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# Particle size behavior, dielectric properties, and electromagnetic modeling of aqueous starch suspensions during microwave heating

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The Pennsylvania State University
College o f Agricultural Sciences
PARTICLE SIZE BEHAVIOR, DIELECTRIC PROPERTIES,
AND ELECTROMAGNETIC MODELING
OF AQUEOUS STARCH SUSPENSIONS
DURING MICROWAVE HEATING
A Thesis in
Food Science
by
Johnny Casasnovas
Submitted in Partial Fulfillm ent
o f the Requirements
for the Degree o f
Doctor o f Philosophy
August 2006
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: 3229005
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The thesis o f Johnny Casasnovas was reviewed and approved* by the follow ing:
Ramaswamy C. Anantheswaran
Professor o f Food Engineering
Chair o f Committee
John D. Floras
Professor o f Food Science
Head o f the Department o f Food Science
Michael T. Lanagan
Associate Professor o f Engineering Science and Mechanics
Raymond J. Luebbers
President, Remcom Inc.
Special Member
Donald B. Thompson
Professor o f Food Science
Gregory R. Ziegler
Professor o f Food Science
♦Signatures are on file in the Graduate School
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Abstract
Microwaves are notable in their ability to penetrate the interior o f foods, heating them more
quickly than conventional methods used in both domestic and industrial applications. However, at present,
microwave heating is not completely understood. It is often d iffic u lt to obtain w ith microwave heating the
characteristics expected o f food cooked w ith conventional heating. Research studying the differences
between conventional and microwave heating has frequently been compromised by differences in
time/temperature treatment, heating patterns, or both, used in the two heating methods. Some authors
speculate that the differences between the heating methods result from differences in heat and mass
transfer, and not from any “ athermal effect” or “ microwave effect.”
Better knowledge o f both heat and mass transfer during microwave heating would be needed to
take fu ll advantages o f microwave heating. Currently, commercial computational tools are able to predict
heat and mass transfer during conventional heating. These models, however, depend on a p rio ri
knowledge o f the nature o f the heat generation and its intensity distribution in space and time. The
microwave heat generation is best understood by using M axw ell’s equations, which require knowledge o f
the dielectric properties that govern the interaction. Many o f the models in the literature have used
sim plifications to these equations that lim it their accuracy, and cannot correctly predict the intensity and
distribution o f heat generation during microwave heating. Therefore, these models have lim ited usefulness
fo r the development o f products and processes using microwave technology.
Microwave heat generation is a function o f the dissipated power produced when microwaves are
absorbed by the system in question. Dissipated power follow s directly from application o f M axw ell’s
equations, but the application o f these equations to a real system usually requires the use o f a numerical
method like the Finite Difference Time Domain (FDTD) method, which was used in this work.
The general objective o f this work was to study the behavior o f starch in dilute aqueous
suspension during microwave heating, while examining the effect o f starch type and temperature on the
dielectric behavior o f the suspensions. This research focused on water-starch and water-starch-sodium
chloride systems, using different types o f com starch and also potato starch. Another important
contribution o f this work was an extensive literature review covering different aspects o f electromagnetic
theory and molecular polarization, and models that have been proposed to model the dielectric behavior o f
substances.
In the first part o f this study, the swelling behavior o f dilute aqueous suspensions o f common com
starch, waxy maize starch, and cross-linked waxy maize starch was measured in-line in a microwaveheated, well-m ixed system. The heating rate was controlled (5 °C/min) to match the rate used in a previous
study conducted w ith conventional heating in order to allow comparison between the results o f both
studies. No significant difference was found between the two studies in the maximum diameters and the
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temperature o f maximum swelling. The plot o f median diameters against temperature showed the same
features for both studies. Thus, no “ athermal effect” or “ microwave effect” was observed in this study.
In view o f the sim ilarities observed between the two studies, it seemed reasonable that microwave
heating o f dilute starch suspensions could be modeled using available numerical methods, i f heat
generation data could be obtained. Heat generation for microwaves is a function o f the dielectric
properties. In the second part o f this work, a method to measure the dielectric properties o f dilute starch
suspensions in-line during microwave heating was developed. The suspensions were heated from 30 to
80°C at 5°C/min and the dielectric properties ( e/ and er") were measured at frequencies between 0.3 and 3
GHz. Additional experiments were conducted at frequencies between 45 M Hz and 25.6 GHz. The results
show that the dielectric properties o f dilute suspensions (3% w /w ) o f waxy maize, common com, and
potato starches in general follow the properties o f water. Among the differences observed, a lower e / may
be caused by a dilution effect o f the starch at least at temperatures lower than gelatinization. The dielectric
loss (s /') was higher fo r the starch suspensions than for water, and it is hypothesized that this is caused by a
higher main relaxation time, the contribution o f conductivity due to ions associated w ith the starch, and,
possibly, to the presence o f a second relaxation peak at frequencies lower that those measured. Additional
measurements w ith waxy maize suspensions (3% w /w ) w ith added NaCl (2% w /w) show a greatly
increased e," and a decrease in s/. The decrease o f e/ is lower than that predicted for water-NaCl solutions
by the Hasted-Debye model, suggesting that starch decreases the effect o f conductivity in this system.
Anomalous curvature at low frequencies for e/ is hypothesized to correspond to electrode polarization. The
results o f these measurements were used to fit a model consisting o f a Debye-Hasted model w ith an
The third and last part o f this study used the measured dielectric properties w ith the FDTD method
to model the electromagnetic field distributions inside a bowl fille d w ith dilute starch suspensions as it was
heated in a domestic microwave oven. In addition to the electromagnetic field distribution, the results o f
the model included the specific absorption rate (SAR), which is a quantity related to dissipated power. This
model was meant as a first step in the development o f a fu ll model, capable o f predicting both heat and
mass transfer during microwave heating o f food systems.
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V
List o f Tables ......................................................................................................................................ix
List o f Figures......................................................................................................................................x
List o f Symbols................................................................................................................................... xv
List o f Equations................................................................................................................................. x v iii
Acknowledgments............................................................................................................................... xxv
1 IN TR O D U C TIO N....................................................................................................................... 1
1.1
B rie f History o f Microwave Ovens......................................................................... 1
1.2
Industrial Uses o f Microwave Technology.............................................................. 2
1.3
Justification o f Microwave Research....................................................................... 3
1.4
Objectives o f this Research Project......................................................................... 7
2 D YN AM IC MEASUREMENT OF STARCH PARTICLE SIZE BEHAVIO R DURING
M ICROW AVE H EATIN G .......................................................................................................... 9
2.1
Introduction............................................................................................................. 9
2.2
Review o f the Literature.......................................................................................... 11
2.2.1
Sources, Uses and Composition o f Starch............................................... 11
2.2.2
Starch Granules........................................................................................14
2.2.3
Transformations o f Starch During Cooking............................................ 15
2.2.4
Starch M odifications............................................................................... 20
2.2.5
Methods o f Evaluating Gelatinization..................................................... 22
2.2.6
The Use o f LSPSA D uring Conventional Heating.................................. 26
2.2.7
A B rie f Summary o f Microwave Heating Mechanisms..........................27
2.2.8
Survey o f Research on the Microwave Heating o f Starch....................... 29
2.3
Objectives................................................................................................................ 37
2.4
Materials and M ethods............................................................................................ 39
2.4.1
Starches....................................................................................................39
2.4.2
Scanning Electron M icroscopy (SEM ).................................................... 39
2.4.3
Heating and Flow System ....................................................................... 40
2.4.4
Particle Size Analysis.............................................................................. 44
2.4.5
Statistical Procedures.............................................................................. 44
2.5
Results and Discussion.............................................................................................46
2.5.1
Starch M icrographs................................................................................. 46
2.5.2
Particle Sizes........................................................................................... 46
2.5.3
Comparison w ith Conventional Heating................................................. 50
2.6
Conclusions............................................................................................................55
3 IN -LIN E MEASUREMENT OF THE DIELECTRIC PROPERTIES OF AQUEOUS STARCH
SUSPENSIONS DURING M ICROW AVE H E A TIN G ............................................................. 56
3.1
Introduction............................................................................................................56
3.1.1
Chapter O verview ................................................................................... 56
3.1.2
The Nature o f Electromagnetic Radiation.............................................. 57
3.1.3
The Nature o f M olecular Interactions w ith Electromagnetism...............61
3.2
Review o f the Literature...........................................................................................66
3.2.1
M axwell’ s Equations............................................................................... 66
3.2.2 M axwell’ s Equations in the Frequency Dom ain......................................74
3.2.3
Materials in a Microwave: Conductors and Insulators............................77
3.2.4 Polarization and Dielectric P erm ittivity.................................................. 83
3.2.5
Dielectric Relaxation............................................................................... 88
3.2.6
The Debye M odel....................................................................................93
3.2.7
The Cole-Cole M odel............................................................................. 96
3.2.8
The Dielectric Behavior o f Pure W ater.................................................. 99
3.2.9 Total System Frequency-Dependent Conductivity...................................102
3.2.10 The Hasted M odel...................................................................................106
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vi
3.2.11
3.3
3.4
3.5
Heat Transfer in a Microwave O ven......................................................... 110
3.2.11 .a The “ Molecular Friction” E ffect...............................................110
3.2.1 l.b The Role o f Ionic Conductionin Microwave H eating..............113
3.2.1 l.c D iffusion o f Heat...................................................................... 113
3.2.12 Penetration Depth o f Electromagnetic Radiation...................................... 114
3.2.13 How Dielectric Properties Are Measured in Literature............................. 116
3.2.14 Survey o f the D ielectric Properties ofF oods............................................ 122
3.2.14.a Fats............................................................................................ 123
3.2.14.b Proteins..................................................................................... 124
3.2.14.c Carbohydrates and Starches......................................................125
3.2.14.d In fluence o f Salts on the Dielectric Properties ofFoods.......... 130
3.2.14.e Mixtures and Solutions............................................................. 134
3.2.15 How Some Researchers Have Modeled Dielectric Properties................. 137
Objectives................................................................................................................ 148
Materials and M ethods............................................................................................149
3.4.1
M aterials..................................................................................................... 149
3.4.2
Apparatus...................................................................................................150
3.4.3
Calibration.................................................................................................. 154
3.4.4
Water Reference Measurements.................................................................155
3.4.5
Starch Suspension Measurements..............................................................155
3.4.6 Determination o f Model Parameters..........................................................156
3.4.6.b “ interpolate” .............................................................................. 160
3.4.6.C “ correcttable” ............................................................................ 162
3.4.6.d “ th2o” ........................................................................................ 163
3.4.6.e Bounds...................................................................................... 163
3.4.6.f Convergence C rite ria ................................................................ 164
3.4.6.g “ recursivehc” ............................................................................. 165
3.4.6.h “ LSQDHTP” ............................................................................. 167
