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Analysis of temperature redistribution in model food during pulsed microwave heating

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ANALYSIS OF TEMPERATURE REDISTRIBUTION IN MODEL FOOD
DURING PULSED MICROWAVE HEATING
by
Huai-Wen Yang
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
(Agricultural Engineering)
at the
UNIVERSITY OF WISCONSIN-MADISON
2002
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C o m m ittee’s Page. This page is not to be hand-written except for
A dissertation entitled
ANALYSIS OF TEMPERATURE REDISTRIBUTION IN MODEL
FOOD DURING PULSED MICROWAVE HEATING
submitted to the Graduate School of the
University of Wisconsin-Madison
in partial fulfillment of the requirements for the
degree of Doctor of Philosophy
by
HUAI WEN YANG
Date of Final Oral Examination:
C om m ittee’s Page. This page is not to be hand-written except for the signatures
Month
April 18, 2002
&Year Degree to be awarded: December
May
August 2002
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Approval Signatures of Dissertation Committee
U).
Signature, Dean of Graduate School
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ANALYSIS OF TEMPERATURE REDISTRIBUTION IN MODEL FOOD
DURING PULSED MICROWAVE HEATING
Huai-Wen Yang
Under the supervision of Professor Sundaram Gunasekaran
At the University of Wisconsin-Madison
ABSTRACT
Two-percent agar gel was treated as a model food to study the temperature
distribution (TD) during continuous and pulsed microwave heating. Microwave power,
derived from Maxwell’s equations and/or the Lambert’s law, was applied to a heat
transfer model which was solved numerically to predict the TD within the 3.5-cm and 4cm radius agar gel samples. The sample temperatures were measured and compared to
the numerical predictions. The TDs predicted based on the Maxwell’s equations were
more accurate than those predicted based on the Lambert’s law, especially around the
sample edges. This is because, unlike the Lambert’s law, the power solution based on
Maxwell’s equations accounts for the standing wave effect. The predicted and measured
TDs were more uniform under pulsed than continuous microwave heating.
When microwave energy is applied to a food material, the TD within the material
depends on the heating duration, pulsing ratio (ratio of total microwave heating time to
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11
power-on time under pulsed mode), sample size, and power level. These effects
were examined using a 3 x 2 x 2 factorial experimental design. The pulsed and
continuous microwave applications were maintained at the same average power based on
the oven settings. Analysis of variance indicated that all variables affect the sample TD
significantly. The interactions among the experimental parameters were also determined.
The results show that pulsed microwave heating is preferred when temperature
uniformity is a major concern.
Pulsed microwave heating was applied to mashed potato cylinders to determine
an optimal set of operating parameters. The effects of sample radius, power level,
processing duration and temperature constraints during the microwave application were
evaluated. The sample radius and temperature constraints were critical. Depending on the
dielectric properties of the sample, the pulsed microwave heating is best suited only over
a certain range of sample sizes (about 1-2 times of penetration depth). The simulation
model presented in this thesis is suitable for evaluating optimal pulsed microwave heating
of other solid foods.
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Ill
ACKNOWLEDGEMENTS
I would like to sincerely thank my advisor, Professor Sundaram Gunasekaran, for
his guidance during the course of this research. His encouragement and constructive
criticism made the completion of this study possible. I am also grateful for Professors
Richard W. Hartel, John W. Mitchell, Ronald J. Vernon, and Robert J. Witt for their
suggestions and serving on my thesis committee. For the financial support, I am indebted
to Department of Biological Systems Engineering. Finally, this thesis is dedicated to two
persons who are thousands of miles away from the United States, my parents, Mr. and
Mrs. H. Y. Yang for their perpetual support over the years.
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TABLE OF CONTENTS
ABSTRACT........................................................................................................................ i
ACKNOWLEDGEMENTS
....................................................................................iii
LIST OF TABLES............................................................................................................ x
LIST OF FIGURES........................................................................................................xii
CHAPTER 1 INTRODUCTION......................................................................................1
1.1 Development of Microwave O ven................................................................................ I
1.2 Pulsed Microwave Application......................................................................................1
1.3 Objectives........................................................................................................................ 3
1.4 Limitations......................................................................................................................5
1.5 References.......................................................................................................................6
CHAPTER 2 LITERATURE REVIEW....................................................................... 11
2.1 Microwave Frequencies............................................................................................... 11
2.2 Microwave Food Process............................................................................................. 11
2.3 Characteristics of Microwave Heating........................................................................13
2.3.1 Methods o f Heating Food......................................................................................13
2.3.2 Theory o f Microwave Heating.............................................................................. 14
2.3.2.1 Dielectric Properties........................................................................14
23.2.2 Term Derived from Dielectric Properties...................................... 17
2.3.2.3 Basic Mechanisms of Microwave Heating.................................... 18
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2.4 Microwave Power Formulations.................................................................................20
2.4.1 Lambert’s Law....................................................................................................... 20
2.4.2 Maxwell’s Field Equations................................................................................... 22
2.5 Temperature Modeling.................................................................................................25
2.5.1 Finite-difference Formulations............................................................................25
2.5.1.1 Explicit Formulation.........................................................................27
2.5.1.2 Implicit Formulation.........................................................................28
2.5.2 Temperature Rise Analysis...................................................................................28
2.6 Pulsed Microwave Application...................................................................................29
2.7 References.....................................................................................................................31
CHAPTER 3 GENERAL METHODOLOGY............................................................. 35
3.1 Laboratory Microwave O ven...................................................................................... 35
3.2 Preparation of 2% Agar Gel Sample Cylinders.........................................................35
3.3 Temperature Measurement.......................................................................................... 36
3.4 Average Absorbed Microwave Power........................................................................37
3.5 Average Surface Heat Transfer Coefficient............................................................... 38
3.6 Azimuthal Wave Assumption..................................................................................... 39
3.7 One-dimensional Heat Transfer Assumption............................................................. 40
3.8 References.....................................................................................................................42
CHAPTER 4 TEMPERATURE PROFILES IN A CYLINDRICAL MODEL FOOD
DURING PULSED MICROWAVE HEATING.......................................................... 56
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4.0 Notations.......................................................................................................................57
4.1 Abstract.........................................................................................................................60
4.2 Introduction...................................................................................................................61
4.3 Mathematical M odel....................................................................................................63
4.3.1 Implicit Finite-difference M odel..........................................................................63
4.3.2 Evaluating Microwave Absorbed Power.............................................................65
4.4 Methods and Materials.................................................................................................67
4.4.1 Microwave Heating Process................................................................................ 67
4.4.2 Data Analysis........................................................................................................ 68
4.5 Results and Discussion.................................................................................................70
4.5.1 Model Validation................................................................................................... 70
4.5.2 Temperature Profiles............................................................................................. 70
4.6 Conclusions...................................................................................................................72
4.7 References.....................................................................................................................73
CHAPTER 5 COMPARISON OF TEM PERATURE PROFILES IN A
CYLINDRICAL MODEL FOOD BASED ON MAXWELL’S AND LAM BERT’S
L A W ........................................................................................................................................ 87
5.0 Notations.......................................................................................................................88
5.1 Abstract.........................................................................................................................91
5.2 Introduction...................................................................................................................92
5.3 Theory and Analyses....................................................................................................93
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5.3.1 Heat Transfer Equation......................................................................................... 93
5.3.2 Wave Propagation..................................................................................................94
5.3.3 Power dissipation...................................................................................................97
5.3.4 Maxwell’s equations fo r a cylinder...................................................................... 98
5.3.5 Analytical Solution fo r Absorbed Power........................................................... 100
5.3.6 Temperature Distribution Prediction.................................................................101
5.4 Methods and Materials................................................................................................ 101
5.4.1 Microwave process...............................................................................................101
5.4.2 Temperature Prediction...................................................................................... 102
5.5 Results and Discussion................................................................................................ 103
5.6 Conclusions..................................................................................................................105
5.7 References....................................................................................................................114
CHAPTER 6 EFFECT OF EXPERIMENTAL PARAMETERS ON
TEMPERATURE DISTRIBUTION DURING CONTINUOUS AND PULSED
MICROWAVE HEATING........................................................................................... 116
6.0 Notations....................................................................................................................117
6.2 Introduction..................................................................................................................119
6.3 Methods and Materials................................................................................................120
6.3.1 Microwave Heating Processes...........................................................................120
6.3.2 Statistical Analysis...............................................................................................121
6.4 Results and Discussion................................................................................................122
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6.4.1 Microwave Absorbed Power.............................................................................. 122
6.4.2 Temperature Distribution (TD) in Samples Under the same average OSP
Condition....................................................................................................................... 123
6.4.3 General Linear Model Under.............................................................................124
6.4.4 Effects o f Different Pulsing Ratios on Sample Temperature Distribution Under
The Same Average Absorbed Condition......................................................................125
6.5 Conclusions................................................................................................................. 127
6.6 References................................................................................................................... 128
CHAPTER 7 OPTIMIZATION OF PULSED MICROWAVE HEATING.......... 137
7.0 Abstract....................................................................................................................... 138
7.1 Introduction................................................................................................................. 139
7.2 Methods and Materials............................................................................................... 140
7.2.1 Sample Preparation............................................................................................. 140
7.2.2 Physical and Dielectric Properties................................................................... 140
7.2.3 Microwave Heating............................................................................................. 141
7.2.4 Analysis o f Optimal Process.............................................................................. 141
7.3 Results and Discussion............................................................................................... 144
7.4 Conclusions................................................................................................................. 147
7.5 References................................................................................................................... 148
CHAPTER 8 SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK
......................................................................................................................................... 158
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8.1 Summary..................................................................................................................... 158
8.2 Recommendations for Future Work..........................................................................159
APPENDICES....................................
160
Appendix A Engineering Equation Solver (EES, F-Chart Software Co., Middleton,
WI) program code for calculating temperature profiles in continuous and pulsed
microwave heated 2% agar gel cylinders........................................................................161
APPENDIX B.l MATLAB (MathWorks Inc., Natick, MA) program code for the
calculating of temperature profile within 4-cm radius agar gel cylinders according to
the Maxwell’s equations.................................................................................................. 169
APPENDIX B.2 PREDICTED AND MEASURED TEMPERATURE
DISTRIBUTION OF 3.5- AND 4-cm RADIUS 2% AGAR GEL CYLINDERS AT
250-W OVEN SETTING..................................................................................................176
APPENDIX C.l General linear model with time and covariate interaction terms using
MiniTab (Minitab Inc., State Park, PA) analysis............................................................183
APPENDIX C.2 General linear model without time and covariate interaction terms
involved using MiniTab (Minitab Inc., State Park, PA) analysis (including the
Probability Plot of the Residuals and Residuals versus the Fitted Values).................. 186
APPENDIX D Determination o f the experimental error due to 30-s lag during
temperature measurement.................................................................................................191
APPENDIX E The evaporation cooling effect during microwave heating............... 196
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X
LIST OF TABLES
Table 1.1 Summary of experimental sample R/H (radius/height) ratios of food used for
temperature distribution (TD) simulation studies...........................................................8
Table 2.1 Microwave Food Processing Applications (from Decareau and Peterson,
1986).................................................................................................................................13
Table 2.2 Methods of Heat Processing of Foods (from Heldman and Singh, 1981)........ 14
Table 3.1 Absorbed power and energy-transfer efficiency in the microwave oven at 2.45
GHz..................................................................................................................................43
Table 3.2 Effect of sample placement (1 and 2 cm from the center of turntable in the
microwave oven) on the absorbed power at 2.45 GHz-statistical analysis................ 44
Table 3.3 Temperatures at different angular locations in a 3.5-cm agar cylinder..............45
Table 3.4 Effect of angular locations (0, 120 and 240 °) on the measured temperatures in
agar gel cylinders-statistical analysis............................................................................46
Table 3.5 Predicted and measured temperatures for 5 x 7 cm (radius x height) agar gel
cylinders (from Mudgett, 1986)..................................................................................... 47
Table 3.6 Measured temperatures for 3.5 x 7 cm (radius x height) agar gel cylinders in
the microwave oven at the 250-W oven setting........................................................... 48
Table 3.7 Measured temperatures for 4 x 7 cm (radius x height) agar gel cylinders in the
microwave oven at the 250-W oven setting................................................................. 48
Table 3.8 Paired /-test for temperatures at mid-plane (MP, depth=3.5 cm) compared to
other locations along z-axis........................................................................................... 49
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XI
Table 4.1 Explicit and implicit schemes for heat fluxes (qn and qi+i) for finitedifference analysis...........................................................................................................76
Table 4.2 Dielectric, physical, and thermal properties of 2% agar gela..............................76
Table 4.3 Parameter values of an infinite cylinder used for validation of the implicit finite
difference model..............................................................................................................77
Table 5.1 The Chi-square values for Maxwell’s and the Lambert’s predictions compared
to the measured temperatures during microwave heating..........................................106
Table 6.1 Analysis of Variance for the Factorial Experiment in a General Linear Model.
........................................................................................................................................ 136
Table 7.1 Thermal and dielectric properties of mashed potatoes with 82.7% moisture
content (M, %) and different temperatures (T, °C)..................................................... 149
Table 7.2 Power-on (PO) to total processing (TP) time ratios under the ATon (ATd) =20(3)
°C and ATon (ATd) =15(3) °C constraints at different final average sample
temperature (Tfas, °C).....................................................................................................150
Table 7.3 Microwave pulsing sequences applied to different radius samples and comply
the AT0„ (ATd)=20(3) °C criteria...................................................................................150
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LIST OF FIGURES
Figure 1.1 Temperature equalization in roast beef sample just after microwave
application (0 min) and after a 30-min rest time (30 min) (from Poliak and Foin,
I960).................................................................................................................................. 9
Figure 1.2 Two-dimensional microwave power absorbed in a 3-cm radius (r), 5-cm high
(z) alginate gel cylinder at 30 °C (from Lin et al. 1995).............................................. 10
Figure 2.1 The electromagnetic spectrum (from Brennan et al., 1969).............................. 12
Figure 3.1 Schematic of the microwave oven with arrows indicating the directions of
circulating air.................................................................................................................. 50
Figure 3.2 Pictures of the fans and waveguide in the microwave oven............................. 51
Figure 3.3 Type-T thermocouple, datalogger and cardboard temperature guide used for
temperature measurements............................................................................................ 52
Figure 3.4 Absorbed power at 250-W oven setting in the microwave oven by placing the
sample at 0, 1 and 2 cm from the center of the turntable.............................................53
Figure 3.5 Cylindrical coordinate system and corresponding unit vectors for the case of
electromagnetic radiation incident normal to the surface............................................54
Figure 3.6 Temperature measurement locations in 3.5 x 7 cm (radius x height) agar gel
cylinders for validating one-dimensional heat transfer assumption........................... 55
Figure 4.1 Schematic of a typical inner node of a cylindrical object for one-dimensional
analysis (Ri=0 along the sample axis)...........................................................................78
Figure 4.2 Power-on/-off periods employed during microwave applications for pulsing
ratios (PR) of 1, 2 and 3, with total microwave power on-time of 3 min. The solid
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xiii
segments represent power-on periods, dotted segments represent poweroff periods, and vertical dash segments represent the power on/off transition
79
Figure 4.3 Analytical and implicit finite-difference numerical temperature profiles in an
infinite cylinder (see Table 3.3 for parameter values used in analytical
approximation)................................................................................................................ 80
Figure 4.4 Comparison of temperature profiles according to implicit fmite-difference
model (FD) and that of the temperature-rise (TR) model by Padua (1993), shell
thickness increments are 0.1, 0.15, and 0.2 cm........................................................... 81
Figure 4.5 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in
radius and 7 cm in height) after 1 min of total microwave power application at
different pulsing ratios (PR). The power incident is from the sample outer periphery
(the radial distance of 3.5 cm from the centerline)......................................................82
Figure 4.6 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in
radius and 7 cm in height) after 2 min of total microwave power application at
different pulsing ratios (PR)...........................................................................................83
Figure 4.7 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in
radius and 7 cm in height) after 3 min of total microwave power application at
different pulsing ratios (PR)...........................................................................................84
Figure 4.8 Measured (M) and finite-difference model predicted (P) temperature profiles
at different pulsin j ratios (PR) at a radial distance of 0 cm (center line)..................85
Figure 4.9 Measured (M) and finite-difference model predicted (P) temperature profiles
at different pulsing ratios (PR) at a radial distance of 1 cm from the central line... 86
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XIV
Figure 5.1 A cylindrical sample exposed to plane waves normal to the surface
with sub-shell sections for numerical modeling..........................................................107
Figure 5.2 Microwave power absorbed in 2 % agar gel cylinders (3.5-cm and 4-cm
radius) as a function of radial distance from sample center. The electric field is
oriented along the vertical z-axis.o f the cylinder........................................................108
Figure 5.3 Microwave power density absorbed in 2% agar gel cylinders (3.5-cm and 4cm radius) as a function of radial distance from sample center. The electric field is
oriented along the vertical z-axis of the cylinder........................................................109
Figure 5.4 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP)
models compared to the measured (M) temperature profile in 4-cm radius 2 % agar
gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 1 (i.e.
continuous)..................................................................................................................... 110
Figure 5.5 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP)
models compared to the measured (M) temperature profile in 4-cm radius 2 % agar
gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 3.....111
Figure 5.6 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP)
models compared to the measured (M) temperature profile in 3.5-cm radius 2 % agar
gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 1 (i.e.
continuous power)........................................................................................................ 112
Figure 5.7 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP)
models compared to the measured (M) temperature profile in 3.5-cm radius 2 % agar
gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 3...... 113
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XV
Figure 6.1 Temperature distribution in 2 % agar gel cylinders, 4-cm radius,
after 1, 2 and 3 min of microwave heating with a same average microwave output
power of different pulsing ratios (PR=1 @ 250-W setting and PR=2 @ 500-W
setting)............................................................................................................................130
Figure 6.2 Temperature distribution in 2 % agar gel cylinders, 3.5-cm radius, after 1, 2
and 3 min of microwave heating with a same average microwave output power of
different pulsing ratios (PR=1 @ 250-W setting and PR= 2 @ 500-W setting)
131
Figure 6.3 Mean sample temperature vs. processing time at two pulsing ratios (PR).... 132
Figure 6.4 Mean sample temperature vs. processing time for samples of different radii
(R)...................................................................................................................................133
Figure 6.5 Mean sample temperature vs. pulsing ratio for different sample radii (R).... 134
Figure 6.6 Temperature distribution in 2 % agar gel cylinders, 4-cm radius, after 4 min of
heating by using an average microwave absorbed power of 225 W under continuous
(Mode A) and pulsed (Mode B) microwave applications.......................................... 135
Figure 7.1 Measured and predicted temperature profiles in a 4-cm radius mashed potato
cylinder after 1 min of continuous microwave heating at the 250-W oven setting. 151
Figure 7.2 Predicted temperature profiles in 1.6,2.4,2.8 and 3.2-cm radius mashed potato
cylinders after 30 s of microwave heating at the 250-W oven setting. Measured data
for 2.4-cm radius sample is also shown...................................................................... 152
Figure 7.3 Average sample temperature profiles in 2.4-cm radius, 7 cm-long, mashed
potato cylinders heated by pulsed microwave at the 250-W oven setting. Power-on
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XVI
constraints (A TJ were 20 and 15 °C and power-off temperature
difference constraints (ATJ were 5 and 3 °C (i.e. ATon(ATd)=15(5), 15(3), 20(5) and
20(3)).............................................................................................................................. 153
Figure 7.4 Average sample temperature profiles in 2.8-cm radius, 7-cm long, mashed
potato cylinders heated by pulsed microwave at the 250-W oven setting. Power-on
constraints (A TJ were 20 and 15 °C and power-off temperature difference
constraints (ATJ were 5 and 3 °C (i.e. ATon(ATd)=15(5), 15(3), 20(5) and 20(3)). 154
Figure 7.5 Average sample temperature profiles in 3.2-cm radius, 7-cm long, mashed
potato cylinders heated by pulsed microwave at the 250-W oven setting. Power-on
constraints (A T J were 20 and 15 °C and power-off temperature difference
constraints (ATd) were 5 and 3 °C (i.e. ATon(ATd)=15(5), 15(3), 20(5) and 20(3)). 155
Figure 7.6 Temperature distribution in 2.4-, 2.8- and 3.2-cm radius potato cylinders under
the ATon(ATd)=20(3) criteria after average sample temperature is about 60 °C at the
250-W oven setting........................................................................................................156
Figure 7.7 Time and average sample temperature of a 3.2-cm mashed potato cylinder
with 20 °C power-on constraints and a power-off differential of 3 °C lower than the
power-on constraint (i.e. ATon(ATd)=20(3)) at the 250- and 500-W oven settings. 157
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1
CHAPTER 1
INTRODUCTION
1.1 Development of Microwave Oven
In February 1940, two British scientists, Randal and Boot, developed an
electronic magnetron tube that generated large amounts of microwave energy. This
equipment enabled the British military to develop smaller and more powerful radar
systems to detect German aircrafts during World War n. In September 1940, two other
British scientists, Lizard and Cockroft, visited the Massachusetts Institute of Technology
(MIT Cambridge, MA), and demonstrated a magnetron microwave generator. Originally,
these two scientists planned to have an American company build radar systems for them.
In 1945, Dr. Percy Spencer purportedly used microwave radiation to prepare popcorn.
This was the first known application of microwave radiation for heating food (Reynolds,
1989). The first commercial microwave oven was introduced into the restaurant and
institutional market in 1947. In 1967, Amana Inc. introduced the first “countertop”
microwave ovens to the U.S. households (Reynolds, 1989; Happel, 1992), and more than
90 % of the U.S. households owned a microwave oven in 1997 (Liegey, 2001). The
Association of Home Appliance Manufacturers (AHAM, 2000) forecasted that the annual
sale of home microwave ovens in the U.S. will exceed 12 million units in 2002.
1.2 Pulsed Microwave Application
The major reason for infrequent usage of microwave heating by the food industry,
which also echoes complaints made by home consumers, is the uneven heating of food
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2
materials. This is caused by the characteristics of the food materials that absorb
microwave energy.
