrer and the
continuous interaction of the field with food load during microwave
heating make it very difficult to compute the solution of the electric
45
field by Maxwell equations; therefore, an empirical technique was
developed to characterize the oven power distribution.
3.1.1
Empirical measurement of the Power Distribution
To determine the microwave power distribution quantitatively, the
microwave power at any location in the oven needs to be measured
independently.
Presence of more than one load in the oven during the
power measurement will dis'tort the mode pattern.
The procedures for the
determination of microwave power distribution were described as follows.
3.1.1.1
Microwave Power Measurement
A Tappan 500 (Model 562077) microwave oven operating at 2450 MHz
was used in this research.
The line voltage on the power supply was
monitored and maintained by the Powerstat (Model 3PN126, Superior
Electric Co. Bristol, CT) at 115 V.
Power output was determined
calorimetrically according to the standard procedure of the
International Electrotechnical Committee (IEC) (Schiffmann, 1987).
Deionized water weighing 1000 g and equilibrated at a temperature of 5°C
below room temperature, was heated in the microwave oven at full power.
Heating was continued for a period of time until the final temperature
of the water reached about 5 °C above room temperature.
Temperatures of
the water before and after heating were measured by using an Omega type
K thermocouple probe (Omega, Inc., Stamford, CT). Water was thoroughly
46
mixed with a spatula before temperature measurement.
The microwave
power absorbed was calculated by Eq. 3.1.
. mCp& T
(3>1)
At
where
m
is the mass of the water, [kg]
Cp
is the specific heat of water, [4.184 kJ/kg°C],
AT
is the temperature difference between the initial
temperature and the final temperature, and
At
is the duration of microwave heating time.
The time needed for the magnetron to reach a steady state was
determined using a Soar ME-530 digital multimeter.
The time period for
the filament in the magnetron to warm up to the operating temperature is
the come-up time of a magnetron.
When the magnetron was operating at
its full power, line voltage dropped from 115 V to about 112.8 V.
come-up time was recorded by a stop watch.
This
The magnetron come-up time
was subtracted from the time setting of microwave heating to give the
net time of duration of the microwave heating.
3.1.1.2
Measurement of Power Distribution
Power distribution was measured by determining the rate of heating
of water in a plastic cup at different locations and at different
heights of the oven cavity.
The dimension of the microwave oven cavity
was measured to be 29.5 cm x 28.5 cm x 16.5 cm.
The horizontal plane of
the oven cavity was divided into grids of 3 cm x 3 cm and spaced 1.25 cm
47
vercically as illusCraCed in Fig. 3.1.
To standardize the operation of
the oven for each test, a 1000 mL of water was heated in a glass beaker
5 min before the first test of the day.
Microwave power was also
measured periodically according to the standard procedure to make sure
that Che oven performance had not changed.
I OVEN CAVITY
(0 ,0 ,0)
Fig. 3,1
Discretization of the power measurement points for microwave
power distribution in a Tappan 500 microwave oven
A small polyethylene cup (4 cm radius by 3 cm height) was used to
contain the water for microwave power measurement at each junction of
Che grid shown in Fig. 3.1.
According to Ohlsson (1989), small
cylinders of dielectrical materials will exert little interference to
the microwave field.
In each test, 100 g of deionized water at about
15°C ± 3°C was heated in the oven at each of the 245 junctions on the
grid for 20 s at full power setting with Che line voltage maintained at
115 V.
Temperature measurements at points located at elevated levels
48
were conducted by elevating the cup with styrofoam blocks, which are
transparent to microwaves.
Before the temperature measurement, water
was thoroughly mixed with a plastic spatula to get an average
temperature reading.
Temperature measurement by a thermocouple probe
was done within approximately 8 s after the microwave heating.
Complete
randomized experimental design was used and three replications were
performed,
3.1.2
Model Development for the Power Distribution
The temperature increase at different locations in the oven cavity
was calculated from the initial and final temperatures obtained from the
power distribution measurement.
Temperature increase values of four
replications at all the locations were analyzed by Statistic Analysis
System (SAS Institute Inc., 1985) on an IBM 3090.
Stepwise procedure
was used to find the polynomial model that would best fit the
experimental data by maximizing the coefficient of determination (R2).
Temperature increase, as a dependent variable, was expressed as a
polynomial function of the coordinates (x, y and z) of the oven cavity
up to the eighth order.
The origin of the coordinates of the grid was
set at (6.75 cm, 2.25 cm, 5.5 cm) of the oven as shown in Fig. 3.1.
The
ratio of the temperature increase from the regression model to the mean
temperature increase for the region, where the load was located, was
used as an index of power distribution at any location in the
mathematical model.
49
3.2
Determination of Microwave Power in Relationahip to Loading
Microwave power absorbed by the load inside the oven is affected
by the nature of impedance of the load, such as the size, the
geometrical configuration, and the chemical composition of the load.
Two sets of experiments were conducted to determine the relationship of
microwave power to the nature of loading.
Microwave power absorbed by
deionized water ranging from 25 g to 2500 g in four different sized
cylindrical containers and rectangular containers at the center of the
oven were determined calorimetrically.
The time of microwave heating
for each sample was determined according to the size of the load so that
the final temperature would not exceed the room temperature by 10°C.
The microwave power absorption was calculated from Eq. 3.1,
3.3
Derivation of Heat Transfer Model with Microwave Heat Generation
The transient heat transfer equation consists of heat conduction,
rate of gain of volumetric thermal energy, and microwave power
absorption.
Cylindrical and rectangular coordinate systems were used in
this study for the prediction of temperature distribution in cylindrical
and rectangular shaped samples using the finite element method.
The
model was developed based on the assumptions that the model food system
used to simulate the food product was homogenous and its properties were
isotropic.
50
3.3.1
Model Development
For an Incompressible gel undergoing a constantpressure process,
the thermalenergy equation
can be described as:
p cpj ¥ - V-(JcVT) + Voix + *
(3.2)
where the V u :t term Is the irreversible rate of energy change due to
viscous dissipation, and can be assumed to be zero when considering the
gel as a solid (i> - 0).
The thermal energy equation for the cylindrical
coordinate system thus becomes:
pcjj - 1 3
r p 3t
(r^ | T ) + i a
z dz dz
z 30
r 30
dz
a (ic| r ) + 4
( 3 .3)
dz
In the first approach, material properties and the normal incident
microwave power were assumed constant and symmetrical; therefore, there
is no circumferential variation in the thermal and physical properties
and microwave power.
The terms with d/dO are equal to zero; therefore,
the governing equation reduces to:
ar
i a
, a
t
* £<*£> * *
(3-‘ >
The heat conduction equation in a cube shaped material (slab) was
derived based on the same procedure and is expressed as shown below:
51
(3.5)
3.3.2
Equation for Microwave Power Source (ft)
The microwave power source term, ft, represents the microwave power
absorption density.
For use in the FEM, ft at any location must be
specified as a differential equation for use by the finite element
package.
Such an equation has not been reported in literature and,
hence, was derived as part of this research.
The exact forms of
absorbed power density terms for cylindrical coordinates and rectangular
coordinates can be obtained by shell balance as described below.
The microwave power attenuates exponentially from the surface of
incident following Lambert's law (von Hippel, 1954).
The electric field
strength, expressed as a function of distance from the surface of a
material, is Ex - E0e*sx.
Since power is proportional to the square of
the electric intensity, the power unabsorbed at a distance from the
surface is
(3.6)
where PQ is the surface power in [W].
3.3.2.1
Cylindrical Coordinate
It was assumed that the incident power at the surface of the
material was uniform and normal to the surface.
The surface power per
52
unit surface area is F" - P0/A>l where A# is the total surface area of
the product.
Therefore, for a cylinder Pc" - P^^trlUL+R)),where R Is
the radius of the cylinder and L is the height of the cylinder.
The
total power incident in the radial direction is equal to PR - 2jrRLPc".
Substituting PQ by PR in Eq. 3.6, the power propagating in the radial
direction from the surface of a cylinder is equal to
Pr - PRe‘2a<“'r> - 2trRLPc"e'2a(R'r>
(3.7)
Dividing both sides of Eq. 3.7 by 2TrrL, the radial surface area at
any given r can be expressed as
p" . J ± _
Px
2^1
„ 2*RLPJ
/e-“ '”-''
2M L
p ri'R) e -2,{*-r>
(3.8)
r
where Pr" is the power per unit area at a distance r from the origin in
the r direction [W/cm2].
Consider the power incident at the ends of a cylinder axially,
PL/2 -
jtR2Pc"
and therefore,
p!> „ J k
*
nR*
= *X*P,
c0 '3''*~') .
nR2
(3-9)
c
where Pz" is the power per unit area at a distance z from the center in
the z direction [W/cm2).
By doing shell balance on the cylinder as seen in Fig. 3.2, the
power absorbed per unit volume by the finite shell in the radial
direction is equal to the difference of the power at r+Ar and the power
reaching at r divided by the volume of the shell; therefore,
53
rjH
_
Pi*i,r i 2 i t ( r + A x ) A z > - P t ( 2 n r A z )
=
2vzAiAz
(3 .10 )
where P"ghen.r is the absorbed power density of the shell 2trArAz in the
r direction [W/cm3].
The shell balance of the power absorbed between z+Az and z per
unit volume of the shell in the axial direction becomes
p'',Ag(2xxAr) - P % (2wrAr)
(3.11)
2xrArAz
where P'ghgit.g Is the absorbed power density of the shell 2irArAz in the
z direction [W/cm3].
z - Lz
<k
L
♦r
z- O
z - - Lz
Fig. 3.2
Shell balance within a cylindrical model
54
The totalpowerabsorbed
quantities
inEq. 3.10
by the shellequals the sum
and Eq. 3.11.
Taking limits, as
ofthe
Ar and
Az
approach zero, the total power absorbed by an infinitesimal volume at r
and z becomes:
„/// _ dPr+dP, _ dip" . BPfJ
c
~dv
~ zs r
(3. 12)
Tz
where P"c is the absorbed power density at r and z in a cylinder
[W/cra3].
Differentiating Pr" and Pzn in Eq. 3.8 and 3.9 against r and z
respectively, they become
2*1
-
<*-!■) + 2aRp,c
3r
r*
2*1
dz
(3.13)
-r
-
<3 1 4 >
Substituting Eq. 3.13 and 3.14 into Eq. 3.12, power absorbed per volume
in a cylinder becomes
_ dip"
c ”
-
3.3.2.2
rdr
, dp'j _ P'J , dP?
dz
2 a R P ° 0 -2m(M-ri
~ z
dP?
dr
dz
(3,15)
+ 2 a p f ' e im{i ' a)
Rectangular Coordinate
For a rectangle with dimensions L, x 1^ x 1^, the power per unit
surface area is P8" - P ^ 2 (L1L^+LjL^+L^L^).
direction is
Surface power along x and y
55
Pl. .
P l ,,
L*L >P°
2 (LjLj+LjLj+LjLj) '
■=
L^ p o
2 [ L lL 2*LiL i*LiLi) ‘
respectively; therefore, microwave power propagating in the direction of
x and y from the surface is:
Px “ Pl,©
a
respectively.
and y
-a»ti -*)
-a<<h -y>
2 , Py ■ Pr,e
2 ,
a
To obtain the expression for unit surface power in the x
direction, divide the Px and Py by their surface areas,LgL^
and
LjL^, respectively:
p" -
-
p?e'a*<:£"x)
(3 -16)
£2£j
p"
-
- l z -
.
P " e ~ 2 m i ^ * ~ y)
( 3
. 1
7
)
LjLj
Therefore, the power absorbed per unit volume becomes
p/// = U y L ^ U x - ^ y L ^ ) + (AxL3PytAy-AxL3Py)
* "
A xAyLj
(3.18)
„ P'Ux-LP" + PyUy-APy
Ax
Ay
Taking limits as Ax and Ay approach zero, Eq. 3.18 becomes a derivative
of absorbed power density in x and y direction in the unit of [W/cm3].
•
art
dx
_
ap1
;
dy
« f
= 2aPite
_a «
(3.19)
(~-x)
-2*{h-y)
2
♦ e
2 ]
There is a considerable amount of power transmitted from the
surface in the negative directions of the x and y axes if the size of
56
the product is on the order of magnitude of the wavelength in the
material.
Therefore, the absorbed power density equations for cylinders
and rectangles become:
(3.20)
4>(x,y>
(3.21)
Also, because the permittivity of the material heated in the microwave
is significantly higher than that of the ambient air, internal
reflection should be included. In the case of the first internal
reflection, additional power can be added to the source surface power
per unit surface area as, P" + P"e'2al-.
For multiple (n times) internal
reflection as in the case of a small sized product, P"-P"(l + e‘2<lL +
e-4«L+....+e-2n«Lj
jn this study, only the first internal power
reflection term was considered.
3.3.3
Boundary Conditions
The boundary conditions for the heat transfer problem in this
study are of three kinds.
They are: (1) no heat flux (i.e. thermal
insulation), (2) convective heat loss when the surface exchanges heat
with the surroundings, and (3) evaporative heat loss at the surface
57
where moisture exchange with the surroundings took place (i.e. when the
surface was not covered with materials such as Saran wrap).
(1) Insulated boundary:
q - 0
(3.22)
(2) Convective heat loss:
q - h (T - Ta)
(3.23)
(3) Evaporative heat loss: q «
where
(3.24)
ih y A H y /A
h is the convective heat transfer coefficient, [W/m2>C],
Ta is the ambient temperature,
niy is the mass flow rate of water vapor, [kg/s],
A H y
is the latent heat of vaporization, [kJ/kg*C] and
A is the surface area.
The surface heat transfer coefficient, h, was assumed to be 37.05 W/m2,C
for slab and 39.44 W/m2*C for cylinder (Swami, 1982).
The values for mv
and AHy as functions of temperature were evaluated from Swami (1982) and
Heldman and Singh
as follows:
( 1 9 8 1 )
n i y / A
-
k y ( T )
P y ( T )
k y ( T )
-
k y
+
A H y ( T )
-
2 5 0 1
P V ( T )
-
1
. 1
[ ( T
7
. 8
8
6
2
7
-
3
( R H - R H a ) / 1
) / ( T
2
. 3
9
a +
2
7
3
0
0
( 3
. 2
5
)
) ] 1 *5
( 3
. 2
6
)
( 3
. 2
7
)
( 3
. 2
8
)
T
e ( 0 - W 9 8 T )
where ky is the mass transfer coefficient of water vapor in air,
[kg/s'm2-kPa],
Py is the water vapor pressure in [kPa],and
RH is the equilibrium relative humidity.
The mass transfer coefficient is a function of temperature due
to the
temperature dependency of diffusivity of water vapor toair (Bird
al., 1960).
et
The equilibrium relative humidity of the sample was assumed
to be 99% and the RHa of ambient air was assumed to be 60%.
Foods may
58
have water activity lower than 99%, and the humidity in the oven cavity
may also vary depending on the amount of water vapor driven out of the
food during microwave heating.
The fan inside the oven ventilated the
air during heating and, therefore, kept the relative humidity of ambient
air at a relative constant during microwave heating.
3.3.4
Material Properties
Thermal conductivity of the gel was assumed to be constant within
the temperature range in this research (Swami, 1982), though it may vary
considerably with temperature in the case of thawing or freeze-drying.
Density of gel was determined from weighing 100 mL gel in a graduated
cylinder before the gel was solidified.
Specific heat of the gel was
determined from the Dickerson equation Cp - 1.675 + 0.025 w
(Singh and
Heldman, 1984).
Density, p - 1010 kg/m3
Thermal conductivity, k - 0.8374 W/m°C
Specific heat, Cp - 4.12 kJ/kg°C
Dielectric loss of the ionic solution increases because of the
increased ionic conductivity, since o - o«r" (Nelson, 1973).
Dielectric
properties of sodium chloride solutions at different concentrations have
been measured by Mudgett (1986) as a function of temperature.
Dielectric properties of the gel in this research were found to be
similar to that of aqueous ionic solutions based on selected
determination of dielectric loss and dielectric constant of the gel
samples by an outside laboratory.
In order to determine the equivalent
59
sodium chloride concentration for the gel, the conductance of the gel
samples was measured by a conductivity bridge (Model FM-70CB,
Sybron/Barnstead).
Standard curves were obtained by measuring
conductance of NaCl solution of 0, 0.05 M, and 0.1 M concentrations from
20°C to 80°C.
The conductance of gel measured from 23°C to 77°C was
compared with the standard curves, and the equivalent NaCl concentration
was thus determined.
NaCl concentration.
The gel was found to be equivalent to the 0.05 M
Dielectric properties data from Mudgett (1986) for
0.05M NaCl solution were used for the gel in this study.
From these
data, dielectric properties were expressed as functions of temperature
as shown in Eq. 3.29 and Eq. 3.30.
3.4
er' - 81.79 - 0.299 T
(3.29)
er" - 22.6 - 0.378 T +0.00293 T2
(3.30)
Finite Element Method and TWODEPEP
TWODEPEP, a commercial finite element software package available
on the Penn State IBM 3090-ES mainframe was used.
The software is in
FORTRAN and distributed by the International Mathematical and
Statistical Libraries (IMSL) Inc.
TWODEPEP can solve transient heat
conduction equation with internal heat generation.
TWODEPEP solves the parabolic (time-dependent) partial
differential equation of heat conduction with heat generation in twodimensional and axisymmetric regions.
It uses six-node (and higher
order) triangular elements, with the quadratic basis shape function, and
the Crank-Nicolson scheme to discretize time domain (IMSL, 1984).
The
60
initial triangulation was made with only enough triangles to define the
region, where the parameter NTF defines the number of final triangles
desired.
A function D3EST(x,y) was used to refine the triangulation by
setting the appropriate density of final triangles for the region of
interest.
Fig. 3.3 shows the initial triangulation of all the
geometries studied.