3.4.6.1 “tauw” ....................................................................................... 168
3.4.6.j
“ SDHTP” .................................................................................. 168
3.4.6.k “ DHTP” ..................................................................................... 168
3.4.6.1 “ SaltDens” ................................................................................ 168
3.4.6.m “ conductivity” ..........................................................................169
3.4.6.n “ recursiveresults” ...................................................................... 171
3.4.7 Error Analysis and Goodness o f F it...........................................................171
3.4.8 Data Analysis for waxy maize-NaCl suspensions...................................... 173
3.4.9 Broad Spectrum Dielectric Measurements................................................. 175
3.4.10 Conductivity Measurements.......................................................................176
3.4.11 Time-Temperature Superposition............................................................... 177
3.4.12 Statistical Procedures................................................................................. 177
Results and Discussion........................................................................................... 179
3.5.1 Recognizing the Differences Between the Debye Model and the Data.... 179
3.5.2 Selection Process for In itia l Estimates....................................................... 184
3.5.3 Comparison o f the Dielectric Properties o f Starch andW ater.................... 188
3.5.3.a Relative Perm ittivity Curves..................................................... 189
3.5.3.b Relative Loss Curves................................................................ 190
3.5.4 The F it o f the Debye-Hasted Model w ith Two Peaks................................ 190
3.5.5 Parameters fo r the Debye-Hasted Model w ith Two Peaks......................... 193
3.5.5a High Frequency Parameters...................................................... 198
3.5.5b Low Frequency Parameters...................................................... 202
3.5.5.C Using the Parameters to Predict Dielectric Behavior at
Extended Frequencies............................................................ 207
3.5.6 Equivalent Sodium Chloride Concentration and the Effect o f
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Electrolytes.............................................................................................. 210
3.5.6.a Equivalent Sodium Chloride Concentration Parameter........... 210
3.5.6.b Effect o f Added Sodium Chloride.............................................211
3.5.6.C The Importance o f Taking Conductivity into Account
when M odeling....................................................................... 213
3.5.7
Broad Spectrum D ielectric Measurements.............................................. 217
3.5.8
Conductivity Measurements.................................................................... 220
3.5.9
Time-Temperature Superposition............................................................ 222
3.6
Conclusions............................................................................................................226
4 FIN ITE DIFFERENCE TIM E D O M AIN M ODEL OF THE ELECTROMAGNETIC FIELD
DISTRIBUTIO N IN A DOMESTIC M ICROW AVE OVEN LOADED W ITH AN AQUEOUS
STARCH SUSPENSION AT DIFFERENT TEMPERATURES.............................................. 228
4.1
Introduction............................................................................................................228
4.1.1
Power Transmitted by a W ave................................................................ 229
4.1.2
Classification o f Models o f Food Processes............................................ 233
4.2
Review o f the Literature.........................................................................................234
4.2.1
Survey o f Some Attempts to Model Microwave H eating....................... 234
4.2.2
The Yee C e ll............................................................................................238
4.3
Objectives...............................................................................................................243
4.4
Materials and M ethods...........................................................................................244
4.4.1
Hardware and Software............................................................................244
4.4.2
Determining Cell Size..............................................................................245
4.4.3
Modeling the Oven...................................................................................248
4.4.4
Running the M odel...................................................................................255
4.4.5 Representing the D ata..............................................................................256
4.5
Results and Discussion........................................................................................... 258
4.5.1
Confirmation o f Steady State Using Electric Field Am plitudes.............. 258
4.5.2 Steady-State Electric Field Magnitudes in the Entire O ven..................... 260
4.5.3 Electric Field Magnitudes in the B o w l.................................................... 261
4.5.4 SAR Values in the B o w l..........................................................................266
4.6
Conclusions............................................................................................................ 272
5 CONCLUSIONS AN D SUGGESTIONS FOR FUTURE RESEARCH................................... 273
5.1
Conclusions............................................................................................................273
5.2
Suggestions for Future Research............................................................................ 276
5.2.1
Suggestions fo r Dynamic Measurement o f Starch Particle Size Behavior
During Microwave Heating..................................................................... 276
5.2.2
Suggestions for In-line Measurement o f the Dielectric Properties o f Aqueous
Starch Suspensions During Microwave H eating................................... 277
5.2.2.a Test Materials and Sample Preparation.....................................277
5.2.2.b Experimental Method................................................................278
5.2.2.C M odel........................................................................................279
5.2.3 Suggestions for the FDTD Model o f the Electromagnetic Field
D istribution............................................................................................ 280
5.2.3.a Im proving how the model is constructed and run in XFDTD.280
5.2.3.b Applying the results to obtain information about microwave
heating................................................................................... 280
B ibliography..................................................................................................................................... 282
Appendix A:
Derivation o f Electric and Magnetic Field Wave Equations.................................. 298
A. 1.
Derivation o f the Electric Field Wave Equation...................................... 299
A.2
Derivation o f Penetration Depth............................................................... 302
A.3
Sim plification for a Slightly Conducting M edium ................................... 306
A.4.
Derivation o f the Magnetic Field Wave Equation.................................... 307
A.5
Calculation o f the Wavelength in a Medium............................................ 310
Appendix B:
Reconciling Different Versions o f kj and kR.......................................................... 311
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Appendix C:
Appendix D:
Appendix E:
Appendix F:
Appendix G:
Appendix H:
Appendix I:
Appendix J:
Appendix K:
Computer Program to Collect Time-Temperature Data from the Data Logger
318
Flowcharts and Code fo r Computer Programs Used in Processing Data from
Chapter 3 ................................................................................................................. 325
D .l
D.2
“ interpolate” ...............................................................................................334
D.3
“ correcttable” .............................................................................................336
D.4
“ th2o” .........................................................................................................340
D.5
“ recursivehc” ..............................................................................................342
D .6
“ LSQDHTP” ..............................................................................................350
D.7
“ tauw” ........................................................................................................359
D .8
“ SDHTP” ...................................................................................................361
D.9
“ DHTP” ......................................................................................................363
D.10
“ SaltDens” ..................................................................................................366
D. 11
“ conductivity” ........................................................................................... 3 69
D. 12 “ recursiveresults” ...................................................................................... 375
D.13
“ saltrecursivehc” ....................................................................................... 377
D.14
“ saltLSQDHTP” ........................................................................................ 383
Recursive Results for Waxy Maize Starch.............................................................. 390
Recursive Results for Common Com Starch........................................................... 397
Recursive Results for Potato Starch........................................................................ 400
Comparing Dielectric Properties o f StarchSuspensions w ith Those o f Water
405
Plots o f Dielectric Data and Fitted Curves for Starch Suspensions.........................412
Coordinates o f Microwave O ven............................................................................ 421
Additional Results For Finite Difference Time Domain M odeling.........................423
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ix
List o f Tables
Table
2.1
2.2
2.3
2.4
2.5
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.1
4.2
4.3
E .l
F .l
G .l
J .l
Caption
Phosphorus Content (% w /w ) in Starches Reported by Kasemsuwan and Jane (1986)
Starches Used in Particle Size Analysis Experiments
Heating rates fo r starches
Comparison o f in itia l and maximum D [v,0.5] for microwave and conventional heating
Comparison o f Tmax Values
Conductivities at 20°C
Radiation required for orientational polarization to occur
Factors used to convert from mks units to electrostatic units
Representative Samples o f Relaxation Parameters Used in the Literature
Data about Starches Used in Experiments
Summary o f the Details o f the Chosen Combinations
Equivalent Sodium Chloride Concentrations (% w /w ) for a ll Starch Suspensions
Calculated from M atrix o f Parameters
Summary o f Details o f the Broad Spectrum Experiments
Effect o f Changes o f Parameters on Temperature Profile A fter Microwave Heating
(from van Remmen et al., 1996)
Dielectric Properties o f Materials at 2.45 GHz (except as noted) and Calculated
Wavelengths
Settings for Calculations fo r the Model
Recursive Results for Waxy Maize Starch
Recursive Results for Common Com Starch
Recursive Results fo r Potato Starch
Locations o f oven features and sample points in terms o f cell numbers
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List o f Figures
Figure
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
C aption
Fischer projection o f D-glucose.
Pyranose form o f a-D-glucose.
Hypothetical amylopectin chain.
Fringed micelle structure o f a polymer w ith crystalline and amorphous regions.
Schematic o f LSPSA apparatus.
Schematic diagram o f the stages o f baking and the types o f ovens in which they occur.
Test tube samples at various starch-water ratios exhibiting chalky, gelled, and watery
regions (adapted from Zylema et al., 1985).
Experimental set-up for PSA o f starch granules during microwave heating.
Inside view o f Tappan microwave oven w ith heating vessel, plastic tubing, and beaker o f
water.
PSA heating curves for: a) CC; b) W M ; c) CWM.
Close-up schematic view o f Y connector.
SEM pictures o f starches: a) CC; b) W M ; c) CWM.
PSA surface for CC.
Particle size distribution for CC at three representative temperatures.
PSA surface for C W M .
Particle size distribution for C W M at three representative temperatures.
D[v,0.5] vs. T and calculation o f Tmax for CC. Colored lines represent repetitions o f
experiments.
D[v,0.5] vs. T and calculation o f Traax for W M . Colored lines represent repetitions o f
experiments.
D[v,0.5] vs. T and calculation o f Tmax for C W M . Colored lines represent repetitions o f
experiments.
Electromagnetic radiation from a point source.
Traveling plane wave.
Traveling pulse.
Standing wave.
Schematic illustrations o f molecular stretching and bending vibrations.
Polar molecule w ith dipole moment vector.
Water molecule.
A nonpolar molecule can undergo vibrational transitions i f its dipole moment changes
during the vibration.
An electromagnetic field can cause the permanent dipole moment (arrow) to oscillate.
Magnetic and electric field components o f an electromagnetic wave perpendicular to
each other and to their direction o f propagation.
D efinition o f curl.
D efinition o f divergence.
Three-dimensional representation o f relationship between time, frequency, and
amplitude.
Electromagnetic radiation may be either reflected, transmitted, or absorbed by a material.
Illustration o f valence and conduction bands at 0 K: a) conductors and b) insulators.
Basic block diagram o f microwave oven power system.
Orientation o f polar molecules.
Atomic polarization.
Electronic polarization.
Ionic polarization.
Interfacial polarization.
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3.22 Graph o f dielectric perm ittivity and loss as functions o f frequency, w ith dominant types
90
o f polarization indicated.
3.23 Loss tangent diagram.
92
3.24 Relative dielectric perm ittivity and loss curves showing es, fcnt, and &*,.
94
3.25 A plot o f s' verses s o " yields a slope o f - t.
95
3.26 Argand diagram o f substance behaving according to Debye’s theory.
96
3.27 Argand diagram o f substance that does not obey Debye’s theory.
97
3.28 Three-dimensional plot depicting e / and Sr" as functions o f frequency.
98
3.29 Dielectric spectrum o f water showing orientational polarization.
99
3.30 Convection current density: the product o f the charge density and velocity o f the cloud
102
particles.
3.31 Conduction current density.
103
3.32 Electric field vector as a function o f z, parallel to the x-axis.
114
3.33 Coaxial probe method.
117
3.34 Transmission line method.
118
3.35 Network analyzer block diagram.
120
3.36 Diagram o f experimental set-up for the measurement o f dielectric properties during
151
heating.
3.37a View o f experimental setup w ith network analyzer on the left.
152
3.37b View o f experimental setup w ith microwave oven on the right.
152
3.38 Interior o f Tappan microwave oven w ith heating/cooling vessel, tubing, and beaker o f
152
ballast water.
3.39a Close-up o f dielectric measurement cell.
152
3.39b Dielectric measurement cell w ith insulating cover.