The idea of pulsed drying was first investigated by Edholm in 1933 (Hamoy and
Radajewski, 1982). In this technique, the convection energy for drying is periodically
turned on and off. Intermittent drying improves energy efficiency and product quality
(Farkas and Rendik, 1997). Poliak and Foin (1960) measured the temperature distribution
(TD) of roast beef immediately after microwave cooking and after a 30-min rest period.
The temperature redistribution they observed is presented in Fig. 1.1. This is a good
illustration of temperature redistribution within foods during power-off times. The
temperature equalization during power-off times occurs due to thermal diffusion
(Chamchong and Datta, 1999). The redistribution effect of pulsed microwave heating
implies a remedial approach to obtain a uniform temperature profile within a heated food.
Microwave heating produces large temperature variations within a food sample
due to penetrating nature of microwaves. However, if a proper pulse mode is employed,
thermal diffusion can reduce temperature variations during power-off periods.
Temperature redistribution can be reliably predicted and explained via numerical models.
In this thesis, microwave power within the oven cavity was assumed to be one­
dimensional, which is often employed to study microwave heating and thawing of foods
(Ayappa, et al., 1991; Pangrle et al., 1991; Padua, 1993; Barringer et al., 1995).
In reality, microwaves are three-dimensional and can be described as transverse
electromagnetic components. The absorbed power in a 6 x 5 cm (diameter x height)
alginate gel cylinder at 30°C resulted in more uniform temperature along the sample axis
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than along the sample radius (Fig. 1.2). This was supported by a two-dimensional finite
element analysis of microwave heating reported by Lin et al. (1995). The TD along the
sample vertical depth (z-axis) has also been reported to be more uniform than that along
the sample radial depth (r-axis) in a 3.5 x 7 cm (radius x height) potato cylinder (Chen et
al. 1993). In Table 1.1 different experimental R/H (radius/height) ratios of food samples
used for TD simulation studies are summarized, the R/H ratios range from 0.36 to 0.71.
These studies assumed and/or reported that the TD along sample vertical depth (z-axis) is
not significant compared to that along sample radius.
The microwave power solution is either derived from the Lambert’s law or
Maxwell’s equations. The Lambert’s law is a simplified solution representing a multitude
of boundary conditions. Standing waves, with their attendant nodes and anti-nodes, are
not represented in the Lambert’s law. Maxwell’s equations better represent the standing
wave effects encountered with microwaves.
1.3 Objectives
The overall objective of this thesis was to evaluate the temperature distribution in
a model food material during pulsed and continuous microwave heating. The specific
objectives of this thesis are listed below under Chapter numbers in which they addressed.
The objectives are to:
[Chapter 2]
•
Review literature related to pulsed microwave heating.
[Chapter 3]
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•
Describe the microwave oven used for the experiments.
•
Provide the overall experimental procedures.
[Chapter 4]
•
Develop a heat transfer model for predicting temperature distribution in
2% agar gel cylinders heated by continuous and pulsed microwave
energy.
•
Validate the model predictions with experimental data.
[Chapter 5]
•
Investigate internal temperature distributions in 2% agar gel cylinders
with different sample radii heated by continuous and pulsed microwave
energy.
•
Predict sample temperature based on the solutions of the Lambert’s law
and Maxwell’s equations of microwave power absorbed by the sample.
•
Compare the measured and predicted temperature distribution for both
continuous and pulsed microwave energy.
[Chapter 6]
•
Examine the effect of continuous and pulsed microwave heating at the
same average power setting.
•
Evaluate the effects of microwave processing duration, power mode,
power level, and sample size on the temperature distribution within 2%
agar gel cylinders.
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[Chapter 7]
•
Evaluate proper sample dimension which pulsed microwave energy can be
efficiently employed to create heating in a uniform manner.
•
Investigate the interior sample temperature of mashed potato cylinders
affected by different pulsed modes of microwave energy and temperature
constraints.
1.4 Limitations
The quantitative results obtained and presented in Chapter 6 are based on data
collected using the Labtron 500 (Zwag Inc., Epone, France) microwave oven. The
General Linear Model for the statistical analysis is valid only for the range of
experimental conditions specified.
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6
1.5 References
AHAM. 2000. Forecasts. Appliance Manufacturer. 48(6): 11.
Ayappa, K.G., Davis, H.T., Crapiste, G., Davis, E.A. and Gordon, J. 1991 Microwave
heating an evaluation of power formulations. Chem. Engineering Science. 46 :
1005-1016.
Barringer, S. A., Davis, E. A., Gordon, J., Ayppa, K. G., and Davis, H. T. 1995.
Microwave- heating temperature profiles for thin slabs compared to Maxwell and
Lambert law predictions. J. Food Sci. 60(5): 1137-1142.
Chamchong, M. and Datta, A. K. 1999. Thawing of Foods in a Microwave Oven: I.
Effect of Power Level and Power Cycling. J. Microwave Power and
Electromagnetic Energy. 34(1): 9-21.
Chen, D. D., Singh, R. K., Haghighi, K. and Nelson, P. E. 1993. Finite Element Analysis
of Temperature Distribution in Microwaved Cylindrical Potato Tissue. J. of Food
Engineering. 18:351-368.
Farkas, I. and Rendik, Z. 1997. Intermittent Thin Layer Com Drying. Drying
Technology. 15 (8): 1951-1960.
Happel, M.E. 1992. Consumer attitudes + marketing statistics = the consumer equation.
Microwave World. 12(2):7-8.
Hamoy, A. and Radajewski, W. 1982. Optimization of Grain Drying-With Resting
Period. J. Ag. Eng. Research 27(4): 291-308.
Liegey, P. R. 2001. Hedonic Quality Adjustment Methods For Microwave Ovens In the
U.S. CPI. U.S. Bureau of Labor Statistics.
R eproduced with permission of thecopyright owner. Further reproduction prohibited without permission.
7
Lin, Y. E. Anantheswaran, R.C. and Puri, V.M. 1995 Finite Element Analysis of
Microwave Heating of Solid Foods. J. of Food Engineering 85-112
Padua, G.W. 1993. Microwave heating of agar gels containing sucrose. J. Food Sci.
58(60): 1426-1428.
Pangrel, B.J. Ayappa, K.G. Davis, H.T. Davis, E.A. and Gordan, J. 1991 Microwave
thawing of cylinders J. AIChE 37(12): 1789-1800.
Poliak, G.A. and Foin, L.C. 1960. Comparative Heating Efficiency of a Microwave and
Convection Electric Oven. Food Technology 14:454-457.
Reynolds, L.R. 1989. The history of the microwave oven. Microwave World 10(5): 7-11.
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8
Table 1.1 Summary of experimental sample R/H (radius/height) ratios of food used
for temperature distribution (TD) simulation studies.
Sample
R/H ratio
Agar gel
0.71
Salted ice
NDa
Agar gel
0.36-0.42
Potato
0.5
Alginate gel
0.6
a Not described
bOne-dimensional
cTwo-dimensional
d Not significant
e Not evaluated
Model used
1 Db
ID
1D
2 Dc
2D
TD along
sample axis
NSd
NEe
NE
NS
NS
Reference
Mudgett (1986)
Pangrle et al. (1991)
Padua (1993)
Chen et al. (1993)
Lin et al. (1995)
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9
Temperature,
80
40
30 min
0 min
0
0
5
10
Thickness, cm
Figure 1.1 Temperature equalization in roast beef sample just after microwave
application (0 min) and after a 30-min rest time (30 min) (from Poliak
and Foin, 1960).
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10
ab8 •'€«.
I ■so
oo
IcbJ
0
■
00
r fen]
so
Figure 1.2 Two-dimensional microwave power absorbed in a 3-cm radius (r), 5-cm
high (z) alginate gel cylinder at 30 °C (from Lin et al. 1995).
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11
CHAPTER 2
LITERATURE REVIEW
2.1 Microwave Frequencies
Electromagnetic waves are classified by their frequencies. The electromagnetic
spectrum is shown in Fig. 2.1. Microwave radiation ranges from 300 MHz to 300 GHz,
which lies between radio and infrared waves on the spectrum. Microwave frequencies of
915 MHz, 2.45 GHz, 5.8 GHz, and 24.2 GHz are approved for industrial and scientific
applications by the U.S. Federal Communications Commission. Industrial food
equipment uses only 915 MHz and 2.45 GHz. At 2.45 GHz, the penetrability is up to 10
cm and has characteristics similar to that of radar waves. At 915 MHz, the penetrability
in a food material is up to 30 cm and has characteristics similar to that of radio waves.
Most food applications, including cooking and heating in home microwave ovens, are at
2.45 GHz with a few at 915 MHz (Brennan et al., 1969; Clary, 1994; Copson, 1975;
Barringer, 1994).
2.2 Microwave Food Process
The most common applications of microwave food processing are drying, freeze
drying, vacuum drying, baking, pasteurizing, sterilization, cooking, tempering, roasting,
blanching and rendering (Decareau, 1985; Decareau and Peterson, 1986). These are
summarized in Table 2.1.
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1021
COSMIC RAYS
1020
10
19
GAMMA RAYS
10 1*
10
17
10
16
X-RAYS
1015
1014
ULTRA VIOLET
1013
VISIBLE
10
12
10
11
THERMAL (infrared)
10 10
109
RADAR
10*
107
106
mm
^m
mm
mm
mm
mm
m b
■
MICROWAVE HEATING
mm
^m
^m
mm
mm
mm
mm
m
105
104
103
102
RADIO
101
(Hz or cycle/s)
Figure 2.1 The electromagnetic spectrum {from Brennan et al., 1969).
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13
Table 2.1 Microwave Food Processing Applications (from Decareau and Peterson,
1986).
Process
Drying
Freeze drying
Vacuum drying
Baking
Pasteurizing
Sterilization
Cooking
Tempering
Roasting
Blanching
Rendering
Products
Pasta, onion, rice, cakes, egg yolk, snack foods, seaweed
Meat, vegetables, fruits
Orange juice, grains, seeds
Bread, Doughnuts
Bread, yogurt
Pouch-packed foods
Bacon, meat patties, sausage, potatoes, chicken
Meat, fish, poultry
Nuts, cocoa beans, coffee beans
Corn, potatoes, fruits
Lard, tallow
2.3 Characteristics of Microwave Heating
2.3.1 Methods o f Heating Food
There are two methods used to heat food materials: direct and indirect. The major
difference between direct and indirect heating is the need for a heat exchanger or
medium. Indirect heating requires heat transfer media such as steam, air, water or other
exchangers. The major examples of indirect heating are conduction, convection, and
thermal radiation. Direct heating and microwave heating do not require any intermediate
heat exchangers to transfer the heat into foods and results in a more rapid heating rate
(Brennan et al., 1995; Copson, 1975; Decareau, 1985). The methods describing the
application of heat to food materials have been summarized in Table 2.2 (Heldman and
Singh, 1981).
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Table 2.2 Methods of Heat Processing of Foods (from Heldman and Singh, 1981).
Major category
Indirect heating
Methods
By vapor or gas such as steam or air.
By liquid such as water or organic heat exchange
Liquids.
By electricity in resistance and radiation heating system
(ohmic heating).
Direct heating
Using electricity, by dielectric or microwave.
2.3.2 Theory o f Microwave Heating
2.3.2.1 Dielectric Properties
Electric field behavior basically classifies materials into three categories:
conductors, semiconductors, and insulators. The electrons in the outermost shell within
atoms of conductors are loosely held, and tend to migrate easily from one atom to
another. Most metals are conductors. The electrons in the outermost shell of insulators
(dielectrics) are confined to the specific orbit. Under normal circumstances, they cannot
be liberated. Most biological materials, including foods, are dielectrics. The field
behavior of semiconductors falls between that of conductors and dielectrics (Cheng,
1992).
When a conductor is placed in an external electric field, the free charges move to
the surface and the electric field in the interior vanishes. However, the bound charges of a
R ep rod uced
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15
dielectric in the presence of an external electric field cause small displacements of both
positive and negative charges in opposite directions due to the lack o f free charges. These
small displacements result in the polarization of a dielectric material.
Water molecules, which are polar, possess permanent dipole moments. Under
normal conditions, the individual dipoles are randomly oriented. The net dipole moment
is zero in the macroscopic view. The application of an electric field will exert torque on
the individual dipoles, and partially align them with the field.
The fundamental postulate of electrostatics in free space specifies the divergence
of E (electric field intensity):
( 2. 1)
where p = volume charge density of free charges, and Eo = permittivity of free space.
Since a polarized dielectric gives rise to an equivalent volume charge density pp, the
divergence must be modified as:
V « E = — (p + p )
£0
( 2 .2 )
where
(2.3)
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16
and P = polarization vector.
Equation 2.2 could be expressed as:
V • (e0E + P) = p
or
V »(D ) = p
(2.4)
where D = electric displacement.
When the dielectric properties of the media are linear and isotropic, the
polarization is directly proportional to the field intensity. The proportionality constant Xe
is a dimensionless quantity called susceptibility. The polarization vector in terms of %e is:
P=eoX*E
(2-5)
Substituting equation 2.5 in equation 2.4 yields:
£>=eo(l+Xe)£ or
=8oK’£=£’£
(2.6)
where
and
k’
= dimensionless quantity known as the relative permittivity or dielectric constant
of a medium, and Eo = absolute permittivity or simply permittivity. The dielectric
constant s ’ of a material describes its ability to store electrical energy.
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17
With the application of an alternating electric field, the inertia of charged particles
tends to prevent particle displacements, and hence causes a frictional damping that leads
to power dissipation. Power dissipation in a dielectric is expressed by the loss factor e”,
and is included in the imaginary part of the complex permittivity e*:
(2.7)
where
2.3.2.2 Term Derived from Dielectric Properties
The ratio e’Ve’ is the loss tangent, tan 5 = e”/e \ The tan 8, along with dielectric
properties, is useful for the calculation of some terms that are of interest in microwave
heating.
The attenuation constant ((3) is important in determining the penetration depth of
microwave in a sample. At higher frequencies, (3 increases, and thus the penetration depth
of microwaves in the sample decreases. The reciprocal of (3, known as the penetration
depth (Dp), is the distance at which the incident field intensity decreases to i/e of its
incident value. The (3 in terms of dielectric properties and tan 5 is:
( 2 .8 )
where c = speed of light a n d /= frequency of radiation used.
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18
The phase constant (a) represents the change in phase of the propagating wave.
a
2nf jfc'i/l + itanS)2 +1
=
—
—
c
2
(2.9)
The a is related to the wavelength of radiation in the sample (Xs) by:
( 2 . 10)
The k s is very important when considering standing wave interactions in the microwave
oven.
2.3.2.3 Basic Mechanisms o f Microwave Heating
When a material is placed within a coil carrying a high frequency alternating
current, it is heated by the current induced in that material. This is analogous to the action
within a transformer when the application of a current on the primary side induces a
current in the secondary winding (Copson, 1975, Brennan et al., 1969; Copson, 1975).
Dielectric heating, on the other hand, heats materials due to power losses within those
materials when subjected to alternating current electric fields (Brennan et al., 1969;
Decareau, 1985). Copson (1975) also described dielectric heating as the heating of poor
dielectric materials such as air, paper, and foods.
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19
Dielectric heating of substance is generally achieved by placing the sample
between two parallel plates (electrodes). The plates are connected to a source of high
frequency alternating current (Brennan et al., 1969; Copson, 1975; Decareau, 1985).
Heating is brought about by molecular friction due to the rapid orientation of the electric
dipoles within the materials (Brennan et al., 1969).
The basic concepts of dielectric heating can be applied to microwave heating.
However, microwaves, a radiation phenomenon, differ from induction heating between
parallel plates in the following ways:
1. Higher energies associated with the higher frequencies used in microwave
heating allow for the same energy input to be achieved through the application
of a lower voltage (Brennan et al., 1969; Copson, 1975; Decareau, 1985).
2. Loss factors are higher as the frequency becomes higher (Copson, 1975).
The electromagnetic field in the microwave region affects the orientation of free
water molecules and ionic polarization, and generates the heating in a food product.
Dipole rotation of free water molecules is the major force in producing heat within foods,
due to the fact that water molecules generally make up the largest portion of raw food
materials. Water molecules themselves are randomly polarized in the absence of external
forces. In the presence of an electric field, water molecules align themselves in the
direction of the field. An alternating current electric field, such as microwaves, causes
water molecules to repeatedly change their orientation according to the direction of the
field. This is the way dipole rotation generates heat. The rate of rotation corresponds to
the frequency of the alternating field.
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20
The other major mechanism is ionic polarization. The movement of both
dissolved anions and cations in solutions can accelerate in response to an electric field.
The accelerated ions collide with each other and convert kinetic energy into heat
(Brennan et al., 1969; Clary, 1994; Copson, 1975).
Food components classified by proximate analysis are moisture, protein, lipids,
carbohydrates and ash. Among these components, free water or moisture is associated
with dipole rotation, while dissolved ions of ash are associated with ionic polarization.
Other components are relatively inert to microwaves.
2.4 Microwave Power Formulations
Power formulations for microwave heating are related to the electric field in the
oven cavity. When the microwaves contact the sample surface or at interfaces within the
sample, some of the radiation is transmitted/absorbed and some of it is reflected. The
amount of transmitted and reflected radiation depends on the dielectric properties of
materials at both sides of the interface or surface. The electric field intensity entering the
sample is a critical parameter to be determined. The electric field intensity and power
formulation at a given depth can be determined by using either the Lambert’s law or
Maxwell’s field equations.
2.4.1 Lambert’s Law
Von Hippel derived the Lambert’s law (1954). It is a semi-infinite solution for
electric field intensity and power terms. The Lambert’s law predicts an exponential decay
of radiation with increasing depth into the sample. The value of transmitted power must
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21
be determined in order to simulate the power term at a given depth within the sample.
Different methods for the determination of transmitted power have been proposed. They
are: monitoring gas breakdown (MacLathy and Clements, 1980), toner particle melting
(Washisa and Fukai, 1980), electrical discharge in fluorescent lights (Bosisio et al., 1975)
and relative temperature of a tray of water segmented into cubes (Watanabe et al., 1978).
However, these methods do not give an absolute value of power that is needed for model
simulation.
In 1987, the International Electrotechinal Commission (IEC) accepted a standard
method for determining the power rating of a microwave oven. This is based on the
assumption that the power absorption of a beaker of water is equal to the energy
transmitted into the sample. It is one of the first and the most common methods for
measuring the power delivered in a microwave oven. Generally, a beaker containing
certain amount of water is placed in the microwave oven and heated. The temperature
rise over a certain time is measured. The absorbed power (PabS), which causes the
temperature to rise, is calculated by energy balance.
Pabs=mCp(AT/At)
(2.11)
where AT = temperature rise, At = microwave heating time, m = sample mass, and Cp =
specific heat capacity of water. Ohlsson and Bengsston (1971), Stuchly and Hamid
(1972), Mugett (1986) and Padua (1993) used this standard method to determine the
absorbed power. The knowledge of transmitted power flux is beneficial for calculating
the power distribution within the sample, P(z), at a given depth (Mudgett, 1986; Padua,
1993).
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22
P(z) = 2 I0Pe-2pz
( 2 . 12)
where Io= transmitted power flux, P = attenuation constant, z = given sample depth.
Equation 2.12 is the Lambert’s law which predicts an exponential decay of radiation
with increasing depth into the sample. This semi-infinite power formulation does not take
the reflected radiation at the exit interface into account. The reflected radiation is
evidenced by center focusing in cylinders and causes local hot spots. Even though this
produces some errors in model simulation, because of its simplicity, the Lambert’s law is
still often used in food research.
2.4.2 Maxwell's Field Equations
Power formulation can also be determined by solving Maxwell’s field equations
(Cheng, 1992). The equations are more comprehensive than the Lambert’s law, and are
capable of modeling the exact electric field intensity at any point within a microwave
oven. The knowledge of electric field intensity can be incorporated with the Poynting
vector, a power density vector associated with an electromagnetic field. The power
dissipated per unit of volume is:
P M(Z) = - Re(V • S) = ^(o e0K" E ■Ec
(2.13)
where Re = real part of a complex number, S = Poynting vector, to = angular velocity, So
= permittivity of free space,
k”
= dielectric loss factor and Ec = conjugate electric field
intensity at a given point.
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23
Equation 2.13 is the local power formulation in unit of power per volume. Jia and
Jolly (1992) and Ofoli and Komolprasert (1988) have modeled the exact theoretical
electric field at any point within a specific microwave oven. These studies are complex,
and applicable only to their specific systems.
Ayappa et al. (1991 a and b) developed models according to the solution of
Maxwell’s field equations for infinite slab, composite slabs and cylinders. The models
were based on two major assumptions about the microwave radiation: (1) it is incident
perpendicularly on the sample surface, and (2) it can be treated as uniform plane wave. A
plane wave is a one-dimensional spatial dependence wave. A uniform plane wave is a
particular solution of Maxwell’s field equations which assumes the same direction,
magnitude and phase in infinite planes perpendicular to the direction of propagation.
Strictly speaking, a uniform wave does not exist in practice because a source infinite in
extent would be required to create it. However, it is a common assumption made by food
engineers and others.
During modeling with uniform plane wave assumptions according to the solution
of Maxwell’s field equations, reflected and transmitted radiation are taken into account.
In the case of normal incidence at a plane dielectric boundary, two parameters are very
important for modeling. They are reflection coefficient (Ru+i) and transmission
coefficient (Tu+i). In terms of intrinsic impedance (q), they are:
,
/+1
~ n
Hi+1 o. «
(2.14)
and
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They can be also be expressed as a function of the attenuation constant (3 and the
phase constant a , they are:
Jsu-i
( 2 . 16)
where
\R
i m
~
■
(a, - a l+l)2+CP, - f$M)z
(a, +aMy- + U3, +i3l+l)2
S i.m = Tan
-i
2(a/+, f l -<*,PM )
(a,2 + p y ) - ( f x M 2 + PM 2)
(2.17)
( 2 . 18)
and
T
—\T \a
*/./+1 —*/./+! F
(2.19)
where
\1 i
m
“
4(ar
l
+ P,2)
7 "
m
’ («i + « i+i ) + ( A
^ 7 7
U -2 U )
+ A +I-)
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25
*l.M = tan"‘
(off2 + 0 t2 ) + ( a la M + 0, p M )
( 2 .21)
The power dissipated within a sample of single slab geometry (one- dimensional)
with radiation is:
( 2 .22 )
Ayappa (1992) presented a more complete derivation of the power formulation.