Table 3.1 shows the element sizes and time steps of
cylindrical and slab geometries determined by minimizing the percent
error between the FEM prediction and the experimental results.
The
element sizes and time steps were examined to be within the constraint
of the following equation (Puri, 1989):
At <
(3.31)
2l\
9D(1 -0)
where At is the time step,
1
is the time coefficient,
which equals to Cl or pCp,
A
is the element size,
D
is the material property or thermal conductivity,
0
is a time-shape function, which equals to 0.5.
k,
In every case, the algebraic equations are solved by Newton's
method and the linear system (which must be solved to do a Newton
iteration) is solved directly by Gaussian elimination (IMSL, 1984).
reverse Cuthill-McKee algorithm and a special bandwidth reduction
algorithm are used by TWODEPEP to number the nodes and to give this
linear system a banded structure.
The frequency of updating the
The
61
R**2cm
H*»2.5cm
R“3cra H"2.5cm
R=4cm
H“2.5cm
R“3cm
R-4cm
H“5cm
R=6cm
H**2.5cm
1 t—
R=2cm
H-5cm
H“5cm
R*6cra
H»5cm
* Drawing not according to scale
Fig. 3.3
Initial triangulations for cylinders of various sizes
62
Table 3.1 Element Sizes and Time Steps Used in TWODEPEP Programs for
Cylinders and Slabs of Different Sizes
Geometry*
2 cm rad. X 2.5 cm
2 cm rad. X 5 cm
3 cm rad. X 2.5 cm
3 cm rad. X 5 cm
4 cm rad. X 2.5 cm
4 cm rad. X 5 cm
6 cm rad. X 2.5 cm
Element Sizes2 [cm2} Time Step3 [s]
0.469-0.625
5
0.312
5
0.469-1.406
5
0.938
5
0.938-1.562
5
1.000
5
0.469-1.406
5
1.000
5
4 cm x 4 cm x 5 cm
0.160
1
6 cm x 6 cm x 5 cm
0.360
1
8 cm x 8 cm x 5 cm
0.640
1
6 cm rad. X 5 cm
* Dimensions of cylinders and slabs in cm.
2 Numbers show the range of final triangle element sizes used.
3 Element size (A) and time step (At) relationship falls in the criteria
described in Eq. 3.31.
63
Jacobian matrix was determined adaptively.
The general equation used in
TWODEPEP is;
CHi.y.u.t)™
.-3.oxx(*.y.$:.fy .u.C)
*
(3
32)
* «
where Cl is a material property parameter,
U
is the dependent variable,
OXX
is the material property * dU/dx,
OXY
is the material property * dU/dy and
Fl
is the element forcing function.
For heat dissipation problems, Cl equals pCp; the material
property in OXX and OXY is the thermal conductivity, k; and Fl is the
rate of volumetric heat generation.
Slab;
♦ £<*§> *
•<*■»
<3-33)
>* *<*•*>
(3.34)
Cylinder:
I? “
^
(JrS ) + * <r,Z>
where pCp can be a constant or can be
expressed as functions of x, y, T,
and t.
When TWODEPEP is used to solve axisymmetric problem, the
(k/r)dT/dr term in the cylindrical heat conduction equation was added to
the element forcing function.
The r-axis can be replaced by the x-axis
64
and Che z-axis by Che y-axis, and Chus Che problem of cylindrical
specimen can be solved using Che CarCesian coordinate syscem.
For slabs, the region of inCerest is one quarter of Che
horizontal slice through the center of the slab because of the symmetry
in the x and y direction.
plane passing through r-0.
cylindrical sample.
The region of interest in the cylinder is any
Radial symmetry was assumed in the
These regions were divided initially into eight
triangles, and the number of final triangles varies according to the
geometry and size of the specimen.
The TWODEPEP programs for slab and
cylinder and the specifications of the input statement are shown in the
Appendix A.
The solutions of temperature and heat flux at final node locations
at each time step (NOUT) were produced as an output by TUODEPEP.
The
scalar and vector fields for a specific time step were also generated as
part of the graphical output.
3.5
Experimental Design
To verify the time-temperature profile simulated by TWODEPEP, a
series of experiments were designed using sodium alginate gel as a model
food system.
The gel simulates a high moisture solid food product.
The
advantage of using model food systems is that they are highly
reproducible; and their thermal, physical, as well as electrical
properties are well defined as a function of temperature.
A thermally
irreversible sodium alginate gel, Hanucol DM (Kelco, San Diego, CA) was
used.
The formulation of the gel is listed below:
65
Hanucol DM gel
1.5%
CaCl-2H20
0 .2%
Hexa meta sodium phosphate
0.12%
Citric acid
0 .2%
Water
97.98%
All of the dry ingredients, except citric acid, were mixed together, and
then vigorously mixed into heated water in small quantities.
When the
whole mixture reached 70-80°C, the dissolved citric acid solution was
added in with gentle mixing.
If gel crumbs were not dissolved properly,
the gel was blended in a high speed blender to break up the crumbs
before citric acid was added.
The gel was poured into ready made sample containers and allowed
to set over night until it equilibrated with the ambient temperature
(25°C).
Sample containers were made with materials transparent to
microwave.
Rigid plastic films Scotch T-501 (3M, St. Paul, MN) and
styrofoam blocks were used to make rectangular and cylindrical
containers of different sizes.
The experimental design and the configurations of the samples are
shown in Table 3.2 and Fig. 3.4.
Heating time of each sample was
determined according to the size of the sample such that the final
temperature after heating would not exceed room temperature by 10°C.
66
Table 3.2 Experimental Design for the Effect of Size and Shape of the
Model Food Systems during Microwave Heating
Cylinder
Radius
Slab
height
Length
Width
Height
[cm]
[cm]
[cm]
[cm]
[cm]
3
5
6
6
5
4
5
8
8
5
3.5.1
Effects of Size and Shape
Rectangular and cylindrical samples of various sizes were prepared
by Manucol DM gel.
Rectangular samples of 4 cm, 6 cm and 8 cm widths
and lengths and 5 cm height and cylindrical samples of 2 cm, 3 cm, 4 cm
and 6 cm radii and 2.5 cm and 5 cm heights were the dimensions of
samples used in the study of effect of size and shape.
Cylindrical gel
samples were heated from all the directions in a Tappan 500 microwave
oven for different time Intervals according to the size of the sample.
Rectangular gel samples were also heated at different time intervals,
but the gels were microwave shielded on the top and bottom surfaces by
aluminum foil in the z direction during heating.
3.5.2
Effect of Shielding of Power
Double layers of aluminum foil were used to shield microwaves from
different directions of the oven by covering different faces of the
cylindrical samples (4 cm radius x 2.5 cm height).
Aluminum foil is a
good conductor; therefore, microwaves cannot transmit through and are,
67
Z= 0
z = — L,
x
Cylinder
z=0
z=
Slab
Fig. 3.4
Geometrical configurations of cylindrical and
rectangular gel samples
68
instead, totally reflected.
Blocking the microwave with aluminum foil
on the axial ends of the cylinder was equivalent to radial microwave
heating, while shielding in the radial direction simulated axial
microwave heating.
The arrangement of this experiment is shown in Fig.
3.5.
Radial Shielding
Top Surface Shielded
Fig. 3.5
Axial Shielding
Bottom Surface Shielded
Arrangement of microwave shielding using aluminum foil
(shaded areas represent surfaces shielded with aluminum
foil)
69
3.5.3
Effect of Salt Content
Dielectric properties, as affected by the chemical composition,
were studied by adding sodium chloride to the gel solution during sample
preparation.
Unbound ions such as sodium chloride will hydrate with
free water molecules thus depressing the dielectric constant.
Dielectric loss, however, is increased due to electrophoretic behavior
of the free ions.
Uithout additional salt added, the total salt used in the gel
formulation was 0.32% (w/w). According to Liu (1990), the dielectric
properties of sodium chloride solution change drastically when salt
concentration increases up to 2% (w/w), therefore, two levels of salt
concentration, 1% (w/w) and 2% (w/w), were used in this study.
NaCl
powder was added to the desired total salt concentration in making the
gel.
The NaCl mole equivalent concentrations of these gel samples were
determined from the conductance measurement as described in Section
3.3.4.
3.6
Temperature Measurement
To verify the time temperature history of the product during
microwave heating as predicted by the mathematical model, gel samples
were heated in the microwave oven for different time periods.
The
temperature measurements were performed using both fluoroptic probes
during the heating and thermocouple probes before and after microwave
heating.
70
3.6.1
Temperature Measurement by Fluoroptlc Probes
Four fluoroptlc probes (Luxtron Thermometry System 750, Mountain
View, CA) were used to measure sample temperatures during the
microwave
heating, because they do not Interfere with the electromagnetic field In
the oven.
Temperatures at different locations relative to the center of
the gel, and on the gel surface, were measured.
to an IBM-PC via an RS 232 serial port.
The unit was Interfaced
Time-temperature data were
collected using a BASIC program on PC for data analyses.
The maximum
number of channels for temperature measurement was limited to four
fluoroptlc probes.
Considering the variation of temperature with
location due to factors such as geometry and field distribution,
thermocouple probes were constructed to monitor temperature over a wider
number of locations in the sample.
3.6.2
Temperature Measurement by Thermocouple Probes
Jigs to hold thirty-seven type K thermocouple probes to measure
temperatures in rectangular and cylindrical samples at different
locations were constructed as shown in Fig. 3.6.
The arrangements of
thermocouple probes for temperature measurements at different locations
of the rectangular and cylindrical samples are shown in Fig. 3.7 (a) and
(b).
During microwave heating, samples were positioned at the center of
the microwave oven as shown in Fig. 3.8.
Temperatures at the locations
shown in Fig. 3.7 (a) and (b) were measured before and after microwave
71
□
A.
B.
C.
D.
E.
F.
G.
H.
□□ □
Luxtron fluoroptlc temperature measurement system
Microwave oven cavity
Gel sample
Styrofoam block
Thermocouple probe assembly
AM32 multiplexer
2IX micrologger
Microcomputer data acquisition network
Fig. 3.6
Gel sample temperature measurements using thermocouple
assemblies and fluoroptlc probes with data acquisition from
a PC
72
tBACK
6 CB
4 CB
3 CB
2 CB
RIGHT
left
•4-
I FRONT
tBACK
L=8 c m
L=6 c m
L=4 c m
RICHT
LEFT
t FRONT
Fig.
.7 Positioning of thermocouple probes on templates for
temperature measurements in (a) cylindrical gels, and (b)
rectangular gels of different sizes
73
Center of Sam ple @
Center of Oven Cavity
Gel
Sam ple
Oven JBa ckf)Ve t
Oven Front
Fig. 3.8
heating.
Position and the orientation of sample with
respect to microwave oven during heating
Probes were aligned according to the orientation of the sample
inside the oven as indicated in Fig. 3.8.
All temperature data were
collected by a data acquisition system consisting of a Campbell 21X
Micrologger, an AM32 Multiplexer and a SM192 Storage Module (Campbell
Scientific, Logan, UT). The time-temperature data measured by
thermocouple probes were stored in the storage module.
At the end of
measurement, data in the storage module were downloaded unto a floppy
disk using PC 208 Datalogger Support Software (Campbell Scientific,
Logan, UT). The schematic diagram of the circuitry and data acquisition
program are illustrated in Appendix B.
The measurement always began
from the top level (z - Lz) of the gel sample since the top surface was
subjected to surface cooling due to convective and evaporative heat
loss.
Then the probes were brought down to the center level (z - 0) and
then the bottom level (z - -Lz) for the temperature measurement at these
levels.
Each run/level took 5 a for execution time.
Therefore, the
74
time used to complete three runs for temperature measurement In a sample
at three levels was around 20 s.
3.7
Data Analysis
Temperature measurements were taken at all the locations In each
experimental unit In four replications.
The Initial and final
temperatures, and the heating time obtained from the measurement, were
used to calculate the rate of temperature rise, RTR, which is defined by
Eq. 3.35.
RTR -
(3.35)
The RTR at different locations for all the experiments were calculated
according to Eq. 3.35.
The effect of initial temperature from different
experiments was normalized, to enable the temperature data measured from
thermocouples and fluoroptlc probes to be compared with the model
prediction.
The mean and standard deviation of the RTR were evaluated for each
point of measurement, and they are listed in Appendix C.
The mean RTR,
based on the study of different factors, were plotted against sample
geometrical locations from the center point at Lj, 0, and -Lj— three
levels along the z direction.
The RTR data within a sample, measured at
different orientations from the center of the oven toward the right,
front, left and back sides of the oven, were also presented to show the
75
effect of field distribution.
The layout of this orientation during
temperature measurement Is Illustrated In Fig. 3.8.
3.8
Model Verification
Data predicted from the model were compared with the means of the
experimental results by t-test.
The statistical analysis verifies
whether the model predictions of the RTR values at different locations
in the sample were significantly different from the experimental results
(testing H0: RTRpredicted - R T R ^ ^ )
at 0.01 significance level.
Response surfaces from model predictions were plotted in threedimensional graphics by the G3GRID procedure in SAS/GRAFH1CS software by
Statistic Analysis System (SAS, 1985).
The graphs plotted a continuous
surface of temperature distribution by interpolation and extrapolation
between the measured points using a cubic spline process.
The slowest
heating points during microwave heating can be located from these
graphs.
3.9
Sensitivity Analysis of the Model Prediction
Sensitivity analysis is a technique to examine the extent of
variability due to different material properties on the response
variables model prediction.
Sensitivity analysis for each of the
material properties and system conditions was performed.
The key
parameters included thermal diffusivity, attenuation factor as
calculated from dielectric constant and dielectric loss of the material,
76
microwave power output, convective heat loss as affected by the heat
transfer coefficient, and evaporative heat loss due to the humidity
gradient between the food and ambient air in the oven.
This was done by
running the base simulation and then varying the parameters with
different percentage changes according to the realistic situation
encountered.
The variability of the response due to the changes in the
parameters reflects the sensitivity of the model.
The relative
sensitivity is an index to determine the influence of a parameter on the
model prediction.
It was calculated as the ratio of the variability of
the model response to the percent change of the parameter as shown in
Eq. 3.36,
AX
R e l at i v e S e n s i t i v i t y *
(3.36)
~P
where Y is the model response and P is the value of the parameter
chosen.
77
Chapter 4
RESULTS AND DISCUSSION
Results of mathematical modeling and the experimental data for the
heating characteristics of model foods during microwave heating were
analyzed and compared based on the rate of the temperature rise (RTR).
Microwave heating of foods is affected by the oven parameters and the
properties of the food product.
The latter consists of material
properties and geometrical configuration, while the oven parameters
include the field distribution and the power-load relationship.
Material properties and oven characteristics interact with each other
and cause different microwave heating behavior of the food product under
different conditions.
In discussion of the results, the results from measurements and
the finite element model prediction are presented in sections 4.1 and
4.2.
The finite element model predictions were then compared with the
means of the experimental measurements.
4.1
Experimental Results
All of the experiments were conducted based on four replications
in order to obtain information on the range of experimental error.
means and the standard deviations of sample means are presented.
The
78
4.1.1
Mapping the Power Distribution
Since the dimensions of the microwave oven cavity are of the same
order of magnitude as the wavelength of the microwaves, a number of
standing waves exist inside the microwave oven.
The development of the
field distribution depends on the oven type and the geometry and
composition of the food inside the oven.
Due to the lack of information about the mode patterns that exist
in the test oven (Tappan 500), the field distribution cannot be solved
mathematically.
instead.
The field distribution was determined empirically
Different oven power mapping procedures have been proposed in
the literature (Wilhelm and Satterlee, 1973; Washisu and Fukal, 1980 and
Schiffmann, 1987).
The use of egg white, pancake batter or tonic powder
as described in the literature does not determine the field distribution
quantitatively.
Presence of a large amount of sample or numerous
individual samples may also interfere with the development of mode
patterns.
The method used in this study results in minimal field
interference and gives quantitative data on the power distribution.
4.1.1.1
Three-Dimensional Presentation of Power Distribution
Temperature difference between the initial and final temperatures
(AT) in the three dimensional oven cavity were plotted using the G3GR1D
procedure in SAS/GRAPHICS. The three-dimensional plots are shown in
Fig. 4.1.
The origin represents the front left corner of the oven.
is proportional to microwave power absorbed at each location.
AT
The power
79
absorbed is a function of the microwave field intensity at that
particular position.
Therefore, AT at different positions of the cavity
was used to describe the power distribution within the microwave cavity.
The results showed that power distribution patterns are of sinusoidal
shape and are composed of high and low nodes.
The field Intensity of a
level close to the cavity ceiling is higher than that at lower level.
This could be due to microwaves that are directly incident from the
waveguide and mode stirrer at that location.
At the lower level, fewer
direct incident and more reflected waves from the side walls and floor
strike the sample.
The wave may also be attenuated depending on the
conductivity of the oven walls and the humidity within the cavity.
A.1.1.2
Multiple Regression Model
According to the general form of the standing wave equation (Eq.
2.14), the standing wave is a sine and cosine function of the
coordinates.
In the development of the regression model for the power
distribution, these trigonometric functions and the combinations of
coordinates up to the eighth order were included.
Models with different
orders of polynomial functions were tested by the STEPWISE procedure in
the Statistical Analysis System (SAS, 1982) to select terms which
yielded the maximum R2.
In determining the number of terms to retain in
the model, a compromise was made between increasing R2 and decreasing
the number of terms.
A polynomial of the eighth order with
trigonometric functions (43 terms in total) was chosen, which yielded an
R2 of 83.8%.
The entire model and the estimates of parameters are shown
in Table 4.1.
The coefficient of variation for the standard error is
2.65% of the overall mean from the analysis of variance.
The F value
from the test of variance is 37.93, which indicates that the variance
due to the power distribution is significant.
The three dimensional
standing wave patterns developed quickly in the oven cavity after the
magnetron was turned on.