152
3.40 Circulating bath.
153
3.41 Parameter definitions for the Debye-Hasted Model w ith Two Peaks.
158
3.42 Diagram showing computer programs and their subfunctions.
159
3.43 Interpolation o f temperature at 3 different frequencies.
161
3.44 Correction factor determined by water data.
162
3.45 Illustration o f the definitions o f statistical quantities.
172
3.46 Comparison o f the a) relative dielectric perm ittivity and b) relative dielectric loss o f
waxy maize starch suspension and water during the heating cycles. Inset details the
181
relative dielectric perm ittivity at 30, 35 and 40°C.
3.47 Data for water theoretically m odified to take into account the addition o f electrolytes in
183
the concentration o f 10‘2 % (w /w ) sodium chloride.
3.48 Data for water theoretically m odified to take into account an additional relaxation at
183
lower frequencies w ith
- e,i ~ 7 and t 2 ~1.5 ns.
3.49 Data for water theoretically m odified to take into account an additional relaxation at
184
lower frequencies w ith Es2- esi ~ 16 and t 2 ~ 16 ns.
3.50 Graphical summary presentation o f the selection o f in itia l estimates for a) waxy maize;
186-187
b) common com starch; andc) potato starch.
3.51 Comparison o f the fitted curve w ith the measured data for the a) relative dielectric
perm ittivity and b) relative dielectric loss during the heating cycle for 3% waxy maize
192
starch suspension.
3.52 Diagram illustrating the variables affecting the influences on the relative dielectric
195
properties o f the starch suspensions.
3.53 Parameters o f importance at high frequencies: a) e^; b) t2.
196
3.54 Parameters o f importance at low frequencies: a) es,- e^; b) Tt.
197
3.55 Parameters o f importance at low frequencies: £*,- es2; and Ti for waxy maize starch
198
3.56 Extended results: a) changes in the relative dielectric perm ittivity and b) changes in the
208
relative dielectric loss during the heating cycle for 3% waxy maize starch suspension.
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3.57 Extended results: a) changes in the relative dielectric perm ittivity and b) changes in the
relative dielectric loss during the cooling cycle for 3% common corn starch
suspension.
3.58 Comparison o f the fitted curve w ith the measured data for the relative dielectric
perm ittivity during the a) heating and b) cooling cycles waxy maize w ith N aC l.
3.59 Comparison o f the fitted curve w ith the measured data for the relative dielectric loss
during the a) heating and b) cooling cycles waxy maize w ith NaCl.
3.60 Extended frequency results for waxy maize w ith N aC l during the heating cycle: a)
relative dielectric perm ittivity and b) relative dielectric loss.
3.61 Comparison o f the fitted curves w ith the measured relative dielectric perm ittivity and loss
data from the broad frequency experiments fo r waxy maize starch suspension during
the a) heating and b) cooling cycles.
3.62 Comparison o f parameters for both the original data and the broad frequency data: a) es1
- £52 and i i curves; b)
and x2.
3.63 Conductivity o f a waxy maize starch suspension during heating and cooling. Colored
contour lines and color bar indicate equivalent NaCl concentration in % (w/w).
3.64 Time-temperature superposition curves o f the relative dielectric perm ittivity data for the
waxy maize starch suspension narrow spectrum experiments during the a) heating and
b) cooling cycles.
3.65 Time-temperature superposition curves o f the relative dielectric loss data for the waxy
maize starch suspension narrow spectrum experiments during the a) heating and b)
cooling cycles.
3.66 M ultipliers for the frequency and relative complex perm ittivity for the time-temperature
superposition curves fo r the waxy maize starch suspension broad spectrum
experiments.
4.1 Change in intensity o f radiation passing through a material o f thickness z.
4.2 Yee cell showing locations o f the indexed fie ld components used in FDTD.
4.3 Sharp Carousel microwave oven.
4.4 View o f microwave oven cavity showing recessed back, location o f waveguide (metal
plate on right), oven floor, and rotational coupler (center o f floor).
4.5 Model o f the back, floor, rotational coupler, and rotating ring inside the microwave oven
cavity.
4.6 View o f the floor, rotational coupler, and rotating ring o f the microwave oven.
4.7 Glass turntable (shown upside down).
4.8 Model o f glass turntable.
4.9 View o f inside o f microwave oven and waveguide without magnetron.
4.10 View o f magnetron from Sharp Carousel oven.
4.11 Cross-sectional view o f magnetron taken from McDunn and Quinn (2003).
4.12 Cross-section o f model o f magnetron in waveguide.
4.13 W ater-filled bowl inside oven cavity.
4.14 Model o f w ater-filled bowl inside oven cavity.
4.15 Location o f sample points.
4.16 Electric field amplitudes at 10 sample points versus real oven time at 30°C fo r water.
4.17 Electric field amplitudes at 10 sample points versus real oven time at 30°C fo r 3% waxy
maize starch suspension.
4.18 Electric field magnitudes in the entire oven at the end o f the calculation w ith w ater as the
load at a) 30°C and b) 75°C. Scale given in V/m.
4.19 Electric fie ld magnitudes in the entire oven at the end o f the calculation w ith 3% waxy
maize starch suspension as the load at a) 30°C and b) 75°C. Scale given in V/m.
4.20 Electric field magnitudes in the bowl at the end o f the calculation w ith w ater as the load:
a) bowl position in oven, b) 30°C, c) 75°C. Scale given in V/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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x iii
4.21
4.22
4.23
4.24
4.25
5.1
A .l
B .l
D .l
D.2
D.3
D.4
D.5
D .6
D.7
D .8
D.9
D.10
D .ll
D.12
H .l
H.2
H.3
H.4
H.5
H .6
1.1
1.2
1.3
1.4
1.5
Electric fie ld magnitudes in the bowl at the end o f the calculation w ith 3% waxy maize
starch suspension as the load: a) bowl position in oven, b) 30°C, c) 75°C. Scale given
in V/m.
SARs in the bowl at the end o f the calculation w ith w ater as the load: a) bowl position in
oven, b) 30°C, c) 75°C. Scale given in W /kg.
SAR s in the bowl at the end o f the calculation w ith 3% waxy maize starch suspension
as the load: a) bowl position in oven, b) 30°C, c) 75°C. Scale given in W /kg.
1-g SARs in the bowl at the end o f the calculation w ith w ater as the load: a) bowl
position in oven, b) 30°C, c) 75°C. Scale given in W/kg.
1-g SARs in the bowl at the end o f the calculation w ith 3% waxy maize starch
suspension as the load: a) bowl position in oven, b) 30°C, c) 75°C. Scale given in W/kg.
Sketches o f the general trends in graphs o f e' and e" vs. frequency for different dielectric
models: a) Hasted-Debye w ith an extra Debye relaxation; b) simple Debye model; c)
Cole-Cole model; d) two Debye relaxations; e) Hasted-Debye model.
Magnetic fie ld vector as a function o f z, parallel to the y-axis.
Representation o f the angle 8.
Flowchart fo r program to read data files.
Flowchart fo r “ interpolate” program.
Flowchart fo r program to correct the data.
Flowchart fo r “ th2o” function.
Flowchart for “ recursivehc” .
Flowchart for “ LSQDHTP” function.
Flowchart for “ tauw” function.
Flowchart for “ SDHTP” function.
Flowchart for “ DHTP” function.
Flowhchart for “ SaltDens” function.
Flowchart for “ conductivity” function.
Flowchart for “ recursiveresults” program.
Comparison o f the relative dielectric perm ittivity o f waxy maize starch suspension and
water during the a) heating and b) cooling cycles.
Comparison o f the relative dielectric loss o f waxy maize starch suspension and water
during the a) heating and b) cooling cycles.
Comparison o f the relative dielectric perm ittivity o f common corn starch suspension
and water during the a) heating and b) cooling cycles.
Comparison o f the relative dielectric loss o f common corn starch suspension and water
during the a) heating and b) cooling cycles.
Comparison o f the relative dielectric perm ittivity o f potato starch suspension and water
during the a) heating and b) cooling cycles.
Comparison o f the relative dielectric loss o f potato starch suspension and water during
the a) heating and b) cooling cycles.
Comparison o f the fitted curve w ith the measured data for the relative dielectric
perm ittivity during the a) heating and b) cooling cycles for waxy maize starch
suspension.
Comparison o f the fitted curve w ith the measured data for the relative dielectric loss
during the a) heating and b) cooling cycles fo r waxy maize starch suspension.
Comparison o f the fitted curve w ith the measured data for the relative dielectric
perm ittivity during the a) heating and b) cooling cycles for common corn starch
suspension.
Comparison o f the fitted curve w ith the measured data for the relative dielectric loss
during the a) heating and b) cooling cycles for common corn starch suspension.
Comparison o f the fitted curve w ith the measured data for the relative dielectric
perm ittivity during the a) heating and b) cooling cycles for potato starch suspension.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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xiv
1.6
1.7
1.8
K .l
K.2
K.3
K.4
K.5
K .6
K.7
K .8
K.9
K.10
K . ll
K.12
K.13
K.14
K.15
K.16
Comparison o f the fitted curve w ith the measured data for the relative dielectric loss
during the a) heating and b) cooling cycles for potato starch suspension.
Comparison o f the fitted curve w ith the measured data for the relative dielectric
je rm ittivity during the a) heating and b) cooling cycles for the waxy maize starch
suspension w ith sodium chloride.
Comparison o f the fitted curve w ith the measured data for the relative dielectric loss
during the a) heating and b) cooling cycles for the waxy maize starch suspension w ith
sodium chloride.
Electric field amplitudes at 10 sample points versus real oven time w ith w ater as the loac
at a) 30°C, b) 40°C, c) 50°C.
Electric fie ld amplitudes at 10 sample points versus real oven time w ith w ater as the loac
at a) 60°C, b) 70°C, c) 75°C.
Electric field amplitudes at 10 sample points versus real oven time w ith 3% waxy maize
starch suspension as the load at a) 30°C, b) 40°C, c) 50°C.
Electric field amplitudes at 10 sample points versus real oven time w ith 3% waxy maize
starch suspension as the load at a) 60°C, b) 70°C, c) 75°C.
Electric field magnitudes in the entire oven at the end o f the calculation w ith w ater as the
load at a) 30°C, b) 40°C. Scale given in V/m.
Electric field magnitudes in the entire oven at the end o f the calculation w ith w ater as the
load at a) 50°C, b) 60°C. Scale given in V/m.
Electric field magnitudes in the entire oven at the end o f the calculation w ith w ater as the
load at a) 70°C, b) 75°C. Scale given in V/m.
Electric field magnitudes in the entire oven at the end o f the calculation w ith 3% waxy
maize starch suspension as the load at a) 30°C, b) 40°C. Scale given in V/m .
Electric field magnitudes in the entire oven at the end o f the calculation w ith 3% waxy
maize starch suspension as the load at a) 50°C, b) 60°C. Scale given in V/m.
Electric field magnitudes in the entire oven at the end o f the calculation w ith 3% waxy
maize starch suspension as the load at a) 70°C, b) 75°C. Scale given in V/m.
Electric field magnitudes in the bowl at the end o f the calculation w ith w ater as the load:
a) bowl position in oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, f) 70°C, and g) 75°C.
Scale given in V/m.