Equation 2.22 is an evidence of the power formulation according to the solution of
Maxwell’s field equations calculating reflection at front and back of the sample. The
internal standing waves are predicted, and therefore account for focusing and edge
heating.
2.5 Temperature Modeling
2.5.1 Finite-difference Formulations
Microwave radiation delivers energy into the sample and, thus, heat transfer
occurs within the sample. Microwave power is treated as a volumetric internal power
generation (P) that can be incorporated with transient (unsteady state) heat transfer.
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26
d 2T d 2T d 2T p
1 dT
T T + T V + -r^ - + — =
rdx2 dy2 dz2 k a H dr
For a solid, the energy equation is:
(2.23)
For a cylinder, the energy equation is:
d 2T d 2T I d T p
I dT
- r r - + ^ r ^ + — — + — = ------—
dz2 dr2 r dr
k a H dr
(2.24)
where T = temperature, x, y, z and r are length along dimension, k = thermal
conductivity, p = power or heat generation, and a H = thermal diffusivity and x = time.
Although the partial differential equation can be solved analytically, it is complicated
when the heat generation is nonlinear. Alternatively, numerical techniques such as finitedifference, finite-element, or boundary-element methods can be used. The first step in
any numerical analysis is to select nodal points in the discretized sample. Each nodal
point represents a certain region around that point. The temperature for that nodal point is
an average temperature of the region. The heat transfer equation at each nodal point as an
exact differential equation is reduced to an approximate algebraic equation. The
approximate finite-difference form of the heat equation may be applied to any interior
node that is equidistant to the neighboring nodes. The finite-difference equation for a
node may also apply conservation of energy to a control volume about the node. Since
the actual direction of heat flow (inward or outward) is often unknown, it is convenient to
formulate the energy balance by assuming that all the heat flows are into the node. Such a
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27
condition is, of course, impossible, but if the rate equation is expressed in a manner
consistent with this assumption, the correct form of the finite-difference equation is
obtained (Incropera and DeWitt, 1995; Arpaci, 1966). For an unsteady state condition
with heat generation, the general form is:
E „ + E '= E „ = ^
(2.25)
where the subscript “in” = energy inward, “g” = heat generated, and “st” = heat store.
Equation 2.25 indicates that time dependent temperature information will be available
when solving the unsteady state equation in finite difference form.
2.5.1.1 Explicit Formulation
The explicit finite-difference form is used to solve for the unknown nodal
temperature for a new time with known nodal temperature at previous time. The explicit
method is not conditionally stable because of the numerically-induced oscillations. To
prevent such erroneous results, the time interval chosen must be maintained below a
certain limit, which depends on the spatial properties of the system. The limit is termed a
stability criterion, and is discussed by Incropera and Dewitt (1996). For a given space
increment of explicit formulation, it is frequently indicated that an extremely small time
step must be chosen. In other words, a very large number of time intervals may be
necessary to obtain a reasonable solution.
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28
2.5.1.2 Implicit Formulation
An implicit formulation, rather than explicit scheme, evaluates all other nodal
temperatures at new time. It provides a backward difference approximation, instead of a
forward one obtained by explicit formulation. The new temperature of any given node
depends on the new temperature of its adjoining nodes, which are generally unknown.
This implies that to determine the unknown nodal temperature at new time, the
corresponding nodal equations must be solved simultaneously. The implicit formulation
has the important advantage of being unconditionally stable. That is, the solution remains
stable for all space and time intervals. The oscillating results induced from the explicit
method can then be avoided by using this implicit approach. Since a larger time interval
may be used with the implicit method, computation time may often be reduced with little
loss of accuracy. Nevertheless, a sufficiently small time interval must be chosen to ensure
maximum accuracy (Incropera and DeWitt, 1996).
Ohlsson and Bengtsson (1971) developed microwave-heating profiles using a
finite-difference technique. Kirk and Holmes (1975) predicted the temperature profile of
a high moisture substance (1 % ion-agar gel in water) at any given time by using the
finite-difference method. Nykvist et al. (1976) investigated microwave beef roasting
using the finite-difference method by considering the case of non-perpendicular wave
incidents.
2.5.2 Temperature Rise Analysis
Thermal conductivity of a biological material is small compared to other material
like metals. In the case of large volumetric heat generation like microwave heating of
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food materials, the thermal conduction may be neglected for simplicity. The energy
balance equation (equation 2.11) of this type of analysis is termed the temperature rise
method.
Padua (1993) proposed a temperature analysis for 2 % agar gel cylinders using the
Lambert’s law of power formulation. Barringer et al. (1995) proposed a temperature
analysis for 2 % agar gel using the Lambert’s law, Maxwell’s field equations, and a
combined power formulation for slab geometry with microwave radiation from one side
of the slab. Barringer (1994) used temperature rise method to analyze microwave heating
of emulsions. The emulsion system, either oil-in-water or water-in-oil, is generally a
liquid or semi-liquid, and it is very difficult to measure the temperature at given point
within the sample because a convection effect occurs. The bulk (average) temperature
rise is a better alternative to obtain some additional information for microwave
application.
2.6 Pulsed Microwave Application
Despite the many seemingly excellent reasons for adopting microwave energy,
the non-uniform microwave heating of foods is recognized as a major drawback to
microwave heating. Ohlsson and Riman (1978) conducted a study on the non-uniform
temperature distribution in meat and potato cylinders; central heating effects (local hot
and cold spots) were found in small sample cylinders. One of the approaches to improve
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30
the temperature uniformity in samples heated by microwave is pulsed (intermittent)
operation. Pulsed microwave drying operation can also reduce energy loss through
flowing air, which is often evidenced in continuous microwave application. That is,
pulsed microwave application results in a lower energy requirement. However, the total
heating or drying time increases. In spite of the time factor, pulsed microwave application
is still worthwhile from product quality and energy cost consideration.
Shivhare et al. (1992 a, b, and c) studied continuous and intermittent microwave
drying of com with various duty cycles. Yongsawatdigul and Gunasekaran (1996 a and b)
investigated microwave drying of cranberries with at different vacuum and power levels,
and duty cycles. The drying efficiency improved under pulsed application compared to
continuous application. Tulasidas et al. (1994) evaluated pulsed microwave drying of
grapes in a single mode resonant cavity. The cost (energy consumption) of pulsed
application was substantially lower than that of continuous application.
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31
2.7 References
Arpaci, V. S. 1966. Conduction Heat Transfer. Addison-Wesley Publishing Company,
Reading, MA.
Ayappa, K.G. 1992. Analysis o f microwave heating o f dielectric material. Ph. D.
Dissertation, University of Minnesota, St Paul, MN
Ayappa, K.G., Davis, H.T., Crapiste, G., Davis, E.A. and Gordon, J. 1991a. Microwave
heating an evaluation of power formulations. Chem. Engineering Science. 46 :
1005-1016.
Ayappa, K.G., Davis, H.T., Davis, E.A. and Gordon, J. 1991b. Analysis of microwave
heating of material with temperature-dependent proties. AIChE J 37(3): 313-322.
Barringer, S.A. 1994. Experimental and predictive heating rates o f microwaved food
systems. Ph. D. Dissertation, University of Minnesota, St Paul, MN.
Barringer, S.A., Davis E. A., Gordon, J., Ayappa, K. G. and Davis, H. T. 1995.
Microwave- heating temperature profiles for thin slabs compared to Maxwell and
Lambert Law prediction. J. Food Science. 60(5): 1137-114.
Brennan, J.G., Butters, J.R., Cowell, N.D., and Lily, A.E.V. 1969. Food Engineering
Operation. Elservier Publishing Company Limited, New York, NY
Bosisio, R. G., Nachman, M. and Nobert, R. 1975. A simple method for determining the
electric field distribution along a microwave applicator. J. Microwave Power
11(1): 3-24.
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32
Cheng, D. K. 1992. Field and Wave Electromagnetics. Addison-Wesley Publishing
Company, Reading, MA
Clary, D.C. 1994. Application o f microwave vacuum and liquid media dehydration fo r
the production o f dried grapes. Ph. D. Dissertation, Michigan State University.
Copson, D. A. 1975. Microwave Heating. The AVI Publishing Company, Inc., Westport,
CT.
Decareau, R.V. 1985. Microwave Application in Food Processing Industry. Academic
Press, Inc., Orlando, FL
Decareau, R.V., and Peterson, R.A. 1986. Microwave processing and Engineering. Ellis
Horwood Ltd, Chichester, England.
Heldman, D.R., and Singh, R.P. 1981. Food Processing Engineering. The AVI
Publishing, Westport CT.
Incropera, F. P., and DeWitt, D.P. 1996. Introduction to Heat Transfer. John Wiley &
Sons, Inc., New York, NY.
Jia, X. and Jolly, P. 1992. Simulation of microwave field and power distribution in a
cavity by three-dimensional finite element method. J. Microwave Power and
Eletromagnetic Energy 27(10): 11-22.
Kirk, D. and Holmes, A. W. 1975. The heating of foodstuffs in a microwave oven. J.
FoodTechnol., 10: 375-384.
MacLatchy, C.S. and Clements, R.M. 1980. A simple technique for measuring high
microwave electric field strengths. J. Microwave Power 15(1): 7-14.
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33
Mudgett, R. E. 1986. Microwave propertiesand heating characteristics of foods. Food
technol. 40(6): 84-93.
Nykvist W. E., and Decareau R. V. 1976. Microwave meat Roasting. J. Microwave
Power 11(10): 3-24.
Ofoli, R. Y., and Komolprasert, V. 1988. On the thermal modeling of foods in an
electromagnetic field. J. Food Process. Preserv. 12: 219-241.
Ohlsson, T. and Bengtsson, N. 1971. Microwave heating profile in Foods- a comparison
between heating and computer simulation. Microwave Energy Application New
Letter 6: 3-8.
Ohlsson, T,and risman, P.O. 1978. Temperature distrubution of microwave heatingspheres and cylinders. J. Microwave Power 13(4): 303-309.
Padua, G. W. 1993. Microwave heating of agar gels containing sucrose. J. Food Science
58(60): 1426-1428
Shivhare, U.S., Raghavan, G.S. and Bosisio, R.G. 1992a. Microwave drying of com I.
Equilibrium moisture content. Transactions of the American Society of
Agricultural Engineers. 35(3): 947-950.
Shivhare, U.S., Raghavan, G.S. and Bosisio, R.G. 1992b. Microwave drying of com II.
Constant power, continuous operation. Transactions of the American Society of
Agricultural Engineers. 35(3): 951-957.
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Shivhare, U.S., Raghavan, G.S., Bosisio, R.G., and Mujumdar, A.S. 1992c. Microwave
drying of com HI. Constant power, intermittent operation. Transactions of the
American Society of Agricultural Engineers. 35(3): 959-962.
Stuchly, S.S. and Hamid, M.A. K. 1972. Physical Parameters in microwave heating
processes. J. Microwave Power 7(2): 117-137.
Tulasidas, T.N., Raghavan, G.S.V., Hudra, T. Gariepy, Y, and Akyel, C. 1994.
Microwave drying of grapes in a single mode resonant cavity with pulsed power.
Paper No. 94-6547, present at Ann. Mtg. Of the American Society of Agricultural
Engineers.
von Hippel, A. R. 1954. Dielectric and Waves. The MIT Press. Cambridge, MA
Washisu, S. and Fukai, 1980. A simple method for indication of the electric field
distribution in a microwave oven. J Microwave Power 15(1): 59-61.
Watanabe, M., Suzuki, M. and Ohkawa, S. 1978. Analysis of power density distribution
in a microwave oven. J. Microwave Power 13(2): 173-181
Yongsawatdigul, J. and Gunasekaran, S. 1996a. Microwave drying of cranberries: part I.
Energy and Efficiency. J. Food Processing and Preservation 20: 121-143.
Yongsawatdigul, J. and Gunasekaran, S. 1996b. Microwave drying of cranberries: part II.
Quality evaluation. J. Food Processing and Preservation 20: 145-156.
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35
CHAPTER 3
GENERAL METHODOLOGY
3.1 Laboratory Microwave Oven
A laboratory microwave oven (Labotron 500, Zwag Inc., Epone, France) was
used for all the experiments. The microwave oven operates at 2.45 GHz and has two
continuous output power settings, 250 and 500 W. The oven cavity is 33 x 22 x 35 cm
(width x height x depth) and houses a 25 x 3.5 cm (diameter x height) turntable that
rotates at 15 rpm. A of 6.5 x 6.5-cm waveguide (microwave energy source) is positioned
on the left cavity wall. The edges of the waveguide are 12.5 cm from the bottom and 6
cm from the front edge of the cavity. There are two mode fans in the oven cavity. The
horizontal mode fan, 6.5 x 6.5 cm, is positioned right across the waveguide on the cavity
wall; the vertical mode fan, 5 x 5 cm, is on the top wall, 18 cm from the front and 8.5 cm
from the left edge. The third fan, 6 cm in diameter, is an air outlet circulator. Its center
was positioned at 15 cm from the bottom and 23 cm from the left edge on the rear wall
(Figs. 3.1 and 3.2).
3.2 Preparation of 2% Agar Gel Sample Cylinders
Agar gel samples were prepared by dissolving 40 g (2%) of agar powder (Bacto
Agar, Difco Inc., Detroit, MI) in 1,960-mL of warm (~ 40°C) distilled water in a 2,000mL pyrex glass beaker. The agar-water mixture was heated until agar powder was totally
dissolved and the gel solution was clear. It was then poured into 600-mL or 400-mL
pyrex glass beakers and cooled to room temperature into solid sample cylinders. The radii
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36
of samples prepared in 600-mL and 400-mL beakers are 4 and 3.5 cm, respectively. The
sample cylinders were stored at 4 °C for 16 h to ensure uniform initial sample
temperature for microwave heating. The plastic film wrap (Saran Warp, The Dow
Chemical Company, Indianapolis, IN) was used to cover the beakers during the heating,
cooling and storage to prevent moisture loss.
3.3 Temperature Measurement
For each heating experiment, one agar gel cylinder was placed at the center of the
turntable in the microwave oven. Temperatures were measured across the horizontal mid­
plane at the radial distances of 0, I, 2, and 3 cm for 3.5-cm radius samples and 0, 1, 2, 3
and 4 cm for 4-cm radius samples. The sample cylinder was removed from the
microwave oven after every minute of microwave heating, and temperature
measurements were made. A type-T thermocouple probe (Omega Engineering Inc.,
Stamford, CT) connected to a datalogger (Model 34970A, Hewlett Packard, Beaverton,
OR, see Fig. 3.3) was used to measure the temperature. A 0.5-cm thick cardboard with
holes drilled 1 cm from each other along the radius (Fig. 3.3) was placed on top of the
agar gel cylinder matching the sample edge. The thermocouple probe was then inserted
into the sample via the holes in the cardboard until it reached the mid-plane. Single
thermocouple was used for all measurements. The readings were taken starting from the
sample center (0 cm) and moving outward. The temperature at each location was
measured in triplicates using three samples. The average and the standard deviation of the
three temperature measurements were calculated to represent the temperature distribution
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37
in the sample cylinder. For each sample, all temperature measurements were taken within
30 s after the heating process was completed and the sample was removed from the
microwave oven. The temperature variation at each location due to the 30-s delay during
the measurements was determined (Appendix D) not to affect the temperatures measured
significantly (P >0.01, Table D).
3.4 Average Absorbed Microwave Power
The average absorbed power (PabS) in the microwave oven was determined
experimentally. The 250- and 500-W oven settings were employed to heat distilled water
in different volumes for 180 s. The efficiency of energy transfer between the microwave
oven and the food sample is related to the sample’s dielectric properties. The efficiency is
also related to the nature and volumetric load of the sample (Mudgett, 1986). The average
temperature rise (ATav), the difference between the temperatures before and after
microwave heating, were measured. The temperatures were measured after stirring the
heated water for 10 s using a hand mixer (KitchenAid, KTM-7) at the lowest speed (250
rpm) to ensure temperature uniformity. The distilled water can be lost during the
microwave heating due to the moisture evaporation. The effect of evaporative cooling
was determined by measuring the difference in mass of distilled water (Aw) before and
after microwave heating; Aw multiplied by latent heat of water (hfg=2,338 U/kg at 27 °C
and at atmospheric pressure) then divided by the total heating time (t) is the power
correction term for the evaporative cooling. The average temperature rise (ATav) can be
calculated from the following energy balance relationship:
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38
p V C A T + h f &w
- ' ■ t
*
(3-1),
where p = density of distilled water (1,070 kg.m'3), V = volume of the distilled water, Cp
= specific heat capacity of distilled water (4.18 kJ.kg'l.°C'1), ATav= average temperature
rise (°C) and t = microwave heating period (180 s). Three replications of temperature rise
measurements were performed for each microwave power setting.
The variation in the absorbed microwave power at the center and in the vicinity of
the turntable was also determined. The absorbed power at the 250-W oven setting in the
microwave oven was measured by placing the samples at I and 2 cm from the center of
the turntable in triplicates and compared to the absorbed power measured at the center
(Fig. 3.4). The absorbed powers and the energy-transfer efficiencies in the microwave
oven for all experimental conditions are listed in Table 3.1. The variation in absorbed
power at the range between the center and at 2 cm from center was (3.7 W) and was not
statistically significant (P >0.01, Table 3.2). The energy-transfer efficiency (ratio of
absorbed power to oven setting) increased as the sample volume increased.
3.5 Average Surface Heat Transfer Coefficient
The average surface heat transfer coefficient was determined by measuring the
temperature of a cylindrical aluminum block, 3.5 x 7 cm (radius x height). The aluminum
block was cooled to 4°C and placed at the center of the microwave oven and allowed to
warm by the ambient temperature in the oven. The temperature change (at center of the
aluminum block) was recorded every 10 s using a fiberoptic sensing system (MetricCor,
Model 1400, Woodivelle, WA). The average surface heat transfer coefficient was
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calculated according to Rizvi and Mittal (1992). Since the thermal conductivity of the
aluminum block is very high (i.e. Biot number «
1), it was assumed that a lumped
capacitance method could be used. The experimental parameters are: air temperature =
23°C, initial temperature of the aluminum block = 4°C, surface area = 0.0194 m2, mass
of the block = 0.72 kg, and specific heat capacity = 0.930 kJ/kg."C. The measured
average surface heat transfer coefficient was 41.7 ± 0.56 W .m '^ C '1(mean ± standard
deviation).
3.6 Azimuthal Wave Assumption
In this research, microwave radiation was assumed to be incident normal to the
cylinder surface (Bowman, 1988; Pangrle et al., 1991). For propagation of waves in polar
(cylindrical) coordinates, the electric E,(r) and the magnetic H<t>(r) components lie along
the cylindrical surface of uniform intensity varying only in the direction of wave
propagation along the radial r-axis (azimuthal condition, Fig. 3.5).
The azimuthal microwave heating also means that the sample temperatures at the
same radial distance but at different angular locations should be the same. To verify this,
3.5-cm radius agar gel sample cylinders were heated in the microwave oven for 30 s at
the 250-W oven setting. The temperatures at every 120° angular location (0°, 120° and
240°) in the same sample were measured centimeter-wise in triplicates. At the sample
center, only one angular location is available. The resulting temperature measurements
with respect to the angular variation are shown in Table 3.3. The temperature variation at
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40
different spatial angles is not significant (P >0.01, Table 3.4). Therefore, the assumption
of azimuthal wave behavior is acceptable.
3.7 One-dimensional Heat Transfer Assumption
The heat transfer equation was solved by finite-difference approximation, with
intemal-heat-generation terms expressed as volumetric heating rate based on the levels of
microwave power absorption in successive shell volumes. The transient finite-difference
model used in this thesis is one-dimensional, temperature variation was considered only
along r-axis, i.e. the predicted temperature of each incremental shell represents the
average temperature of the entire shell.
Mudgett (1986) used a one-dimensional model to predict the temperatures within
agar gel cylinders ( 5 x 7 cm, radius x height) heated by microwaves (750 W, 1.5 min).
The predicted and measured temperatures at different vertical (z-axis) and radial (r-axis)
locations (Table 3.5) indicated that one-dimensional heat transfer assumption is
reasonably valid.
Temperature variation along the z-axis in 3.5 x 7 and 4 x 7 cm (radius x height)
agar gel cylinders (used as sample in Chapters 4, 5 and 6) after being heated in the
microwave oven used in this thesis was evaluated. The heating duration was 3 min at the
250-W oven setting and the temperature distribution in the agar gel cylinders was
measured (method described in Section 3.3). The measured points are shown in Fig. 3.6.
The temperature distributions in the 3.5- and 4-cm radius sample cylinder are shown in
Tables 3.6 and 3.7 respectively. Each set of temperatures measured at the same horizontal
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41
plane along z-axis was compared to that at the mid-plane (depth = 3.5 cm) using a paired
/-test. The temperature sets at all horizontal planes are not significantly different from
that at the mid-plane (P >0.01, Table 3.8). The temperatures at 6.5-cm depth are the
highest because the bottom of the agar gel cylinder is in direct contact with the turntable.
The dielectric boundary causes the reflected and transmitted radiation. The reflection and
transmission coefficients are related to the power dissipation in terms of dielectric
properties of both materials across the boundary and described in Section 2.4.2. The
reflected waves cause higher power dissipation near the contact surface of the agar gel
cylinder. However, the results indicated that the mid-plane temperatures can represent the
average temperatures along the z-axis (P > 0.01, Table 3.8). Therefore, one-dimensional
heat transfer assumption is statistically acceptable.
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42
3.8 References
Bowman, F. 1988. Introduction to Bessel functions. Dover Publications Inc., New York,
NY.
Mudgett, R. E. 1986. Microwave properties and heating characteristics of foods. Food
technol. 40(6): 84-93.