This was seen in the straight line
relationship of time-temperature profile in each power measurement at
different locations.
A time-temperature profile obtained by Luxtron
temperature measurement system is shown in Fig. 4.2 to illustrate the
time-temperature relationship in a gel at four different locations
during microwave heating.
All of the temperature profiles were found to
be linear with respect to time, hence, RTR was used as a way to describe
the microwave heating.
Fig. 4.3 shows the three-dimensional graphs of the predicted
temperature difference at the same horizontal levels as in Fig. 4.1
within the region of the sample location in the oven.
These graphs are
based on the multiple regression model characterized the power intensity
pattern in the Tappan 500 oven.
The power distribution regression model
was thus used in the microwave power absorption term in the model to
predict the relative power intensity at different locations in the oven.
Based on the dimension of the sample, an average value was first
calculated from the regression model at the locations r - 0, 1 c m
for every n/6 and z - ± Lz within the domain.
R
Then the ratio of the
temperature difference calculated from the power distribution model at a
particular location Inside the sample to the average value is used as an
index of the power intensity at that point within the sample.
The
Fig. 4.1
Measured power distribution in a Tappan 500 microwave oven at
three levels (a) z - 2.5 cm, (b) z - 0, and (c) z - -2.5 cm.
82
Table 4.1
Multiple Regression Model for Field Distribution
Parameter
Estimate
Intercept
Y
XY
YZ
X2Z
Y4
Y3Z
YZ3
x 2y 2
X2Z2
X5
X4Z
y 3z 2
y 2z 3
x 2y 2z
XY5
18.033
-2.953
0.167
0.952
-0.044
1.128E-3
-2.673E-3
-3.562E-2
1.690E-3
3.324E-3
-3.502E-5
4.359E-4
-1.180E-3
4.206E-3
-2.241E-3
-2.479E-6
6.545E-4
-4.890E-5
-4.617E-4
8.554E-4
-1.952E-7
7.854E-7
-7.971E-5
9.144E-6
-3.790E-6
-1.601E-5
3.224E-6
3.296E-5
4.464E-6
-1.050E-4
2.326E-5
1.025E-6
8.792E-9
2.189E-7
3.123E-6
6.416E-7
-1.543E-6
-7.285E-8
-1.843E-6
3.200E-6
-4.784E-1
1.223E-1
2.485E-1
1.658E-1
xz5
X4Z2
x 2y z 3
x 2y 2z 2
Y7
Y*Z
XZ6
YZ6
X2Y5
y 2z 5
x 4y z 2
x 2y ^z
x y 3z 3
x 2y 3z 2
x 2y 2z 3
z8
x 7y
X2y6
XYZ6
X3Z5
x 2y 5z
X*YZ
XY2Z5
X^Z2
COS X
COS X * SIN Y
SIN Y * COS Z
SIN Z * COS Y
PR > |T|
0.0
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0007
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0412
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0463
0.0346
0.0002
83
(a) 3 cm rad.x 5 cm ht.
o
r=0
*
■
1 60
2
.
'
r=2R/3
r=R/3
r=R
0
20
40
Time, s
(b) 4 cm rad. x 2.5 cm ht.
80
o
$
| 6°-
r=R/4
IQ
6
r=0
•
r=R/2
40
r=R
20
0
Fig. 4.2
20
Time, s
40
Time-temperature profiles of cylindrical gels of sizes
(a) 3 cm radius by 5 cm height, and (b) 4 cm radius by 2.5
cm height measured by fluoroptic probes at different
distances from the center at z - 0 level
o • and
r"****
85
microwave absorption term was modified by this ratio as shown in the
finite element program in the Appendix A.
4.1.2
Effect of Loading on Energy Coupling Efficiency
The efficiency of microwave power absorption is affected by the
load within the oven.
The microwave power absorption may vary depending
on the material properties, total mass and geometrical configuration of
the product load.
Fig. 4.4 shows the effect of loading on the microwave
power absorption by different amounts of rectangular and cylindrical
loads placed at the center of the oven.
The best-fit equations for the
loading effect of rectangular and cylindrical loads were determined to
be exponential as shown in Eq. 4.1 and 4.2.
Rectangle:
PBbs “ 50<27 + 56•6 ln<wt>
R2-91.5%
(4.1)
R2-95.4%
(4.2)
Cylinder:
P«bs “ 40-1 + 59•8 ln<wt>
where wt is the weight of the test water, [g].
These two equations were used in the mathematical model to determine the
power absorbed by samples of different sizes and shapes.
For small size loads, the microwaves penetrate through the load
without much absorption.
Since the load size is small, temperature
increases at a faster rate and the dielectric properties change rapidly
during the heating cycle.
As explained in section 2.3, when temperature
increases, dielectric constant and dielectric loss decrease.
The loss
tangent or dissipation factor decreases with the increasing temperature.
Therefore, the efficiency of microwave absorption is lowered at higher
86
temperatures.
This may partially explain the lower energy coupling
efficiency from smaller loads, and the Increased power absorption by
increasing load size.
When the load size Increased to a point that most
of the microwave power was absorbed, further increase In the size of the
load did not Increase the coupling efficiency.
Cylinder
Rectangle
600
500
4 0 0
§300
o
£
3200
100
0
500
1000
1500
2000 2500 0
Load, g
Fig. 4.4
4.1.3
500
1000
1500
2000 2500
Load, g
Effect of loading on the microwave power absorption by
cylindrical and rectangular shaped loads
Effect of Salt Concentration
Temperature distribution in the 4 cm radius by 2.5 cm height
cylinders with total added salt concentration ranging from 0.32% (w/w)
to 2% (w/w) was measured as described in sections 3.5.3 and 3.6.
The
0.32% of salt according to the conductivity measurement described In
Section 3.5.3 is equivalent to 0.05 H NaCl concentration.
Whereas 1%
87
and 2% total salt in the gel are equivalent to 0.172 M NaCl and 0.369 M
NaCl, respectively.
RTR values at different radial locations and at
half the cylinder height (z - 0) were calculated and are shown in Pig.
4.5.
The curves in Fig. 4.5 are the means from four different
experiments.
Surface heating phenomena was observed in all of the
samples at all salt levels.
concentration increased.
At r/R < 1/2, the RTR decreased when salt
In the region close to the surface of the
cylinder, the higher the salt level, the steeper the RTR with respect to
radial distance from the center (r/R).
This can be rationalized by the
concept of penetration depth as influenced by the dielectric properties
of the material.
The higher the salt content, the more microwave power
is attenuated or the shorter the penetration depth.
Hence, more
microwave power is absorbed closer to the surface of the sample.
Hudgett (1986) investigated the effects of salt concentration and
temperature on the dielectric constant and loss of aqueous sodium
chloride solution at 2450 MHz.
The attenuation factors, a, calculated
from dielectric properties are plotted against temperature in Fig. 4.6.
The graph supports the results for this part of the study.
the salt content, the higher the attenuation factor.
The higher
At low temperature
region (below 25°C), the attenuation factor of salt solutions decreased
with increasing temperature.
Beyond 50°C, the attenuation factor of
salt solutions increased with increasing temperature.
However, the
attenuation factor decreased with increasing temperature for deionized
water (zero salt content). The degree to which the dielectric constant
was depressed by salt concentration was not as large as the increase of
the dielectric loss.
temperature.
However, they are both affected by the change in
This combined effect can be seen more readily in the
Q
1/4
1/2
3/4
1
r/R
Fig. 4.5
RTR values In 4 cm radius by 2.5 cm height cylindrical gels ae
different salt concentrations versus radial distance from
the center at z - 0 level
2
RT R , Deg
C/s
1 5
1
0.5
□
0
25
SO
75
100
Temperature, Deg C
Fig. 4.6 Effect of NaCl concentration and temperature on the
attenuation Jactors of aqeous ionic solutions (Mudgett,
1986)
89
attenuation factor that changes with the salt concentration and
temperature.
Because of the increased microwave attenuation at higher
temperatures and at high salt concentration, the sample containing 2%
salt (0.369 M NaCl equivalent) had a significant drop in RTR near the
surface.
The sample with 1% salt (0.172 H NaCl equivalent) attenuated
less compared to the sample with 2% salt, therefore implying more
microwave absorption and higher temperature increase than the sample
with 2% salt.
The 0.32% of salt (0.05 M NaCl equivalent) is the amount
of salt (calcium chloride and sodium phosphate) required for gelling of
the Manucol gel and this may be considered as bound ions.
They
contributed to the least amount of deviation in dielectric properties
from pure water.
Therefore, less microwave power was absorbed at the
surface, which then implied that more power was available at a distance
away from the surface.
As shown in Fig. 4.5, the rate of temperature increase at the
surface was slowest for 0.32% salt samples, yet the same sample at the
center had the highest RTR compared with other samples.
This phenomenon
may also explain how microwave power is coupled differently by materials
of the same geometrical configuration but different dielectric
properties.
Temperature distributions within these samples were
different, but the total power absorbed in all the samples of different
salt content was approximately the same (Liu, 1990).
Samples containing
2% salt did not show higher RTR at the surface than samples containing
1% salt, and this may be due to excessive heat loss from surfaces of
higher temperatures.
90
LEFT - RIGHT
BACK - FRONT
z=1.25 an
1.0
0.5
0.0
RTR[°C/s]
z=0
z=0
1.0
0.5
0.0
RTR [*C/s]
1.5
z=—1.25 an
z=—1.25 an
1.0
0.5
0.0
1
-
1 /2
0
r/R
Fig. 4.7
1 /2
1
1
-
1 /2
0
1 /2
1
r/R
The means and standard deviations of RTR in a 4 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z - 0, and z - -1.25 cm three levels (thermally insulated on
the top and bottom surface)
The distributions of mean RTR and standard deviation of 4 cm
radius by 2.5 cm height cylinder at three levels (z - 1.25 cm, 0 and 1.25 cm) are shown in Fig. 4.7.
The graphs show significant
variabilityin the temperature distribution comparing the back-front
orientation (the set of figures on the left column) to the left-right
orientation (the set of figures on the right column) inside the
microwave oven, which can be accounted for by the mode pattern in the
cavity of the microwave oven.
4.1.4
Effect of Geometrical Configuration on Temperature Distribution
The effect of geometrical configuration on the temperature
distribution of materials heated using microwaves has been studied by
different research groups (Ho et al., 1971; Ohlsson and Risman, 1978 and
Swami, 1982).
In addition to the material properties and power
distribution, geometrical configuration is the major factor affecting
the temperature distribution in a food heated within a microwave oven.
4.1.4.1
Effect of Shape
Rectangular and cylindrical shapes were considered in this study.
Cylindrical samples of 3 cm radius by 5 cm height and rectangular
samples of 6 cm length and width and 5 cm height are of comparable
overall size.
Fig. 4.8 shows the mean cylinder RTR and standard
deviation of the mean at different locations away from the center at z 0 and z - -2.5 cm levels.
Fig. 4.9 shows the mean slab RTR and standard
92
2.0
0.5
0.0
Distance to the center, an
-2.5 cm
2.0
(0
0.0
3
2
1
0
1
2
3
Distance to the center, cm
Fig. 4.8 Mean RTR in a 3 cm radius by 5 cm height cylindrical gel at
different locations from the center at z - 0 and z - -2.5 cm
levels
93
z - 0
.5
to
o
00
0)
0
n
oz
tz
0.5
0.0
3 - 2 - 1 0 1 2 3
Distance to the center, cm
z - -2.5 cm
to
5
u
60
<U
o
0
n
0
0.5
0.0
3 - 2 - 1
0
1
2
3
Distance to the center, cm
z - -2.5 cm (diagonal)
—to
0
.
60
01
n
o4
&
0.5
0.0
-4.25
-2
0
2
4.25
Distance to the center, cm
'
Fig. 4.9 Mean RTR in a 6 cm x 6 cm x 5 cm rectangular gel at z - 0,
z - -2.5 cm (both measured in the directions parallel to x and y
axes, and z - -2.5 cm measured in diagonal directions to the axes
94
deviation of the mean at different locations measured from the center at
(a) z - 0, (b) z - -2.5 cm and (c) diagonal directions of the slab at z
- -2.5 cm, respectively.
Center heating effect was observed at the middle height (z - 0) of
the cylindrical sample.
In the cylindrical sample, RTR increases toward
the center of the sample.
At the same height within the slab,
temperature distribution was fairly uniform throughout the entire cross
section.
However, at the bottom level (z - -2.5 cm) of both the
cylindrical and slab samples, the heating behavior changed drastically.
At this level in the cylinder, surface heating, or edge heating, was
significant; center heating still existed but was less apparent compared
to the edge heating effect.
The slabs at the bottom level also showed
significant edge heating effect.
In both of the cases, edge heating was
evident from r/R - 1/3 to the surface of the sample.
Similar results
were found for cylinders of 4 cm radius by 5 cm height and slabs of 8 cm
x 8 cm by 5 cm height as shown in Figs. 4.10 and 4.11, respectively.
The edge heating phenomenon may be explained by the fact that the
microwave power is transmitted into the sample from different directions
(r and z directions in the cylinder, and x, y and z directions in the
slab).
The power from the surface is focused at the edges before it is
attenuated.
The edge heating effect is intensified at the corners of a
slab where microwaves come from all three directions.
This is shown in
Figs. 4.9 and 4.11 for slabs of 6 cm x 6 cm x 5 cm and 8 cm x 8 cm x 5
cm along their diagonals.
95
2 — 0
Cfl
0.5
0.0
Distance to the center, cm
z - -2.5 cm
in 1 * 0
o
*
i 0.5
0.0
4
2
0
2
Distance to the center,
4
cm
Fig. 4.10 Mean RTR in a 4 cm radius by 5 cm height cylindrical gel at
different locations from the center at z - 0 and z - -2.5 cm
levels
96
RTR [°
z - 0
c/r t
1. 0
0.5
I-
0.0
—
I----- 1
-4
I
-2
I----- 1
I-
'"— I----- 1----- 1
0
2
I■
4
•2.5cm
1.0
0.5
0.0
RTR [*C/s]
1.5
4
2
2
4
0
Distance to the center, cm
z - -2.5 cm (diagonal)
Distance to the center, cm
Fig. 4.11
Mean RTR in a 8 cm x 8 cm x 5 cm rectangular gel at z - 0,
z - -2.5 cm (both measured in the directions parallel to x
and y axes, and z - -2.5 cm measured in diagonal directions
to the axes
97
4.1.4.2
Effect of Size
Cylindrical gels were used to study the effect of size on
temperature distribution during microwave heating.
Four radii (2 cm, 3
cm, 4 cm and 6 cm) and two heights (2.5 cm and 5 cm) were studied.
Figs. 4.12 to 4.15 show the RTR for 2.5 cm height cylinders, whereas
Figs. 4.16 to 4.19 show the results for 5 cm cylinders.
As expected, the smaller the sample size, the faster the average
rate of temperature rise.
The average RTR decreased with increasing
radius and increasing height.
For example, temperature at the center of
2 cm radius by 2.5 cm height cylinder rose at a rate of 3.5 °C every
second; the temperature increased at a rate of 1.8 °C per second at the
center of a 3 cm radius by 2.5 cm height cylinder while the average RTR
for a 3 cm radius by 5 cm height cylinder was 0.8 °C.
The curve in each figure shows the RTR within the sample at
different radial directions with respect to the axes of the oven
(parallel to the x or y axes).
These figures also show significant
effect of power distribution on the RTR results.
In samples of larger
sizes, it was observed from the figures that the RTR in the directions
of front and right with respect to the oven had higher value than did
the RTR in the back and left directions in the oven.
It may be due to
the specific power distribution present in this type of oven.
This is
evident from Fig. 4.1.
At z - 2.5 cm and z - -2.5 cm levels, the absorbed power at areas
3 to 4 cm away from the center of the oven is very high.