Electric field magnitudes in the bowl at the end o f the calculation w ith 3% waxy maize
suspension as the load: a) bowl position in oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, f)
70°C, and g) 75°C. Scale in V/m.
SARs in the bowl at the end o f the calculation w ith w ater as the load: a) bowl position in
oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, 0 70°C, and g) 75°C. Scale given in W/kg.
SAR s in the bowl at the end o f the calculation w ith 3% waxy maize suspension as the
load: a) bowl position in oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, 0 70°C, and g) 75°C.
Scale given in W/kg.
1-g SARs in the bowl at the end o f the calculation w ith w ater as the load: a) bowl
position in oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, 0 70°C, and g) 75°C. Scale given
in W/kg.
1-g SARs in the bowl at the end o f the calculation w ith 3% waxy maize suspension as
the load: a) bowl position in oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, f) 70°C, and g)
75°C. Scale given in W/kg.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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XV
List of Symbols and Abbreviations
M eaning
Symbol
G reek letters
Cole-Cole parameter
a
symbol used by other researchers for ^
a
symbol
used by other researchers for kR
P
V
divergence operator
v.
Vx
AV
voltage applied to a capacitor
a volume element in a dielectric
Ao
U nits
dimensionless
inverse meters
inverse meters
8
loss angle, tan' 1(e'Ve')
8+
hydration number for positive ion
volts
meter3
dimensionless
8~
hydration number for negative ion
dimensionless
average hydration number
complex perm ittivity
relative complex perm ittivity
dielectric loss
dipole loss or loss factor
infinite perm ittivity = optical dielectric constant
relative ionic loss
relative dielectric loss
dielectric perm ittivity
perm ittivity o f free space = 8.85 x 10' 12 F/m
relative dielectric perm ittivity = dielectric constant
static dielectric constant (perm ittivity)
solvent viscosity or microscopic internal friction
Complex intrinsic impedance
a generic angle
absorption coefficient o f an absorbing solution
dimensionless
8
8*
8r*
8"
o
"
& dinole
Boo
F
.
°r ionic
Er"
8'
So
Sr'
Bs
h
ri*
e
K
K'
X
X0
p*
Mdm
Fc
fi
Fo
P
Pv
^charge
other authors’ notation for dielectric constant
wavelength
wavelength in a vacuum
complex permeability
dipole moment
m obility o f the charges
permeability
permeability o f a vacuum = 1.26 x 10'6H/m
density
electric charge density
conductivity due to moving charges
CT
dielectric conductivity term
X
relaxation time
m
..
-i
V
m
F/m
dimensionless
F/m
F/m
dimensionless
dimensionless
dimensionless
F/m
F/m
dimensionless
dimensionless
poise
P
ohm
n
units of (concentration.length)'1 l/(mol.m
)
dimensionless
meters
m
meters
m
henries/meter
H/m
debyes or coulomb.meters
D or
Cm
meter2/(volt.second)
m2/(Vs)
henries/meter
H/m
henries/meter
H/m
kilograms/meter3
kg/m3
coulomb / meter3
C/m3
siemens/meter or
S/m or
mhos/meter
G'Vm
siemens/meter or
S/m or
mhos/meter
Q'Vm
seconds
s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xvi
Xe
0)
phase angle
electric susceptibility
angular frequency = 2 n f
English letters
A
amplitude o f a wave or
area o f capacitor plates
a
radius o f a sphere, ion, or molecule
a™A, B, generic coefficients o f a third order polynomial
C
B
magnetic flu x density vector
phasor notation o f magnetic flu x density vector
B
c
concentration
speed o f light in a vacuum, 3 x 108 m/s
CC
CWM
D
D
D
D |v, 0.51
Dr4,21
Df
Di
Dmax
d
dD
E
E
E (ti)
E(t2)
Ex
E*
E1
Er
Ephoton
Ex
e
f
fcrit
21
H
H
H*
H1
Hr
h
Io
common com starch
polynom ial for median particle size diameter
electric flu x density vector
phasor notation o f electric flu x density vector
median diameter o f the volume distribution
mean diameter o f the volume distribution
final D [v, 0.5]
in itia l D [v, 0.5]
maximum D [v, 0.5]
distance vector pointing from negative charge to
positive charge in an electric dipole
penetration depth
electric field vector
phasor notation o f electric fie ld vector
magnitude o f the electric fie ld vector
at times 1 and 2
magnitude o f the electric fie ld vector in the xdirection
complex conjugate o f E
incident electric wave
reflected electric wave
energy o f a photon
magnitude o f the x-component o f the electric field
base o f the natural logarithm ,« 2.71828
frequency
critical frequency
frictional retarding force on an ion
magnetic field strength vector
phasor notation o f magnetic field strength vector
complex conjugate o f H
incident magnetic wave
reflected magnetic wave
Planck’s constant = 6.625 x 10"34 J s
in itia l intensity o f radiation at the surface o f a
medium
dimensionless
meter2
meters
2
m
m
Webers/meter2
W /m2
moles/liter
meters/second
m ol/t
m/s
micrometers
coulombs/meter2
pm
C/m2
micrometers
micrometers
micrometers
micrometers
micrometers
meters
pm
pm
pm
pm
pm
m
meters
volts/meter
m
V/m
volts/meter
V/m
volts/meter
V/m
volts/meter
V/m
Joules
volts/meter
dimensionless
seconds"1= Hertz
seconds"1= Hertz
kilogram.meter/second2 = 1
Newton
amps/meter
J
V/m
amps/meter
Joule second
power/unit solid angle or
watts
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
s"1= Hz
s"1= Hz
kg.m/s2
= IN
A/m
A/m
Js
W
xvii
Iz
intensity o f radiation at a distance z in the medium
J
electric current density
conduction current density
convection current density
source current density
square root o f-1
Boltzmann’s constant, 1.38 x 10'23J/K or
linear absorption coefficient fo r a medium
complex propagation constant
imaginary part o f k*
real part o f k*
Laser Scattering Particle Size Analysis
molecular weight or
a molecule in its ground state
a molecule in an excited state
a general number or exponent
number o f time lapses in finite difference method
index o f refraction
polarization
power at the surface o f a medium
power at a distance z in the medium
square root o f the generic constant p
particle size analysis
a generic constant
electric charge
radius o f a complex number in polar notation
coefficient o f determination
abbreviation for the real part o f a complex equation
Poynting vector
complex Poynting vector
time-average Poynting vector
speed o f a moving ion
temperature or
period o f time-harmonic signal = 1/ f
glass transition temperature
transmittance o f a material
m elting temperature o f crystalline regions
temperature o f maximum swelling rate
time
energy stored in a electrical field
energy stored in a magnetic field
voltage applied to a capacitor
a volume element o f a dielectric
velocity o f charged particles
speed o f propagation o f radiation in a medium
waxy maize
a generic time-harmonic wave
distance electromagnetic wave travels or
a generic complex number
a generic vector
Jcond
Jconv
Jo
j
k
k*
ki
kR
LSPSA
M
M*
n
P
P„
Pz
^(p)
PSA
q
Q
r
r2
Re
S
S*
<s>
s
T
Tg
Tm
Tmax
t
UE
UH
AV
AoV
V
WM
w (t)
z
Z
power/unit solid angle or
watts
amps/meter2
amps/meter2
amps/meter2
amps/meter2
dimensionless
joules/Kelvin
meters' 1
inverse meters
inverse meters
inverse meters
grams/mole
W
A /m 2
A /m 2
A /m 2
A/m 2
J/K
-1
m
-i
m
-i
m
m -i
g/m ol
dimensionless
coulombs/meter2
watts
watts
C/m2
W
w
c
coulombs
dimensionless
dimensionless
dimensionless
watts/meter2
watts/meter2
watts/meter2
meters/second
degrees Celcius or Kelvin
seconds
degrees Celcius or Kelvin
dimensionless
degrees Celcius or Kelvin
degrees Celcius
seconds
joules/meter
joules/meter
volts
meter3
meters/second
meters/second
°C or K
°C
s
J/m3
J/m3
V
m
m/s
m/s
meters
various
m
various
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
W/m2
W /m2
W /m2
m/s
°C or K
s
°C or K
XV111
List o f Equations
Equation
2.1
2.2a
2.2b
D = ao + aiT + a2T2 + asT*
<t< _ '-p i 2
f max
np _
max
f
44
45
d D/dt =0
^2
3a3
45
Maximum swelling pow er-
/
1
V
J)
T. max
J-'initial
\ 3
1
/
50
3.1
3.2
<
II
2.3
Page
Eohoton = h f = h(v/A.)
58
61
3.3
M + photon —►M *
62
3.4
mm = Qd
dipole moment
3B
_
at
3.5
V xE
3.6
V xH
3.7
3.8
V • B = 0
V • D = pv
„
d
d
d
V'
1 Sx + 51 ay + z 3z
„
/ 3EZ
8Ey \
( SEx 3EZ \
V x E = I ( av
&
) + n a T "a x
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
^
at
M axw ell’s adaptation o f Ampere’s Law
V X H - x f 8" ^ 3y
dz ^
determinant definition o f curl
'd z
3x ^
Gauss’ Magnetic Law
Gauss’ Electric Law
67
67
67
67
67
( 3Ey
^ 3x
3EX \
" a y ->
67
3H l 1
dy ^
68
68
68
(V X E ) • x =~ ^ ~
dy
dz
dEz
(Ez) on b - (Ez) on d
68
dy
5Ev
dz
Ay
_
(Ey) on c - (Ey) on a
Az
(Ey) on a Ay + (Ez) on b Az + (-Ey) on c Ay + (-Ez) on d Az
( V X E ) ,X
AyAz
V XH =J
Ampere’s Circuital Law
V • B = 3Bx/3x + 3Bv/3y + 3Bz/3z
V • D = 3Dx/3x + 3Dy/3y + dDJdz
3BX _
{B x} at right - {B x} at left
68
69
70
71
71
71
X
3.20
J+
>
3.9
~
63
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
v t D
D
_
{B x} at right AyAz - {B x} at leftAyAz
volume
e'E
B = pH
Er' = s7e0
w (t) = Arcos (cot + cp) + j sin (cot + <p)l
co = 2n f
e*9 = cos 0 + j sin 0
fft) = Ae’(“ 1+q,) = A eiq>e’0”
=
71
constitutive relation
constitutive relation
relative dielectric perm ittivity
angular frequency
Euler’s Identity
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
72
72
73
74
74
74
74
xix
3.29
3.30
3.31
3.32
3.33
3.34
3.35
3.36
3.37
3.38
3.39
3.40
3.41
3.42
3.43
3.44
3.45
3.46
3.47
3.48
3.49
3.50
w = x + jy
x = A cos(©t + cp)
x = R e{A cos(©t+p) + A .j sin(cot+(p)}
x = Ref A e * ei°>* }
A cos(cot + <p) *-* A = A e19
V x £ = - j© B
Frequency domain version o f M axw ell’s equations
V x H = j© D + J 0
V • B=0
V • D = pv
D = e*E
B = u*H
Jcond OcharaesE
Ohm S LUW
resistivity = l/a chari!es
nAir
P=
lim
(l/A u) IHdmi
Ao -> 0
i= l
D = s0E + P
P = XeSoE
Xe = (bound charge density)/(free charge density)
D = e0E (1 +Xe)
3.52
3.53
3.54
3.55
3.56
3.57
Er = Xe + 1
D = e08rE
x = 1/cOcnt = 1/(2ti fait)
e"
loss current
tan 8 = --------- - — :----- :-------------------------e'
charging current
t = 4nr]C/(kT)
Er' = Eco + [(e. - e „)/(l + © V )]
Er*= E „ + (Ss-S00)/(l + i©T)
rsr' - (£S- s„)/212+ e/ ' 2 = Tfe-Ej/212
3.58
Er* - £«,
3.51
3.59
3.60
3.61
3.62
3.63
3.64
3.65
3.66
3.67
3.68
3.69
3.70
= --------- ^
—-v. .-----n + (i© x0) 1 i
(es- zm Xl+Crato)1^ (sin an/2)l
81 S°°
(E s - £00 )(©To)l<t (cos art/2)
Sr
Jconv = PvV
V —P charges E
r^charees —Pv M-charees
*1 do ^ Jcond
V X H = J 0 + Jcond + 0D/3t
V X H = J 0 + Jcond + d/dt (e'E)
V x H 1 /0 "T Jcond j©E £
relaxation time
complex dielectric perm ittivity
loss tangent
Debye equations
Cole-Cole equations
75
75
75
75
75
76
76
76
77
77
77
79
79
87
87
87
88
88
88
88
89
91
92
93
93
93
93
95
96
98
98
98
convection current density
O —Ocharces ^dioole
V x H = J „ +cr£ + j© s'£
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
103
103
103
104
104
104
105
105
105
XX
3.71
3.72
3.73
3.74
: a/m____________
= o/((Q£0)_________
apparent ~ E r
Sr
4jtoe.