Pangrel, B.J. Ayappa, K.G. Davis, H.T. Davis, E.A. and Gordan, J. 1991 Microwave
thawing of cylinders J. AIChE 37(12): 1789-1800.
Rizvi, S.S.H. and Mittal, G.S. 1992. Experimental Methods in Food Engineering. AVI
Book Co., New York, NY.
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43
Table 3.1 Absorbed power and energy'transfer efficiency in the microwave oven at
2.45 GHz.
Oven setting
(W)
250
500
Sample
radius
(cm)
1.6
2.4
2.8
3.2
3.5
4.0
3.2
3.5
4.0
Sample
volume
(cm3)
56
127
160
225
269
352
225
269
352
Absorbed power
(W)
110.2 ±4.07
130.7 ± 3.90
144.4 ± 4.63
167.7 ±3.12
223.5 ± 3.70
233.9 ± 2.89
279.9 ± 4.03
331.2 ±3.61
344.1 ±3.12
Energy-transfer
Efficiency
(%)
44.1
52.3
57.6
67.1
89.4
93.6
55.8
66.2
68.8
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44
Table 3.2 Effect of sample placement (1 and 2 cm from the center of turntable in
the microwave oven) on the absorbed power at 2.45 GHz-statistical
analysis.
Source of variation
Placement
Replication
Error
Total
Degrees of Freedom
2
2
4
8
F-ratio
2.21
3.09
P-value
0.226
0.309
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45
Table 3.3 Temperatures at different angular locations in a 3.5-cm agar cylinder.
Location from center
0°
120°
240°
0
14.9 ± 0.61°C
--
--
1
8.7 ± 0.17°C
8.4 ± 0.25°C
8.2 ±0 .1°C
2
10.5 ± 0.38°C
10.1 ±0.35°C
10.3 ± 0 .3 1°C
3
11.7 ±0.40°C
12.6 ±0.25°C
12.8 ± 0.40°C
(cm)
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Table 3.4 Effect of angular locations (0,120 and 240 °) on the measured
temperatures in agar gel cylinders-statistical analysis.
Source of variation
Radial location
Replication
Angular location
Error
Total
Degrees of Freedom
3
2
2
28
35
F-ratio
596.67
2.52
2.02
P-value
<0.01
0.098
0.151
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47
Table 3.5 Predicted and measured temperatures for 5 x 7 cm (radius x height) agar
gel cylinders (from Mudgett, 1986).
Temperature (°C) along r-axis
Depth (cm)
2 cm
(Midpoint)
45
4 cm
(Surface)
49
Predicted
All
0 cm
(Center)
57
Measured
1
57
45
44
2
63
47
46
3
62
48
50
4
60
45
50
5
60
46
46
6
55
44
52
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48
Table 3.6 Measured temperatures for 3.5 x 7 cm (radius x height) agar gei
cylinders in the microwave oven at the 250-W oven setting.
Average
Temperature (°C) along r-axis
1 cm
2 cm
Depth
(cm)
All
45.3 ±3.14
35.4± 3.49
33.1 ±3.50
32.9 ± 2.77
0.5
40.7 ± 1.27
31.5± 0.35
29.6 ±0.65
31.4 ±0.78
2.0
43.7 ± 1.01
34.2± 0.45
31.7 ±0.87
30.8 ±0.40
3.5
46.4 ± 1.32
34.0± 1.44
31.5 ±0.25
31.9 ±0.78
5.0
46.6 ±0.61
35.7± 0.46
33.6 ± 0.50
32.4 ± 0.46
6.5
49.1 ±0.9
41.4± 0.76
39.3 ±0.32
38.1 ±0.25
Measured
0 cm
3 cm
Table 3.7 Measured temperatures for 4 x 7 cm (radius x height) agar gel cylinders
in the microwave oven at the 250-W oven setting.
Average
Measured
Depth
(cm)
All
0.5
2.0
3.5
5.0
6.5
0 cm
35.9 ± 2.59
32.8 ±0.67
33.3 ±0.80
35.3 ± 1.25
36.3 ± 1.00
39.5 ± 1.13
Temperature (°C) along r-axis
1 cm
2 cm
3 cm
23.2 ± 3.20
20.2 ± 1.42
20.8 ± 1.21
21.7 ± 1.44
24.2 ±1.15
27.5 ±0.75
24.2 ±3.19
21.0 ± 1.01
21.9 ± 1.17
23.5 ±0.85
25.3 ± 0.90
28.4 ± 1.16
26.5 ±2.41
24.5 ±0.55
24.5 ± 0.87
25.7 ± 1.23
27.4 ± 1.21
29.7 ± 1.05
4 cm
30.9 ± 1.96
31.3 ±0.47
30.3 ±0.85
31.6 ± 1.07
30.1 ± 1.21
31.7 ±0.85
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Table 3.8 Paired Mest for temperatures at mid-plane (MP, depth=3.5 cm)
compared to other locations along z-axis.
3.5-cm radius sample
4-cm radius sample
Depth (cm)
r-value
P
r-value
P
MP vs. average
1.24
0.323
1.42
0.229
MP vs. 0.5
MP vs. 2.0
MP vs. 5.0
MP vs. 6.5
2.37
1.20
2.46
5.17
0.098
0.455
0.091
0.014
3.83
3.60
1.62
3.87
0.019
0.023
0.181
0.018
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50
Figure 3.1 Schematic of the microwave oven with arrows indicating the directions of
circulating air.
Vertical mode fan
Air outlet fan
Horizontal mode fan
Waveguide
0
Turntable
Top view of turntable
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Figure 3.2 Pictures of the fans and waveguide in the microwave oven.
19
f r
Vertical mode fan
(on oven ceiling)
Air outlet fan
(on back wall)
itiunrnt
Horizontal mode fan
(on right wall)
Waveguide
(on left wall)
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52
Figure 3.3 Type-T thermocouple, datalogger and cardboard temperature guide used
for temperature measurements.
g a e w r ■■■
Type-T thermocouple and datalogger
3.5-cm radius marked cardboard
4-cm radius marked cardboard
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53
250
240
U
1 230
2
TJ
2w
o
CO
A
220
<
210
200 40
0.5
1
1.5
2
2.5
Distance from center of turntable (cm)
Figure 3.4 Absorbed power at 250-W oven setting in the microwave oven by placing
the sample at 0,1 and 2 cm from the center of the turntable.
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54
Figure 3.5 Cylindrical coordinate system and corresponding unit vectors for the
case of electromagnetic radiation incident normal to the surface.
Z
X
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55
Figure 3.6 Temperature measurement locations in 3.5 x 7 cm (radius x height) agar
gel cylinders for validating one-dimensional heat transfer assumption.
u
Top view
7 cm
i r r
•— »
"T T cm
•—
Mid-plane
»—
A
T
^ 3.5 cm
*-
O' Q 0
A
5 cm
0
6. 5 cm
~9----♦----•----9~
9------ * ------♦ -
Turntable
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7 cm
•—
56
C H A PT E R 4
TEMPERATURE PROFILES IN A CYLINDRICAL MODEL FOOD DURING
PULSED MICROWAVE HEATING*
*
This chapter has been published in Journal of Food Science: Yang, H.W. and S. Gunasekaran. 2001.
Temperature Profiles in a Cylindrical Model Food During Pulsed Microwave Heating. J. of Food Sci.
66(7): 998-1004.
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4.0 Notations
As:
surface area exposed to ambient air, m2
Bp
Biot number
C:
coefficient of the analytical solution for an infinite cylinder
Cp:
specific heat capasity, kJ/kg.°C
dR:
increment of between two nodal points, m
F0:
Fourier number
h:
heat transfer coefficient, W/m2.°C
Jo:
zero order Bessel function of first kind
J.:
First order Bessel function of first kind
k:
thermal conductivity, W/m.°C
P:
power generation from microwave, W/m3
P(x): power dissipated at depth x, W/m3
Po:
incident power or power at the surface, W/m2
Pp
power generation with volume
V j, W
Pto tal- total microwave energy absorbed by the heated body, W
PR:
pulsing ratio of microwave application
Q i-i:
heat flux from an inner nodal point, W/m2
Q i+ l:
heat flux from an outer nodal point, W/m2
r:
radial distance, m
R:
radius of the sample, m
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58
Rj:
t:
radial distance o f a nodal point, m
processing time, s
toff: microwave power-off time, s
ton: microwave power-on time, s
T:
temperature, °C
Tajr: temperature of ambient air, °C
Tj:
predicted temperature at nodal point i, present time, °C
Tj.t: predicted temperature at the inner nodal point i-1, present time, °C
Ti+j: predicted temperature at the outer nodal point i+1, present time, °C
Tint:
uniform initial temperature, °C
T„,i: predicted temperature at nodal point i, at new (next) time step, °C
T„,j.i: predicted temperature at the inner nodal point i-1, at new (next) time step, °C
Tn.j+i: predicted temperature at the outer nodal point i+1, at new (next) time step, °C
V:
total volume of the sample, m3
V;: volume of sub-shell i for numerical model, m3
x:
distance from sample surface to the center, m
Z:
height of the sample, m
z:
axial distance, m
an:
thermal diffusivity, m2/s
(3:
attenuation constant
X2:
chi square (statistical table) value
5:
loss angle, rad
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59
ATav- average temperature rise in the sample, °C
n:
series term involved in analytical heat transfer solution for an infinite cylinder
K’:
dielectric constant
K”:
dielectric loss factor
Xq:
incident wavelength, m
u:
angle in cylindrical coordinates, rad
P:
density of the sample, kg/m3
dx:
time increment, s
&
eigenvalue or positive root of the transcendental equation
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60
4.1 Abstract
Agar gel cylinders (2 %) were heated by pulsed and continuous microwave
energy. The total microwave application time of three minutes was maintained for all
experiments. Sample temperature distribution (TD) was measured at various radial
distances along the mid-plane as a function of heating time and heating mode. A local hot
spot was observed around the sample center during the continuous microwave
application. This hot spot was less significant during pulsed microwave applications,
especially when longer intermittent power-off times were employed. An implicit finitedifference simulation based on the Lambert’s law was used to predict TD in the sample
during microwave heating. The predicted TD matched the experimental values
statistically.
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61
4.2 Introduction
The prediction of temperature distribution (TD) in food and biological materials
heated by microwave radiation is critical for developing food processes and medical
treatments. Regarding medical treatments, Ho et al. (1971) reported microwave-heating
patterns of human limbs (cylindrical models) associated with human arms and thighs. In
their study, the limb tissue was exposed to a direct contact energy source in which the
waveguide and transmission lines were coupled through small apertures. They found
differences in the heating patterns of the tissue cylinders when aperture size was
different. Kritikos and Schwan (1975) reported microwave-heating patterns of spheres
representing human and animal heads of various sizes (radii).
Regarding food heating process using microwave energy, it is important to
simulate the time-dependent TD and design the process related to product quality and
microbial safety (Mudgett, 1986). Finite-difference approximations have been used to
obtain reasonable predictions of TD in the sample during microwave heating. Ohlsson
and Bengtsson (1971) offered a one-dimensional numerical solution for a finite slab to
approximate the temperature profiles in meat blocks heated by microwave radiation.
Nykvist and Decareau (1976) developed a two-dimensional model for cylinders
representing meat roasts. Padua (1993) developed a temperature rise model for agar gel
cylinders containing sucrose in terms of dielectric properties and total absorbed power.
Barringer et al. (1995) reported another one-dimensional model representing thin slabs
for temperature prediction of agar gels in terms of dielectric properties and various
formulations of microwave absorbed power.
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62
One of the major drawbacks concerning microwave-heated foods is the nonuniform TD. Ohlsson and Riman (1978) studied the non-uniform TD inside meat and
potato cylinders. Local hot spots at the sample center were observed in their study. Pulsed
(intermittent) microwave energy has been reported to result in lower energy requirement
and more uniform TD in food materials compared to continuous microwave energy.
Shivhare et al. (1992 a, b, and c) studied com drying using pulsed microwave energy.
They reported that the pulsed microwave energy is more efficient than conventional hot
air drying. Yongsawatdigul and Gunasekaran (1996 a and b) investigated the pulsed
microwave drying of cranberries. They found that a pulsed application (30-s power-on,
150-s power-off) under vacuum (5.33 kPa) resulted in maximum drying efficiency.
However, in these studies temperature uniformity in the heated sample were not
considered.
The objectives of this Chapter were to:
•
Develop and verify a finite-difference heat transfer model in an implicit form
that is capable of simulating TD in microwave heated model food cylinders
(2% agar gel).
•
Compare sample TD under the heating of continuous and pulsed microwave
energy using the heat transfer model.
•
Evaluate the temperature equalization effect via thermal conduction during
pulsed microwave heating.
•
Validate the model predicted TD using experimental data and a statistical
model.
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63
4.3 Mathematical Model
4.3.1 Implicit Finite-dijference Model
One-dimensional finite-difference models have been used under the assumption
that the incident microwave radiation is normal to the material surface (Padua, 1993;
Barringer et al., 1995; Pangrle et al., 1991). For a cylinder, the unsteady state (transient)
differential equation can be solved considering term-by-term difference approximation of
the differential equation:
BZT
1 BT
+
dr~ r dr
BZT
P
BT
------T
k
a Hdt
(4-1)
+ ~ ^~ r + ~ i
Bz
where T = temperature, r = radial distance, z = axial distance, P = power generation, k =
thermal conductivity, t = time, and an = thermal diffusivity.
In the case of one-dimensional heat transfer, the third term of equation 4.1 on the
left drops out and it becomes a 2nd-order partial differential equation (Incropera and
DeWitt, 1996). Arpaci (1966) illustrated a finite-difference (FD) formulation for
cylindrical geometry. For the cylindrical geometry corresponding to a typical inner point,
the first law of thermodynamics yields:
V't^ ' dT‘
=
-
y
) 2 , ! Z + « ■ * '( R ' +
Y )27e* p ~ p'
( 4 '2 )
where Ri = distance from the sample center, V; = volume between Rj±dR/2, Pj = power
generation in Vj, p = density of the material, Cp= specific heat capacity,
t
= time
increment, dTj = temperature difference at nodal point i between present time (t) and new
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64
time step (t+At) and Pc = power correction term due to the evaporation cooling (see
Appendix E for the evaluation of Pc). The heat flux terms,
and qj+i. are related to
temperature by Fourier’s law of conduction. Fig. 4.1 shows the schematic of a typical
inner node (nodal point) of a cylindrical object for finite-difference simulation. In an
explicit finite-difference scheme, the temperature of any node at t+At can be calculated
from the knowledge of temperature at the same and neighboring nodes for the preceding
time t. This method suffers from limitations on the selection of At to ensure compatibility
with stability requirements. In an implicit finite-difference scheme, the temperature of
neighboring nodes relates to new (next) time. For the implicit method, the temperature of
each node at new time step (t+At) depends on new temperatures of its adjoining nodes,
which are generally unknown. Hence, to determine the unknown temperature at (t+At),
the corresponding nodal equations must be solved simultaneously. The marching solution
would then involve simultaneously solving the nodal equation at each time t= At, 2A t,...
until the desired final time is reached. Compared to the explicit method, the implicit
formulation has the advantage of being unconditionally stable, that is, it remains stable
for all space and time intervals (Incropera and DeWitt, 1996).
Thus, the implicit method was used to evaluate the TD in the model food, 2%
agar gel cylinders. The heat flux terms for the explicit and implicit schemes are listed in
Table 4.1. For a typical boundary nodal point, the second term on the right side in
equation 4.2 is substituted by the boundary condition: hAs(Ti-Tajr), where h = average
surface heat transfer coefficient, As = surface area at the boundary and Tajr= temperature
of ambient air.
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65
4.3.2 Evaluating Microwave Absorbed Power
The average temperature rise in a material within time t depends on the total
microwave energy absorbed by the heated material (Pabs) (Padua, 1993; Barringer et al.,
1995). The energy balance gives:
A7*
“v
(4.3)
VpCp
where ATav = average temperature rise, Pabs = total absorbed power (evaluated using the
method described in Section 3.4), t = heating time and V = sample volume, p = sample
density and Cp = sample specific heat capacity.
If one-dimensional simulation is considered and the incident radiation is assumed
to be normal to the surface, the power dissipated at a certain sample depth is given by the
exponential decay of the incident power along that direction. The power term was derived
by von Hippel (1954) and is often referred as the Lambert’s law:
P(x) = P0e '2*
(4.4)
231 (—)(Vl + tan25 - l )
2
(4.5)
where
'
and
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66
K
tan S = —
k
(4.6)
'
where x = depth or distance from surface along the radial axis (in the reverse direction of
radial axis), P(x) = power dissipated at the depth x, PQ= incident power or power at the
surface, (5 = attenuation constant in terms of dielectric properties, incident wavelength
(Xo), k = dielectric constant and
’
k”
= loss factor.
To calculate the incident power (P0), the total absorbed power (Pabs) is expressed
as the volume integral of the P(x) function. For a cylindrical sample in an azimuthal
wave Field:
ZZnR
Pat, = j P ( x ) d V = j j j P 0e - iPxdxdudz
(4.7)
0 0 0
where Z and R are the height and radius of the test sample, respectively. Integrating
between limits and solving for Po gives:
PpVCAT
(48)
The following expression can be used to calculate power absorbed by each
annular shell (Pj) in the sample (Padua, 1993).
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67
P0e '2^dxdudz = ^ — {e'2Pxi ~ e 2Px' )
0 O r,
(4.9)
P
where the subscripts “ 1” and “2” refer to the outer and inner peripheries of the shell. In
the case of pulsed microwave application, the power term for each node (Pj) is applied as
a heat generation term in equation 4.2 during power-on periods and is zero for those time
intervals during which the microwave energy source was not powered. This means that
during every power-off period, there is no heat generation within the sample, and there is
presumably only thermal conduction occurring in the sample and convective heat
transfer across the boundary between the sample and ambient air.
4.4 Methods and Materials
Agar gel cylinders, 3.5 cm in radius, were prepared (method described in Section
3.2). The sample TDs were measured (method described in Section 3.3). The microwave
absorbed power in the microwave oven (PabS) was evaluated (method described in Section
3.4). The power absorbed by each cylindrical shell (Pj) during each microwave power-on
period was calculated using equations 4.8 and 4.9. The average surface heat transfer
coefficient in the microwave oven was determined (method described in Section 3.5).
4.4.1 Microwave Heating Process
The microwave oven (described in Section 3.1) was used as the microwave
energy source. Microwave power-on and -off times were adjusted to obtain different
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68
pulsing ratios (duty cycles) as follows (Fig. 4.2): 1 (continuous), 2 (30-s power-on, 30-s
power- off), and 3 (20-s power-on, 40-s power-off). The pulsing ratio, PR, is defined as:
p R _ (to* + toff'>
(4.10)
where ton and toff = duration the microwave power is on and off per duty cycle,
respectively. The total microwave power-on time was maintained 3 min for each PR.
However, the total heating (process) times were 3 ,6 and 9 min respectively for PR=1, 2
and 3.
4.4.2 Data Analysis
The physical, dielectric, and thermal properties of 2% -agar gel reported by
Barringer et al. (1995) were used for calculation (Table 4.2). The attenuation constant (P)
was calculated using equation 4.5. The implicit transient finite-difference simulation was
used to predict the sample TDs heated by both continuous and pulsed microwave energy.
The convection effect at the surface was also considered.
In the case of an infinite cylinder, which is initially at a uniform temperature and
experiences a change in convective boundary condition with one-dimensional transient
heat conduction, the exact solution is given by (Incropera and DeWitt, 1996):
T.-T
(4.11)
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69
where F 0
=
a Mt ,
^
^
„
—£y-, the coefficient Cn is Cn
R~
2
■/,(£,)
C, y2o(C,)+y2.(C,)
and the discrete values (eigenvalues) of £n are positive roots of the transcendental
equation which are related by the Biot number ( B j ) . The quantities J! and J0 are Bessel
functions of the first kind. For F0 > 0.2, the series analytical solution (equation 4.11) can
be approximated by single term:
(4.12)
The implicit finite-difference (FD) scheme was validated first by considering a
non-power generation case (i.e. P=0) and comparing the model evaluation to the
analytical solution of equation 4.12 for an example situation with parameter values listed
in Table 4.3. The effect of shell interval on the proposed implicit FD model including
microwave power was then validated by comparing the results to the temperature rise
model of Padua (1993) under the same conditions: continuous microwave heating for 15
s, sample radius = 3 cm, sample height = 8.6 cm, initial temperature = 23°C, power level
= 1500 W and ATav = 4°C. The predicted internal temperature profiles were calculated
using Engineering Equation Solver (Klein and Alvarado, 1996). The X test was used to
accept or reject the hypothesis that measured and predicted temperatures are the same, at
P = 0.01 and degrees of freedom (n-1) = 35, where n = 3 PRs x 3 time intervals x 4
measured locations (Bender et al., 1981).
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70
4.5 Results and Discussion
4.5.1 Model Validation
The analytical and numerical solutions for the infinite cylinder case matched very
well, though the analytical solutions were slightly lower than the corresponding
numerical values (Fig. 4.3). This is because only the first term of the series solution was
used (i.e. used only ^ instead of £n=i,2 , 3 .... ~) in equation (4.11) for the analytical
solutions. The temperature predictions of the implicit FD model also agreed well with the
solution of the temperature-rise model of Padua (1993). Different shell increments used
in FD model (0.1,0.15, and 0.2 cm) did not affect the predictions significantly (Fig. 4.4).
These results indicated that the implicit FD model was unaffected by the increment
chosen. The nodal increment of 0.1 cm was chosen for ongoing simulation.
4.5.2 Temperature Profiles
The measured sample TDs after the total microwave applications of 1, 2 and 3
min with PR = 1, 2 and 3 are shown in Figs. 4.5,4.6 and 4.7, respectively. The center (R
= 0 cm) temperature of all samples was the highest among temperatures at all locations.