The rate of
temperature increase was lower in the case of the 4 cm radius cylinders
RTR[*C/s]
BACK - FRONT
LEFT - RIGHT
4
z=l.25 an
z=l.25 an
z=0
z=0
3
2
1
0
3
2
1
0
RTR [°C/s]
4
z=-l.25 an
z=-1.25 an
3
2
1
0
1
0
r/R
Fig. 4.12
1 /2
1
1
0
1
r/R
The means and standard deviations of RTR in a 2 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z «■ 0, and z - -1.25 cm three levels
99
RTR[° C/ s ]
2
BACK - FRONT
LEFT - RIGHT
z=l.25 an
z=l.25 an
1
0
RTR[°C/s]
2
z=0
z=0
1
0
RTR[°C/s]
z=-l.25 an
z=-1.25 an
1
0
-1 -2/3 -1/3 0 1/3 2/3
r/R
Fig. 4.13
1
1 -2/3 -1/3
0 1/3 2/3
r/R
1
The means and standard deviations of RTR in a 3 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z - 0, and z - -1.25 cm three levels
100
RTR[° C/s]
BACK - FRONT
LEFT - RIGHT
z=1.25 cm
z=l.25 cm
1.0
0.5
0.0
RTR[°C/s]
z=0
2=0
1.0
0.5
0.0
RTR[°C/s]
z=-l.25 cm
z=-l.25 cm
1.0
0.5
0.0
-
1/2
1 /2
r/R
Fig. 4 .14
1
-
1 /2
0
r/R
1 /2
1
The means and standard deviations of RTR in a 4 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z - 0, and z - -1.25 cm three levels
RTR[°C/s]
BACK - FRONT
LEFT - RIGHT
z=1.25 on
z=l.25 an
101
1.0
0.5
0.01
RTR[°C/s]
z=0
z=0
1.0
0.5
0.0
z=-l.25 an
z=-l.25 an
1.0
0.5
0.0
1
Fig. 4.15
-
1 /2
0
r/R
1/2
1
1
-1/2
0
r/R
1/2
1
The means and standard deviations of RTR in a 6 cm radius by
2.5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 1.25 cm,
z - 0, and z - -1.25 cm three levels
C/s]
BACK - FRONT
LEFT - RIGHT
z=2.5 cm
z=2.5 cm
102
4
3
2
1
0
m
C/s]
4
z=0
z=0
3
2
1
0
m
4
C/s]
z— 2 •5 cm
3
2
1
0
-1
-
1/2
0
r/R
4.16
1/2
1
-
1 /2
0
r/R
1
The means and standard deviations of RTR in a 2 cm radius by
5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 2.5 cm,
z - 0, and z - *2.5 cm three levels
RTR[*C/s]
2.0
BACK - FRONT
LEFT - RIGHT
z=2.5 an
z=2.5 cm
103
1.0
0.5
0.0
RTR[*C/s]
2.0
RTR[°C/s]
2.0
z=0
z=-2.5 cm
-1 -2/3 -1/3 0 1/3 2/3
r/R
Fig. 4.17
z=-2.5 cm
1
1 -2/3 -1/3
0 1/3 2/3
r/R
1
The means and standard deviations of RTR in a 3 cm radius by
5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 2.5 cm,
z - 0, and z - -2.5 cm three levels
104
BACK - FRONT
LEFT - RIGHT
z=2.5 cm
z=2.5 cm
1.0
0.5
0.0
RTR[°C/s]
1.5
z=0
z=0
z-~2.5 cm
z=-2.5 cm
1.0
0.5
0.0
RTR[°C/s]
1.5
1.0
0.5
0.0
1
Fig. 4.18
-1/2
0
r/R
1/2
1
1
-1/2
0
r/R
1/2
1
The means and standard deviations of RTR In a 4 cm radius by
5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 2.5 cm,
z - 0, and z - -2.5 cm three levels
RTR[°C/s]
BACK - FRONT
1.5
LEFT - RIGHT
z=2.5 cm
105
z=2.5 cm
0.5 •
0.0
RTR[0C/s]
1*5
z=0
z=0
1.0
0.5
0.0
RTR[° C/s]
z=-2.5 cm
z=-2.5 cm
0.5
0.0
-1/2
1
r/R
Fig. 4.19
-1/2
0
1/2
1
r/R
The means and standard deviations of RTR in a 6 cm radius by
5 cm height cylindrical gel measured at different
orientations and distances from the center at z - 2.5 cm,
z - 0, and z - -2.5 cm three levels
106
as shown In Figs. 4.7 and 4.14, as compared to the samples of smaller
radii, the sample had a more uniform temperature distribution.Therefore,
4 cm radius by 2.5 cm height cylindrical gel samples were used in
studying other aspects of microwave heating.
Edge heating at z - 1^
level became significant in geometries of 3 cm radius and larger due to
the reasons discussed in section 4.1.4.1.
The RTR curves at the middle level (z - 0) in cylinders of 2 cm
and 3 cm radius show that RTR peaked at the center of the cylinder for
both 2.5 cm and 5 cm height cylindrical gels.
the center to the surface.
The RTR decreased from
For cylinders of larger radii (4 cm and 6
cm), the center heating phenomenon was not observed; instead,
temperatures at surface areas were higher than the center.
The sharp
decrease of the RTR close to the surface of the cylinders of 2.5 cm
height was partly due to the surface heat loss from convection, because
of higher surface temperatures as compared to the samples of 5 cm
height.
The other possibility for this pattern of RTR is the unique
power distribution at z - 1.25 cm and z - -1.25 cm.
However, similar
RTR patterns were observed in a 4 cm radius by 2.5 cm height cylinder
that was shielded by aluminum foil at the z - 1.25 cm (Fig. 4.22).
Therefore, the power distribution at z - 1.25 cm (top surface) had
little effect on the RTR patterns in the 4 cm radius by 2.5 cm height
cylinder, but the RTR patterns were mostly affected by the waves coming
in from the lower part of the oven as seen in Figs. 4.7, 4.14, 4.15,
4.18, and 4.19 at z - -1.25 cm or z - -2.5 cm level.
As for cylinders of 5 cm height (Figs. 4.16 to 4.19), since the
power attenuated from the z - ± 2.5 cm surfaces toward the center, the
influence of the power distribution on the surface power at z - ± 2.5 cm
107
levels may not be as noticeable at center level (z - 0) of the gel.
However, the effect of nonuniform power distribution at the z - -2.5 cm
level on the bottom surface is very distinct.
These observations were
based on the variations of the RTR curves at different orientations with
respect to the oven cavity at z - 0 and z - -2.5 cm in Figs. 4.16 to
4.19.
Fig. 4.1 shows that the variability of the power distribution at
z - 2.5 cm and z » 0 levels is less than that from the z - -2.5 cm
level.
Since the top surface was uncovered, there were evaporative and
convective heat losses during the heating cycle and during the
temperature measurement by thermocouple probes.
This probably is
another reason why the distinct patterns due to power distribution in
cylinders of 5 cm height at z - 2.5 cm level were not as evident as
those at the z - -2.5 cm level.
4.1.5 Effect of Shielding the Surface Microwave Power
Nonuniform microwave field distribution within the microwave oven
results in a nonuniform temperature distribution within the food.
The
power intensity within the oven varied with respect to both vertical and
horizontal directions.
Examination of this variation can be conducted
by electrically shielding different sides of the gel samples with
aluminum foil.
The dimensions of the cylindrical gel samples used in
this part of the study were 4 cm radius by 2.5 cm height.
108
4.1.5.1 Axial Shielding
By shielding microwave power incident on the top and bottom of the
sample (axial direction), the sample is subject to microwave radiation
merely in the radial direction.
The RTR by radial microwave heating of
gel samples of 4 cm radius by 2.5 cm height cylinders at three levels
are shown in Fig. 4.20.
They are compared to the data from the samples
without microwave shielding in Fig. 4.7.
The radial microwave heating
resulted in a more uniform temperature distribution at all radial
locations irrespective of the orientation relative to the oven cavity.
As expected, the variations of RTR in the z direction were minimal
because of the microwave shielding in the z direction.
Surface heating
was predominant, and there was a moderate center heating.
Comparing
Fig. 4.7 and Fig. 4.20, the microwave power absorption apparently was
higher when there was no shielding on the top and bottom surfaces.
Without shielding, the variations of the RTR at different orientation
with respect to the oven were greater.
This is attributed to the uneven
vertical field intensity in the microwave oven.
An attempt was made to verify the induced temperature variations
due to the presence of vertical power distribution.
Separate studies
were conducted with gel cylinders of 4 cm radius by 2.5 cm height with
shielding just at the bottom (z - -1.25 cm) and at the top (z - 1.25
cm).
Results from the two experiments are shown in Figs. 4.21 and 4.22.
It is evident that the variability of the temperature distribution
at radial locations with respect to orientation in the oven is higher
when the top of the gel was shielded by the aluminum foil.
This
indicates that the microwaves primarily incident from the top of the
RTRfC/s]
BACK - FROMT
z=l.25 cm
LEFT - RIGHT
109
z=l.25 cm
0.5
0.0
RTR[° C/s ]
z=0
0.5
0.0
RTR[° C/s ]
z=-l.25 cm
z=-l.25 cm
0.5
0.0
-1/4
r/R
r/R
Fig. 4.20 The means and standard deviations of RTR measured at
different orientations and distance from the center at z - 1.25
cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5
cm height cylindrical gel shielded with aluminum foil on the z ±1,25 cm surfaces
RTR[°C/s]
BACK - FRONT
LEFT - RIGHT
z=l.25 cm
z=l.25 cm
z=0
z=0
z=-l.25 cm
z=-l.25 cm
.5
1.0
0.5
0.0
RTR[°C/s]
.5
.0
0.5
0.0
RTR[°C/s]
.5
.0
.5
0.0
1
-1/2
0
r/R
1/2
1
1
-1/2
0
1/2
1
r/R
Fig. 4.21 The means and standard deviations of RTR measured at
different orientations and distance from the center at z - 1.25
cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5
cm height cylindrical gel shielded with aluminum foil on the z - 1.25 cm surface
RTR[°C/s]
BACK - FRONT
LEFT - RIGHT
z=l.25 cm
z=l.25 cm
z=0
z=0
0.5
0.0
RTR[°C/s]
0.5
0.0
RTR[°C/s]
z=-l.25 cm
1.0
0.5
0.0
1
-1/2
0
r/R
1/2
1
1
-
1 /2
0
r/R
1/2
1
Fig. 4.22 The means and standard deviations of RTR measured at
different orientations and distance from the center at z - 1.25
cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5
cm height cylindrical gel shielded with aluminum foil on the z 1.25 cm surface
112
oven are more uniform than the secondary waves reflecting from the oven
floor.
Comparing Figs. 4.21 and 4.22 with Fig. 4.7 (where there was no
shielding), it can be seen that the temperature distributions in Fig.
4.22 are similar to those of Fig. 4.7.
This is because of the presence
of nonuniform secondary reflected waves from the lower level of the
oven.
The temperature distributions in Fig. 4.21 at z - 1.25 cm look
similar to Fig. 4.7 at the same level since the primary waves incident
at the top of the oven are more uniform.
Blocking microwave power at
the bottom of the sample made less microwave power available to the
bottom surface of the gel, and similarly less power was available to the
top surface when the gel was shielded at the top.
Hence, temperatures
behind a surface that was not shielded was higher.
However, when the
bottom surface was shielded, the RTR values at the top surface level of
the gel were not found to be much higher than those at the bottom level.
This could be due to the heat loss from the top surface during the
thermocouple temperature measurement.
This was verified by the
significantly higher RTR values calculated from the fluoroptic
temperature measurement during microwave heating than the RTR values
from the thermocouple temperature measurement.
4.1.5.2
Radial Shielding
As expected from the discussion in the previous section, when
samples were shielded radially, the axial microwave heating resulted in
a nonuniform temperature distribution in the gel.
The effect of uneven
horizontal power distribution is readily demonstrated by results in Fig.
113
4.23.
The microwave power coupling during axial heating seemed to be
higher than with both radial and axial heating (without radial microwave
shielding). This may be due to the electrical conducting environment
provided by the aluminum foil which causes the induction of the
secondary field in the cylinder known as near field phenomenon (Risman
et al. 1987).
For most of the points away from the center, the RTR rose
and then dropped at a distance close to the surface.
This is due to the
shading effect from the shielding of the radial microwave power from the
surface.
It was also noticed that there was a consistently low RTR
toward the right of the oven at all levels, which could be due to the
weak field distribution in that area of the oven cavity.
Thermocouple
probes used in the experiment were examined for calibration to verify
that this was not an artifact.
4.2
Mathematical Modeling
In this section, results from mathematical modeling of microwave
power absorption and temperature distribution in the model food systems
under different conditions are discussed.
The finite element model was
used in this study. Variability in the predication due to changes of
estimated parameters was tested by sensitivity analysis, and the model
validation was done by comparing the predicted and measured values.
RTR[°C/s]
BACK - FRONT
LEFT - RIGHT
z=l.25 cm
z=l.25 cm
0.5
0.0
RTR[°C/s]
z=0
z=0
0.5
0.0
RTR[°C/s]
z=-l.25 cm
z=-l.25 cm
1. 0
0.5
0.0
1
-
1 /2
0
r/R
1 /2
1
-
1 /2
0
r/R
1 /2
Fig. 4.23 The means and standard deviations of RTR measured at
different orientations and distance from the center at z - 1.25
cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5
cm height cylindrical gel shielded with aluminum foil on the r - 4
cm surface
115
4.2.1 Microwave Power absorption
The critical step In mathematical modeling of microwave heating
lies in the formulation of the term for internal microwave power
generation.
In the differential equation for heat transfer this term
was derived as the localized microwave power absorption per unit volume.
Four crucial variables in the microwave power absorption terms are
dielectric constant, dielectric loss of the element, location of the
element and total microwave power absorbed by the load.
Dielectric
constant and loss can be lumped into attenuation factor as shown in Eq.
4.3.
(4.3)
O
where a is the attenuation factor in [cm'1]; er* and er" are the
dielectric constant and the dielectric loss, respectively.
4.2.1.1
Effect of Geometrical Configuration on Microwave Power
Absorption
The absorbed microwave power density for cylinders and slabs are
given by Eqs. 3.20 and 3.21, respectively.
They are functions of the
location and the attenuation factor which in turn is a function of the
dielectric properties of the material.
116
The effect of geometrical configuration on microwave power
absorption can be understood by Eqs. 3.20 and 3.21.
For rectangular
geometry, the surface microwave power attenuates exponentially into the
food while in the case of a cylinder, the wave attenuates exponentially
from the surface but also concentrates at regions close to the center.
This is because of the 1/r term in the power absorption equation.
In
two-dimensional microwave heating of a slab, any location inside the
food is subject to the microwaves from the positive and negative x and y
directions.
This is also true for the cylinder where microwaves are
incident from both the positive and negative radial and axial
directions.
The microwave absorption of a cylinder in the z direction
attenuated exponentially (similar to a slab) as shown in Eq. 3.20.
Since there is no attenuation of the microwave power at the surface, the
microwave power is maximum at this location.
The microwave power at the
line where two surfaces meet, that is at the edge or corner of the
product, is the sum of the power from both directions.
Microwave power
at an edge or a corner is thus much higher than that at other locations
upon the surface.
Microwave power absorbed by a sample with rectangular geometry is
the result of attenuation of surface microwave power from x and y
directions as shown in Fig. 4.24.
In this figure, microwave power
incident at the surfaces of a 6 cm x 6 cm x 5 cm slab perpendicular to
both x and y directions at 30°C constant temperature was calculated by
Eqs. 3.20 and 3.21,
High microwave power distribution at corners is due
to the net absorption of microwave power from both of the surfaces at
-3
Fig. 4.24
the corner.
00
Two-dimensional microwave power absorbed ac z - 0
in a 6 cm x 6 cm x 5 cm slab gel at 30°C
As microwave power travels into the slab, the absorbed
power density decreases exponentially as shown in Fig. 4.24.
Fig. 4.25 shows the absorbed microwave power distribution in a 3
cm radius by 5 cm height cylinder.
Absorbed power attenuated from both
the radial (sides) and axial surfaces (top and bottom) toward the
center.
Edge heating effect discussed earlier can be found along the
circumferences at both ends of the cylinder (z - ± 2.5 cm).
However,
because of the decreasing control volume as microwave energy approaches
the center along the radial direction, power density increased
dramatically close to the center.
This is also indicated by the
equation for radial microwave power density absorption (Eq. 3.20).
An
increase in the denominator r would cause the calculated microwave power
absorption to increase drastically at regions close to the center.
118
12
9
6
3
a
3
•3
00
Fig. 4.25 Two-dimensional microwave power absorbed In a 3 cm
radius by 5 cm height cylindrical gel ac 30°C
4.2.1.2
Effect of Sample Temperature on Microwave Power Absorption
Figs. 4.24 and 4.25 were both calculated from the microwave power
absorption terms In Eqs. 3.20 and 3.21 based on a constant temperature
(30°C) to show the differences in microwave power absorption between a
slab and a cylinder.
Since temperature affects the material dielectric
properties, it will affect the microwave power distribution during
heating.
Effects of temperature and distance (from the surface) on the
microwave power absorption in a cylinder (3 cm radius by 5 cm height) in
the axial and radial direction are shown in Figs. 4.26(a) and (b),
respectively.
119
From the conductivity measurements It was determined that the gel
samples used have an equivalent NaCl concentration of 0.05 M.
Referring
to Fig. 4.6, the attenuation factor varies with respect to temperature;
it first decreases with temperature from 0°C to 50°C and then increases
after 50°C.
Looking at Fig. 4.26(a), the power density at the surface
varies similar to the variation of attenuation factor with respect to
temperature as shown in Fig. 4.6.
they attenuate exponentially.
As microwaves travel into the food,
At temperature regions where the
attenuation factor is high, microwave power absorbed is lower than that
at room temperatures (low attenuation factor).
Therefore, in Fig.
4.26(a) the surface microwave power decreases toward the center (z-0) at
a faster rate up to 50°C.
Then at higher temperatures (low attenuation
factor), the decrease of the microwave power absorption is at a slower
rate.
In Fig. 4.26(b) the effect of temperature on the microwave power
absorption due to the temperature dependency of the attenuation factor
is more evident.
The power absorption term in the radial direction in a
cylinder as seen in Eq. 3.20 has an r term in the denominator.
The
microwave power absorption becomes very high as r approaches zero
(radial center).
At these temperatures, when the attenuation factor is
low, there is more power available toward the center.
This is indicated
by the high microwave power at a temperature range from 45°C to 75°C.
di<eC
dltec cioti
* /c i»
aV>®
16^
Fig,
26 Power
3 cmz direct
J-*calabsorption
gel alongin
thea (a)
*4<f€erent temperatures
4 .cylindrical
20
_
Bdirection at diffe-
121
It also shows that power absorption at the center becomes less efficient
as temperature increases because of the high attenuation factor.
At the
latter stages of heating, the microwave absorption is more of a surface
phenomenon because of the increased attenuation factor toward the
surface.
The microwave power absorption decreases from the surface up
to half the radius and then increases.
This would indicate the location
of the slowest heating point in the sample to be at half the radius from
the surface of a cylinder.
4.2.2
Effective Microwave Power Gain
The volumetric concentration with the increase in microwave power
absorption at areas close to the center of the cylinder was defined by
Kritikos and Schwan (1975) in Eq. 2.36.
Fig. 4.27.
This is also illustrated in
When the effective gain of microwave power becomes greater
than one, the concentration heating effect occurs.
The distance from
the radial center where this phenomenon occurred was calculated for
cylinders with radii ranging from 2 cm to 6 cm and with attenuation
factors of 0.3, 0.4 and 0.5.
Fig. 4.27 showed that this concentrated
heating effect is predominant in cylinders of smaller radii and at low
attenuation factors.