105
106
106
4 7 t g esu/ ( 0
47tomks • 9 x 109
=
gmks
tO
S p inks *• 36n
3 6 7 1 X 109______ (OSom ks
to Somlcs
® Boesu
3.75
3.76
3.77
Er
3.78
t*r ionic solution
3.79
c ionic solution =
3.80
t*r * ionic solution
3.81a
3.81b
3.82
3.83a
3.83b
3.84
3.85
3.86
3.87
M * —►M + heat
M * —>M + Enhoton
V E + (o |i0 e* E = 0
E = \E X
E = xEne"Jk' z
107
107
108
108
apparent" Br
tTmks^(tO£o mks)
i ^ c _
_I
Es waterr+ 2 8c
5 = (8 + S' )/2
Es water "^"2 8 C
Sop v
Sqq
1+ co2t 2
(g Swater
2 5 C
109
^charge
me„
Sqqwater) tOT
l+coV
(Ss
+ 2 8 c —So,
1+jCQT
_
+ Ea
109
JCTcharge
toe0
112
112
complex propagation constant
k* = <o(HqS*)'/2
k* = kR- jk ]
^ (z,t) = E0 e'kiz cos (<nt-kRz)
dD= l/k !
3.88
k * = to [P o (s '-je ")]
3.89
k * = 0 (p oE') 1/2 [ l - j - f ]
109
114
114
114
114
114
115
115
115
1/2
115
UT~
3.90
3.91
3.92
115
= tan 8
08'
m 2i 1/4,
-1
ki = m (p0s ')1/2{ 1 + |a/(cos')]2} ‘/4{sin ( 1/2 tan 1 [ct/(os')1)}
3.93
<V
3.94
= •* =
o (p 0e') 1/2 (1 + [CT/(oE')12} 1/4{sin ( l/ 2tan 1 fg /(o e ')1)}
(E - E2) /( l + jO Ti) + (s2 - £«,)/(! + .jOT2) + Eoc
_
3.95
(Esl
^ s2 )
E i+qm n)]
3.96a
3.96b
3.97a
3.97b
3.98
3.99
3.100
3.101
3.102
3.103a
3.103b
COrrh
£w t
Ewmeash
C O rrc
Ewt
Ewmeasc
gh c i = Esmhi
Ecci
+ Eoo2+ -
( Es 2 - 2 8 C -£ a o 2
)
J°
ri+ (jO T 2)l_________ SpO
115
115
general equation for
penetration depth
116
Double Debye Model
144
Debye-Hasted Model
w ith Two Peaks
157
+ corrh
Esmci
C O lT c
SSEConv - standard deviation o f the previous SSE and the current SSE
mean o f the previous SSE and the current SSE
standard deviation o f previous amean and current amean
convMean =
mean o f previous amean and current amean
density = c0 + cx l*T + c ^C T *12) + cMC + cM,*(C eMI) + cM2*(CeM2) + cT2M.*(TelJ)*'(Cm )
A
=
10 k /C
= A C x 10
SSTO = SSR + SSE
SSR = SSTO — SSE
k
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162
162
162
162
166
166
169
170
170
171
171
XXI
3.104
3.105
3.106
3.107
3.108
4.1
4.2a
4.2b
4.2c
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
SSTOem —SSEexp
_
S S R ^ IXD
exp
SSTO,exp
SSTOevn
No. o f observations = (No. o f frequencies) * (No. o f dielec properties) * (No. o f temps)
number o f parameters = 5 * (number o f temps) + 1
degrees o f freedom = number o f observations - number o f parameters
SSE
mse = degrees o f freedom
TA m = ±P7!/P
An
Transmittance o f a material
I, = I0e
Lambert’s Third Law
Iz = I0e-K[clz
Beer-Lambert Law
Power form o f Lambert’ s Third Law
Pz ~ P<>e
W
m
m
Cm
ms
my
m ' i „s
S(x,y,z,t) = E (x,y,z,t) x H (x,y,z,t)
H = y (EfAi*)e~J
,2
<S> = z (1/2)(E07 * I )e '2V cos cp
<S> = z (constant) e
AA
_
A (t2) - A ( t 2)
5A
At
t2- t i
at
dBx
dEz _ aEy
dz
8t
dx_
3B.
dEx
8EZ
=
dt
dz
8x
8BZ
9EX _ aEy
dx.
at
dz_
[Bx
(i, j+1/2, k+1/2) - Bx“ (i, j+1/2, k+1/2)]/ At = [Ey" (i, j+1/2, k+1) Ey- (i, j+1/2, k)1/Az - rEzn(i, j+1, k+1/2) - Ezn(i. j, k+ 1/2)1/ Ay__________
t = nAt
2?t
ci)(tt0s') 1/2 { 1 + [ct/((os')]2} 1/4{cos ( 1/2 tan 1 [cr/ftpe')])}
f(Ax)z + (Ay)" + (Az)211/2> vAt n / ( 6n )i1/2At
c =At <
1
(e„por
Ax
2ti
X=
4.19
A .l
A.2
A.3
A.4
A.5
A .6
A.7
A .8
A.9
A. 10
o = e r EpO)
V x E = - jo g //
V • e *E = pv
229
230
230
231
231
231
232
232
232
239
239
239
239
240
240
241
241
241
Courant stability equation
242
172
245
• ( ¥ ) “ { [ - ( f) '] “+i}
V x H = \( oe*E
V • B=0
V • £=0
V x (V x £ )= - jtogo ( V x H )
V x ( V x E ) = q)gn e*E
V x (V x Z ) = V (V > Z ) -V z Z
V (V • £ ) - V E = to u„ s*E
173
173
173
173
velocity o f light
cV3
4.18
172
vector identity
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246
299
299
299
299
299
299
300
300
300
300
xxii
A . ll
A. 12
V 2£ + (02nos* E = 0
&EX
&EX
&EX
5x2
Sy2
dz2
300
| m 2u r *
p
® ^°E Ex
_
0
0
A.13
Ex
5z2
A. 14
Ex = eqz
dEx . - a e qz
dz
(?Ex
_
2qz
az2
qe
301
A.17
A.18
A.19
A.20
A.21
A.22
A.23
A.24
A.25
q = ± jca(p0s* ) 1/2
k * = (»(|j.0e*)'/2
k * = kR- jk ,
q = - jk *
Ex = e‘Jk’ z
Ex = E0 e'Jk' z
E = xEx = xE0e'Jlc' z
E = xE0e"kz e'-’V
301
301
302
302
302
302
302
302
302
A.26
Ex (z,t) = Re{E„e'kiz
303
A.27
Ex (z,t) = Re{E0 e'V [cos (cot - kRz) + j sin (cot - kRz)]
303
A.28
A.29
A.30
Ex (z,t) = E0 e'kz cos (cot- kRz)
dD= l/k r
k * = co[p0( e '- js " ) l1/2
o
1/2
k * = co[h0(e' - j — )]
a
303
303
303
A.15
A.16
A.31
, . 2 c+ cr n
+ ® PoE t=x- 0
301
301
301
301
303
1/2
A.32
k
A.33
A.34
A.35
A.36
z“ = r”(cos n0 + j sin n0)
zm = r 1/2(cos 0/2+ j sin 0/2)
r 1/2 = {1 + ro /(a e ')l2} 1/4
0 = tan-1 f-a /(a e ')l
- ^
[i-i
r
l , ]
304
304
304
304
304
1/2
A.37
A.38
(1 - j ^ ; )
= {1 + [o /(a s ')]2} 1/4{cos ( 1/2 tan-1 [ - cj/ ( cos ' ) ] ) + j sin ( 1/2 tan-1 [-a/(cos')])}
tan' 1(-A ) = -tan' 1(A )
304
305
1/2
A.39
( 1-j^ ; )
= {1 + [o/(coe')]2} 1/4{cos (- 1/2 tan-1 [a/(coe')]) + j sin (-1/2 tan-1 [a/(coe')])}
A.40
A.41
cos (-A ) = cos A
sin (-A ) = -sin A
A.42
( 1-
A.43
A.44
A.45
k * = oo(p0e')1/2{ 1 + ro /(a e ')l2} 1/4{cos ( 1/2 tan’ 1 [a/(coe')l) - j sin ( 1/2 tan-1 ro /(a e ')l)}
kR = o)(p0s ')1/2{ 1 + ra/(coe')l2} 1/4{cos ( 1/2 tan-1 [a /(a e ')])}
ki = co(p0e')1/2{ 1 + [a/(coe')]2} 1/4{sin ( 1/2 tan-1 [cr/(coe')])}
,
1
p
ffl(n 0s')1/2 {1 + ra /(o e ')l2} 1/4{sin ( l/ 2tan"’ [a/(coe')l)}
r 1/2 = { 1 + [a /(a e ')l2} 1/4 ® 1
tan 0 ~ 0
cos 0 « 1
305
305
305
1/2
A.46
A.47
A.48
A.49
)
= {1 + [cr/(coe')]2} 1/4{cos ( l/ 2tan_1 [a /(c o s ')])-j sin ( 1/2 tan-1 [a/(cos')])}
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305
305
305
305
305
306
306
306
XX111
A.50
A.51
A.52
A.53
A.54
A.55
A.56
A.57
A.58
A.59
A.60
A.61
A.62
A.63
sin 9 « 0
0 = tan-1 ( - ct/ ( oos' ) ) » -a/fras')
[1 - j ( ct/ cos') 1 1/2~ { c o s 1/2 r-a/(a>e')l + j sin l/ 2f-a /(ffls')1}
f l - j (a/cos' ) ] 1/2 ~ (cos [a /(2© e')l - j sin [a /(2a>e')]}
n - i (a/<»s')l1/2» 1 - j cr/(2a>e')
k* * co(u0e ')1/2{ l- . i cr/(2coe')}
kR= fl)(Uoe') I/2
k, = (CT/2)(Po/e') 1/2
d n = (2/a)(s7p0) 1/2
V • H= 0
V x (V x W ) = joos*(V x £ )
V x (V x W ) = co2p0 e* H
A.64
^ l
A.65
Hv = e,z
8Hy
_
V (V • W ) -V2 H = <d2u„ e* H
0 = V2 H + ffl2n „ e* H
A .6 6
1 CD | I 0G
Sz
308
qz
308
_
2 qz
A.69
A.70
Sz2
qC
H , = e -^
H v = H„e'J
H = y H v = yH 0e1*’ z
A.71
JWFoHo C
A .6 8
•
A.72
308
U
qC
Hy
A. 67
tly
-
308
308
308
309
-ik*z
it
- jk * E0e‘jk*z =
309
jco|i0H0e'jk*z
309
k ^ ,
A.73
309
Ho = (e*/Uo) 1/2E„
n *= (u „/E * ) 1/2
Ho= E„/ri*
H = y H v = y ( E J x ) * ^ '2
t|* = Ti* e19
A.74
A.75
A.76
A.77
A.78
A.79
A.80
A.81
309
309
309
309
310
B .l
B.2
B.3
H = y (Eq/ | n * 1) e'kiz e'jkRz e‘j9
v = ra/kR
X - 2jt/k R
. _
2n
0s') 1/2 {1 + [a/(o)e')l2} 1/4{cos ( 1/2 tan-1 [cj/(gje')])}
Ex = Exoe'Jkz = Exoe'aze'J,5z
Hayt and Buck’s (2001) notation
jk = a + jP
Hayt and Buck’s (2001) notation
k = P - ja
Hayt and Buck’s (2001) notation
B.4
k*
A.82
306
306
306
306
306
306
306
306
307
307
307
307
307
308
_n
= G )( p 0e ' ) 1/2
[
1_J
E.