This local hot spot became significant within a short time of continuous microwave
heating (PR = 1). The measured temperature profiles showed a difference of 9.1°C
between radial distances of 0 and 1 cm at the end of 1 min with PR = 1. As the
continuous microwave processing time increased, the local hot spot became more
significant. The measured sample temperature profiles showed a difference of 13.4°C
between radial distances of 0 and 1 cm after 3 min with PR = 1. The uneven TD within
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71
continuous microwave heated food materials is not desirable for both process safety and
quality.
Under pulsed microwave applications (PR = 2 and 3), the unevenness of
temperature distribution decreased substantially. After 3 min of a total microwave
application (TMA) with PR = 2 and 3, the differences between temperatures at R=0 and I
cm were 8.0 and 6.3°C, respectively. These results indicate that pulsed microwave
application is preferable to continuous application in avoiding the development of local
hot spots. The higher the pulsing ratio, the better temperature uniformity should be
expected during microwave heating.
The convective heat transfer affects temperature distribution within the sample
during heating. The air temperature inside the microwave oven was around 22 °C. Once
the sample surface temperature is higher than the ambient temperature, air convection
becomes a cooling effect.
The measured and FD predicted temperature profiles for different pulsing ratios at
sample radial distances of 0 and 1 cm are shown in Figs. 4.8 and 4.9, respectively. The
predicted TD at the sample center (Fig. 4.8) indicated that the temperature decreased
during the power-off periods. This implies that the direction of heat flux was outward
because of the lower neighboring temperature. The predicted TD at a radial distance of 1
cm shows that the temperature held steady during the power-off period. The pulsed
applications for either PR = 2 or PR = 3 minimized the sample center temperatures
significantly. The conductive temperature equalization during the power-off times is a
major reason for a more uniform TD during pulsed heating compared to continuous
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72
microwave heating. The chi square value, x2, was determined for the difference between
predicted and measured temperatures. After 3 min of total microwave application, the
predicted temperatures for PR = 1, 2 and 3 were not significantly different (P >0.01)
from the corresponding measured temperatures (calculated x2 = 1185 and x2 = 50.7 at P
= 0.01 with degrees of freedom = 35, Bender et al., 1981).
4.6 Conclusions
Temperature uniformity in 2% -agar gel cylinders was improved during pulsed
microwave application compared to continuous application. The local hot spot observed
at the sample center during continuous microwave application was minimized during
pulsed applications. The longer the power-off period (i.e. the higher the pulsing ratio), the
better the expected temperature uniformity. The conductive temperature equalization
during the power-off periods leads to better temperature uniformity in samples heated by
pulsed microwave energy. The predicted temperature distribution based on one­
dimensional finite-difference heat transfer simulation in the sample matched with the
corresponding measured temperature detebution statistically.
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4.7 References
Arpaci, V. S. 1966. Conduction Heat Transfer, Addison-Wesley Publishing Co.,
Reading, MA.
Barringer, S. A., Davis, E. A., Gordon, J., Ayppa, K. G., and Davis, H. T. 1995.
Microwave- heating temperature profiles for thin slabs compared to Maxwell and
Lambert law predictions. J. Food Sci. 60(5): 1137-1142.
Bender, F.B., Douglass, L.W., and Kramer, A. 1981. Statistical Methods fo r Food and
Agriculture, Food Products Press Inc., Binghamton, New York.
Ho, H.S., Guy, A.W., Sigelman, R.A., and Lehmann, J.F. 1971. Microwave heating of
simulated human limbs by aperture sources. IEEE Trans. Microwave Theory &
Techniques 19(2): 224-231.
Incropera, F. P. and DeWitt, D.P. 1996. Introduction to Heat Transfer, 3rd Ed. John
Wiley & Sons Inc., New York, NY.
Klein, S. A. and Alvarado, F. L. 1996. Engineering Equation Solver fo r Microsoft
Windows Operating Systems. F-Chart Software Co., Middleton, WI.
Kritikos, H.N. and Schwan, H.P. 1975. The distribution of heating potential inside lossy
spheres. IEEE Trans. Biomed. Eng. 22(6): 457-463.
Mudgett, R. E. 1986. Microwave properties and heating characteristics of foods. Food
Technol. 40(6): 84-93.
Nykvist W. E. and Decareau R. V. 1976. Microwave meat Roasting. J. Microwave Power
11(10): 3-24.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
74
Ohlsson, T. and Bengtsson, N. 1971. Microwave heating profiles in foods- a comparison
between heating and computer simulation. Microw. Energy Appl. Newsletter 6: 38.
Ohlsson, T. and Risman, P.O. 1978. Temperature distribution of microwave heatingspheres and cylinders. J. Microwave Power 13(4): 303-309.
Padua, G.W . 1993. M icrowave heating of agar gels containing sucrose. J. Food
Sci. 58(60): 1426-1428.
Pangrle, B.J., Ayappa, K.G., Davis, H.T., Davis, E.A. and Gordon, J. 1991.
M icrowave thawing of cylinders. Journal AIChE 37(12): 1789-1800.
Shivhare, U.S., Raghavan, G.S., and Bosisio, R.G. 1992a. Microwave drying of com I.
Equilibrium moisture content. Trans, of the ASAE 35(3): 947-950.
Shivhare, U.S., Raghavan, G.S., and Bosisio, R.G. 1992b. Microwave drying of com II.
Constant power, continuous operation. Trans, of the ASAE 35(3): 951-957.
Shivhare, U.S., Raghavan, G.S., Bosisio, R.G., and Mujumdar, A.S. 1992c. Microwave
drying of com HI. Constant power, intermittent operation. Trans, of the ASAE
35(3): 959-962.
von Hippel, A. R. 1954. Dielectric and Waves. The M IT Press. Cambridge, MA.
Yongsawatdigul, J. and Gunasekaran, S. 1996a. Microwave drying of cranberries: Part I.
Energy and efficiency. J. Food Proc. and Pres. 20: 121-143.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
Yongsawatdigul, J. and Gunasekaran, S. 1996b. Microwave drying of cranberries: Part II.
Quality evaluation. J. Food Proc. and Pres. 20: 145-156.
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Table 4.1 Explicit and implicit schemes for heat fluxes (qM and qi+i) for finitedifference analysis
Explicit scheme
qi.1=k(Ti.r Ti)/dR
Implicit scheme
q j. i=K).5k(Tj. i-Tj+Tn,j.i-Tn,i)/dR
qi+l=k(Ti+l-Ti)/dR
qi+l=0.5k(Ti+1-Ti+Tn,i+i-Tn.i)/dR
Table 4.2 Dielectric, physical, and thermal properties of 2% agar gela
Property
Specific heat capacity, Cp (kJ/kg.°C)
Thermal conductivity, k (W/m.°C)
Density, p (kg/m3)
Dielectric Constant, k
Dielectric loss, k”
a Data From Barringer et al. (1995)
'
Value
4.2
0.60
1070
73.6
11.5
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77
Table 4.3 Parameter values of an Infinite cylinder used for validation of the implicit
finite difference model.
Parameter
Value
Heat transfer coefficient, h
40
0.03a
Radius, R (m)
0 .T
Height, Z (m)
Thermal conductivity, k (W/m.°C)
0.6
2
Biot num ber5
1.34
Coefficient, Ct
1.6°
Eigenvalue, Ci
4
Initial uniform temperature, Tjnt
60
Ambient temperature, Ta;r( °C)
100
T im e, s
1.5* 10'5
Thermal diffusivity, an (m2/s)
0.3d
Fourier number, F0
a Chosen to designate an infinite cylinder Z/R>10
b Bi=hR/k
c Table value from Incorpera and DeWitt (1996)
d Chosen to designate conditions can be approximated Fo>0.2
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78
Sample axis
Node i
dR=0.1cm
Heat flu x from
node i+1 to node i
Heat flux from
node i-1 to node i
Figure 4.1 Schematic of a typical inner node of a cylindrical object for one­
dimensional analysis (R,=0 along the sample axis).
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79
ON
PR=3
OFF
n
PR=2
OFF
ON
PR=1
0
1
2
3
4
5
6
7
8
9
Tim e, min
Figure 4.2 Power-on/-ofT periods employed during microwave applications for
pulsing ratios (PR) of 1,2 and 3, with total microwave power on-time of 3
min. The solid segments represent power-on periods, dotted segments
represent power-off periods, and vertical dash segments represent the power
on/off transition.
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10
80
Temperature (°C)
50
40
30
20
approximate analytical solution
10
numerical solution
0
0
0.5
2
1.5
Radial distance from center (cm)
1
2.5
3
Figure 4.3 Analytical and implicit finite-difference numerical temperature profiles
in an infinite cylinder (see Table 3.3 for parameter values used in
analytical approximation).
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81
60
—
-
A
55 -{
FD increment=0.1 c m
FD in c re m e n ts . 15 cm
50
Temperature (°C)
TR
FD increm ent=0.2 cm
45 i
30 25 -
50
4
2
0
Radius distance from center (cm)
Figure 4.4 Comparison of temperature profiles according to implicit finitedifference model (FD) and that of the temperature-rise (TR) model by
Padua (1993), shell thickness increments are 0.1,0.15, and 0.2 cm.
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82
30
PR=1
Temperature (°C)
25
PR=2
PR=3
20
15
10
5
0
1
2
3
4
Radial distance from center (cm)
Figure 4.5 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in
radius and 7 cm in height) after 1 min of total microwave power
application at different pulsing ratios (PR). The power incident is from the
sample outer periphery (the radial distance of 3.5 cm from the center line).
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40
35
PR=1
Temperature (°C)
PR=2
PR=3
30
25
20
15
0
0.5
1
1.5
2
2.5
3
3.5
Radial distance from center (cm)
Figure 4.6 Measured temperature profiles in cylindrical agar gel samples (3.5 cm
radius and 7 cm in height) after 2 min of total microwave power
application at different pulsing ratios (PR).
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84
50
PR=1
Temperature (°C)
45
PR =2
PR =3
40
35
30
25
0
0.5
1
1.5
2
2.5
3
3.5
Radial distance from center (cm)
Figure 4.7 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in
radius and 7 cm in height) after 3 min of total microwave power
application at different pulsing ratios (PR).
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85
60
50
40
9
£3
2
30
2L
E
PR=1
20
— - P R -2
PR=3
10
•
PR=1
■
PR=2
•
PR=3
0
0
100
200
300
400
500
6 00
Time (s)
Figure 4.8 Measured (M) and finite*difference model predicted (P) temperature
profiles at different pulsing ratios (PR) at a radial distance of 0 cm
(center line).
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86
40
35
30
(0 o) a jn ie ja d u ia i
25
20
PR=1 (P)
PR =2 (P)
15
PR=3 (P)
PR=1 (M)
10
PR = 2(M )
PR = 3(M )
5
0
0
100
200
300
400
500
600
Time (s)
Figure 4.9 Measured (M) and finite-difference model predicted (P) temperature
profiles at different pulsing ratios (PR) at a radial distance of 1 cm from
the central line.
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87
CHAPTER 5
COMPARISON OF TEMPERATURE PROFILES IN A CYLINDRICAL MODEL
FOOD BASED ON MAXWELL’S AND LAMBERT’S LAW
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5.0 Notations
As = surface area, m2
B = magnetic induction, Wb. m '2
c = velocity of radiation, m.s'1
ci, c2, c3 and c4: parameters involved in analytical solution of Maxwell’s equations
C,,C2 and C3: parameters involved in the evaluation of electric field intensity
Cp = specific heat capacity, J.g'l.°C'1
D = electric displacement, C.m'2
E = electric field intensity, V.m'1
Ec = conjugate electric field intensity, V.m’1
/ = frequency of incident radiation, Hz
h = heat transfer coefficient, W .m'VC*1
H = magnetic field intensity, A .m '1
Hc = conjugate magnetic field intensity, A.m'1
J = current flux A.m'2
Jo = Bessel function of first kind in zero order
Ji = Bessel function of first kind in first order
k = thermal conductivity, W .m'l.° C l
p = microwave source power term, W .m 2
P = dimensionless microwave source power term
r = radial distance of the sample, m
r* = dimensionless radial distance of the sample
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89
R = radius, m
Re=real part of a complex number
Im=imaginary part of a complex number
S = Poynting vector, W.m'2
t = time, s
T = sample temperature, °C
Ts = surface temperature of the sample, °C
To. = ambient temperature, °C
u = dimensionless electric field intensity
v = dimensionless real field component
w = dimensionless imaginary field component
Y0 = Bessel function of second kind in zero order
Yi = Bessel function of second kind in first order
Greek letters
a = wave number, m '1
(Xo = free space wave number, m '1
P = attenuation constant, m 1
%: parameter involved in analytical solution of Maxwell’s equations
5 = Kroncker delta
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90
e = permittivity, F.m '1
£o = free space permittivity, F.m'1
e’ = dielectric constant, F.m '1
£ = dielectric loss factor, F.m '1
k=
relative permittivity
k’ =
k”
relative dielectric constant
= relative dielectric loss factor
Xm= wave length in sample, m
(A= permeability, H.m '1
|io = free space permeability, H.m'1
a = electric conductivity, mho, m '1
a) = angular frequency, rad.s'1
V= parameter involved in analytical solution of Maxwell’s equations
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5.1 Abstract
Two-percent agar gel cylinders were heated using pulsed and continuous
microwave energy. Temperature distribution (TD) inside the sample was measured
and compared with numerical predictions based on the Lambert’s law and Maxwell’s
equations'. The Maxwell’s equations account for the standing wave effect inside the
sample, the Lambert’s law does not. The results show that the predictions based on
the Maxwell’s equations are statistically more accurate than those based on the
Lambert’s law, especially around the sample edge. The measured TDs and the
corresponding predictions using both models indicate better temperature uniformity
in the agar gel cylinders under pulsed microwave heating than under continuous
microwave heating.
1 Also referred as “Maxwell’s field equations”
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92
5.2 Introduction
Heating food materials by electromagnetic radiation is widely used in
commercial, industrial, and household applications. These applications rely on the
internal heat generation due to interaction between the material and electromagnetic
radiation. The propagation fields based on Maxwell’s equations are associated with a
power flux, namely, Poynting vector of harmonic fields (Ayappa et al., 1991; Chen,
1992).
The oscillating power distribution within the material is useful for predicting the
locations of hot spots (Fu and Metaxas, 1992). The oscillation power absorbed by thin
slabs of food is related to dielectric properties. This is the key to several patents claimed
by a research group at Pillsbury Corp., Minneapolis, MN (Atwell et al.,1990, 1992;
Peschek et al., 1991a,1991b and 1992). The basic principle of these patents is that the
heating rate of a given food layer is a combined function of the layer’s dielectric
properties, thickness and thermal mass.
In a sphere or a cylinder, overheating frequently occurs at the focal points in the
sample during continuous microwave application (Ohlsson and Risman, 1978). Pangrle et
al. (1991) proposed cylindrical models to study phase changes of frozen brine (salted
water) thawed by microwave energy. In Chapter 4, the Lambert’s law was used to
describe the electric field behavior. However, the Lambert’s law does not account for
node/anti-node formation (Barringer et al., 1995). In this Chapter, Maxwell’s equations,
related to space and time dependence, were solved to determine the absorbed microwave
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93
power. The absorbed power was applied intermittently to simulate the temperature
distribution in 2 % agar gel cylinders heated by pulsed microwave energy.
The objectives of this Chapter were to:
•
Investigate the temperature distribution (TD) in 2% agar gel cylinders heated by
continuous and pulsed microwave energy.
•
Predict the sample TD by solving the Lambert’s law and Maxwell’s equations of
microwave energy absorbed by the sample.
•
Compare the measured and predicted TD in the sample under continuous and pulsed
microwave heating.
5.3 Theory and Analyses
5.3.1 Heat Transfer Equation
The heat equation in a material is:
St
pCp — = V •(kVT) + p(r,t)
at
(5.1)
where p, Cp and k are density, specific heat capacity, and thermal conductivity,
respectively. In the case of microwave heating, the absorbed power, p(r, t) is determined
from real and imaginary components of the electromagnetic field. The transient one­
dimensional heat equation is:
P C ” TOt~ = ~r dr
T ( k r lor
T )+ p ( r ’°
The boundary conditions for the heat equation are:
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(52)
94
(5.3)
dr
and
(5.4)
The initial condition is:
T(t=0) = Tj fo rO < r< R .
(5.5)
The absorbed microwave power is obtained by solving the Maxwell’s equations.
5.3.2 Wave Propagation
A propagating electromagnetic wave is composed of oscillating electric (£) and
magnetic (B) Field components. Maxwell’s equations describing their space and time
variations are:
and
(5.6)
where E and H are the electric and magnetic fields, J = current flux, D = electric
displacement and B = magnetic induction. The constitutive relations relating J, D and B,
to E and H are:
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95
J = a((o)E(t),
D = £(co)E(t) and
B = fi( 0))H (t)
(5.7)
where E=E*e'l(ut and H=HVIUJt. Alternatively e',aican be used to express the time
dependence. Equations 5.6 and 5.7 yield:
V x £ * =ia)ii((o)H'
(5.8)
and
V x H * = \a((o)-i(DE((o)]p' = -icoe' E ' ,
(5.9)
where the complex dielectric constant, e\ is defined as:
e ’ (co) = £(cu) +
= £' ((D) + i£" ((D)
(5.10)
(0
The material’s ability to store and dissipate electric energy is represented by the real and
imaginary parts of £*, e’=Re(e*) and e”= Im(E*) respectively, and e” accounts for energy
losses through dissipation. The electric conductivity a(co), dielectric constant e((d), and
magnetic permeability p((0 ) are generally complex functions of frequency of radiation, u).
Neglecting magnetic effects, the magnetic permeability |n(co) is approximated by
its value po in free space. The time derivative of a and £ can also be neglected, since the
time scale of electromagnetic propagation is smaller than the time scales for thermal
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96
diffusion. With these assumptions and the condition of electroneutrality of the medium,
which implies V- (e*E) =0, equations 5.8 and 5.9 can be combined to give:
~
) + V 2E ' + kl2E m
(5.11)
where kt2 =a>2pi0e0(fc'+iK")
(5.12)
The relative dielectric constant k ’ and the relative dielectric loss
K '= e’/ e Q and
K"=£"/e0
k”
are:
(5.13)
To simplify the notations, the superscript “stars” on E and H will be dropped from this
point on. The propagation constant ki is a complex quantity:
kt = a + iP,
(5.14)
where a and (3 are related to the dielectric properties of the material and frequency of
radiation by:
and
where
tan<5=— .
(5.17)
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97
where c is the speed of light and a) is replaced by I n f , f is the frequency of radiation.
The phase constant a is related to the wavelength of radiation in the material (Am) by:
which in free space reduces to Ao=cI f . The attenuation constant P is a rate constant for the
decay of the incident field of intensity E 0. For instance, in a semi-infinite sample the
interior field obeys the equation:
£ = £„<?■*.
(5.19)
5.3.3 Power dissipation
The power flux associated with a propagating electromagnetic wave is
represented by the Poynting vector (S) and the time average flux for harmonic fields:
S= ^E xH c
(5.20)
The power dissipated per unit volume is:
p(r) = —Re(V- S) =±(oe0K"(E £ r ).
(5.21)
Therefore, given the electric field intensity in the material the local power dissipated is
calculated by equation 5.21. The subscript “c” is the conjugate field intensity.
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98
5.3.4 Maxwell's equations for a cylinder
Microwaves are assumed to be incident on a cylinder as shown in Fig. 5.1. For
propagation of an azimuthal condition in polar coordinates, the electric and magnetic
components orient along a cylindrical surface of a uniform intensity varying only in the
direction of propagation along r-axis (Section 3.6). The wave equation, Equation 5.9 is:
d ' E 1 dE
r-H
— 1
- k f E = 0,
d r 2 r dr
(5.22)
fo r
0 < r< R
Maxwell’s equations are rendered dimensionless by substituting the relationships:
r
E
r* = — and u = —
R
E0
(5.23)
Noting that u = v+iw, equation 5.22 yields to the following two equations that are solved
with their appropriate boundary conditions:
(5.24)
and
dr*2
r* dr*
+ ynv + %v=0,
(5.25)
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where \j/ = R2a)2Ho£oK’ and
x= R2a)2Ho£oK”. The boundary conditions at r* = 0 are:
dv _ dw
dr* dr*
(5.26)
while boundary conditions at r* = 1 are:
dv
dr *
+
C .V +
C-,
w = c.
(5.27)
and
dw
+ c,w - c 2v = c 4,
dr *
(5.28)
where
c | —Ret q
J l ( a 0R ) J 0(ccQR ) + Y l(cc0R)Y0(ccQR )
J 02( a 0R) + Y02( a 0R)
c-, =
n [ j 02{ a 0R) + y02( a 0Ji)]’
* J 0' ( a 0R) + Y0' ( a 0R)
and
J 0(ct0R)
n J 0- ( a 0R) + Y0- ( a 0R)
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100
where J and Y are Bessel functions, subscripts 0 and 1 are zero- and first-order of Bessel
functions and Oo = free space wave number. Eo is the intensity of the incident field
related to power term Pqby:
„
c e nEo~
/ >o = - ^ - r - .
n a 0R
(5.29)
The power term as a function of r* is calculated from v and w:
P(r*,t) = R 2Q)£0K"E0(v2 + w 2)
fo r
0 < /? * < !.
(5.30)
5.3.5 Analytical Solution for Absorbed Power
For the case of constant properties, the coupled equations 5.24 and 5.25 can be
solved with the help o f :
C ____________________________ C 3 + / C 4_____________________________
1 - (C, + iC, )/, (C2 + iC3) + (c,+ ic2)JQ(C2 + /C3) ’
(531)
where
C, + iC3 = yj\}f + i%.
The analytical solutions of the coupled equations are:
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101
v(r*) = Re{C, J 0[(C, + i'C, )r *]}
(5.32)
and
w(r*) = Im{C| y 0[(Cl + iC 2)r * J
(5.33)
5.3.6 Temperature Distribution Prediction
Applying the analytical solutions of the electromagnetic equations to the heat
equation (equation 5.2) by a finite-difference heat transfer method described in Chapter 4,
the interior temperature profile of a microwave-heated sample can be predicted. For
pulsed microwave heating, the absorbed power = 0 in the heat transfer equation during
power-off periods.