For instance, center heating effect occurs at a
cylinder of 2 cm radius over two thirds of the area from the center when
a equals to 0.3 while it occurs less than one thirds of the area if a
equals to 0.5.
As the radius increases, this center heating effect
becomes insignificant because heating occurs in a very narrow region
close to the center of the cylinder.
122
1.4
2
3
4
5
6
CyIinder radlI, cm
V<. : EZJO.3
E30.4
E S I 0.5
Fig- 4.27 Effect of attenuation factor on the radial distance of
concentrated heating from the center in cylinders of various radii
4.2.3
Sensitivity Analysis
Listings of TUODEPEP finite element programs used for cylinders
and slabs are listed in the Appendix A.
selected in the sensitivity analysis.
Five parameters used were
They are thermal diffusivity (a(
— k/pCp), convective heat transfer coefficient, evaporative heat loss,
microwave surface power and attenuation factor.
Two of these
parameters, i.e. thermal diffusivity and attenuation factor, are
123
material properties.
Others are related to the environment and
operating conditions.
The parameters were varied in ranges according to realistic
situations.
They are listed in Table 4.2.
The justification of the
selected percent variation in the parameters is as follows.
Thermal
diffusivity lumps together thermal conductivity, density and specific
heat of the material.
Thermal diffusivity of high moisture foods varies
within the range of ± 20% from the value used in the model (Choi and
Okos, 1983).
The convective heat transfer coefficient as used by Swami
(1982) may vary due to the degree of ventilation caused by the fan
during microwave heating in different ovens.
The heat transfer
coefficient of the moving air may change from 11.3 to 55 W/m2oC . The
relative humidity of the ambient air may vary significantly from day to
day.
Since the water activity of gel is very high and close to that of
water, fluctuation in the relative humidity in the air may account for
about ± 60% from the assumed 39% difference in the relative humidity
between the sample and the air.
According to Gerling (1987), variability of the voltage from the
power supply may cause ± 10% change in the power output from the
magnetron.
The variations in the attenuation factor were derived based
on the attenuation factor of water and 0.1 M NaCl solution.
Cylinders of sizes 2 cm and 4 cm radii by 5 cm height were used
for sensitivity analysis.
They represented a small size cylinder (2 cm
radius) which has pronounced center heating and a larger size cylinder
(4 cm radius) with little center heating.
The changes of temperature
124
Table 4.2 Variations of the characteristic parameters of the
microwave heating model for sensitivity analysis
Parameters1
1 at
hc
Values
Variation
“t
2.011E-3 fcm2/®]
± 20 %
hc
3.944E-3 (w/cm2oC)
± 40 %
ARH
39 «
± 60 %
Po
40.1+59.81n(wt) [w]
± 10 %
a
f(er\
± 30 %
er") [cm’1J
is thermal diffusivity, k/pCp,
is convective heat transfer coefficient,
ARH is the difference of relative humidity between sample and the
ambient air,
P0
a
is the total microwave power absorbed,
is the attenuation factor calculated from Eq. 4.3.
125
prediction due to the variation of the various parameters in cylinder of
2 cm radius by 5 cm height were significant.
The temperature
distribution as a result of a change in the various parameters was
simulated by the finite element program.
The time-temperature
relationships at one point in the cylinder due to changes of parameters
are shown in Figs. 4.28 to 4.32.
Temperature variations due to the
change of the material's thermal diffusivity, attenuation factor and
microwave power output were compared at the center point of the
cylinder.
2 can rad. x 5 cm ht.
r=0
4 cm rad. x 5 cm ht.
z=0
r=0
z=0
40
80 ■
10%
10%
40
5
10
Time, s
Fig. 4.28
15
20
10
20
30
Time, s
Sensitivity analysis for 2 cm and 4 cm radii cylinders of
5 cm height by varying microwave power output ± 5% and ± 10%
126
T [°C]
T [°C]
100
2cm rad. x 5cm ht.
r-0
z-0
4cm rad. x 5cm ht.
-30:
80-
-3 0 %
40
30%
60
40-
Time , 8
Fig. 4.29
Time, s
Sensitivity analysis for 2 cm and 4 cm radii cylinders of
5 cm height by varying the attenuation factor ± 30%
T [°C]
T [°C]
100
2cm rad. x 5cm ht
r-0
4cm rad. x 5cm ht
z-0
r-0
80
z-0
40
60-
20%
10%
20%
10%
30
-10%
-20%
20
5
10
15
Time, s
20
10
20
30
Time, s
Fig. 4.30 Sensitivity analysis for 2 cm and 4 cm radii cylinders of
5 cm height by varying the thermal diffusivity ± 10% and ± 20%
T [°C]
127
T [«C]
100
2 cm rad- x 5 cm ht.
r=2 cm
4 cm rad. x 5 cm ht.
z=2.5 on
r=4 cm
z=2.5 cm
40
80
's -40%
♦ - 20%
»
0
* 20%
x 40%
60
40
30
i i i1V "i— i
— i
— i
— i— i— i
— i— i
— r
10
15
20
20
20
10
Time, s
Fig. 4.31
20
Time, s
30
Sensitivity analysis for 2 cm and 4 cm radii cylinders of
5 cm height by varying the heat transfer coefficient ± 20%
and ± 40%
T [°C]
T [°C]
100
2 an rad. x 5 an ht
r=2 an
4 an rad. x 5 cm ht
z=2.5 an
r=4 cm
z=2.5 cm
40
80
60
30
40
20
20
5
10
15
Time, s
Fig. 4.32
20
10
20
Time, s
30
Sensitivity analysis for 2 cm and 4 cm radii cylinders of
5 cm height by varying the difference of relative humidity
between sample and ambient air ± 30% and ± 60%
128
The effects of changes of the boundary conditions (heat transfer
coefficient and relative humidity) on temperature predictions were
evaluated at r - R and z - 2.5 cm surface.
The relative sensitivity was
calculated as the ratio of the change in temperature with respect to the
change of parameters to the original value as shown in Eq. 3.36.
The
results of this part are presented in Table 4.3.
Table 4.3
Relative Sensitivity Calculated from Sensitivity Analysis1.
2 cm rad. x 5 cm
4 cm rad. x 5 cm
“t
53.4
32.2
hc
1.4
1.8
ARH
4.7
7.2
Po
59.8
30.3
40.0
10.9
Parameters
a
'Calculated based on Eq. 3.36 expressed in percentage.
The effects of ± 40% change in heat transfer coefficient and ± 60%
change in relative humidity were not as significant as compared to the
variations in other parameters.
This is due to the fact that these
parameters are only affecting the boundary condition.
The model was
equally sensitive to changes in thermal diffusivity and power
absorption.
Variations in attenuation factor also lead to considerable
changes in temperature predicted by the finite element model for 2 cm
radius by 5 cm height cylinder.
129
4.2.4
Model Verification
The temperature distribution during microwave heating of
cylindrical and rectangular gels under different conditions were
predicted by the finite element program TWODEPEP. The results of model
predictions were verified using experimental measurements.
Comparisons
of the predicted RTR and the measured RTR were performed at 0.01
significance level by t-test.
4.2.4.1
Temperature Distribution in Slabs and Cylinders with Thermal
Insulation
Cylinder
Fig. 4.33 shows (a) the predicted RTR in a half cylinder
of 4 cm radius and 2.5 cm height using the finite element program and
(b) the RTR distribution obtained from the experimental results,
respectively.
Duration of heating was 30 s in a microwave oven.
When
the magnetron come-up time (3 s) was subtracted, the exact heating time
was 27 s.
The element size of 1.562 cm2 and 5 s time step were
determined for the finite element program that resulted in the least
percent error when compared with the predicted RTR with the experimental
RTR.
Assuming uniform surface microwave power (uniform field
distribution all around the cylinder), the temperature distribution
within the cylinder has radial and axial symmetry.
The diagrams show
from the center to the bottom part of cross sectional area of a
cylinder.
The boundary and initial conditions were set as 25 °C initial
temperature and no heat loss from the boundaries (thermally insulated).
130
(a) Predicted
1.0
u
60
01
Q
0.5
I
0.0
0.0
-0.625'J'Ujv
z [cm]
r [cm]
-1.25
(b) Experimental
u>
o 1. 0
60
0)
<=>
(A 0.5
0.0
-0.625
z [cm]
-1.25
Fig. 4. 33
(a) Predicted RTR and (b) experimental RTR in a 4 cm radius
by 2.5 cm height cylinder with no heat loss from the
boundaries
131
Center heating was observed in the predicted RTR.
The variation
of RTR in the z direction indicated the attenuation of bottom surface
power from z - -1.25 cm and then concentrated at the center.
Fig.
4.33(b) shows the asymmetrical variation of RTR when compared to the
predicted RTR, which reflected the nonuniform microwave power
distribution in the area where the sample was heated.
Because of the
imperfect insulation on the radial surface, the experimental RTR
decreased at r - ±4 cm surface areas due to convective heat loss.
Slab
Figs. 4.34(a) and 4.34(b) show the RTR values from the
finite element program prediction and experimental measurements in slabs
of 8 cm x 8 cm x 5 cm.
The initial and boundary conditions are 25 °C
initial temperature and no heat losses at the boundaries.
of microwave heating was 35 s.
cm2 while the time step was 1 s.
The duration
The element size was determined as 0.64
The microwave radiations from x and y
directions only were taken into account due to the two-dimension
limitation of the finite element program.
The figures show RTR results
over the cross sectional area of the slab at z - 0 level.
distribution of RTR was observed in both figures.
Symmetrical
Again, the effect of
microwave power distribution is evident in the experimental result shown
in Fig. 4.34(b).
Predicted Edge heating and corner heating effects are
shown in both the RTR prediction and the experimental RTR.
The
predicted RTR values were shown higher than the measured RTR in many
locations at surface of the gel, which may be caused from imperfect
insulation and heat loss during temperature measurements.
132
(a) Predicted
u>
o
00
4>
0
p? 0.5
1
0.0
y [cm]
x [cm]
(b) Experimental
ID
U
t>0
4)
P
. 0.5
I
0.0
y
x [cm]
[cm]
-4 -4
Fig. 4. 34
(a) Predicted RTR and (b) experimental RTR in an 8 cm x 8 cm
x 5 cm slab at z - 0 with no heat loss from the boundaries
133
4.2.4.2
Temperature Distribution in Cylinders and Slabs with Heat Loss
from the Boundaries
Cylinder
RTR as predicted by the model for cylinders of different
sizes are plotted in Figs. 4.35 to 4.42.
The model considered
convective heat loss radially, and convective and evaporative heat loss
on the top surface of the gel.
For the most part, the predicted
temperature distributions in cylinders of different sizes were
comparable with the results from the experimental data.
Results from
finite element model predictions and the experimental measurements were
compared at 0.01 significance level and are listed in Appendix C.
At
most of the locations the model predictions were not significantly
different from the experimental results.
Locations where model
predictions were significantly different from the experimental results
were at the center point or at regions close to the center line of the
cylinder ( r - 0 ) , especially at the bottom level (z - -Lz) of the
cylinder.
It was found that the model predictions at the center points
were mostly higher than the experimental results, while lower than the
experimental results at regions near by the center line.
This is
possibly due to the time delay from when the samples were taken out of
the microwave oven and the temperature measurements were taken.
134
4.5
z=l.25 can
co
CJ
3.0.
i
0.0
4.5
z=0
m
3.0'
o
S
0.0
4.5
z=-l.25 cm
to
o
*
0.0
r/R
measured
Fig. 4.35
—
predicted
The means and standard deviations of predicted and measured
RTR at various radial distances from the center (r/R - 0) in
a 2 cm radius by 2.5 cm height cylindrical gel at z 1.25 cm, z - 0, and z - -1.25 cm levels
135
3.0
z=l.25 cm
§
rj> 2.0
a
0.0
3.0
z=0
1.0
0.0
3.0
z=-l.25 cm
(0
u 2.0
S
1.0-
0.0
r/R
measured
Fig. 4.36
|—
predicted
The means and standard deviations of predicted and measured
RTR at various radial distances from the center (r/R - 0) in
a 3 cm radius by 2.5 cm height cylindrical gel at z 1.25 cm, z - 0, and z - -1.25 cm levels
136
z=l.25 cm
0.5-
0.0
1.5
z=0
0.0
1.5
z=-l.25 cm
1. 0
0.5
0.0
1/2
3/4
r/R *
measured
Fig. 4.
1—
predicted
The means and standard deviations of predicted and measured
RTR at various radial distances from the center (r/R - 0) in
a 4 cm radius by 2.5 cm height cylindrical gel at z 1.25 cm, z - 0, and z - -1.25 cm levels
137
z=l.25 an
(0
£
0.5-
0.0
1.5
z=0
(0
0.0
1.5
z=-l.25 an
CO
I
0.5-
0.0
|—
Fig. 4.38
measured
r/R
*— |—
predicted
The means and standard deviations of predicted and measured
RTR at various radial distances from the center (r/R - 0) in
a 6 cm radius by 2.5 cm height cylindrical gel at z 1.25 cm, z - 0, and z - -1.25 cm levels
138
4.5
z=2.5 an
3.0
0.0
4.5
z=0
S
0.0
4.5
z=-2.5 an
$
0.0
1 /2
r/R
measured
Fig. 4.
— I—
predicted
The means and standard deviations of predicted and measured
RTR at various radial distances from the center (r/R - 0) in
a 2 cm radius by 5 cm height cylindrical gel at z - 2.5 cm,
z - 0, and z - -2.5 cm levels
139
3.0
z=2.5 an
0.0
3.0
2=0
*
°
2.0
0.0
3.0
z=-2.5 an
§
$
0.0
2/3
r/R
measured
Fig. 4.40
— f—
predicted
The means and standard deviations of predicted and measured
RTR at various radial distances from the center (r/R - 0) in
a 3 cm radius by 5 cm height cylindrical gel at z - 2.5 cm,
z - 0, and z - -2.5 cm levels
140
z-2.5 cm
§
a
0.5
0.0
1.5
z=0
(0
\
a
0.0
1.5
z=-2.5 cm
&
0.5
0.0
1/4
1 /2
r/R
measured
Fig. 4.41
3/4
—
1
predicted
The means and standard deviations of predicted and measured
RTR at various radial distances from the center (r/R - 0) in
a 4 cm radius by 5 cm height cylindrical gel at z - 2.5 cm,
z - 0, and z - -2.5 cm levels
141
.5
z=2.5 an
.0
.5
0.0
1.5
z=0
I
0.0
1.5
z=-2.5 an
$
0.5 -
0.0
r/R
measured
Fig. 4.42
predicted
The means and standard deviations of predicted and measured
RTR at various radial distances from the center (r/R - 0) in
a 6 cm radius by 5 cm height cylindrical gel at z - 2.5 cm,
z - 0, and z - -2.5 cm levels
142
The field distribution was incorporated in the power absorption
term.
However, this did not effectively improve the prediction by the
model.
This may result from the fact that the technique used to
determine the field distribution cannot accurately predict the local
field intensity in the presence of a load, and thus the mode patterns
within the microwave oven cannot be accurately described.
It is also
possible that the regression model tends to smooth the high and low
nodes as was observed by comparing the experimental data (Fig. 4.1) with
data from the regression model (Fig. 4.2).
In all cases, except 6 cm radius cylinders, center heating
phenomenon was observed.
As was discussed in the previous section, the
area where center heating occurred decreases as the radius increases.
The temperature increase predicted from cylinders of 5 cm height is
about half of that from cylinders of 2.5 cm height.
Vertical
temperature distribution is consistently uniform in cylinders of 2.5 cm
height.
For 5 cm high cylinders, the temperature predicted at the
center plane (z - 0) is lower than the temperatures at the top and
bottom surfaces.
This was found in all the 5 cm high cylinders.
Table 4.4 compares the calculated RTR from model prediction with
the thermocouple measurement and the fluoroptic probe measurement.
Limited by the number of channels for the temperature measurement by
fluoroptic probes, the locations which gave rise to large temperature
changes were selected.
The predicted RTR values were compared with
results of experimental measurements at 0.01 significance level.
Most
of the disagreement between the model prediction and the experimental
results came from samples of smaller diameters where center heating was
143
Table 4.4 The Means and Standard Deviations of RTR In Cylindrical Gels
of Various Sizes Measured by Thermocouples and Fluoroptic Probes
and Compared to the Model Predicted RTR
r [cm]
z [cm]
Measured RTR
Thermocouple
Fluoroptic
Predicted RTR
2 cm rad, x 2.5 cm ht.
0
1.25
1.25
2
0
0.00
0
-1.25
1.65110.311
1.22110.200
3.48810.307
2.51110.335
2.01110.649
1.47410.357
3.63210.500
1.34210.397
3.087
1.615
3.187
3.307
2 cm rad. x 5 cm ht.
0
2.50
2
2.50
0
0.00
0
-2.50
1.15810.128
1.49611.894
2.39010.352
1.25510.177
1.30410.340
1.89410.320
3.22510.699
0.78410.169
2.947
0.810
3.079
2.645
3 cm rad. x 2.5 cm ht.
0
1.25
3
1.25
0
0.00
3
-1.25
0.80610.124
1.02110.100
1.75210.134
1.75910.173
0.82110.180
1.35610.208
1.97110.594
2.12210.383
1.494
0.826
2.110
0.863
3 cm rad. x 5 cm ht.
0
2.50
3
2.50
0
0.00
3
-2.50
0.44110.091
0.95810.285
0.84110.119
0.90110.081
0.51310.134
1.60410.476
0.78210.212
1.10710.178
1.004
0.519
0.900
0.569
4 cm rad. x 2.5 cm ht.
0
1.25
3
1.25
4
1.25
4
-1.25
0.57510.047
0.64310.063
0.65810.089
1.38210.173
0.72610.120
0.59710.063
0.58210.097
1.49010.410
0.703
0.499
0.592
0.635
4 cm rad. x 5 cm ht.