310
310
310
310
312
312
312
1/2
]
312
i«
B.5
B .6
B.7
(
1 - i - V )
6 '
=
( 1+
[e"/e ']2} 1/4{cos ( 1/2 tan' 1 [E " /s '])-j sin ( 1/2 tan' 1 [e "/s '])}
k * = ® (u„e')1/2{ l + r£"/E'l2>1/4{cos (1/2 tan"1rs'Ve'D- j sin (1/2 tan' 1 \e"/e’])}
8
( 1 - cos 8 \ 1/2
sm 2 (
2
)
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312
312
313
xxiv
B.8
B.9
B.10
8
, / 1 + cos 8
cos
2
- + (
2
cos 8 = 1 / V l + [e"/e']2
k*
\ 1/2
)
313
313
= co(n0e'/2) 1/2{ l + [e"/e']2} 1/4
j
[ 1+ /
1
1/2
1 + [e"/s']2 ]
1
1-
V l + [e"/e']2 ]
1/2
}
313
B . ll
kR = to(ji 0e'/2)1/2 [ {1 + [e"/e']2} 1/2 + 1]
B.12
ki = co(p 0s'/2)1/2
B.13
B.14
B.15
dp = l/(ot)
a = (2nA.n) r(l/2 )K '(l + tan2 8)1/2- \ ] m
K' = e7s0
314
314
314
B.16
tan28 + 1 “ (
k
315
B.17
B.18
B.19
B.20
B.21
B.22
[ {1 + [e"/e']2} 1/2 - 1]
—
m
314
2 \
cos 8 /
■ = « [ ( <
A.0 = c /f
f = <b/(2ji)
X0 = In c /a
2rr
314
{
‘2T
315
}“ ]
/
315
315
315
\ 1/2
(0 (£0P-o)
315
,
o /2 r , .si/7 / l- c o s 8 x 1/2 -1
ra(e0n„) [( k )
j 2cosg j J
a
,
B.23
,0/2
316
( l- c o s 8 -» 1/2
{
2cos6
}
316
B.24
tan 8
B.25
8 = tan _1 ( - ^ 7)
vcoe '
316
B.26
kI = o>(Moe ')1/2 [ l + ( ^ ) 2 ] 1/4 sin ( 1/2 8)
316
B.27
k, = o ( W 0 “
B.28
k ,-
B.29
.
B.30
k, - 0,
B.31
,k, -
B.32
a
»
316
(OS
r
/
0 \
[ l + ( - ^ )
21
J
1/4
/
(
1-
cos 8
2
\
)
1/2
GwO“ [ l + ( W S ) ]
,
, 0/2
k i = co(n„e')
,
, 0/2
)
,
,\l/2
CO(H.S )
T - ....1
Lcos28
1 1/4
J
/
316
1-cosS
X
1/2
V 2 )
r
1
i 1/2 / 1 - cos 8 \
[ cosSJ
(
j
)
- COS 8 jX
{f 1 2cosS
316
316
1/2
1/2
= k!
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
317
317
317
XXV
Acknowledgments
I was a grateful recipient o f the follow ing scholarships and awards: W illiam B. Rosskam
Mem orial Scholarship (1996-97 and 1997-98), John E. H etrick Scholarship (1995-96), and M inority
Scholars Award (1995-96). This financial support is gratefully acknowledged.
Many people have helped me in a great many ways during the years I have been working on this
thesis. I would like to acknowledge a few o f them, fu lly knowing that many others w ill go unmentioned,
but not forgotten.
A ll members o f my committee have helped me in various ways throughout this work. M y advisor,
Dr. Ramaswamy C. Anantheswaran, has provided support and guidance for many years. His long
commitment to my education is gratefully acknowledged, and his fam ily’s hospitality is fondly
remembered. Dr. Gregory R. Ziegler allowed me to use his laboratory for the laser scattering particle size
analysis experiments presented in Chapter 2. Dr. Donald B. Thompson carefully and patiently reviewed
Chapter 2 o f this thesis and provided me w ith numerous insightful suggestions. Both Drs. Ziegler and
Thompson have instructed me and have positively influenced my academic development throughout my
graduate education. Dr. K arl S. Kunz (requiescat in pace) introduced me to the subject o f
electromagnetism. Dr. Raymond J. Luebbers furthered my training in electromagnetism, introduced me to
the subject o f Finite Difference Time Domain modeling, and kindly supplied the XFDTD software used in
this work. Dr. Michael T. Lanagan agreed to become one o f my committee members w ith very short
notice. Dr. Lanagan also made his lab and his broad frequency Network Analyzer available to me for
experiments. A ll m y doctoral committee members also showed great patience w ith seemingly endless
scheduling conflicts, and an equally endless thesis manuscript.
The faculty and staff at the Department o f Food Science have been a blessing and a most helpful
resource throughout a ll these years. I am particularly grateful to Juanita M . W olfe and Melissa A. Strouse
fo r going w ell beyond the call o f duty. Their amazing patience and willingness to help me out o f so many
administrative tangles w ill always be remembered w ith deep appreciation. Dr. John D. Floros, the head o f
the Department o f Food Science, has shown great patience in letting me continue in the doctoral program
for a ll these years. The staff at the University Creamery was also most helpful during the hours I spent
using their equipment and workspace. Martha, in the night cleaning staff, gave me her company, kind
words o f wisdom, and her example o f an unwavering passion for work carefully and lovingly done.
M y fellow graduate students, especially those w ith whom I shared a laboratory were a continual
source o f inspiration, help and camaraderie. Among those graduate students, Vikram Ghosh, Suresh
Jambunathan, Ravikumar Sunkara, Hiquiang Chen, Jennifer Swanderki and Liping Liang shared w ith me
an important part o f their time talents and support, and are gratefully remembered. Tanuj Motwani, my
new acquaintance in my old lab, helped me w ith the broad frequency dielectric measurements and w ith the
conductivity measurements. Thutran Tran, a high school student in the Pennsylvania Governor’s School
fo r the A gricultural Sciences, spent several weeks one summer working in our laboratory and taking the
SEM micrographs shown in Chapter 2 o f this thesis.
Other people working at Penn State also helped me in many ways. Rosemary Walsh provided her
help a good few times w ith her SEM and light microscopy knowledge. Tineke J. Cunning guided me
through the stressful process o f job hunting. Steve Perini set up the network analyzer and helped w ith the
dielectric measurements at the Materials Research Laboratory. Donna Lucas, in the same building, helped
me w ith Dr. Lanagan’ s schedule. The campus ministers at the Penn State Catholic Center, especially Fr.
Fred Byrne, Fr. Conan Feigh, and Fr. Jim Krieger (requiescat in pace), were always a source o f spiritual
strength, consolation, and grace, and their m inistry was always a keystone o f my life at Penn State.
M y friends at Penn State, especially M aria DiCola, Jim Havington, Tom&s Fajardo and Jazmir
Hernandez, Lumane Claude, Dr. Madeline A. Feltus, Bob Hirsch, Frank Abate, and Claire Chou were a
most important part o f my life. They were my constant company, my eyes and hands in moments o f need,
and my encouragement in my most d iffic u lt days. You w ill always be gratefully remembered in my heart,
and in my prayers o f thanksgiving.
M y coworkers and managers at K raft Foods Research and Development have encouraged me to
continue my struggle to finish this work for many years. Among them, Julie Simonson, Jimbay Loh, Erik
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xx vi
Whalen-Pedersen, Tico Rivero, John Topinka, Ron Meibach, Cathy Sullivan, Christine Kw iat, T im Stubbs,
Tom Nosek, Warren Zaug, C olin Crowley, and Jerry Fountain, have been most persistent and helpful in
their encouragement. Andrew McPherson was especially generous w ith his starch chemistry experience,
serving as a sounding board, directing me to pertinent literature, and lending me some valuable books.
M y friends and neighbors around Grayslake, IL , especially Tom (requiescat in pace) and Audrey
Rixie, have also provided me w ith their encouragement, hospitality and friendship. The sta ff and members
o f the Santa M aria del Popolo Church music m inistry, especially Gloria, Maria Teresa, Teresa, Leticia,
David, Manuel, Debbie, M arita, Geoff, Fr. Bob Aguirre, Fr. B ill Shields, Fr. Edgar Rodriguez, Fr. D avid
Arcila, and Fr. Edgar Cardenas, have been my source o f spiritual w ork and life in this present stage o f my
life.
M y parents Alfonso and Luisa Casasnovas have supported me during a ll my life w ith their love,
their example and their prayers. M y brothers and sisters have in their own way kept the fire o f my hope
shining through all these years.
It would be unfair to simply acknowledge the help o f my w ife, Jacqueline M . Casasnovas. This
has been as much a part o f her life as it has been a part o f mine. Her endless work, patient editing, most
kind and pertinent criticism , but most im portantly her love, have been the most important part o f this thesis.
Her loving kindness has supported my throughout the years. Your work and your love are in these pages,
and they w ill be one w ith mine as my life is one w ith yours for as long as the one Love unites us.
Lord my God, you are my life , my strength, my hope, and my faith. M y life and my work are
yours, and so is this thesis. M ay they be instruments o f Your glory.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
1.