5.4 Methods and Materials
5.4.1 Microwave process
Two-percent agar gel cylinders, 3.5 and 4 cm in radius, were prepared (method
described in Section 3.2). The microwave oven operating at 250-W oven setting was used
as the heating source. Microwave pulsing ratios (duty cycles) of 1 (continuous), 2 (30-s
power-on, 30-s power-off) and 3 (20-s power-on, 40-s power-off) were applied. The
pulsing ratio, PR, is defined as:
pp _
+ 1off )
(5.34)
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102
where to„ and W = duration of the microwave power-on and -off per duty cycle,
respectively. The total microwave power-on time for every pulsing ratio was 3 min in the
experiments. The TD in the samples was measured (method described in Section 3.3).
The microwave power absorbed by the samples (Pabs) and average surface heat transfer
coefficient in the microwave oven were determined (method described in Section 3.4 and
3.5, respectively).
5.4.2 Temperature Prediction
The physical and thermal properties (Table 4.2) of agar gel reported by Barringer
et al. (1995) were used for the calculation. The finite-difference model proposed in
Section 4.3.1 was used to solve the heat transfer equation numerically to predict the TD
in the agar gel cylinders. The convection effect inside the microwave oven (ambient air
effect that contributes to the heat transfer boundary conditions of the agar gel cylinders)
was also included in the model. The absorbed microwave power applied to the heat
transfer equation was based on either Maxwell’s equations (Section 5.3.2 and 5.3.3) or
the Lambert’s law (Section 4.3.2). Maxwell’s absorbed power was solved using the
MATLAB software (MathWorks Inc., Natick, MA). The X2-test was used to accept or
reject the hypothesis that measured and predicted temperatures were the same, at P = 0.01
(Bender et al., 1981).
__ v r
X ““
(Tmeasured - T* predicted'
_
•,
J
sc n c \
predicted
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103
5.5 Results and Discussion
Maxwell’s and Lambert’s microwave power distributions in the 3.5- and 4-cm
radius agar gel cylinders as a function of radial distance are shown in Fig. 5.2. The field
components were assumed to orient along the z-axis (this assumption was verified to be
valid in Section 3.6). The Maxwell’s power distributions present oscillating patterns
because of the space and time dependence of electromagnetic fields. Due to the
node/anti-node effect of waves is not considered, the Lambert’s power distributions
decay exponentially from the sample surface to the center. The exclusion of standing
wave effect also caused the Lambert’s power at the outer portion of the sample along the
r-axis being significantly larger than the Maxwell’s power. The power distribution in a
3.5-cm radius sample cylinder was generally larger than that in a 4.0-cm radius sample
cylinder at the same radial distance. Theoretically, for smaller cylindrical samples, their
oscillating power patterns are stronger due to the penetrating nature of the microwaves
(Fu and Metaxas, 1992).
In the case of a cylindrical body, the annular incremental volume
(V j)
along the r-
axis converges as a function of 1/r2 from the sample surface to the center (the same
sample height). However, the microwave power (P) decreases exponentially along the
same direction (surface to center). Due to the combined effect of power and incremental
volume (P/Vj), the volumetric power (power density) represents the center focusing effect
of microwave energy along the r-axis (Fig. 5.3). The existence of the focusing volumetric
power at the sample center depends on the sample critical length. For a very large
sample, the center focusing of microwave power density may not be observed.
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104
The measured and predicted sample TDs at the end of 1- and 3- min heating by
microwave energy with PR = 1 (continuous) and PR = 3 are shown as Figs. 5.4 to 5.7 (2min results shown in Appendix B.2). The results of PR = 2 (Appendix B.2) followed the
same trend as PR = 3, only less significant. For all the microwave processes, the largest
temperature difference between any two consecutive measured points occurred at R = 0
and R = 1 cm. As the duration of microwave heating increased, the difference increased.
The pulsed microwave heating resulted in better temperature uniformity and minimized
the central focusing effect.
For the continuous microwave heating process, the sample temperature profiles
predicted by the Maxwell’s equations represented an oscillating pattern and agreed with
the absorbed power pattern (Fig. 5.3). For pulsed microwave heating, the sample
temperature profiles based on Maxwell’s prediction represent smooth curves. These
results are expected because thermal diffusion occurred during the power-off periods of
pulsed microwave applications and equalized the oscillating temperatures established
during microwave power-on periods.
The measured temperatures after 1-, 2- and 3-min of total microwave applications
(continuous and pulsed) were compared to the predicted temperatures based on
Maxwell’s equations as well as that based on the Lambert’s law using the x 2-test (Table
5.1). The x 2 = 45.6 at P = 0.01 with degrees of freedom = 26 (Bender et al. 1981). The
predicted TD based on either Maxwell’s or Lambert’s power is not significantly different
from the measured TD (P > 0.01). However, the Lambert’s prediction resulted in
overestimation of temperatures near the sample surface. The calculated x 2 increased as
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105
microwave heating time increased for the Lambert’s predictions. The calculated x2 values
were fairly consistent as a function of time for the Maxwell’s predictions. These
statistical results indicate that Maxwell’s prediction is more accurate than the Lambert’s
for long heating times.
5.6 Conclusions
Pulsed microwave heating improves the temperature uniformity in heated
samples. The local hot spot observed at the sample center during continuous microwave
application was substantially minimized by pulsed microwave applications. The predicted
temperature distribution in the sample based on either the Lambert’s law or Maxwell’s
equations was statistically accurate as compared to the measured temperature
distribution. The power formulation based on Maxwell’s equations is more accurate than
that based on the Lambert’s law, especially for longer heating time. The Lambert’s law is
based on the assumption that the sample critical length is semi-infinite and does not
consider standing wave effect, therefore, leads to the overestimation of temperatures near
sample surface.
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106
Table 5.1 The Chi-square values for Maxwell’s and the Lambert’s predictions
compared to the measured temperatures during microwave heating
Microwave processing time
(min)
1
2
3
X2of Maxwell’s prediction
5.56
3.86
4.24
%2 of Lambert’s prediction
7.25
10.01
16.06
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107
Incident microwaves
normal to the surface
Top view of
cylindrical sample
Figure 5.1 A cylindrical sample exposed to plane waves normal to the surface with
sub-shell sections for numerical modeling.
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108
25
4 cm, Maxwell's
4 cm, Lambert's
20
3.5 cm, Maxwell's
©
§
Q. 15
T3
(D
X3
W
O
CO
n
as
© 10
>
as
3.5 cm, Lambert's
%
5
0
0
1
2
3
4
5
Radial distance from center (cm)
Figure 5.2 Microwave power absorbed in 2 % agar gel cylinders (3.5-cm and 4-cm
radius) as a function of radial distance from sample center. The electric
field is oriented along the vertical z-axis of the cylinder.
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109
14
Absorbed microwave density (MW/m3)
4 cm, Maxwell's
12
4 cm, Lambert's
3.5 cm, Maxwell's
10
3.5 cm, Lambert's
8
6
4
2
0
0
1
2
3
4
5
Radial distance from center (cm)
Figure 5.3 Microwave power density absorbed in 2% agar gel cylinders (3.5-cm and
4-cm radius) as a function of radial distance from sample center. The
electric field is oriented along the vertical z-axis of the cylinder.
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110
60
MP, 1 min
LP, 1 min
MP, 3 min - - - LP, 3 min
Temperature (°C)
50
40
30
20 <>>'
10
0
0
1
2
3
4
5
Radial distance from center (cm)
Figure 5.4 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 4-cm
radius 2 % agar gel cylinders after 1 and 3 min of microwave heating
with a pulsing ratio of 1 (i.e. continuous).
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111
60
MP, 1 min
LP, 1 min
MP, 3 min - - - LP, 3 min
Temperature (°C)
50
40
30
20
0
2
3
4
Radial distance from center (cm)
1
5
Figure 5.5 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 4-cm
radius 2 % agar gel cylinders after 1 and 3 min of microwave heating
with a pulsing ratio of 3.
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L12
60
MP, 1 min
Temperature (°C)
MP, 3 min
■
- -L P , 1 min
M, 1 min
" LP, 3 min
M, 3 min
40
0
1
2
3
Radial distance from the center (cm)
4
Figure 5.6 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 3.5cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating
with a pulsing ratio of 1 (i.e. continuous power).
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113
MP, 1 m in ................. LP, 1 min
MP, 1 m in -
•
M, 1 min
■ LP, 3 min
SO
40
s
s
3
-
* -
30
20
.I-
0
1
2
3
4
Radial distance from the center (cm)
Figure 5.7 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 3.5cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating
with a pulsing ratio of 3.
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114
5.7 References
Atwell, W.H., Peschek, P., Krawjecki, M. and Anderson, G. 1990. U.S. patent 4926020
May 15.
Atwell, W.H., Peschek, P., Krawjecki, M. and Anderson, G. 1992. U.S. patent 5101084
May 15.
Ayappa, K.G., Davis, H.T., Crapiste, G., Davis, E.A. and Gordon, J., 1991a, Microwave
heating an evaluation of power formulations. Chem. Engineering Science. 46 :
1005-1016.
Barringer, S.A., Davis E. A., Gordon, J., Ayappa, K. G. and Davis, H. T. 1995
Microwave- heating temperature profiles for thin slabs compared to Maxwell and
Lambert Law prediction. J. Food Science. 60(5): 1137-1142.
Bender, F.B., Douglass, L.W., and Kramer, A. 1981. Statistical Methods for Food and
Agriculture, Food Products Press Inc., Binghamton, New York.
Cheng, D. K., 1992. Field and Wave Electromagnetics. Addison-Wesley Publishing
Company. Reading, MA.
Fu, W. and Metaxas, A. 1992 A mathematical derivation of power penetration depth for
thin lossy materials. J. Microwave Power 27(2): 217-222.
Ohlsson, T. and Risman, P.O. 1978. Temperature distrubution of microwave heatingspheres and cylinders. J. Microwave Power 13(4): 303-309.
Pangrel, B.J. Ayappa, K.G. Davis, H.T. Davis, E.A. and Gordan, J. 1991 Microwave
thawing of cylinders J. AIChE 37(12): 1789-1800.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Peschek, P., Atwell, W.H., Krawjecki, M. and Anderson, G. 1991a. U.S. patent 4988841
January 29.
Peschek, P., Atwell, W.H., Krawjecki, M. and Anderson, G. 1991b. U.S. patent 5008507
April 16.
Peschek, P., Atwell, W.H., Krawjecki, M. and Anderson, G. 1992. U.S. patent 5140121
August 18.
Rizvi, S.S.H. and Mittal, G.S. 1992. Experimental Methods in Food Engineering. AVI
Book Co., New York, NY.
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116
CHAPTER 6
EFFECT OF EXPERIMENTAL PARAMETERS ON TEMPERATURE
DISTRIBUTION DURING CONTINUOUS AND PULSED MICROWAVE
HEATING
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117
6.0 Notations
a = Effect of processing time
(3 = Effect of pulsing ratio
Y=
Effect of radius (sample size)
C = Constant of general linear model
a, ^Coefficients of general linear model
Pabs = Microwave absorbed power, W
PR = Pulsing ratio
t = Time, s
toff = Duration of the microwave power off per duty cycle, s
ton = Duration of the microwave power on per duty cycle, s
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118
6.1 Abstract
The temperature distribution (TD) in two-percent agar gel cylinders heated by the
same average microwave power level was evaluated. The same average power level
based on the oven settings (OSP) and the absorbed power (AP) was applied to the
sample. A 3 x 2 x 2 factorial design was used to evaluate the effect of different
experimental variables: (A) heating time (1,2 and 3 min), (B) microwave application
(continuous vs. pulsed by employing the same average OSP) and (C) sample size (3.5and 4-cm radius). Analysis of variance under the same average OSP condition indicate
that all variables affected sample TD significantly. Overall, the pulsed microwave
treatment resulted in a more uniform TD than the continuous treatment under the same
average OSP condition. The interactions among variables under the same average OSP
condition were also observed; the results showed that pulsed microwave heating should
be preferred to the continuous microwave heating when temperature uniformity in the
sample is critical. The agar gel cylinders were also heated under the same average AP
condition and present different TDs when different microwave pulsing ratios were
employed.
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119
6.2 Introduction
Microwave energy can enhance interior heating of foods and other materials.
Such heating mechanism of electromagnetic radiation depends on the dielectric properties
and dimensions of the food material (Zhang and Datta, 1999). The power level applies to
foods is also very important (Decareau and Peterson, 1986). Tulasidas et al. (1994)
reported the drying of Thompson seedless grapes using pulsed microwave energy at
different power levels. They concluded that the quality of dried grapes (raisins) was
highly acceptable in all pulsed microwave duty cycles.
Shivhare et al. (1991) studied the effects of pulsed microwave and hot air drying
characteristics of com. They reported that the use of pulsed microwaves for com drying
resulted in reducing the heat loss through exhaust air and actual time that grains was
exposed to microwaves. They also indicated that the magnitude of microwave power and
the pulsing period affected the drying rate and product quality. Yonsawstdigul and
Gunasekaran (1995) investigated pulsed microwave-vacuum drying of cranberries.
Comparing continuous and pulsed modes at microwave output setting of 250-W, they
concluded that the pulsed microwave energy was more efficient than the continuous
mode. Shorter duty cycles provided more favorable drying efficiency in the pulsed mode.
Yang and Gunasekaran (2001) proposed a model for predicting the interior
temperature distribution (TD) during pulsed microwave heating based on the Lambert’s
law. The uneven interior TD during continuous microwave heating was dramatically
reduced by pulsed microwave heating. However, the effect of the sample dimension and
the microwave power level were not investigated.
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120
The objectives of this part of research were to:
•
Examine the effects of sample size, pulsing ratio and microwave processing
duration on the sample temperature distribution by employing the same
average microwave power level based on the oven settings (OSP).
•
Evaluate the effect of the same average microwave power level based on the
absorbed power (AP) on the sample temperature distribution by employing
different pulsing ratios.
6.3 Methods and Materials
Two-percent agar gel cylinders were prepared (method described in Section 3.2).
The sample cylinders were 3.5 x 7 cm and 4 x 7 cm, radius x height (i.e. volume of the
later was about 30 % more than the former). The sample temperature distributions (TDs)
were measured (method described in Section 3.3). The absorbed power in the microwave
oven (Pabs) at two continuous microwave power output settings, 250 and 500 W, was
determined (method described in Section 3.4).
6.3.1 Microwave Heating Processes
Two microwave heating processes of the same duty cycle of 60s (to„ + toff) under
the same average OSP condition (the same average microwave power level based on the
oven settings) were studied:
1. Pulsed process at the 500-W oven setting using a pulsing ratio of 2 (PR = 2,
defined in Section 4.4.1) with 30-s tmand 30-s toff.
2.
Continuous microwave process at the 250-W oven setting (PR = I).
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121
These two microwave processes were selected to maintain under the same
average OSP condition. The total heating time for both the same average OSP processes
was 3 min. The sample TDs were measured after 1,2 and 3 min of total heating time
(note: the sampie TDs presented in Chapters 4 and 5 were measured based on the total
microwave power-on duration).
Two microwave modes, continuous and pulsed, each under the same average
absorbed power level (AP) o f 225 W were applied to 4-cm radius agar gel cylinders:
A. Continuous microwave power (PR=1, 250-W oven setting, 225-W absorbed
power) for 4 min.
B. Pulsed microwave power (PR=1.47, l63/77s power-on/off, 500-W oven
setting, 331-W absorbed power) for 4 min.
The sample temperatures were measured (method described in Section 3.3) at the
end of the 4-min processing duration. The measured TDs were also compared to the
predicted temperatures based on the Maxwell’s model described in Chapter 5.
6.3.2 Statistical Analysis
Analysis of variance (ANOVA) was performed to evaluate the effect of
experimental variables on sample temperature under the same average OSP condition.
The three main effects, heating time (a), pulsing ratio based on the same average OSP
condition (|3) and sample radius (y) were treated as fixed factors, i.e. levels of each factor
were discrete (non-continuous). The microwave energy was applied to the samples in a
non-continuous manner during the pulsed treatment, therefore, the heating time was
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122
considered to be a discrete factor. Each measured temperature was treated as a response
in a general linear model (GLM) for correlation analysis. The radial distance was treated
as a co-variance (continuous factor). The model was also used to evaluate a 3 x 2 x 2
factorial experimental test: (1) heating time (1,2 and 3 min), (2) microwave application
(continuous vs. pulsed) and (3) sample size (3.5- and 4-cm radius). The significant
analysis included (1) main effects (a, (3 and y), (2) primary interactions (a*(3, a*y and
P*y) and (3) the second-order interaction (a*p*y) (Brender et al., 1981; Fienberg, 1981).
MINITAB 13.0 (Minitab Inc., State Park, PA) was used for the statistical analysis.
6.4 Results and Discussion
6.4.1 Microwave Absorbed Power
The absorbed powers were 225 ± 3.7 W and 331 ± 3.6 W at the oven settings of
250 W and 500 W, respectively, for a 3.5-cm radius sample. 234 ± 2.9 W and 344 ± 3.1
W at the oven settings of 250 W and 500 W, respectively, for a 4-cm radius sample
(Section 3.4). This indicated that heating at the 250-W oven setting under the continuous
mode (PR =1) should result in a higher sample temperature than heating at the 500-W
oven setting with PR= 2. The power delivered (PD) by the microwave oven and the
power absorbed (Pabs) by the sample are not the same. The Pabs/PD ratio is often referred
as the energy-transfer efficiency and it depends on the sample load volume (Table 3.1)
and the sample’s ability to store and dissipate the delivered microwave energy. The
energy-transfer efficiency can range from 18 to 100% at different load volumes in a home
microwave oven; only water samples of 1000 mL or more satisfies a 100 % efficiency
(Mudgett, 1986).
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123
6.4.2 Temperature Distribution (TD) in Samples Under the same average OSP
Condition
Figs. 6.1 and 6.2 present the TD in 3.5- and 4.0-cm radius agar gel cylinders after
1-, 2- and 3-min heating by both microwave processes (PR = 1 and 2 under the same
average OSP condition). The TD at different pulsing ratios resulted in a small difference
at the end of one minute of microwave heating. As the microwave heating duration
increased, the continuous microwave heating (PR=l) resulted in higher interior sample
temperatures. The largest temperature difference occurred at the sample center when it
was heated using PR=1 and 2. For continuous microwave heating (PR=1), temperature at
the sample center was higher than at other locations (radial distances of 1,2, 3 and/or 4
cm). The TD obtained under pulsed microwave heating remained considerably flat, i.e.
TD was fairly uniform. As the microwave heating time increased, the temperature
profiles in the samples heated by continuous microwave energy exhibited increases in
temperature variation between sample center and surface.
The TD in the 3.5-cm radius sample was more uneven compared to the 4-cm
radius sample when heated by continuous microwave power. This can be related to the
penetrating and decaying nature of microwave energy (Mudgett, 1986; Cheng, 1992). As
the penetration depth in the 2 % agar gel (2.6 cm) and microwave power remain the
same, the absorbed power along the radial axis in a 3.5-cm radius agar gel cylinder is
greater than in a 4-cm radius agar cylinder (Figs. 5.2 and 5.3). In the 3.5-cm radius
sample, there was more focusing power at the radial center.
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124
6.4.3 General Linear Model Under
Figs. 6.1 and 6.2 show that the measured temperatures (responses) are nonlinearly correlated to the radial distance (RD) under the same average OSP condition.
Therefore, a quadratic term of radial distance (RD2) was included to fit GLM. The
analysis of variance (ANOVA) including all the interaction terms (a*RD2, P*RD2 and
y*RD2) was programmed (Appendix C.l) using MINITAB 13.0 (Minitab Inc., State Park,
PA). The interaction of a*RD 2 was not significant (P=0.062) and was dropped for a
second trial. The second model is:
T = C + axRD + a zRD 2 + a^qRD + axf}RD + asyRD + a6(JRD2 + a ^R D 2
The second trial appeared to be a good model because:
1. All sources of variation included in the GLM were significant (P <0.01, tvalues in Appendix C.2).
2. Residuals were generally scattered randomly about zero. There were no
special features or patterns, such as non-constant variance (residual plot, i.e.
residuals vs. fitted values, Fig. C .l, Appendix C.2).
3. The responses (temperatures) are normally distributed because the points
come close to form a straight line in the normal probability plot of residuals
(Fig. C.2, Appendix C.2).
According to the second trial, all main effects (F-values of a, (3 and y, Table 6.1)
on sample temperatures are significant (P < 0.01) and each main effect significantly
interacts with the other two main effects because all primary interactions (F-values of
a*P, a*y and P*y, Table 6.1) were significant (P < 0.01).
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125
The interaction plots (Figs. 6.3,6.4 and 6.5) demonstrate the primary interaction
effects on the response (temperature). The relative strength of the effects was also
compared. The effect of pulsing ratio vs. processing time is demonstrated in Fig. 6.3. The
mean temperature (Y-axis) is the average of measured temperatures at all locations under
that specific condition. As processing time increased, the temperature for PR=2 is
significantly lower than that for PR=1. This is mainly due to the power absorbed under
the experimental conditions (as described in Section 6.4.1).
As the processing time increased, the heating rate of a 3.5-cm radius sample was
greater than that of the 4-cm radius (Fig. 6.4). The effect of pulsing ratio is dependent on
the sample radius (Fig. 6.5). For the 3.5-cm radius sample, the sample temperature
increased greatly as PR changed from 1 to 2. Even though the total microwave power
output was at the same average OSP, for the 4-cm radius sample, the temperature was
relatively unaffected by the pulsing ratio. The microwave power is a decaying function
penetrating into the sample from the surface. The decaying function is significantly
related to the sample radius.
6.4.4 Effects o f Different Pulsing Ratios on Sample Temperature Distribution Under
The Same Average Absorbed Condition
Fig. 6.6 shows the predicted and measured TD after the agar gel cylinder heated
by the two microwave modes under the same average absorbed power condition The
measured center temperature of the gel cylinder heated by Mode A and B were 42.7 and
34.2 °C respectively. The sample center temperature heated by continuous microwave
(Mode A) was higher than heated by pulsed microwave (Mode B). The temperatures at
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the mid-annular section along the sample radius were approximately the same for both
Modes. During the power-off period, the temperatures near the surface tended to be lower
due to convective cooling. Pulsed application of microwave energy can produce more
uniform sample TD than the continuous application under the same average absorbed
power (AP) condition.