0
2.50
3
2.50
4
2.50
4
-2.50
0.41310.078
0.43010.084
0.76710.190
0.81610.057
0.50010.030
0.33910.071
1.03210.277
0.69110.147
0.449
0.333
0.408
0.445
6 cm rad. x 2.5 cm ht.
5
1.25
6
1.25
5
0.00
6
-1.25
0.37810.034
0.33810.047
0.41110.029
0.34810.042
0.34310.074
0.39910.091
0.68210.120
0.73210.082
0.302
0.364
0.337
0.410
6 cm rad. x 5 cm ht.
2.50
5
2.50
6
6
0.00
-2.50
6
0.19110.007
0.18710.005
0.13410.003
0.19010.017
0.163+0.041
0.42310.600
0.25110.035
0.43010.081
0.195
0.251
0.198
0.282
144
predominant.
The model predictions at the center of the cylinders were
higher for cylindrical samples with smaller diameters.
The predicted
temperatures at the center of the top and bottom surfaces of the
cylinder were higher than those experimentally determined in all the
samples.
For the most part, the discrepancies at the center of the
cylinder were less than those from the top center and the bottom center
of the cylinder.
At areas close to the center (r - 0) at three levels
in the z direction, the predicted RTR values of most of the cylinders
were found significantly different from the experimental results.
The
RTR calculated from experimental temperature measurement with fluoroptic
probes showed similar trends as those from the thermocouple measurement.
Deviation of the model prediction from experimental results may be
from the inability to sufficiently incorporate the interaction of
microwave power with the gel sample.
Apart from the over-estimation of
the center temperature in cylinders of small radii, the model predicted
the trend in temperature distribution with
respect to different
locations in the
gel reasonably well.
The predicted temperature
distribution for
the slower heating region were in close agreementwith
the experimental
results as shown in Table 4.4.
This will assurea safe
estimation of heating time for microwave thermal processing.
Slab
Fig. 4.43 shows the RTR from the model simulation and
experimental results for slabs of 3 sizes with microwave incident in the
x and y directions.
The slab geometry is essentially a three-
dimensional geometry, but TWODEPEP cannot handle three-dimensional
problems.
Because of the limitation of TWODEPEP in solving only two
(b)predict: 6an x 6cm x 5cm
(e)measure: 6cm x 6cm x 5cm
(c)predict: 8cm x 8cm x 5cm
(f)measure: 8cm x 8cm x 5cm
Fie. 4.43 Model predicted RTR and measured RTR at z - 0 in slabs of (a)
and (d) 4 cm x 4 cm x 5 cm, (b) and (e) 6 cm x 6 cm x
cm, and
(c) a n d (f) 8 cm x 8 cm x 5 cm, respectively
146
dimensional conditions, the experiments were conducted by shielding gel
slabs with aluminum foil on the top and bottom surface.
The figure
shows RTR at the central cross sectional layer of the slab.
predictions were close to the experimental results.
The model
This was also
verified by the comparison of the predicted RTR with experimental RTR
atO.Ol significance level as shown in Appendix C-10.
However, the field
distribution effect incorporated in the power source term did not
improve the model prediction significantly.
For all the slabs, the
model predicted an RTR at the center to be lower than the experimental
results.
As seen in Fig. 4.43(d), the RTR in a slab of 4 cm x 4 cm x 5 cm
was unique, and this was not predicted by the model.
The center of the
slab in this case was not the slowest heating point as predicted by the
model.
This phenomenon is similar to the center heating of the
cylindrical gel.
It is possible that the microwave resonance effect
causes the generation of internal mode patterns in the small gel sample.
In order to understand this further, the true mechanism of microwave
coupling needs to be further elucidated.
Gel with 1 % Total Salt Content
Temperature distribution in a 4 cm radius by 2.5 cm height
cylindrical gel containing 1 % total salt concentration during microwave
heating was predicted using TWODEPEP.
Since the NaCl Molar equivalent
concentration for 1% total salt content in gel was determined to be
0.172 M, the attenuation factor for NaCl solution at 0.172 M
147
(Interpolated from Fig. 4.5) was used in the model.
A multiple linear
regression model was developed to describe the change In attenuation
factor as a function of temperature from 0°C to 100°C as shown In Eq.
4.4.
a - 0.758 - 0.00619T + 0.00012T2
R2-99.4 «
(4.4)
This function was used in the program, and the RTR was calculated
based on predicted temperature data.
The predicted RTR at different
locations In the 4 cm radius by 2.5 cm height cylinder were plotted with
respect to radial distance as shown in Fig. 4.44 and compared with the
experimental RTR.
The model predictions are not significantly different
from the experimental results at 0.01 significance level except for the
points at r - 0 and r - 1 cm at z - 0 and z - 1.25 cm two levels within
the cylinder as shown in Appendix C-5.
Radial Microwave Heating
To simulate radial microwave heating,
the source term in the z direction was eliminated, leaving just the
radial microwave absorption term in the program.
The RTR values based
on the predicted data were compared with the RTR values from the
experimental results of radial microwave heating in a 4 cm radius by 2.5
cm height cylinder.
This is shown in Fig. 4.45.
The model predictions
appeared to be generally lower than the experimental results except at
the center region.
Microwave heating of 4 cm radius by 2.5 cm height cylinder with
electrical shielding from just the top surface was simulated by a finite
element model.
The microwave power coming from the z - 1.25 cm surface
148
z=l .25 can
(0
a
S
0.0
z=0
m
o
$
0.0
1.5
z=-l.25 cm
1. 0
0.5
0.0
1/4
r/R
measured - i - predicted
Fig. 4.44 Effect of 1% total salt on the predicted and measured RTR in
a 4 cm radius by 2.5 cm height cylindrical gel at different
locations at (a) z - 1.25 cm, (b) z - 0, and (c) z - -1.25 cm
levels
149
1.5
z=l.25 an
(0
'u
1.0
g
0.5
0.0
1.5
z=0
5
g
0.5
0.0
1.5
S
z=-l.25 an
1.0-
g
0.0
0
1/4
v
— +—
1/2
3/4
1
r/R— t4— predicted
measrued
Fig. 4.45 Effect of radial microwave heating on predicted and measured
RTR In a 4 cm radius by 2.5 cm height cylindrical gel at different
locations at (a) z - 1.25 cm, (b) z - 0, and (c) z - -1.25 cm
levels
150
was eliminated from Che power source term.
The RTR results were plotted
against the radial distance from the center as shown In Fig. 4.46.
Like
the radial microwave heating simulation, the temperature predictions at
the center of the cylinder were all higher than the experimental results
but were lower at other locations at level z - 0.
Appendix C-7 shows
the comparison of measured RTR and predicted RTR at all the locations in
the cylinder.
The microwave power incident from the z - -1.25 cm was removed
from the power source term to predict the microwave heating of a 4 cm
radius by 2.5 cm height cylinder with bottom surface shielded with
aluminum foil.
The predictions were compared with the experimental
results as shown in Fig. 4.47 and Appendix C-8.
RTR values calculated
from temperature predictions at center areas of z « 1.25 cm and z - 0
levels were higher than the experimental results.
Axial Microwave Heating
When the radial microwave heat source was
taken away from the microwave power absorption term, the model simulated
axial microwave heating.
The prediction was compared with the RTR
experimental results by shielding at the 4 cm radius by 2.5 cm height
cylindrical surface.
The data are plotted in Fig. 4.48 with respect to
radial distance from the center.
At z - 0 and z - -1.25 cm levels the
model predicted lower RTR as compared to the experimental results.
Discrepancy between the experimental and predicted RTR may be due to
induced internal electric field inside the aluminum foil circumference
which was not incorporated in the mathematlc model.
151
1.5
z=l.25 cm
(0
1.0
$
0.5
0.0
1.5
z=0
0.0
z=-l.25 cm
s
s
174
172
374"
r/R
—
measured — |—
predicted
Fig. 4.46 Effect of microwave hearing with the top surface shielded on
predicted and measured RTR In a 4 cm radius by 2.5 cm height
cylindrical gel at different locations at (a) z - 1.25 cm, (b) z 0, and (c) z - -1.25 cm levels
152
1.5
z=1.25 cm
1.0
0.5
0.0
1.5
z=0
$
0.0
1.5
z=-l.25 cm
(0
0.5-
0.0
1/2
3/4
r/R
measured
-- 1—
predicted
Fig. 4.47 Effect of microwave heating with the bottom surface shielded
on predicted and measured RTR In a 4 cm radius by 2.5 cm height
cylindrical gel at different locations at (a) z - 1.25 cm, (b) z 0, and (c) z - -1.25 cm levels
153
1.5
z=l.25 an
(0
^
1.0
s
0.5
0.0
z=0
§
1.0 -
ir
|
0.5!
0.0
1.5
co
o
0.5
0.0
0
1 /2
r/R
measured
— J—
1/4
|—
3/4
1
predicted
Fig. 4.48 Effect of axial microwave heating on the predicted and
measured RTR at different locations in a 4 cm radius by 2.5 cm
height cylindrical gel at (a) z - 1.25 cm, (b) z - 0, and (c) z
- -1.25 cm levels
154
4.2.4.3
Summary
Considering the many variables during microwave heating, the model
simulation of microwave heating compared favorably with the experimental
results for gels of different salt concentration, geometry and size.
In
many cases, especially with cylindrical geometry, the center temperature
predicted by the model was higher than that from the experiments.
The
model predicted lower temperature distribution in cylindrical samples
which were microwave-shielded in the radial direction when compared to
the experimental results.
A thorough understanding on the interaction
of microwaves with metals is needed in order to improve the model by
inputting the functions that better describe the phenomenon.
The asymmetrical temperature distribution observed during
experimental measurements was not effectively predicted by the model.
A
better method to assess the microwave power distribution will enable
simulation closer to the experimental results.
Since microwave power
distribution varies from oven to oven, it is almost impossible to
specify this factor clearly without over complicating the model.
It can
be concluded that this mathematical model can be used to simulate the
temperature distribution for samples of different material properties
and geometrical configurations under different microwave heating
conditions within a reasonably small range of deviation.
155
CHAPTER 5
CONCLUSIONS
The heating characteristics of foods In a microwave oven depend on
the oven parameters and the physical and dielectric properties of the
food.
This research was aimed at understanding the nonuniform heating
of food in a microwave oven by means of experimental measurement and
mathematical modeling the microwave heating process based on the
interaction between the microwaves and the food material.
In many research activities related to microwave heating of
dielectric materials, temperature measurements were often performed at
few locations on the sample and the factor of nonuniform microwave power
distribution was not investigated.
In this research, temperature
measurements were carried out three-dimensionally in samples to better
understand the whole picture of temperature distribution in a high
moisture food during microwave heating.
In order to characterize the
microwave oven, the patterns of microwave power distribution at
different locations in the oven were determined.
The relationships
between the microwave power absorption and the loading in cylindrical
and rectangular containers were also studied.
Microwave power distribution in a Tappan 500 microwave oven used
in this study is composed of high and low nodes.
The patterns are
different from one level to another in the vertical direction.
The
power intensity increases with increasing height within the oven cavity.
The absorption of microwave power with different sizes of load follows a
156
logarithmic function.
The power-load relationship is also dependent on
the shape of the container because of the effect of microwave power
coupling efficiency.
A model food system made of sodium alginate gel was used in
studying the effects of salt content, geometrical configuration, and
shielding of microwaves (at different surfaces of the sample) on
temperature distribution of high moisture foods during microwave
heating.
Surface heating was predominant in gels of high salt content
because of the increased attenuation due to the elevation of dielectric
loss and depression of dielectric constant from increased conductivity.
Cylindrical and rectangular shaped gels of different sizes were
studied.
Center heating was predominant in samples with smaller
diameters (4 cm and 6 cm). Surface heating was observed in cylinders of
diameters from 8 cm and larger.
The center heating or surface heating
can be determined by effective power gain of a cylinder with respect to
the distance from the center.
In small cylinders, the absorbed
microwave power density becomes very large (due to decreasing volume) as
microwaves are transmitted towards the center of the cylinder.
In a slab, microwave power attenuated exponentially from the
surface towards the center.
Therefore, the slowest heating point was
located at the center of the slab.
Corner and edge heating occurred in
the slab due to incident microwave power from two or three surfaces.
Shielding of the microwave incident power in the axial and radial
surfaces on a cylinder resulted in microwave heating from the radial
direction and axial direction, respectively.
The temperature increase
at the surface adjacent to the aluminum foil (used for shielding) was
157
lower.
Shielding the microwaves coming In from the bottom part of the
oven eliminated the nonuniform reflected microwaves from the bottom of
the oven.
Shielding at the radial surface resulted In an uneven
temperature distribution much higher than that from other different
configurations.
A mathematical model was developed to simulate transient heat
transfer In a food during microwave heating.
Boundary conditions were
Included in the model to consider the surface heat loss due to
convective and/or evaporative heat loss.
The volumetric microwave power
absorption term with respect to distance from surface was derived for
cylindrical and rectangular shaped products.
In the power absorption
term, internal power reflection and the microwave power from the
positive and negative direction of the coordinates were included.
The
magnetron come-up-time was determined for Tappan 500 microwave oven and
was incorporated in the program.
The TUODEPEP finite element program
was used to calculate the temperature distribution in different
locations and time step within the model food during microwave heating
in different conditions.
The model prediction agreed well with the
experimental results, except for the samples of small sizes.
In small
size samples, the model generally predicted higher center temperature
than the measured experimental values.
The model was able to
characterize the heating behavior of the food but was not able to
totally describe the nonuniform temperature distribution due to the
uneven field distribution in the oven.
The changes in temperature predictions by the model due to
variations in selected parameters were tested.
These parameters
158
Included the material's dielectric properties, thermal diffusivity,
convective heat transfer coefficient, ambient relative humidity and
microwave oven power output.
The sensitivity analysis indicated that
the model prediction will be subjected to a larger amount of variation
when the material properties change.
Change in power output level
influenced the model prediction significantly.
However, it was found
that model prediction will not be affected by the variations in the two
boundary conditions, i.e. changes in convective heat transfer
coefficient or relative humidity.
This study was limited to the conductive heat transfer of high
moisture solid foods during microwave heating.
Microwave heating of the
liquid foods and the effect of salt concentration and viscosity on the
natural convection heating has been investigated (Liu, 1990).
More work
should be conducted in the area of convective/conductive heat transfer
and mass transfer modeling for foods with different physical properties
and chemical compositions during microwave heating.
Physical properties
of materials such as density, porosity, heterogeneity (e.g. liquid with
particulates) affect the dielectric and transmissive properties, which
in turn influence the microwave power coupling and wave propagation
inside the food.
Heat transfer and moisture migration occur
simultaneously during microwave heating.
Modeling of heat and mass
(moisture) transfer should consider the phase change and the resulting
pressure changes, if a food is not porous enough to let the moist air
diffuse out freely.
Moisture migration measurement techniques need to
be developed for conducting experiments to verify the mass transfer
models.
A better technique to determine the mode pattern within the oven
cavity during microwave heating needs to be developed.
From the
empirical determination of the mode pattern, analytical solutions for
the electric field may be obtained, which can be incorporated into the
microwave power source term.
This can lead to a better prediction of
the temperature distribution as affected by the product parameters and
oven performance characteristics.
160
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165
Appendix A
FORTRAN PROGRAMS USED IN TWODEPEP FOR PREDICTION OF
TEMPERATURE DISTRIBUTION DURING MICROWAVE HEATING
A-l
Sample Program Listings for Rectangular Geometry Used In TWODEPEP
//V03XXXXX JOB
// EXEC TWODEPEP
//TWODEPEP.INPUT DD *
1 50 1
8.37D-3*UX
OXX
OXY
8.37D-3*UY
Cl
1.01D0*4.12D0
FI
FLD(X,Y)*PHEE(U,T,X,Y)
UO
25. DO
ARC-1
GB1
-3.705D-3*(U-28.D0)
ARC-2
-3.705D-3*(U-28.D0)
GB1
ARC-3
-3.705D-3*(U-28.D0)
GB1
NX
4
NY
8
45
NOUT
MUR
6
SYMMETRY
1
2
NUPDT
0.5D0
ALPHA
DT
1.D0
45. DO
TF
1.D0
D3EST
VXY
0.DO,-4.DO, 4.DO,-4.DO, 4.DO,0.DO, 4.DO,4.DO, 0.DO,4.DO,
0.DO,0.DO, 2.D0.-2.DO, 2.DO,2.DO
VXY
IABC
1,2,7, 2,3,7, 3,4,8, 4,5,8, 5,6,8, 6,1,7, 3,6,7, 6,3,8
I
1,2,2,3,4,4,0,0
ADD.