INTRODUCTION
In 1993 Ziegler et al. published the results o f their efforts to dynamically measure the sw elling o f
three different types o f com starch granules as they were subjected to conventional heating in dilute
aqueous suspension. An adaptation o f the method o f particle size analysis presented in that previous work
was applied in the first phase (chapter 2) o f this current w ork to study starch granule swelling during
microwave heating. The research was then extended (chapter 3) to the dynamic measurement o f the
dielectric properties o f the starch granules, also in dilute aqueous suspension. D ielectric properties are
fundamental to the understanding o f the interaction o f electromagnetic waves (e.g., microwaves) w ith
materials. The change o f dielectric properties w ith composition, temperature, and frequency are o f interest,
as they help to explain how microwaves interact w ith food in cooking applications. This second phase o f
the study required the development o f new methods to both collect and analyze the data. Upon completing
that data analysis, a third phase (chapter 4) o f research was undertaken: usage o f the dielectric properties
and a numerical method (Finite Difference Time Domain) to model the electromagnetic fields inside a
domestic microwave oven containing loads o f starch suspensions and to estimate the distribution o f heat
generation inside the loads. Before further describing the details o f the current study, some background
inform ation w ill first be given about the history and usage o f microwave technology. This w ill be followed
by a discussion o f the importance o f microwave research.
1.1 Brief History of Microwave Ovens
The microwave oven is the by-product o f warfare research. The radar equipment developed
during W orld War II had proven to be vital to the A llie s’ defense because o f its a bility to detect unseen
aircraft. In 1945, not suspecting that the equipment had another ability, Percy Spencer continued working
w ith it in the radar laboratory o f Raytheon Company in Waltham, Massachusetts. One day he evidently
had a lo t o f “ food for thought” when he brought a snack into the laboratory, although sources differ on
what that snack was. According to one story, it was a candy bar that turned into a molten mess inside his
pocket as he worked w ith the equipment (Bloom field, 2001; Schiffmann, 1997). In another version o f the
stoiy, it was popcorn kernels that began to pop on their own (Schiffmann, 1997). Although the details o f
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the discovery may be lost to history, the importance o f it was certainly not lost on Percy Spencer. He
focused his efforts on harnessing the microwaves emitted by the radar equipment and direct them to cook
food. Once again, some o f the details seemingly have been either lost or embellished by the passing o f
time, and there are two different stories about how Spencer constructed the first microwave oven. Each
story tells o f a different, but equally ingenious and resourceful ( if not slightly amusing!) container that
Spencer used for the firs t microwave oven’ s cavity: a farmer’ s old m ilk can (Parmley, 1990) or a
galvanized garbage can (Schiffmann, 1997). Despite its humble beginnings, the microwave oven has had
an impact on the culture o f society. Although he m ight be exaggerating slightly, Decareau (1992)
summarized this impact by saying, “ N ot since the discovery o f fire in deep pre-history has a phenomenon
had such a striking influence on life styles as microwave cooking and heating.”
A fter Spencer patented his design, Raytheon Company went into production. In 1947, the
company unveiled its Radarange™, the first microwave oven (Schiffmann, 1997). About the size o f a
refrigerator and very expensive, the early models o f Radarange™ were intended fo r restaurants and food
service businesses (Bloom field, 2001; Decareau, 1992; Schiffmann, 1997). In 1955 the Tappan Company
o f M ansfield, Ohio, sold the first microwave ovens intended for household use (Decareau, 1992;
Schiffmann, 1997). Amana Refrigeration Company (which had been bought by Raytheon in 1957)
introduced small, affordable, counter-top models o f microwave ovens to the Chicago area in 1967, and to
the rest o f the United States in the follow ing year.
In time, the number o f consumers buying microwave ovens and the number o f companies making
them gradually began to grow. Today’s microwave ovens are smaller, less expensive, and more widely
utilized than the original Radaranges ™. Schiffmann (1997) echoed Decareau’ s sentiment by asking, “ Is it
any wonder that 150 m illion microwave ovens exist in the United States — a roughly 95% saturation
level?”
1.2 Industrial Uses of Microwave Technology
The microwave oven has caused a revolution in the food industry, spawning a new sector o f
already-prepared products for the retail and food service segments that simply need to be heated in the
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3
microwave oven. The suitability o f a product fo r microwave heating is a very important marketing issue,
and much time is invested in developing these “ microwave-ready” products.
Microwave technology, however, is not lim ited to consumers quickly heating products in their
homes. Kudra et al. (1991) designed a microwave pasteurization system, and Piyasena et al. (2003)
suggested that radio frequency heating (governed by the same laws o f physics as microwave heating) could
be used to pasteurize or sterilize liquids. Khan, et al. (1979) reported savings in tim e and energy as
compared w ith more conventional methods when they used microwave heating to expedite the production
o f sugar syrup by the hydrolysis o f starch at high temperatures and pressures.
In addition to its uses in food service, home-cooking, and food manufacturing, microwave heating
has found application in diverse fields o f science and industry. For example, Deng and L in (1997) reported
that sorbents and catalysts could be prepared quickly and efficiently by using microwave heating. Renoe
(1994) described how microwave heating can be used in extraction processes to reduce the amount o f
solvents needed and the time required fo r extractions. Komameni et al. (1992 and 1993) found that a
method that used microwaves to assist in the hydrothermal synthesis o f ceramic powders was preferable to
more conventional methods because it was faster and did not produce environmentally harm ful by­
products.
1.3 Justification of Microwave Research
One o f the most notable characteristics o f microwaves is their ability to penetrate the interior o f
foods (as w ill be explained further in chapter 3). By doing so, they cause the entire volume o f the food to
heat more quickly than conventional methods, which rely mostly on conduction and convection. The food
industry recognizes that since microwave heating lowers the amount o f time required to achieve a certain
internal temperature, it can also reduce the amount o f tim e (and money) required for food processing,
without sacrificing quality.
However, at present, microwave heating is not completely understood, and its applications to food
heating are often developed by a trial-and-error method. Both consumers and industry experience
difficulties obtaining the same quality w ith microwave cooking as w ith conventional cooking. One
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4
common example o f such a d iffic u lty is the unappealing texture that starch-based foods (such as rolls and
bagels) often develop when heated in a microwave oven. Another example is the uneven heating that can
occur w ithin the various components o f a frozen entree, such that a sauce may reach its b o ilin g point w hile
the meat remains frozen.
Many researchers have published literature (some o f which w ill be reviewed in chapter 2) about
the differences between conventional and microwave heating o f food products. In these studies the
researchers attempted, w ith varying degrees o f success, to compare microwave and conventional heating
under sim ilar conditions. However, in most cases the comparison was compromised because microwave
and conventional heating were different in the time/temperature treatment, in the heating pattern, or in both.
Many o f these authors speculated without firm proof that the differences they observed between microwave
and conventional heating could result from differences in heat and mass transfer, and not from any intrinsic
difference between the heating methods (Goebel et al., 1984; Zylema et al., 1985; Umbach et al., 1990;
Sakonidou et al., 2003; Y iu et al., 1991; Huang et al., 1990). In view o f this uncertainty, it w ould be
beneficial to design an experiment to resolve unequivocally the nature o f these differences.
Such an experiment could m onitor an appropriate property o f a food or food ingredient that
changes significantly (and measurably) during cooking. This property should also be relevant to the quality
o f the food or to the main function o f an ingredient in a food system. In such a hypothetical experiment,
one sample o f the food (or ingredient) could be subjected to conventional heating, and another sample
could be subjected to microwave heating. The experiment should be designed so that the temperature is
identical for both samples at every position and at a ll times during the experiment. I f the chosen property is
measured during cooking under these conditions, then the difference or sim ilarity o f the property as a
function o f time would be a measure o f the difference or sim ilarity o f the two heating mechanisms.
In the first part o f this thesis, an experiment approaching this ideal experiment is described. The
food systems that were examined were dilute suspensions o f com starch, and the property that was chosen
fo r measurement because it changes during cooking was particle size. Measuring particle size made it
possible to m onitor the swelling behavior o f these starch suspensions “ in-line” during microwave heating.
The experimental setup was designed to produce adequate m ixing, to eliminate non-uniform temperature
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5
distributions, to maintain a controlled heating rate, and to preserve the dynamic nature o f the cooking
process. The results o f this experiment were compared w ith those o f Ziegler et al. (1993), in which the
swelling behavior o f w ell-m ixed suspensions o f the same starches had been measured during conventional
heating w ith the same (controlled) heating rate. The experiments and their results are described and
compared in detail in chapter 2 o f this thesis. The main outcome, however, was that both heating methods
(microwave and conventional) produced the same particle size behavior for dilute suspensions o f the three
starches studied.
Since the particle size behavior was found to be the same, it is reasonable, then, to assume that any
differences observed between conventional and microwave heating o f sim ilar systems o f dilute starches
must be caused by differences in heat and mass transfer. Better knowledge o f both heat and mass transfer
during microwave heating would be needed to make fu ll and practical use o f the potential advantages o f
microwaves. The transport phenomena could then be modeled in order to be optimized.
Currently, commercial models that are able to predict mass transfer (by diffusion and convection)
during conventional heating are available. There also exist commercial software packages that can predict
heat transfer alone or in combination w ith mass transfer. These models, however, depend on a p rio ri
knowledge o f the nature o f the heat generation and its intensity distribution in space and time. The heat
generation during microwave heating is a direct consequence o f the interaction o f electromagnetic energy
and the material being heated. This interaction is best understood by solving a set o f rather complicated
physical equations (M axw ell’s equations), and by having knowledge o f the material properties that govern
the interaction (the dielectric properties). Many o f the models in the literature have used sim plifications to
these equations that lim it their accuracy, so they cannot correctly predict the intensity and distribution o f
heat generation during microwave heating. Therefore, these models have lim ited usefulness for the
development o f products and processes using microwave technology, so product and process developers
often must resort to the very inefficient trial-and-error method.
Chapter 3 o f this study was written w ith the hope o f eventually being able to minimize the amount
o f tria l and error by making five main contributions to the field o f knowledge o f microwave heating. The
first contribution was an extensive literature review. It covered different aspects o f electromagnetic theory
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6
and molecular polarization that is necessary fo r understanding the relationship between M axw ell’ s
equations and dielectric properties, as w ell as several models that have been proposed to model the
dielectric behavior o f substances. The second contribution was the development o f a new method to
measure the dielectric properties o f dilute starch suspensions (like those used in chapter 2) in-line during
microwave heating. The third contribution was the actual measurement o f the dielectric properties o f waxy
maize, common com, and potato starch suspensions. The fourth contribution was observing the interaction
that occurred when sodium chloride was added to the starch in the dynamic system. The fifth contribution
was the derivation o f a theoretical mathematical model to predict the values o f the dielectric properties o f
the starch suspensions as functions o f temperature and frequency. Contributions two, three, and five w ill
be described further in the paragraphs below.
The new method to measure the dielectric properties o f starch suspensions in-line during
microwave heating had three important benefits. First, because the heating rate could be controlled, the
dielectric results could be compared w ith other data (such as the swelling data from chapter 2 or values in
the literature) that had been measured at the same rate o f heating. Second, the dynamic conditions were
closer to real cooking than a system at static temperature conditions. Third, the dielectric data was
collected as both frequency and temperature changed, unlike much o f the data reported in the literature that
was collected when one o f those two important variables had been kept constant during measurement.