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ill
6.5 Conclusions
The pulsed microwave heating resulted in more uniform TD in samples than
continuous microwave heating at the same average microwave output power based on the
oven settings (OSP). There was a significant interaction between microwave power in the
different pulsing ratios under the same average OSP condition and sample radius due to
the non-continuous application of microwave radiation and the penetrating nature of
microwaves. Even though the same average absorbed microwave power was delivered
into the samples of the same dimensions, the pulsed application resulted in different
sample temperature distribution as compared to the continuous application.
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128
6.6 References
Brender, F. E., Douglass, L. W. and Kramer, A. 1981. Statistical methods for food and
agriculture. Food Products Press, New York, NY.
Cheng, D. K., 1992. Field and Wave Electromagnetics. Addison-Wesley Puclishing Co.,
Reading, MA.
Decareau, R. V., and Peterson, R. A., 1985. Microwave Processing and Engineering. Ellis
Horwood Ltd., Chichester, Engilshed.
Fienberg, S. E. 1981. The analysis of cross-classified categorical data. MIT Press
Cambrige, MA.
Mudgett, R. E., 1986. Microwave properties and heating characteristics of foods. Food
Technol. 40(6): 3-24.
Shivhare, U. S., Raghavan, G.S.V., Bosisio R. G., and Mujumdar, A. S. 1991. Drying of
com in a pulsed microwave field. Paper No. 91-3009. ASAE International
Summer Meeting, Albuquerque, NM. June 23-26.
Tulasidas, T. N., Raghavan, G. S. V., Kudra, T., Gariepy, Y. and Akyel, C. 1994.
Microwave drying of grapes in a single mode resonant with pulsed power. Paper
No. 94-6547. ASAE International Winter Meeting, Atlanta, GA. December 13-16.
Yang, H. W. and Gunasekaran, S. 2001. Temperature Profiles in a Cylindrical Model
Food During Pulsed Microwave Heating. J. Food Science 66(7): 998-1004.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
129
Yongsawatdigul, J. and Gunasekaran, S. 1996. Microwave-vacuum drying of cranberries:
Part I. Energy use and efficiency. J. Food Processing and Preservation 20(2): 121144.
Zhang, H. and Datta, A. K. 1999. Distributions of heating potentials inside spherical
shaped foods in electromagnetic fields. Abstract No. 75-1. IFT Book of abstracts
IFT annual meeting, Chicago, IL, July 24-28.
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130
50
P R = 1,1 min
P R = 1,2 min
P R = 1,3 min
PR=2,1 min
- - o - - P R s2 ,2 min
Temperature (°C)
PR =2,3m in
0
1
2
3
4
5
Radial distance from center (cm)
Figure 6.1 Temperature distribution in 2 % agar gel cylinders, 4-cm radius, after 1,
2 and 3 min of microwave heating with a same average microwave
output power of different pulsing ratios (PR=1 @ 250-W setting and
PR=2 @ 500-W setting).
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131
50
P R = 1,1 min
PR=2, 1 min
P R = 1,2 min • - o- • PR=2,2 min
PR=1,3min - • «* - PR=2,3 min
45
Temperature (°C)
40
35 ^
30
25
*1
20
15
10
5
0
0
1
2
3
4
Radial distance from the center (cm)
Figure 6.2 Temperature distribution in 2 % agar gel cylinders, 3.5-cm radius, after
1,2 and 3 min of microwave heating with a same average microwave
output power of different pulsing ratios (PR=1 @ 250-W setting and PR=
2 @ 500-W setting).
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132
PR=1
PR=2
30
3
k-
2L
E
2
«»e
S
10
0
1
2
3
4
Processing time (min)
Figure 6.3 Mean sample temperature vs. processing time at two pulsing ratios (PR).
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133
40
R=3.5 cm
— R=4.0 cm
Mean temperature (°C)
30
20
10
0
1
2
3
4
Processing time (min)
Figure 6.4 Mean sample temperature vs. processing time for samples of different
radii (R).
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134
30
Mean temperature (°C)
25
20
15
0
2
1
3
Pulsing ratio
Figure 6.5 Mean sample temperature vs. pulsing ratio for different sample radii (R).
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135
50
Temperature (°C)
a
M ode A, Predicted
- - - M ode B, Predicted
M ode A, M easured
o
Mode B, M easured
40
0
1
2
3
4
5
Radial distance from center (cm)
Figure 6.6 Temperature distribution in 2 % agar gel cylinders, 4-cm radius, after 4
min of heating by using an average microwave absorbed power of 225 W
under continuous (Mode A) and pulsed (Mode B) microwave
applications.
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136
Table 6.1 Analysis of Variance for the Factorial Experiment in a General Linear
Model.
Source of variation
Sum of
squares
Radial distance (RD)
RD2
a: Time
P: Pulsing ratio
y: Sample radius
a*p
a*y
P*y
a*3*y
a*Radial distance
P*Radial distance
y*Radial distance
P* RD2
y*RD 2
Error
Total
184.86
568.36
7380.09
1041.20
865.64
267.53
277.67
75.26
26.12
167.89
89.79
100.21
113.10
96.15
427.39
11681.25
Degrees
of
freedom
1
1
2
1
1
2
2
1
2
2
1
1
I
1
142
161
Mean square
802.47
606.03
1906.26
824.78
732.27
140.86
99.26
77.11
13.06
83.94
170.15
154.45
113.10
96.15
3.10
F
266.6
201.3
633.3
274.0
243.3
46.80
32.98
25.62
4.34
27.89
56.53
51.31
37.58
31.95
P
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.015
0.000
0.000
0.000
0.000
0.000
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137
CHAPTER 7
OPTIMIZATION O F PULSED MICROWAVE HEATING
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138
7.0 Abstract
A simulation model was used to optimize pulsed microwave heating of precooked
mashed potato cylinders of 82.7% moisture content. The experimental variables were: (1)
sample radius: 2.4, 2.8 and 3.2 cm (or 1.5,1.75 and 2.0 time of the penetration depth of
microwave radiation), (2) microwave power-on temperature constraints (ATon): at 20 and
15 °C, (3) power-off temperature constraints (ATd): at 5 and 3 °C lower than ATon, (4)
total processing time (< 1000 s) and (5) average sample temperature (70 °C). The
evaluation showed that the samples of 2.4- to 2.8-cm radius were heated uniformly and
efficiently. The ATon is very critical for optimum pulsed microwave heating; ATon= 20 °C
is a better choice than ATon =15 °C. ATd affects the total processing time for a large
sample more significantly than for a small sample. The total processing time depends on
both ATon and ATd as well as sample radius. In the case of 2.4-cm radius samples (1.5
times of penetration depth) with ATon(ATd )= 20(3) °C , the pulsed microwave energy can
heat the sample to an average temperature of 60 °C in 336 s. This was the most efficient
process among all evaluated conditions with respect to total processing time.
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139
7.1 Introduction
Temperature uniformity is achieved in foods during pulsed microwave heating
due to thermal equalization via conduction from hot to cold region during power-off
periods (Yang and Gunasekaran, 2001). However, to optimize the pulsed microwave
heating process, certain parameters should be constrained.
Penetration of microwave energy inside a material is a function of dielectric
properties, which can alter the temperature distribution (TD) within the sample. As the
sample size increases, the sample regions away from the surface are not heated
satisfactorily due to decaying microwave energy as it propagates into the sample. Poliak
and Foin (1960) reported, in a microwave-heated beef cylinder (radius = 6 cm,
penetration depth = 2.1 cm), the temperature at the center was lower than at the surface.
For a small sample (relative to microwave penetration depth), the focusing effect of
microwave energy accumulates as a function of time and cause overheating at the sample
center. The pulsed heating is especially suitable for such cases. Therefore, sample
dimension and heating time should be optimized to prevent over and under heating
during pulsed microwave application.
The objective of this study was to evaluate the effect of different levels of
processing parameters on temperature uniformity in a microwave heated food. Precooked
mashed potato was used as the sample food material. The parameters studied were:
•
Sample dimension,
•
Sample temperature distribution,
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140
•
Processing time, and
•
Power level.
7.2 Methods and Materials
7.2.1 Sample Preparation
Mashed potato flakes (Idaho Spuds, The Pillsbury Company, Minneapolis, MN)
were purchased from a local supermarket. Mashed potato was prepared according to the
manufacturer’s information with some exceptions (no butter and salt added). 1800 mL of
distilled water was brought to a rolling boil and 370 g of potato flakes were stirred in
with a hand mixer (KitchenAid KHM-7) at the lowest speed (250 rpm), to a uniform
consistency.
7.2.2 Physical and Dielectric Properties
Sample moisture content was determined in triplicate by an infrared moisture
analyzer (U 16 Mettler Toledo Inc, Switzerland) with a set temperature of 160 °C. About
2 g of sample was placed in the moisture analyzer. The sample mass was measured by a
built-in digital balance capable of reading to 4 decimal places during the infrared heating.
The sample moisture content was automatically calculated by the moisture analyzer. The
measurement was terminated, if after 3 consecutive 10 min intervals, the moisture content
varied less than ±0.2 %.
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141
Thermal and dielectric properties of the mashed potato were determined using
moisture content dependent models available in the literature. The models and
corresponding property values are summarized in Table 7.1.
7.2.3 Microwave Heating
The mashed potato samples were transferred into 4-cm radius pyrex glass beakers
to a height of 7 cm and covered with waxed paper to prevent moisture loss. They were
stored in a refrigerator at 4 °C for at least 16 h to ensure sample temperature uniformity
before microwave heating experiments. The sample along with the glass beaker was then
placed at the center of turntable in the microwave oven cavity and heated for 1 min at
250-W oven setting. The sample temperature distribution (TD) was measured (as
described in Section 3.3) and compared to the model (Maxwell’s as described in Chapter
5) predicted TD (Fig. 7.1). The power correction term due to the evaporative cooling was
not considered in the numerical models because the samples were not exposed to air.
7.2.4 Analysis of Optimal Process
The measured and predicted TDs agreed, and the temperature generally decreased
from the surface to center. This is because the penetration depth (Dp in Equation 7.1) of
microwaves is 1.6 cm for the mashed potato sample (82.7 % moisture content, 4 °C).
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142
where A0 is the incident wavelength (12.24 cm), k” is the dielectric loss factor (20.4), 5 is
the loss angle (0.298 rad).
The pulsed microwave heating is most effective for samples with hot spots around
their centers, i.e. when the center temperature of the sample tends to be higher than that
of the surrounding material. In such cases, the equalization of thermal energy due to
conduction during microwave power-off periods tends to result in a more uniform sample
temperature.
Therefore, to study the optimization of pulsed microwave heating using mashed
potato as the example food, the sample radius should be less than 4 cm (2.5 Dp) and
comparable to Dp. The 3.5- and 4-cm radius agar gel cylinders (used in Chapter 5) are
only appropriate to demonstrate the effect of pulsed microwaving for foods with similar
Dp (Note: for 2% agar gel, Dp = 2.8 cm). Therefore, sample radii of 1.6, 2.4, 2.8 and 3.2
cm (or 1,1.5, 1.75 and 2 Dp) were considered. All samples were 7-cm high. The
corresponding sample volumes and absorbed power are listed in Table 3.1. Thirty
seconds of continuous microwave heating was simulated (using the Maxwell’s model as
described in Chapter 5). The simulation results are shown in Fig. 7.2. The center
temperature in the 1.6-cm radius sample was the highest and is an excellent candidate for
illustrating the temperature equalization under pulsed microwave heating, but was
considered too small to be practical (volume = 56 cm3). Thus, 2.4-, 2.8- and 3.2-cm
radius samples were considered for further analysis using the following criteria:
1. Initial sample temperature is 4 °C;
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143
2. Microwave energy, 250-W oven setting, is applied continuously until the
maximum temperature difference between any two locations (0.1 cm apart for
every two consecutive locations) in the sample (ATon) just exceeds 15 and 20
°C;
3. This is followed by a power-off period until the temperature difference during
the power-off period (ATd) is reduced by 3 and 5 °C (i.e. maximum
temperature difference at any two locations in the sample is 12 and 10 °C for
ATon=15 °C, and 17 and 15 °C for ATo„=20°C);
4. The total processing time should be less than 1000 s;
5. Maximum sample temperature anywhere should not exceed 70 °C, because
the predictive dielectric property models available in the literature (Table 7.1)
are valid only between 0 and 70 °C;
6. 500-W oven setting can be used alternatively to satisfy some constraints, if
necessary.
The simulation model based on Maxwell’s equations was modified as follows to
evaluate the effect of different parameters:
1. Automatically calculate the sample average, maximum and minimum
temperatures.
2. Each power-on period continues until the maximum and minimum sample
temperature differential exceeds the set criterion, then followed by a poweroff period.
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144
3. Each power-off period continues until the interior temperature differential
decreases to the set criterion.
4. At the end of each power-off period, the dielectric properties are evaluated
according to the average sample temperature.
5. Conditional loops repeat the computations until power-on or power-off
temperature differentials or time limit (1000 s) is reached.
7.3 Results and Discussion
The results of the optimization analysis using pulsed microwave power at the 250W oven setting are shown in Figs. 7.3, 7.4 and 7.5 for 2.4-, 2.8- and 3.2-cm radius
samples, respectively. At the beginning, the heating rate is the greatest because
microwave energy is applied continuously. The time of continuous power application
(done at the beginning of the simulation) for ATon = 20 °C is longer than ATon = 15 °C and
resulted in higher average sample temperatures. When ATon = 20 °C, the total processing
time was shorter than for ATon = 15 °C.
The power-on (PO) to total processing (TP) time ratios under the ATon =20 °C and
ATon =15 °C conditions with ATj=5 °C at the same average sample temperature were
calculated (Table 7.2). The PO/TP ratios are higher for ATon = 20 °C than for ATon = 15
°C condition. This is because of longer PO and shorter TP for ATon =20 °C. The
penetrating nature of microwaves causes an uneven power distribution in the material. As
the heating time progresses, non-uniform TD in the sample becomes more pronounced. It
can be resolved by setting proper power-on temperature constraint. In the case of ATon
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145
=15 °C, it obviously takes less time to heat the sample from the starting point of the cycle
than it does to heat the same sample to ATon =20 °C. However, the energy needed to heat
the sample under the ATon =20 °C constraint is greater than that under the AT0ff =15 °C
constraint. The temperature gradient within the sample, which is the driving force for
sample temperature equalization, is greater under the ATon =20 °C condition. Therefore,
under the ATon =20 °C constraint, the on-off cycle is shorter than under the ATon =15 °C
constraint.
The power-off temperature differences (ATd) were 5 and 3 °C. Generally, the
larger ATd, the longer the total processing time needed to achieve the same average
sample temperature. For the case of ATon = 20 °C and sample radius= 2.4 cm (1.5 Dp),
there is only small difference between ATd= 5 and 3 °C constraints with respect to the
time-dependent average sample temperature curves (Fig. 7.3). As the sample radius
increased, the difference became larger (Figs. 7.4 and 7.5), and was dependent on the
ATon constraint.
For sample radius of 2.4 cm (1.5 Dp) with ATon(ATd) = 20(3) °C constraints, the
final sample average temperature (Tfsa) can reach around 60 °C in 336 s. For sample radii
of 2.8 and 3.2 cm (1.75 and 2 Dp), all other parameters remaining the same, Tfsa = 60 °C
was achieved at 449 s and 997 s respectively. Fig. 7.6 shows the sample TD after the
microwave heating under ATon(ATd) =20(3) °C constraints. Under the ATon(ATd) = 15(3)
°C constraints, it takes too long to heat the sample to the average temperature of 60 °C
(443 s and 586 s for 1.5-DP and 1.75-DPradius samples, respectively). Under the
ATon(ATd) =15(3) °C constraints, the average temperature of a 3.2-cm (2-Dp) radius
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146
sample did not reach 60 °C within the preset 1000 s maximum processing time constraint.
All simulation results showed that the sample can be heated to an average temperature of
about 30 °C. However, the larger the sample radius, the longer the difference in total
processing time between the ATon (ATd) = 15(5) and 20(5) °C constraints. The microwave
pulsing sequences are consistent for 2.4- and 2.8-cm (1.5- and 1.75-DP) radius samples,
the sequences that can be applied to the samples to obtain a final average temperature of
60 °C and satisfy AT0„ (ATd) = 20(3) °C criteria are listed in Table 7.3. For the 3.2-cm (2Dp) radius sample, longer power-off periods are needed due to the fact that temperature is
high at the surface than at the center (Fig. 7.6). Therefore, pulsed microwave application
is not as beneficial for the 3.2-cm radius samples as it is for the 2.4- and 2.8-cm radius
samples. Accordingly, it may be concluded that pulsed microwave application is most
effective when the critical sample size (radius) is less than 2 Dp.
The use of 500-W oven setting was also simulated for the 3.2-cm radius sample
and the result was compared to that heated at the 250-W oven setting under ATon(ATd) =
20(3) criterion (Fig. 7.7). The sample average temperature of 60 °C was reached in a
shorter total processing time.
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147
7.4 Conclusions
The pulsed microwave heating is most effective when the sample radius is < 2 Dp.
Maximum microwave power-on and -o ff temperature constraints are very critical for the
optimal application of pulsed microwave heating. Power-on temperature constraint
produces suitable temperature gradient. Power-off temperature constraint allows the
temperature equalization to occur. The power-off temperature constraint affects the total
processing time as the sample radius increases. The most efficient process among all the
cases is the heating of 2.4-cm (1.5-DP) radius precooked mashed potato sample under the
ATon(ATd)=20(3) constraint.
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7.5 References
Poliak, G.A. and Foin, L.C. 1960. Comparative Heating Efficiency of a Microwave and
Convection Electric Oven. Food Technology 14: 454-457.
Mohsenin, N.N. 1980. Thermal Properties o f Foods and Agricultural Materials. Gordan
and Breach Science Publishers, Inc. New York, New York.
Calay, R.K., Newborough, M., Probert, D. and Calay, P.. S. 1995. Predictive Equations
for the Dielectric Properties of Foods. International Journal of Food Science and
technology 29: 699-713.
Tinga, W.R. and Nelson, S.O. 1973. Dielectric Properties of Materials for Microwave
Processing-Tabulated. Journal of Microwave Power. 8(1): 24-65.
Yang, H. W. and Gunasekaran, S. 2001. Temperature Profiles in a Cylindrical Model
Food During Pulsed Microwave Heating. J. Food Science 66(7): 998-1004.
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149
Table 7.1 Thermal and dielectric properties of mashed potatoes with 82.7%
moisture content (M, %) and different temperatures (T, °C)
Parameter
Thermal conductivity, W/m2.°C
Specific heat capacity, J/g.°C
Dielectric constant
Model
0.00493M+0.148a
Temperature (°C)
33.3M+833.3b
Value
0.55
3577
2.14-0.104T+0.808MC
4
10
20
30
40
50
60
70
68.4
67.8
66.7
65.7
64.6
63.6
62.6
61.5
4
10
20
30
40
50
60
70
20.4
20.0
19.4
18.7
18.1
17.5
16.8
16.2
Dielectric loss factor
a from Sweat (1974)
b from Mohsenin (1980)
c from Calay et al. (1995)
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150
Table 7.2 Power-on (PO) to total processing (TP) time ratios under the AToa (ATd)
=20(3) °C and ATon (ATd) =15(3) °C constraints at different final average
sample temperature (Tfas, °C).
Radius (cm)
PO/PT at ATon : Tfas =20: 60
2.4
0.88
2.8
0.66
3.2
0.43
PO/PT at ATon : Tfas =15: 30
0.67
0.47
0.22
Table 7 3 Microwave pulsing sequences applied to different radius samples and
comply the ATon (ATd)=20(3) °C criteria.
Radius
(cm)
2.4
2.8
(Time interval): Pulsing ratio: Number of duty cycles
(0-92): 1: 1
(0-108): 1: 1
(92-336): 1.17 : 7
(108-449): 2.2:10
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151
25
Predicted
Measured
Temperature (°C)
20
15
10
5
0
0
1
2
3
Radial distance from center (cm)
4
Figure 7.1 Measured and predicted temperature profiles in a 4-cm radius mashed
potato cylinder after 1 min of continuous microwave heating at the 250W oven setting.
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152
45
40
R=1.6 cm
R=2.4 cm
■
R=2.8 cm
■ “ *R=3.2 cm
O
R=2.4 cm M easured
35
0 30
O
^
1 25
*
2Q» 20
E
£ 15
10
5
0
0
1
2
3
4
Radial distance from center (cm)
Figure 7.2 Predicted temperature profiles in 1.6,2.4,2.8 and 3.2-cm radius mashed
potato cylinders after 30 s of microwave heating at the 250-W oven
setting. Measured data for 2.4-cm radius sample is also shown.
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153
Average sample temperature (°C)
70
60
50
40
30
20(3)
20
20(5)
15(3)
10
15(5)
0
0
200
400
600
800
1000
Time (s)
Figure 7.3 Average sample temperature profiles in 2.4-cm radius, 7 cm-long,
mashed potato cylinders heated by pulsed microwave at the 250-W oven
setting. Power-on constraints (AT,,,) were 20 and 15 °C and power-off
temperature difference constraints (ATd) were 5 and 3 °C (i.e.
AT„n(ATd)=15(5), 15(3), 20(5) and 20(3)).
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154
Average sample temperature (°C)
70
60
50
40
30
-20(3)
20(5)
•15(3)
•15(5)
20
10
0
0
200
400
600
800
1000
Time (s)
Figure 7.4 Average sample temperature profiles in 2.8-cm radius, 7-cm long,
mashed potato cylinders heated by pulsed microwave at the 250-W oven
setting. Power-on constraints (ATon) were 20 and 15 °C and power-off
temperature difference constraints (ATd) were 5 and 3 °C (i.e.
ATon(ATd)= 15(5), 15(3), 20(5) and 20(3)).