DOUBLE PRECISION FUNCTION FLD(X,Y)
IMPLICIT REAL*8 (A-H.O-Z)
SUM-0.DO
COUNT-O.DO
A-4.D0
B-4.D0
C-5.D0
M-2*A+1
N-2*B+1
DO 10 1-1, M
XI— (A+l. DO)+1+9. DO
166
30
20
10
DO 20 J-l, N
X2--(B+l.D0)+J+9.DO
Z-5.D0
Q-18.033D0+X2*<-2.953D0)+X2*X1*0.167D0+X2*Z*0.952D0+
* Xl*Xl*Z*(-4.358D-2)+DCOS(Xl)*(-4.784D-l)+
* DCOS(Xl)*DSIN(X2)*1.223D-l+DSIN(X2)*DCOS(Z)*2.485D-l+
* DSIN(Z)*DCOS(X2)*0.1658D0+(X2**4)*1,128D-3+
* (X2**3)*Z*(-2.673D-3)+X2*(Z**3)*(-3.562D-2)+
* X2*X2*X1*X1*1.69D-3+Xl*Xl*Z*Z*3.324D-3+
* (Xl**5)*(-3.502D-05)+(Xl**4)*Z*4.359D-4+
* (X2**3)*Z*Z*(-1.18D-3)+X2*X2*(Z**3)*4.206D-3+
* X2*X2*Xl*Xl*Z*(-2.241D-3)+(X2**5)*Xl*(-2.479D-06)+
* X1*(Z**5)*6.545D-4+(Xl**4)*Z*Z*(-4.89D-05)+
* X2*X1*X1*(Z**3)*(-4.617D-4)+X2*X2*Xl*Xl*Z*Z*8.554D-4+
* <X2**7)*(-1.952D-07)+(X2**6)*Z*7.854D-07+
* Xl*(Z**6)*(-7.971D-05)+X2*(Z**6)*9.144D-06+
* (X2**5)*X1*X1*(-3.79D-06)+X2*X2*(Z**5)*(-1.601D-05)+
* X2*(X1**4)*Z*Z*3.224D-06+(X2**4)*X1*X1*Z*3.296D-05+
* (X2**3)*Xl*(Z**3)*(4.464D-06)* (1.05D-4)*(X2**3)*X1*X1*Z*Z+
* X2*X2*X1*X1*(Z**3)*2.326D-05+(Z**8)*1.02 5D-06+
* X2*(X1**7)*8.792D-09+(X2**6)*X1*X1*2.189D-07+
* X2*<X1**6)*Z*(-7.285D-08)+X2*Xl*(Z**6)*3.123D-06+
* (Xl**3)*(Z**5)*(6.416D-07)+(X2**5)*X1*X1*Z*
* (-1.543D-06)+X2*X2*Xl*(Z**5)*(-1.843D-06)+
* (X2**4)*X1*X1*Z*Z*(3.2D-06)
SUM-SUM+Q
COUNT-COUNT+1.DO
CONTINUE
CONTINUE
CONTINUE
QM - SUM/COUNT
Xl-(-l.DO)*X + 9.DO
X2=Y + 9.DO
Q-18.033D0+X2*(■2.953D0)+X2*X1*0.167D0+X2*Z*0.952D0+
* X1*X1*Z*(-4.358D-2)+DC0S(Xl)*(-4.784D-1)+
* DC0S(X1)*DSIN(X2)*1,223D-l+DSIN(X2)*DCOS(Z)*2.485D-1+
* DSIN(Z)*DCOS(X2)*0.1658D0+(X2**4)*1.128D-3+
* (X2**3)*Z*(-2.673D-3)+X2*(Z**3)*(-3.562D-2)+
* X2*X2*X1*X1*1.69D*3+Xl*Xl*Z*Z*3.324D-3+
* (Xl**5)*(-3.502D-05)+(Xl**4)*Z*4.359D-4+
* (X2**3)*Z*Z*(-1.18D-3)+X2*X2*(Z**3)*4.206D-3+
* X2*X2*X1*X1*Z*(-2.241D-3)+(X2**5)*X1*(-2.479D-06)+
* X1*(Z**5)*6.545D-4+(Xl**4)*Z*Z*(-4.89D-05)+
* X2*X1*X1*(Z**3)*(-4.617D-4)+X2*X2*Xl*Xl*Z*Z*8.554D-4+
* (X2**7)*(-l.952D-07)+(X2**6)*Z*7.854D-07+
* Xl*(Z**6)*(-7.971D-05)+X2*(Z**6)*9.144D-06+
* (X2**5)*X1*X1*(-3.79D-06)+X2+X2*(Z**5)*(-1.601D-05)+
* X2*(Xl**4)*Z*Z*3.224D-06+(X2**4)*X1*X1*Z*3.296D-05+
* (X2**3)*Xl*(Z**3)*(4.464D-06)* (1.05D-4)*(X2**3)*X1*X1*Z*Z+
* X2*X2*X1*X1*(Z**3)*2.326D-05+(Z**8)*l.025D-06+
*
*
*
*
*
X2*(X1**7)*8.792D-09+(X2**6)*Xl*Xl*2.189D-07+
X2*(X1**6)*Z*(-7.285D-08)+X2*Xl*(Z**6)*3.123D-06+
(Xl**3)*(Z**5)*(6.416D-07)+(X2**5)*X1*X1*Z*
(-1.543D-06)+X2*X2*Xl*(Z**5)*(-1.843D-06)+
(X2**4)*X1*X1*Z*Z*(3.2D-06)
FLD - Q/QM
RETURN
END
DOUBLE PRECISION FUNCTION PHEE(U,T,X,Y)
IMPLICIT REAL*8 (A-H.O-Z)
X1-4.D0
X2-4.D0
X3-5.DO
WT-2.D0*X1*2.DO*X2*X3
P0-50.27D0+56.6DO*DLOG(WT)
PS-PO/(2.DO*(2.D0*Xl*X3+2.D0*X2*X3))
DC-81.7935D0-0.299D0*U
DL-22.6D0-0.378D0*U+0.00293D0*U*U
ATU-2.D0*3.142D0*DSQRT(DC*(DSQRT(1.D0+
*
(DL/DC)**2)-1.D0)/2.D0)/12.245D0
C— 2.D0*ATU
EPX-DEXP(C*(XI-X))+DEXP(C*(Xl+X))
EPY-DEXP(C*(X2-Y))+DEXP(C*(X2+Y))
Pl-2.DO*ATU*PS*(EPX+EPY)
P2-2.DO*ATU*PS*(DEXP(C*2.D0*X1)*EPX+
*
DEXP(C*2.D0*X2)*EPY)
PWR-P1+P2
TL^3.D0
IF (T.LT.TL) PHEE-0.DO
IF (T.GE.TL) PHEE-PWR
RETURN
END
DOUBLE PRECISION FUNCTION TT(U)
IMPLICIT REAL*8 (A-H.O-Z)
TT-(U+273.D0)**1.5D0
IF(U.GE.105.DO)TT-7 349.16D0
RETURN
END
DOUBLE PRECISION FUNCTION VP(U)
IMPLICIT REAL*8 (A-H.O-Z)
VP-1.1779D0*DEXP(.04498D0*U)
IF(U.GE.105,DO)VP-273.13D0
RETURN
END
DOUBLE PRECISION FUNCTION H(U)
IMPLICIT REAL*8 (A-H.O-Z)
H-2501.86DO-2.39D0*U
IF(U.GE.100.D0)H-2250.91D0
RETURN
END
168
A-2
Sample Program Listings for Cylindrical Geometry Used In TWODEPEP
//V04XXXXX JOB
// EXEC TWODEPEP
//TWODEPEP.INPUT DD *
1 20 1
OXX
8.37D-3*UX
OXY
8.37D-3*UY
1.01D0*4.12D0
Cl
FI
8.37D-3*UX/X+FLD(X,Y)*PHEE(U,T ,X ,Y)
UO
25. DO
ARC-1
-2.7D-4*(U-28.D0)/2.5D0
GB1
ARC-2
-3.944D-3*(U-28.D0)
GB1
ARC-3
-3.944D-3*(U-28.D0)-(1.251D-9)*TT(U)*VP(U)*H(U)
GB1
NX
4
NY
2
NOUT
1
MWR
6
NUPDT
2
ALPHA
0.5D0
DT
5.DO
TF
35. DO
D3EST
1.D0
0.D0.-2.5D0, 4.D0.-2.5D0, 4.DO,0.DO, 4.D0,2.5D0,
VXY
0.D0.2.5D0, 0.DO,0.DO, 2.0D0,-1.25D0, 2.0D0.1.25D0
VXY
IABC
1,2,7, 2,3,7, 3,4,8, 4,5,8, 5,6,8, 6,1,7, 3,6,7,
IABC
6,3,8
I
1,2,2,3,4,4,0,0
ADD.
DOUBLE PRECISION FUNCTION FLD(X.Y)
IMPLICIT REAL*8 (A-H.O-Z)
SUM-0.DO
COUNT-O.DO
R-4.D0
ZL-5.OD0
ZH-ZL/2.DO
M-R+l
DO 10 1-1, 12
F-(I-1.)/6.D0
A-F*3.14159D0
DO 20 J-l, 3
Z-(J-2.D0)*ZH+5.DO
DO 30 K-l, M
RX-K-1.D0
Xl-RX*DCOS(A)+9.DO
X2-RX*DSIN(A)+9.D0
169
Q-18.033D0+X2*(-2.953D0)+X2*X1*0.167D0+X2*Z*0.952D0+
X1*X1*Z*(-4.358D-2)+DC0S<Xl)*(-4.784D-1)+
DCOS(XI)*DSIN(X2)*1.223D-1+DSIN(X2)*DCOS(Z)*2.485D-1+
DSIN(Z)*DCOS(X2)*0.1658D0+(X2**4)*1.128D-3+
(X2**3)*Z*(-2.673D-3)+X2*(Z**3)*(-3.562D-2)+
X2*X2*X1*X1*1.69D-3+Xl*Xl*Z*Z*3.324D-3+
(Xl**5)*(-3.502D-05)+(Xl**4)*Z*4.359D-4+
(X2**3)*Z*Z*(-1.18D-3)+X2*X2*(Z**3)*4.206D-3+
X2*X2*X1*X1*Z*(-2.241D-3)+(X2**5)*Xl*(-2.479D-06)+
XI*(Z**5)*6.545D-4+(Xl**4)*Z*Z*(-4.89D-05)+
X2*X1*X1*(Z**3)*(-4.617D-4)+X2*X2*Xl*Xl*Z*Z*8.554D-4+
(X2**7)*(-1.952D-07) KX2**6)*Z*7.854D-07+
X1*(Z**6)*(-7.971D-0a)+X2*(Z**6)*9.144D-06+
(X2**5)*Xl*Xl*(-3.79D-06)+X2*X2*(Z**5)*(-1.601D-05)+
X2*(X1**4)*Z*Z*3.224D-06+(X2**4)*Xl*Xl*Z*3.296D-05+
(X2**3)*Xl*(Z**3)*(4.464D-06)(1.05D-4)*(X2**3)*X1*X1*Z*Z+
X2*X2*X1*X1*(Z**3)*2.326D-05+(Z**8)*l.025D-06+
X2*(XI**7)*8.792D-09+(X2**6)*X1*X1*2.18 9D-07+
X2*(X1**6)*Z*(-7.285D-08)+X2*Xl*(Z**6)*3.123D-06+
(Xl**3)*(Z**5)*(6.416D-07)+ (X2**5)*X1*X1*Z*
(-1.543D-06)+X2*X2*X1*(Z**5)*(-1.843D-06)+
(X2**4)*X1*X1*Z*Z*(3.2D-06)
SUM-SUM+Q
COUNT—COUNT+1.DO
CONTINUE
CONTINUE
CONTINUE
QM - SUM/COUNT
X1-X*DCOS(3.14159DO*1,ODO) + 9.DO
X2-X*DSIN(3.14159D0*1.0D0) + 9.DO
Z-Y+5.DO
Q-18.033D0+X2*(-2.953D0)+X2*Xl*0.167D0+X2*Z*0.952DO+
* X1*X1*Z*(-4.358D-2)+DC0S(Xl)*(-4.784D-1)+
* DC0S(Xl)*DSIN(X2)*l.223D-l+DSIN(X2)*DCOS(Z)*2.485D-1+
* DSIN(Z)*DCOS(X2)*0.1658D0+(X2**4)*1.128D-3+
* (X2**3)*Z*(-2.673D-3)+X2*(Z**3)*(-3.562D-2)+
* X2*X2*X1*X1*1.69D-3+Xl*Xl*Z*Z*3.324D-3+
* (Xl**5)*(-3.502D-05)+(Xl**4)*Z*4.359D-4+
* (X2**3)*Z*Z*(-1.18D-3)+X2*X2*(Z**3)*4.206D-3+
* X2*X2*X1*X1*Z*(-2.241D-3)+(X2**5)*Xl*(-2.4790-06)+
* X1*(Z**5)*6.545D-4+(Xl**4)*Z*Z*(-4.89D-05)+
* X2*X1*X1*(Z**3)*(-4.617D-4)+X2*X2*Xl*Xl*Z*Z*8.554D-4+
* (X2**7)*(-l.952D-07)+(X2**6)*Z*7.854D-07+
* X1*(Z**6)*(-7.971D-05)+X2*(Z**6)*9.144D-06+
* (X2**5)*X1*X1*(-3.79D-06)+X2*X2*(Z**5)*(-1.601D-05)+
* X2*(Xl**4)*Z*Z*3.224D-06+(X2**4)*X1*X1*Z*3.296D-05+
* (X2**3)*Xl*(Z**3)*(4.464D-06)* (1.05D-4)*(X2**3)*X1*X1*Z*Z+
* X2*X2*X1*X1*(Z**3)*2.326D-05+(Z**8)*1.025D-06+
* X2*(Xl**7)*8.792D-09+(X2**6)*Xl*Xl*2.189D-07+
* X2*(X1**6)*Z*(-7.285D-08)+X2*Xl*(Z**6)*3.123D-06+
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
30
20
10
* (Xl**3)*(Z**5)*(6.416D-07)+(X2**5)*Xl*Xl*Z*
* (-1.543D-06)+X2*X2*Xl*(Z**5)*(-1.843D-06)+
* (X2**4)*Xl*Xl*Z*Z*(3.2D-06)
FLD - Q/QM
RETURN
END
DOUBLE PRECISION FUNCTION PHEE(U,T,X,Y)
IMPLICIT REAL*8 (A-H.O-Z)
R-4.D0
D-2.D0*R
Z-5.0D0
ZH-Z/2.DO
WT-(R**2)*3.14L6D0*Z
P0-40.1DO+59.8DO*DLOG(WT)
PS-PO/(6.2832D0*(R*Z+(R**2)))
DC-81.7935D0-0.299D0*U
DL-22.6DO-0.378D0*U+0.00293D0*U*U
ATU-2.DO*3.142D0*DSQRT(DC*(DSQRT(1.D0+
*
(DL**2/DC**2))-1.D0)/2.D0)/12.245DO
C— 2.D0*ATU
EPX-DEXP(C*(R-X))+DEXP(C*(R+X))
EPY-DEXP(C*(ZH-Y))+DEXP(C*(ZH+Y))
Pl-2.D0*ATU*PS*(R*EPX/X+EPY)
P2-2.DO*ATU*PS*(DEXP(C*D)*R*EPX/X+DEXP(C*Z)*EPY)
PWR-P1+P2
TI^3.DO
IF (T.LT.TL) PHEE-0.DO
IF (T.GE.TL) PHEE-PWR
RETURN
END
DOUBLE PRECISION FUNCTION TT(U)
IMPLICIT REAL*8 (A-H.O-Z)
TT-(U+273.D0)**1.5D0
IF(U.GE.105.D0)TT-7349.16D0
RETURN
END
DOUBLE PRECISION FUNCTION VP(U)
IMPLICIT REAL*8 (A-H.O-Z)
VP-1.1779D0*DEXP(.04498D0*U)
IF(U.GT.105.DO)VP-132.5067D0
RETURN
END
DOUBLE PRECISION FUNCTION H(U)
IMPLICIT REAL*8 (A-H.O-Z)
H-2501.86DO-2.39D0*U
IF(U.GE.105.D0)H-2250.91D0
RETURN
END
171
A-3
NEQ
NTF
ND1H
OXX
OXY
U
Cl
FI
UO
ARC
GB1
NX
NY
NOUT
MWR
NUPDT
ALPHA
DT
TF
D3EST
VXY
IABC
I
ADD
FLD
PHEE
A,B,C
R
ZL
PO
PS
DC.DL
ATU
Pi
P2
PWR
TL
TT
VP
H
Specifications for the Inputs in the TWODEPEP Programs
Number of differential equation(s) used
Number of final triangles desired
Stiffness matrix storage reservation
3T/3x
3T/3y
Temperature
pCp
Force functions including heat source terms
Initial temperature
Boundary site
Free boundary condition as a heat flux
Number of output points in xdirection
Number of output points in ydirection
Number of output with time
Output logical unit number
Number of iterations (time steps)
Numerical method used to solve the equation
Time step
Final time
Method of discretization of elements
Coordinates of the initial triangle vertexes
Numbering the nodes on each element
Identification of the arc associated with
each element
Subfunctions defined after this line
Polynomial power distribution function
Microwave power absorption function
Width, length and height of a rectangle
Radius of a cylinder
Height of a cylinder
Total microwave power absorbed by the sample
Surface microwave power intensity
Dielectric constant and loss
Attenuation factor
Primary microwave power absorbed
Secondary microwave power absorbed
Microwave power absorbed per unit volume
Magnetron come-up-time
Mass transer coefficient temperature correction
function
Water vapor pressure as a function of temperature
Latent heat of vaporization as a function of
temperature
172
Appendix B
THERMOCOUPLE TEMPERATURE MEASUREMENT
USING 2IX MICROLOGGER AND AM32 MULTIPLEXER
B-l
Circuitry of 21X Micrologger with AM32 Multiplexer
2IX
AM32
Analog In 1
Com n
Ground
Coin L
Digital Port 1
Switch Ex. 1
Reserve
Clock
+ 12 V
+ 12 V
Ground
Ground
173
B-2
Program Instructions for Differential Thermocouple Measurement
Using 2IX with AM32
*A:
Memory allocation (Default-28)
Inst.#
Para.#
1
2
3
4
5
6
7
8
9
10
11
P
1
P
1
2
P
1
2
P
2
3
4
P
1
2
3
4
5
6
7
8
P
P
1
P
1
2
3
4
5
6
7
8
P
1
P
1
P
1
2
Entry
t
17
1
20
1
1
87
0
n
22
1
0
5000
14
1
1
1
1
1
2C
1
0
95
20
0
14
m
1
2
1
1
n+2
1
0
86
10
77
11
70
n+m+1
1
Description
Execution interval
Measure panel temperature
Store ref. temp, in Loc.#l
Enable AM32 to set port
Set port high (5V)
Port #1 (Analog I/O #1)
Loop
Delay
Count (n £ 32)
Excite channel #1
Delay excitation in 0.01s
0 s delay after excitation
Excitation range mV
Measure differential TC at AM32
Repetition
Range code ± 5 mV
1st measurement @ channel #1
TC type (3 for type K)
Reference temperature loc.