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155
Average sample temperature
(°C)
70 -i
-20(3)
20(5)
■15(3)
15(5)
60
50
40
30
10
0
200
600
400
800
1000
Tim e (s)
Figure 7.5 Average sample temperature profiles in 3.2-cm radius, 7-cm long,
mashed potato cylinders heated by pulsed microwave at the 250-W oven
setting. Power-on constraints (ATon) were 20 and 15 °C and power-off
temperature difference constraints (ATd) were 5 and 3 °C (i.e.
ATon(ATd)=15(5), 15(3), 20(5) and 20(3)).
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156
80
70
Temperature (°C)
60
50
40
30
-2.4 cm
2.8 cm
*3.2 cm
20
2.4 cm Measured
10
0
0
0.8
1.6
2.4
3.2
Radial distance from center (cm)
Figure 7.6 Temperature distribution in 2.4-, 2.8- and 3.2-cm radius potato cylinders
under the ATon(ATd)=20(3) criteria after average sample temperature is
about 60 °C at the 250-W oven setting.
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157
70
Average sample temperature (°C)
250
500
40
0
200
400
600
800
1000
Time (s)
Figure 7.7 Time and average sample temperature of a 3.2-cm mashed potato
cylinder with 20 °C power-on constraints and a power-off differential of
3 °C lower than the power-on constraint (i.e. ATon(ATd)=20(3)) at the 250and 500-W oven settings.
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158
CHAPTER 8
SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK
8.1 Summary
Temperature distribution in 2 % agar gel cylinders was studied experimentally
and numerically. The following conclusions can be drawn are:
•
Pulsed microwave heating results in more uniform temperature distribution than
continuous microwave heating.
•
Pulsed microwave heating can be employed to minimize the overheating of the
sample center.
•
Simulation of temperature distribution using absorbed power based on Maxwell’s
equations is more accurate than that based on the Lambert’s law.
•
At the same average output power, the effect of sample size, pulsing ratio, and
power level are statistically significant.
Regarding the optimization of pulsed microwave application to the re-heating of
mashed potato cylinders, the following conclusions can be drawn:
•
Sample radius, power-on and power-off temperature constraints, power level and
processing time are important parameters to be considered.
•
The optimal process depends on the sample radius related to microwave
penetration depth for the sample being heated.
•
The optimal process is also affected by power-on and power-off temperature
constraints and power level.
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159
8.2 Recommendations for Future Work
The following suggestions are for future research regarding the pulsed microwave
heating:
•
Numerical simulation of temperature distribution including phase change in
frozen food.
•
Mass and heat transfer simulation for long duration considering moisture and
temperature simultaneously.
•
Wider scope of optimization applications to different foods with processing
parameter constrained.
•
Numerical simulation of temperature distribution using different geometry such as
slabs and spheres.
•
Optimization of additional samples of varying dielectric properties of different
geometry.
•
‘Runaway’ heating in frozen foods.
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APPENDICES
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161
Appendix A
Engineering Equation Solver (EES, F-Chart Software Co., Middleton, WI) program
code for calculating temperature profiles in continuous and pulsed microwave
heated 2% agar gel cylinders
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162
"General information"
»*»
Y=0.07 "sample height (m)"
R=0.035 "sample Radius (m)"
dT_av=32.73
K=0.6 "thermal conductivity"
h=42 "heat transfer coefficient"
T_air=25 " air temperature C"
t=180" total heating time seconds"
V=Y*RA2*pi" volume o f the sample mA3"
rho=1070 "density of the sample kg/mA3"
C_p=4200 "heat capacity of the sample J/kg.C"
Kappa=75 "dielctric constant"
Kappa_2=13 "loss factor"
tangent_theta=Kappa_2/kappa
lambda_0=3E+8/2.45E+9 " wavelength in free space"
" is * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * "
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163
"Calculation of the attenuation constant for microwave"
»»*»
n * tt
'»** **************afe* ** * * ** *** ** ** *********** *** * ** ** * ** ** •'»
alpha=2*(Pi/lambda_0)*sqrt(0.5*kappa*(sqrt( l+(tangent_theta)A2)~ 1)) " l/m"
"***** ************** ******* ******* lit******
"Calculation of the total power and P_0"
11*11
< t* n
» » * * * * * * * * * * * * * * * jfc* * jft * * * * * * * * * * * * * * * * * jf: j f c j f e * * * * * * * * * * * * * * * * "
P_total=(rho*V*c_p*dT_av)/t
P_0=(alpha*P_total)/(pi*Y*(l-exp(-2*alpha*R)))
A=(pi*Y*P_0)/alpha
••* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * "
"Calculation of the Radius of each nodal point R[i]"
n*»r
tt* ti
"*********************************************************"
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164
dR=0.001
duplicate i=0,35
R[i]=dR*i
end
' I * * * * * * * * * * * * * * 4c 4c * * * 4c * * * *
* * * * * * * % 4c 4c * 4c * % £ * * * * * * * * * * * * * * * * * * '>
"Calculation of distance of the boundary of each subshell form the surface except the
most outer shell"
t t a|e t t
it* "
»**********************************************************11
duplicate i=0,34
X[i]=R-(R[i]+dR/2)
end
»***********************************************************”
Duplicate i=0,35
U[i]=0
end
{"Determine the power absorbed term (U[i]) for each each subshell (i=l to 34)
except the center and the boundary subshell"
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165
''**************************************************************•*
Duplicate i=l,34
U[i]=A*(exp(-2*alpha*X[i])-exp(-2*alpha*X[i-l]))
end
"♦Sts************************************************************"
"Determine the power absorbed term for the center subshell (U[0]) and the boundary
subshell (U[35])"
««*««
«*•*
U[0]=A*(exp(-2*alpha*X[0])-exp(-2*alpha*R))
U[35]=A*(l-exp(-2*alpha*X[34]))
"Determine the Mass of each subshell (i=l to 34) except the central and boundary
subshell"
"*»
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166
»*"
I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
duplicate i=l,34
M[i]=pi*((R[i]+dR/2)A2-(R[i]-dR/2)A2)*Y*rho
end
•>** ** ********$* ****%%****** **** ***** * *3ft************Jit**** ***#* * * »
"Determine the Mass of the central (M[0]) and boundary (M[35]) subshell"
»»* t i
"*"
"**************************************************************"
M[0]=pi*(dR/2)A2*Y*rho
M[35]=pi*(RA2-(R-dR/2)A2)*Y*rho
"**************************************************************"
Tau_d=l "time increment in second"
Tau=index-1 "Total heating time"
"Energy balcnce equation for each nodal point except the central and boundary point"
11*1I
tt* II
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'I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ..
Duplicate i=l,34
(C_p*M[i]*(T_n[i]-T[i]))/Tau_d=2*pi*Y*K*0.5*((T[i-l]-T[i]+T_n[i-l]T_n[i])*(R[i]-dR/2)+(T[i+l]-T[i]+T_n[i+l]T_n[i])*(R[i]+dR/2))/dR+U[i]+h*pi*(T_air-T[i])*((R[i]+dR/2)A2-(R[i]-dR/2)A2)
end
•I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * "
"Energy balance equation for the central and boundary point"
11*11
(C_p*M[0]*(T_n[0]-T[0]))/Tau_d=K*pi*Y*0.5*(T[l]-T[0]+T_n[l]T_n[0])+U[0]+h*pi*(T_air-T[0])*(dR/2)A2
(C_p*M[35]*(T_n[35]-T[35]))nau_d=2*K*pi*Y*0.5*((T[34]-T[35]+T_n[34]T_n[35])*(R[35]-dR/2))/dR+U[35]+h*(T_airT[35])*((2*pi*Y*R[35])+pi*((R[35])A2-(R[35]-dR/2)A2))
•I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * "
"EES's standard programming for unsteady state energy balance equation involving
time step"
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168
" I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ''
Duplicate i=0,35
T[i]=TableValue(index-l, #T_n[i])
end
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169
APPENDIX B.1
MATLAB (MathWorks Inc., Natick, MA) program code for the calculating of
temperature profile within 4*cm radius agar gel cylinders according to the
Maxwell’s equations
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t=inputCEnter a microwave processing time:');
PR=inputCEnter a pulsing ratios (PR):');
Tml=zeros(l,41);
Rm=zeros(l,41);
MPm=zeros(l,41);
am=zeros(l,40);
bm=zeros(l,41);
cm=zeros(l,40);
dm=zeros(l,41);
Xm=zeros(l,41);
C_lm=zeros(l,41);
C_2m=zeros(l,41);
C_3m=zeros(l,41);
KKl=zeros(41,41);
KK2=zeros(41,l);
TXl=zeros(41,l);
R=0.04;
c=3.00e+8;
f=2.45e+9;
omega=2*pi*f;
epsilon=le-9/(36*pi);
mu=4*pi*le-7;
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kappa_const=75;
kappa_loss=13;
KKKl=(RA2*omegaA2*mu*epsilon*kappa_const);
KKK2=(RA2*omegaA2*mu*epsilon*kappaJoss);
alpha_0=omega/c;
xx=R*alpha_0;
xx 1=(besse Ij(0,xx) )A2+(bessel y(0,xx))A2;
kkk3=R*alpha_0*(besselj( 1,xx)*besselj(0,xx)+bessely( 1,xx)*bessely(0,xx))/xx 1;
kkk4=2/(pi*xxl);
kkk5=(-4/pi)*bessely(0,xx)/xx 1;
kkk6=(-4/pi)*besselj(0,xx)/xx 1;
bb=sqrt(KKKl+i*KKK2);
deno=-bb*besselj(l,bb)+(kkk3-i*kkk4)*besselj(0,bb);
KK=(kkk5+i *kkk6)/deno;
dR=0.001;
Y=0.07;
k=0.6;
h=42;
T_air=22.5;
tau_d=l;
C_p=4200;
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172
rho=1070;
V=Y*RA2*pi;
dT_av=27.2;
t_p=180;
P_total=rho*V*C_p*dT_av/t_p;
P_00=P_total/(2*pi *R* Y);
E_0=sqrt(P_00*pi*alpha_0*R/(c*epsilon));
for n=l:41
Tml(n)=4;
Rm(n)=0.001*(n-1);
Xm(n)=R-(Rm(n)+dR/2);
if n = l
C_lm(n)=(pi*(dR/2)A2*Y*rho*C_p)/tau_d;
C_2m(n)=0.5*pi*Y*k;
cm(n)=-C_2m(n);
dm(n)=C_lm(n)+C_2m(n);
RR(n)=(n-l)*dR;
RB(n)=RR(n)+dR/2;
R V(n)=RB (n) A2*pi *Y;
elseif n<41
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C_lm(n)=(pi*((Rm(n)+dR/2)A2-(Rm(n)-dR/2)A2)*Y*rho*C_p)/tau_d;
C_2m(n)=pi*Y*k/dR;
am(n-1)=(-C_2m(n))*(Rm(n)-dR/2);
cm(n)=(-C_2m(n))*(Rm(n)+dR/2);
dm(n)=C_lm(n)+C_2m(n)*2*Rm(n);
RR(n)=(n-l)*dR;
RB(n)=RR(n)+dR/2;
RV(n)=(RB(n)A2-RB(n-1)A2)*pi *Y;
else
C_ I m(n)=(pi *(RA2-(R-dR/2)A2)* Y*rho*C_p)/tau_d;
C_2m(n)=pi*Y*k/dR;
am(n-1)=(-C_2m(n)):,t(Rm(n)-dR/2);
dm(n)=C_ 1m(n)+C_2m(n):(!(Rm(n)-dR/2);
RR(n)=(n-1)*dR-dR/4;
RB(n)=R;
R V(n)=(RB(n)A2-RB(n-1)A2)*pi *Y;
end
RRD(n)=RB(n)/R;
Zr=KK*besselj(0,bb*RRD(n));
MPm(n)=RV(n)*(omega*epsiIon*kappa_loss*E_0A2)*abs(Zr).A2;
end
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174
for n=l:41
KKl(n,n)=dm(n);
if n<41
KK1(n+1,n)=am(n);
KKl(n,n+l)=cm(n);
end
KK1;
for tau=l:t*PR
tau;
index l=rem(tau,60);
index2=60/PR;
if P R = 1
Pm=MPm;
elseif index2~=0 & (indexl>index2 | in d ex l= 0 )
Pm=zeros(l,41);
else
Pm=MPm;
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175
end
for n=l:41
if n = l
bm(n)=C_ 1m(n)*Tm 1(n)+C_2m(n)*(Tm 1(n+ 1)-Tm 1(n))+Pm(n);
elseif n<41
bm(n)=C_lm(n)*T m 1(n)+C_2m(n)*((Tm 1(n-1)-Tm 1(n))*(Rm(n)-dR/2)+(Tm 1(n+1)Tml(n))*(Rm(n)+dR/2))+Pm(n);
else
bm(n)=C_ 1m(n)*Tm 1(n)+C_2m(n)*(Rm(n)-dR/2)*(Tm 1(n-1)Tml(n))+Pm(n)+2*pi*R*Y*(T_air-Tml(n));
end
end
bm;
KK2=bm.’;
TX=KK1\KK2;
TX=TX.’;
Tml=TX;
end
Tml
plot(Rm,Tml)
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176
APPENDIX B.2
PREDICTED AND MEASURED TEMPERATURE DISTRIBUTION OF 3.5- AND
4-cm RADIUS 2% AGAR GEL CYLINDERS AT 250-W OVEN SETTING
•
PR2=2, after 1, 2 and 3 min o f pulsed microwave heating.
•
PR= 1, after 2 min of continuous microwave heating.
•
PR=3, after 3 min of pulsed microwave heating.
2 Pulsing ratio
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177
60
MP. 1 min
M, 1 min
LP. 2 min
'MP. 3 min
M, 3 min
50
Mff. 2 min
M, 2 min
LP, 3 min
£ 40
bo>
5 30
d>
a
|
20
10
0
0
1
2
3
4
Radi al distance from center (cm)
Figure B.l Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 3.5*
cm radius 2 % agar gel cylinders after 1,2 and 3 min of microwave
heating with PR=2.
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178
60
Temperature (°C)
50
MP, 1min
LP, 1 min
M, 1 min
MP, 2 min
- - - LP, 2 min
M, 2 min
'MP, 3 min
LP, 3 min
M, 3 min
40
30
20
10
0
0
1
2
3
4
5
Radial distance from center (cm)
Figure B.2 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 4-cm
radius 2 % agar gel cylinders after 1,2 and 3 min of microwave heating
with PR=2.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
179
60
50
Temperature (°C)
KP
40
20
0
1
2
3
4
Radial distance from center (cm)
Figure B.3 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 3.5cm radius 2 % agar gel cylinders after 2 min of microwave heating with
PR=1.
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180
60
50
Temperature (°C)
MP
40
0
2
3
4
5
Radial distan ce from center (cm)
Figure B.4 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 4-cm
radius 2 % agar gel cylinders after 2 min of microwave heating with
PR=1.
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181
60
50
MP
LP
10
0
0
2
3
Radial distance from center (cm)
1
4
Figure B.5 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 3.5cm radius 2 % agar gel cylinders after 2 min of microwave heating with
PR=3.
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182
60
Temperature (°C)
50
MP
-LP
40
30
20
10
0
0
1
2
4
3
Radial distance from center (cm)
5
Figure B.6 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s
(LP) models compared to the measured (M) temperature profile in 4-cm
radius 2 % agar gel cylinders after 2 min of microwave heating with
PR=3.
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183
APPENDIX C.1
General linear model with time and covariate interaction terms using MiniTab
(Minitab Inc., State Park, PA) analysis.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
General Linear Model: Temperature versus Time, Pulse and Radius
Factor
Time
Pulse
Radius
Type Levels Values
fixed
3 1 2 3
fixed
2 1 2
fixed
2 3 .5 4.0
Analysis of Variance for Temperature, using Adjusted SS for Tests
Source
Radial D
RDA2
Time
Pulse
Radius
Time*Pulse
Time*Radius
Pulse*Radius
Time*Pulse*Radius
Time*Radial D
Pulse*Radial D
Radius*Radial D
Time*RDA2
Pulse*RDA2
Radius*RDA2
Error
Total
DF
1
1
2
1
1
2
2
1
2
2
1
1
2
1
1
140
161
Seq SS
184.86
568.36
7380.09
1041.20
865.64
267.53
277.67
75.26
26.12
167.89
89.79
100.21
16.61
113.10
96.15
410.78
11681.25
Adj SS
802.47
606.03
2931.40
824.78
732.27
281.73
213.13
77.11
26.12
61.84
170.15
154.45
16.61
113.10
96.15
410.78
Adj MS
802.47
606.03
1465.70
824.78
732 .27
140.86
106.57
77.11
13.06
30.92
170.15
154.45
8.31
113.10
96.15
2.93
F
273.50
206.55
499.54
281.10
249.57
48.01
36.32
26.28
4.45
10.54
57.99
52.64
2.83
38.55
32.77
P
0.000
0. 000
0.000
0.000
0.000
0.000
0.000
0.000
0.013
0.000
0.000
0.000
0.062
0.000
0.000
185
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186
APPENDIX C.2
General linear model without time and covariate interaction terms involved using
MiniTab (Minitab Inc., State Park, PA) analysis (including the Probability Plot of
the Residuals and Residuals versus the Fitted Values)
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
°Q.
C
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1
CD
Q.
with permission
of the copyright owner. Further reproduction
General Linear Model: Temperature versus Time, Pulse and Radius
Factor Type
Time
Pulse
Radius
Levels
fixed
fixed
fixed
Values
31 2 3
212
2 3.5 4.0
Analysis of Variance for Temperature, using Adjusted SS for Tests
prohibited without p erm ission.
Source
Radial D
RDA2
Time
Pulse
Radius
Time*Pulse
Time*Radius
Pulse*Radius
Time*Pulse*Radius
Time*Radial D
Pulse*Radial D
Radius*Radial D
Pulse*RDA2
Radius*RDA2
Error
Total
DF
1
1
2
1
1
2
2
1
2
2
1
1
1
1
142
161
Seq SS
184.86
568.36
7380.09
1041.20
865.64
267.53
277.67
75.26
26.12
167.89
89.79
100.21
113.10
96.15
427.39
11681.25
Adj SS
802.47
606.03
3812.53
824.78
732.27
281.73
198.51
77.11
26.12
167.89
170.15
154.45
113.10
96.15
427 .39
Adj MS
802.47
606.03
1906.26
824.78
732.27
140.86
99.26
77.11
13.06
83.94
170.15
154.45
113.10
96.15
3.01
F
266.62
201.35
633.35
274.03
243 .30
46.80
32.98
25.62
4.34
27.89
56.53
51.31
37.58
31.95
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0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.015
0.000
0. 000
0.000
0.000
0.000
188
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R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
(Response is temperature)
Fig. C. 1 Residuals versus the fitted values
189
•V*.
•4
•t
m
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
Fig. C.2 Normal probability plot of residuals
190
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R eproduced with perm ission of the copyright o w n e r Further reproduction prohibited without permission
191
APPENDIX D
Determination of the experimental error due to 30-s lag during temperature
measurement
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The experimental errors due to the 30-s delay during the sample temperature
measurement were determined. The 3 min of continuous microwave power at 250-W
oven setting were applied to 2% agar gel cylinders (3.5-and 4-cm radius).
Temperatures at center and every centimeter from the center were measured in
triplicates and recorded at 5, 10, 15, 20, 25 and 30 s after the microwave heating
(instrument described in Section 3.3).
The collected temperatures are shown in Figures D -l and D-2. The effect of
30-s delay (time effect) on temperature measurement is not significant (P > 0.01,
Table D).
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
193
40
35 O
"jr
3
0
-
3
©
E 25 ©
I—
20
-
P, 0 c m
M, 0 cm
M, 1 c m
A M, 2 cm
M, 3 cm
M,4cm
—
0
5
10
15
20
25
30
35
Time (s)
Figure D-l Temperature variation at 0 ,1 ,2 ,3 and 4 cm from center in a 4-cm
radius 2 % agar gel cylinder (P is predicted data and M is measured
data), during 30 s after 3-min microwave heating.
R eproduced with perm ission of the copyright o w n e r Further reproduction prohibited without perm ission
194
Temperature (°C)
45
40
35
30
25
a
P , 0 cm
M, 0 cm
M, 2 cm
M, 3 cm
0
Time (s)
Figure D-2 Temperature variation at 0,1,2 and 3 cm from center in a 3.5-cm
radius 2 % agar gel cylinder (P is predicted data and M is
measured data), during 30 s after 3-min microwave heating.
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
Table D Effect of experimental variables on 30-s delay of temperature
measurements- statistical analysis
Source of variation
Radial distance (cm)
Sample radius (cm)
Time (s)
Replication
Error
Total
Degrees of Freedom
4
1
5
2
149
161
F ratio
112.83
271.43
0.75
1.81
P value
<0.01
<0.01
0.584
0.167
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX E
The evaporation cooling effect during microwave heating
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
197
The agar gel cylinders were heated as described in Chapters 4 and 5. The
sample mass before and after 3 min of total microwave power input was measured
(XT-3000 DR, Fisher Scientific, Pittsburgh, PA) in triplicates. The decrease in mass
(Aw) during the process indicated the loss of moisture content in the sample. The
latent heat (h/g) of water is 2,438 J/g at 27 °C and at atmospheric pressure. The power
correction term (Pc) due to evaporative cooling was calculated and applied to the five
outermost incremental shells near the sample radial surface.
Aw * h .
________ SJj_
p
c
TMA* 5
The Aw and Pc for different microwave applications were listed in Table E
below.
Table E. Loss of moisture (Aw) and the corresponding power correction (Pc) for
2 % agar gel cylinders during microwave heating
PR
1
2
3
1
2
3
Sample radius (cm)
3.5
3.5
3.5
4.0
4.0
4.0
Aw (g)
0.75
1.03
1.44
0.86
1.27
1.54
Energy loss (J)
1833
2511
3506
2097
3088
3755
Pc(W)
1.45
1.00
0.94
1.45
1.05
0.85
R eproduced with perm ission of th e copyright owner. Further reproduction prohibited without permission.
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