Indexed destinated 1st input loc.
Multiplier
Offset
End
Set port (Reset AM32)
Set port low (0V)
Differential TC measurement at 21X
Repetition
Range code
Measure 1st TC channel
TC type
Reference temp. loc.
Destinate input loc.
Multiplier
Offset
Output along data collection
Set flag 0 to 1 (Output)
Time instruction
Hr:min:sec
Sample
Repetitions
Loc. of 1st input data
174
Appendix C
SUMMARY OF THE STATISTICAL ANALYSIS ON THE RESULTS FROM
EXPERIMENTAL MEASUREMENTS AND MODEL PREDICTIONS
C-l
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations in 2 cm Radius Cylindrical Gels and Compared
at 0.01 Significance Level
r[cm]
z[cm]
Experimental
RTR
STD
Predicted
RTR
STD
^calc
0
1
2
1.25
1.25
1.25
1.651
1.177
1.317
0.311
0.211
0.257
3.087
1.095
1.615
0.004
0.010
0.016
9.233 *
0.773
2.313
0
1
2
0
0
0
3.488
2.844
1.203
0.307
0.421
0.258
3.187
1.030
1.578
0.008
0.016
0.022
1.957
8.606 *
2.899
0
1
2
-1.25
-1.25
-1.25
2.511
1.577
2.049
0.335
0.117
0.336
3.307
1.199
1.800
0.013
0.028
0.039
4.749
6.265 *
1.478
0
1
2
2.5
2.5
2.5
1.158
0.971
1.224
0.128
0.287
0.202
2.947
0.989
0.810
0.003
0.005
0.008
27.947 *
0.124
4.092
0
1
2
0
0
0
2.390
1.192
0.749
0.352
0.229
0.076
3.079
0.850
0.615
0.007
0.009
0.013
3.915
2.971
3.453
0
1
2
-2.5
-2.5
-2.5
1.255
0.817
1.140
0.177
0.024
0.167
2.645
0.519
1.670
0.010
0.021
0.033
15.678 *
18.891 *
6.233 *
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
3> ^0.01,3 ” 5 -8^l
175
C-2
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations in 3 cm Radius Cylindrical Gels and Compared
at 0.01 Significance Level
r[cm]
z[cm]
Experimental
RTR
STD
0
1
2
3
1.25
1.25
1.25
1.25
0.806
0.868
0.735
0.785
0.124
0.087
0.163
0.222
1.494
0.638
0.830
0.826
0.003
0.006
0.010
0.017
11.090 *
5.241
1.169
0.372
0
1
2
3
0
0
0
0
1.752
1.211
1.271
0.803
0.134
0.135
0.281
0.273
2.110
0.411
0.717
0.792
0.006
0.009
0.015
0.018
5.341
11.795 *
3.939
0.073
0
1
2
3
-1.25
-1.25
-1.25
-1.25
1.214
0.941
0.982
1.290
0.062
0.039
0.174
0.306
1.516
0.762
0.865
0.863
0.009
0.017
0.025
0.021
9.635 *
8.332 *
1.333
2.785
0
1
2
3
2.5
2.5
2.5
2.5
0.441
0.504
0.733
0.675
0.091
0.044
0.312
0.196
1.004
0.481
0.488
0.519
0.002
0.003
0.004
0.006
12.365 *
1.049
1.568
1.583
0
1
2
3
0
0
0
0
0.841
0.620
0.420
0.361
0.119
0.076
0.101
0.102
0.900
0.366
0.381
0.396
0.004
0.005
0.008
0.010
0.995
6.689 *
0.783
0.692
0
1
2
3
-2.5
-2.5
-2.5
-2.5
0.436
0.351
0.542
0.763
0.037
0.020
0.123
0.182
1.034
0.548
0.557
0.569
0.006
0.012
0.021
0.026
31.864 *
16.674 *
0.252
2.114
Predicted
RTR
STD
**calc
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
3 - ^0.01,3 “ 5 841
176
C-3
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations in 4 cm Radius Cylindrical Gels and Compared
at 0.01 Significance Level
r[cm]
z[cm)
Experimental
RTR
STD
0
1
2
3
4
1.25
1.25
1.25
1.25
1.25
0.575
0.474
0.501
0.491
0.552
0.047
0.020
0.122
0.145
0.147
0.703
0.421
0.408
0.499
0.592
0.002
0.004
0.007
0.012
0.021
5.429
5.248 *
1.513
0.111
0.533
0
1
2
3
4
0
0
0
0
0
0.716
0.549
0.555
0.832
0.571
0.180
0.051
0.097
0.255
0.236
0.906
0.431
0.453
0.490
0.574
0.003
0.005
0.009
0.012
0.016
2.115
4.613
2.097
2.687
0.019
0
1
2
3
4
-1.25
-1.25
-1.25
-1.25
-1.25
0.604
0.605
0.564
0.668
0.897
0.073
0.058
0.095
0.163
0.426
0.729
0.521
0.490
0.598
0.635
0.005
0.011
0.015
0.014
0.008
3.403
2.873
1.530
0.851
1.228
0
1
2
3
4
2.5
2.5
2.5
2.5
2.5
0.413
0.361
0.299
0.302
0.423
0.078
0.023
0.036
0.089
0.171
0.449
0.289
0.296
0.333
0.408
0.001
0.002
0.003
0.005
0.011
0.911
6.106 *
0.164
0.701
0.181
0
1
2
3
4
0
0
0
0
0
0.371
0.273
0.261
0.267
0.209
0.036
0.025
0.051
0.060
0.036
0.329
0.193
0.191
0.254
0.324
0.002
0.003
0.004
0.006
0.010
2.359
6.502 *
2.742
0.412
6.193 *
0
1
2
3
4
-2.5
-2.5
-2.5
-2.5
-2.5
0.285
0.257
0.284
0.494
0.731
0.010
0.019
0.058
0.236
0.349
0.481
0.345
0.345
0.389
0.445
0.003
0.008
0.014
0.016
0.017
37.109 *
8.730 *
2.082
0.886
1.632
Predicted
RTR
STD
^alc
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
^.01,3 “
177
C-4
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations in 6 cm Radius Cylindrical Gels and Compared
at 0.01 Significance Level
r [cm]
z[cm]
Experimental
RTR
STD
0
1
2
3
4
5
6
1.25
1.25
1.25
1.25
1.25
1.25
1.25
0.204
0.203
0.202
0.201
0.269
0.396
0.409
0.014
0.009
0.017
0.026
0.031
0.026
0.062
0.249
0.218
0.208
0.219
0.251
0.302
0.364
0.002
0.002
0.004
0.007
0.011
0.013
0.011
6.449 *
3.352
0.655
1.338
1.087
6.423 *
1.441
0
1
2
3
4
5
6
0
0
0
0
0
0
0
0.203
0.217
0.230
0.244
0.354
0.510
0.414
0.006
0.011
0.023
0.034
0.048
0.079
0.071
0.276
0,270
0.191
0.247
0.300
0.337
0.386
0.001
0.003
0.004
0.006
0.010
0.015
0.016
24.087 *
8.971 *
3.379
0.183
2.183
4.312
0.787
0
1
2
3
4
5
6
-1.25
-1.25
-1.25
-1.25
-1.25
-1.25
-1.25
0.189
0.202
0.214
0.227
0.276
0.453
0.540
0.012
0.010
0.021
0.031
0.019
0.139
0.197
0.284
0.286
0.291
0.300
0.313
0.388
0.410
0.003
0.006
0.007
0.007
0.005
0.012
0.020
15.343 *
14.066 *
6.968 *
4.590
3.764
0.935
1.310
Predicted
RTR
STD
^calc
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
178
C-4
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations in 6 cm Radius Cylindrical Gels and Compared
at 0.01 Significance Level (continued)
r[cm]
z [cm]
0
1
2
3
4
5
6
2.5
2.5
2.5
2.5
2.5
2.5
2.5
0.206
0.193
0.180
0.167
0.147
0.230
0.274
0.042
0.020
0.039
0.059
0.033
0.092
0.120
0.206
0.126
0.131
0.155
0.155
0.195
0.251
0.001
0.001
0.001
0.003
0.005
0.007
0.008
0.003
6.867 *
2.492
0.390
0.497
0.755
0.372
0
1
2
3
4
5
6
0
0
0
0
0
0
0
0.114
0.116
0.119
0.121
0.165
0.192
0.206
0.009
0.011
0.023
0.034
0.047
0.047
0.045
0.093
0.067
0.067
0.070
0.096
0.154
0.198
0.000
0.001
0.001
0.002
0.004
0.007
0.010
4.615
8.502 *
4.477
2.966
2.904
1.586
0.344
0
1
2
3
4
5
6
-2.5
-2.5
-2.5
-2.5
-2.5
-2.5
-2.5
0.119
0.126
0.133
0.140
0.182
0.370
0.470
0.018
0.010
0.020
0.031
0.071
0.218
0.312
0.218
0.170
0.172
0.181
0.198
0.245
0.282
0.002
0.004
0.006
0.008
0.007
0.008
0.011
11.025 *
8.019 *
3.664
2.597
0.464
1.141
1.202
Experimental
RTR
STD
Predicted
RTR
STD
Cealc
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
" 5 *841
179
C-5
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical
Gels with 1% Total Salt and Compared at 0.01 Significance Level
r [cm]
z[cm]
Experimental
RTR
STD
Predicted
RTR
STD
0
1
2
3
4
1.25
1.25
1.25
1.25
1.25
0.331
0.392
0.432
0.554
0.785
0.047
0.054
0.090
0.165
0.245
0.602
0.608
0.622
0.788
1.121
0.001
0.005
0.009
0.020
0.042
11.635 *
7.915 *
4.217
2.826
2.704
0
1
2
3
4
0
0
0
0
0
0.399
0.478
0.523
0.831
0.940
0.054
0.072
0.119
0.307
0.379
0.220
0.185
0.280
0.355
0.727
0.000
0.002
0.005
0.008
0.023
6.588 *
8.173 *
4.096
3.093
1.120
0
1
2
3
4
-1.25
-1.25
-1.25
-1.25
-1.25
0.598
0.728
0.720
0.991
1.164
0.087
0.123
0.155
0.344
0.498
0.581
0.578
0.597
0.759
1.085
0.003
0.012
0.017
0.018
0.019
0.394
2.410
1.574
1.344
0.317
*"calc
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
180
C-6
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical
Gels with Axial Shielding and Compared at 0.01 Significance Level
Predicted
RTR
STD
r[cm]
z(cm]
Experimental
RTR
STD
0
1
2
3
4
1.25
1.25
1.25
1.25
1.25
0.247
0.174
0.177
0.283
0.509
0.036
0.037
0.019
0.069
0.075
0.358
0.113
0.106
0.218
0.322
0.001
0.001
0.002
0.005
0.011
6.182 *
3.361
7.573 *
1.879
4.957
0
1
2
3
4
0
0
0
0
0
0.426
0.286
0.291
0.469
0.613
0.009
0.019
0.041
0.104
0.112
0.516
0.141
0.130
0.197
0.304
0.002
0.002
0.003
0.005
0.008
19.542 *
15.498 *
7.834 *
5.247
5.497
0
1
2
3
4
-1.25
-1.25
-1.25
-1.25
-1.25
0.286
0.186
0.179
0.259
0.487
0.022
0.010
0.030
0.043
0.114
0.334
0.106
0.103
0.210
0.312
0.003
0.003
0.004
0.005
0.004
4.312
15.214 *
4.984
2.263
3.074
*calc
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
3 - *0.01,3 ” 5 *8^1
181
C-7
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations In 4 cm Radius by 2.5 cm Height Cylindrical
Gels with Shielding on the Top Surface and Compared at 0.01
Significance Level
r[cmj
z[cm]
Experimental
RTR
STD
0
1
2
3
4
1.25
1.25
1.25
1.25
1.25
0.312
0.298
0.355
0.432
0.344
0.069
0.062
0.058
0.059
0.065
0.414
0.174
0.164
0.276
0.368
0.001
0.002
0.003
0.007
0.013
2.947
3.987
6.587 *
5.209
0.724
0
1
2
3
4
0
0
0
0
0
0.544
0.534
0.593
0.746
0.508
0.063
0.077
0.058
0.115
0.110
0.699
0.286
0.281
0.337
0.431
0.002
0.004
0.006
0.008
0.012
4.910
6.419 *
10.741 *
7.109 *
1.397
0
1
2
3
4
-1.25
-1.25
-1.25
-1.25
-1.25
0.621
0.704
0.733
0.738
0.843
0.103
0.079
0.103
0.126
0.272
0.718
0.446
0.433
0.538
0.609
0.005
0.009
0.013
0.012
0.008
1.884
6.490 *
5.801
3.154
1.720
Predicted
RTR
STD
^calc
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
3 ‘ * 0 .01,3 “
5 * 8 41
182
C-8
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical
Gels with Shielding on the Bottom Surface and Compared at 0.01
Significance Level
r[cm]
z[cm]
Experimental
RTR
STD
0
1
2
3
4
1.25
1.25
1.25
1.25
1.25
0.450
0.411
0.383
0.424
0.682
0.015
0.055
0.040
0.075
0.168
0.762
0.469
0.451
0.558
0.629
0.002
0.004
0.007
0.013
0.021
41.410 *
2.130
3.366
3.514
0.620
0
1
2
3
4
0
0
0
0
0
0.567
0.435
0.446
0.610
0.671
0.045
0.034
0.052
0.121
0.162
0.700
0.286
0.282
0.337
0.431
0.002
0.004
0.006
0.008
0.012
5.900 *
8.765 *
6.277 *
4.505
2.962
0
1
2
3
4
-1.25
-1.25
-1.25
-1.25
-1.25
0.376
0.320
0.295
0.362
0.484
0.039
0.022
0.028
0.050
0.139
0.385
0.166
0.158
0.266
0.356
0.003
0.004
0.005
0.006
0.004
0.456
13.552 *
9.762 *
3.805
1.848
Predicted
RTR
STD
^calc
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
183
C-9
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical
Gels with Radial Shielding and Compared at 0.01 Significance Level
r [cm]
z[cm]
Experimental
RTR
STD
0
1
2
3
4
1.25
1.25
1.25
1.25
1.25
0.393
0.485
0.597
0.571
0.372
0.061
0.076
0.085
0.136
0.017
0.411
0.410
0.404
0.406
0.385
0.001
0.003
0.006
0.009
0.013
0
1
2
3
4
0
0
0
0
0
0.573
0.687
0.733
0.663
0.302
0.011
0.113
0.155
0.166
0.021
0.293
0.278
0.299
0.283
0.279
0.001
0.004
0.006
0.007
0.008
50.721 *
7.267 *
5.607
4.581
2.089
0
1
2
3
4
-1.25
-1.25
-1.25
-1.25
-1.25
0.774
0.975
0.938
0.798
0.383
0.027
0.178
0.223
0.244
0.066
0.393
0.391
0.387
0.391
0.373
0.002
0.008
0.011
0.009
0.005
28.166 *
6.571 *
4.943
3.326
0.294
Predicted
RTR
STD
Ccalc
0.603
1.987
4.545
2.418
1.148
1. Number of replications is 4
2. Calculated t values are significant at 0,01 level (*)
184
C-10
Means and Standard Deviations of Measured and Predicted RTR at
Different Locations In Rectangular Gels of Various Sizes Under
Two-Dimensional Microwave Heating and Compared at 0.01
Significance Level
x [cm]
y [cm]
Experimental
RTR
STD
Predicted
RTR
STD
^alc
4 cm x 4 cm x 5 cm
0
1
2
1.114
0.871
1.264
0.216
0.100
0.185
0.616
0.824
1.191
0.001
0.013
0.024
4.613
0.931
0.771
0.283
0,355
0.499
0.612
0.023
0.035
0.079
0.129
0.195
0.268
0.524
0.805
0.004
0.005
0.012
0.020
5.785
4.962
0.614
2.956
0.098
0.125
0.213
0.362
0.506
0.016
0.012
0.030
0.042
0.040
0.066
0.092
0.188
0.402
0.634
0.010
0.002
0.004
0.012
0.020
3.407
5.318
1.607
1.858
5.730
0
1
2
6 cm x 6 cm x 5 cm
0
1
2
3
0
1
2
3
8 cm x 8 cm x 5 cm
0
1
2
3
4
0
1
2
3
4
1. Number of replications is 4
2. Calculated t values are significant at 0.01 level (*)
3 * ^ . 01,3 “ 5 -841
VITA
Yah-Hwa was born on January 26, 1958, in Taipei, Taiwan, the
Republic of China.
In June, 1976, she graduated from Taipei First
Girls' High and enrolled in the Department of Food Science at National
Chung-Hsing University in Taichung, Taiwan.
In 1980, Yah-Hwa obtained
her B.S. Degree in Food Science and worked in Wei-Chuan Food Company and
National Taiwan University untill she came to the United State in 1981.
She started her Master's program with the Department of Food Science and
Technology at Texas A & M University, and then completed an M.S. Degree
in the Department of Agricultural Engineering specializing in Food
Engineering.
She received her Master’s Degree in 1985 and entered a
Doctoral program in the Department of Food Science and Technology at
University of Nebraska.
She has been actively working in the area of microwave heating of
foods and other food engineering related research projects and was
awarded the Widman Distinguished Graduate Assistant Award in 1987.
In
1988, she transferred to the Department of Food Science at Penn State
and continued the research on microwave heating.
During her graduate
study, she published three papers and five abstracts.
She is currently
a member of the Institute of Food Technologists, the American
Association of Cereal Chemists, Sigma XI (the Scientific Research
Society), Phi Tau Sigma (the Honor Society for Food Science), and Gamma
Sigma Delta (the Honor Society of Agriculture).
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