The theoretical mathematical model was derived after the dielectric measurements had been
closely analyzed, and the relevant physical and molecular principles in the literature had been carefully
reviewed. This model (henceforth known as the Debye-Hasted Model w ith Two Peaks), contained six
parameters. An intricate series o f computer programs (in Appendix D) was written to recursively estimate
the values o f the six parameters based on the measured dielectric data. Knowledge o f these parameters at
different temperatures and frequencies makes the model very versatile and able to predict the dielectric
properties o f dilute starch suspensions under various conditions. These parameter results, as w ell as some
insight into their significance, are discussed at the end o f chapter 3.
Returning to the discussion o f heat and mass transfer, once the dielectric properties o f a system are
known, the modeling o f the microwave heat generation can be tackled. The heat generation is a function o f
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the dissipated power produced when microwaves are absorbed by the system in question. This dissipated
power follows directly from the application o f M axw ell’ s equations. The application o f these equations to
a real system w ith any but the most sim plified geometry requires the use o f a numerical method. One such
method that has found wide application in the fie ld o f electromagnetic modeling is the Finite Difference
Time Domain (FDTD) method.
The third and last part o f this study applied the FDTD method to modeling the electromagnetic
field distributions inside a bowl fille d w ith dilute starch suspensions as it was heated in a domestic
microwave oven. The dielectric properties that were measured in chapter 3 were employed in the FDTD
model as the dielectric properties o f the starch suspensions in the bowl. In addition to the electromagnetic
field distribution, the results o f the model included the specific absorption rate (SAR), which is a quantity
related to dissipated power. This model, discussed in chapter 4 o f this thesis, was meant as a first step in
the development o f a fu ll model, capable o f predicting both heat and mass transfer during microwave
heating.
1.4 Objectives of this Research Project
The general objective o f this work was to study the interaction o f starch, a common food
component, w ith microwaves during heat processing, while examining the effect o f its physical and
chemical structure on the dielectric behavior. This research focused on water-starch and water-starchsodium chloride systems, using different types o f com starch and potato starch. The special effort was
made to obtain the data in-line during heating, so as to take into consideration the dynamic nature o f the
cooking processes. This methodology and insight should be applicable to the development o f “ microwaveready” food products and to the design o f new industrial processes using microwave heating.
The four main objectives o f this research were:
1)
To develop a technique to monitor starch granule swelling in-line during microwave heating, and
to compare the swelling behavior o f different types o f com starch w ith that observed during
conventional heating in a previous study, Ziegler et al., 1993 (chapter 2);
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8
2)
To develop a technique to measure the dielectric properties o f liquids in-line during heating, and to
use it to measure the dielectric behavior o f aqueous starch suspensions (both w ith and w ithout
sodium chloride) during gelatinization and to study the interaction o f the starch and the salt
(chapter 3)
3)
To develop a mathematical model, based on known physical principles, to represent the measured
dielectric data at various temperatures and frequencies (chapter 3);
4)
To use the fin ite difference tim e domain method to model the electromagnetic fields inside a
domestic microwave oven containing loads o f starch suspensions at different temperatures, and to
estimate the SAR distribution inside the loads (chapter 4).
W hile pursuing these objectives, the follow ing hypotheses were held:
1)
The swelling behavior o f starch granules in dilute aqueous suspensions is the same fo r microwave
and conventional heating when temperature, heating rates, and processing conditions are the same
(chapter 2);
2)
The dielectric properties o f starch in aqueous suspension change as functions o f temperature,
frequency o f electromagnetic radiation, and type o f starch. The functions describing this behavior
for each starch can be based on parameters w ith a physical meaning, as opposed to purely
empirical predictive models (chapter 3);
3)
The effect o f sodium chloride on the dielectric properties o f water is reduced by the addition o f
starch (chapter 3);
4)
Numerical methods can be used to model the interaction o f electromagnetic radiation w ith an
object i f the geometry o f the object, its dielectric properties, and the nature o f the electromagnetic
radiation o f interest are known.
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9
CHAPTER 2
DYNAMIC MEASUREMENT OF STARCH PARTICLE SIZE
BEHAVIOR DURING MICROWAVE HEATING
2.1 Introduction
Particle size analysis (PSA) refers to indirect methods used to determine the size o f particles that
are too small and too irregularly shaped to be measured directly. In this study, the particles o f interest were
starch granules that swell when they are heated in excess water. This heating may occur through a variety
o f conventional methods (e.g., conventional ovens, jacketed kettles, and domestic “ stove-top” cooking in a
pot) or through the use o f microwave power (e.g., in a microwave oven). Conventional and microwave
heating methods diffe r in how heat is generated and in how heat and mass are transferred. There has been
much interest in the literature about how these differences relate to differences in quality o f foods cooked
using each o f the heating methods. Specifically there has been a number o f studies comparing conventional
and microwave heating o f starch pastes and suspensions, and o f starch-rich foods.
Ziegler et al. (1993) developed a method that used laser scattering particle size analysis (LSPSA)
to dynamically m onitor the size distribution o f three kinds o f com starch in dilute suspension during
conventional heating. That investigation continued in this current study, in which the LSPSA method was
adapted to m onitor the particle size behavior o f the same starches during microwave heating. It was
believed that a comparison o f the results from both investigations could determine whether or not the
method o f heating has a significant effect on the particle size behavior o f these starches.
To aid in the understanding o f the work done in this chapter, a literature review w ill follow this
introduction, beginning in sections 2.2.1 and 2.2.2 w ith some comments about the sources, uses, chemical
composition, and structure o f native starch granules. The transformations that starch undergoes when it is
heated in water (w ith an emphasis on gelatinization) w ill be discussed in section 2.2.3. Section 2.2.4 w ill
describe some starch modifications and their use to improve the performance o f native starches in different
food applications. Some methods to evaluate gelatinization w ill be described in section 2.2.5, and then the
focus w ill turn in section 2.2.6 to the previous study (Ziegler et al., 1993) about the particle size behavior o f
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10
corn starch in dilute suspension during conventional heating. A b rie f summary o f microwave heating
mechanisms w ill be presented in section 2.2.7, and section 2.2.8 w ill contain a survey o f research on the
microwave heating o f starch. Following this literature review, the objectives o f this study w ill be reiterated
in section 2.3, the materials and methods w ill be described in section 2.4, the results w ill be discussed in
section 2.5, and the conclusions w ill be listed in section 2.6.
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11
2.2 Review of the Literature
2.2.1 Sources, Uses, and Composition of Starch
Starch, a carbohydrate, is the major component o f flour (Lineback and Wongsrikasem, 1980) and
the major food component consumed by humans. According to W histler and Daniel (1985), starch
provides 75-80% o f a person’ s total caloric intake. Carbohydrates (including starch, sugars, and fiber) are
found in plant parts such as cereal grains, vegetables, fruits, seeds, and roots. Some o f the most im portant
sources o f edible starch are maize, potatoes, wheat, cassava, and waxy maize (Swinkels, 1985). These sources
are used (after undergoing processing to various extents) in many food items, such as pasta, baked goods, and
prepared foods. Purified starch from these sources is also used in other food applications (such as puddings,
gravies, and sauces) to provide part o f their characteristic texture (Light, 1990; Zallie, 1988). In addition,
starch is an important ingredient in “ fat-free” and “ low calorie” versions o f products like salad dressings,
mayonnaise, and sauces, where it provides viscosity and mouthfeel, helping to m im ic the texture o f the “ fu llfat” products.
Any discussion o f starch structure must first begin w ith an understanding o f the structure o f glucose,
C6H i 20 6. Chemists have classified the form o f glucose that occurs in starch as a-D-glucose. The “ D”
designation indicates that the hydroxyl group in C5 is in the position shown in the Fisher projection in
Figure 2.1. (the hydroxyl group is shown bolded and underlined for emphasis). The “ a ” designation refers
to the fact that the hydroxyl group in C l occurs in the axial position when glucose is in the pyranose form,
1 CHO
H
2 HCOH
3 HO CH
4
HO
CH2OH
HC O H
H
5 HCOH
,
6
I-
OH
CliOH
Figure 2.1 Fischer projection of D-glucose (adapted
ffomMcMurry, 1984).
Figure 2.2 Pyranose form o f a-D-glucose
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12
as shown by the bold underlined OH in Figure 2.2 (M cM urry, 1988). I f this hydroxyl group were in the
equatorial position (as the hydrogen in C l), the molecule would have a “ P” designation. For sim p licity’ s
sake, however, throughout the rest o f this thesis, the molecule shown in Figures 2.1 and 2.2 w ill sim ply be
referred to as “ glucose.”
Starch is composed prim arily o f two
polymers o f glucose: amylose and amylopectin.
The main difference between these two
polymers is the proportion o f branches they
contain. Both are composed o f linear chains o f
o-o
glucose molecules joined by glycosidic bonds.
These glycosidic bonds are formed between the
carbons o f two glucose molecules in a
Figure 2.3 Hypothetical amylopectin chain (adapted from
Manners, 1989).
condensation reaction that removes one
molecule o f water (Isaacs et al., 1999). Usually
these glycosidic bonds are formed between C l o f one molecule and C4 o f another molecule, and it is these
a-1,4-bonds that lin k consecutive glucose molecules w ithin a single chain o f starch. However, one chain
o f starch can be connected to another chain o f a -l,4 -lin ke d glucoses through an a -1,6 linkage form ing a
branch. This type o f linkage is formed from glycosidic bonds between C l o f one glucose molecule and C6
o f another (Hoseney, 1986, M cM urry, 1988). Figure 2.3 illustrates these two types o f linkages.
The range o f amylose content in starch reported in the literature spans between 16 - 30%
(Eliasson and Gudmundsson, 1996; Hizukuri, 1996; Manners, 1989; W histler and Daniel, 1985). Swinkels
(1985) reports an amylose content o f 28% for maize starch, and 21% for potato starch. Amylose has
historically been considered a linear polymer o f glucose linked a -1,4, but it is now known to contain some
long branches linked a-1,6 (Hoseney, 1986; Manners, 1989; Hizukuri, 1996). The molecular weight, chain
length, and amount o f branching present in amylose vary according to the plant source from which the
amylose is derived (Hizukuri, 1996; Takeda et al., 1987). The long, linear segments o f amylose typically
contain several hundred a -1,4 linked glucose residues (Manners, 1989). The group o f Takeda, Hizukuri,
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and their co-workers in Japan studied amylose molecules from various botanical sources. The summary o f
their results (Takeda et a l, 1987) indicates that the degree o f polymerization o f these amylases ranged from
960 for com to 3280 fo r sweet potato. They noticed that the cereal amyloses were smaller than the other
amyloses extracted from seeds, roots, and tubers. Other authors report degrees o f polym erization o f
amylose in the range o f 350-1000 (V/histler and Daniel, 1985), and average molecular weights in the range
o f 105-10s (Shi et al., 1991).
In neutral solutions, amylose molecules exist as random coils (Manners, 1989). In the amorphous
regions o f starch granules, part o f the amylose forms random coils and single helices (H izukuri, 1996), and
possibly some double helices (W histler and Daniel, 1985). The linear chains in amylose can form
complexes that enclose fatty acids, hydrocarbons, iodine, and other molecules. These groups are often
referred to as inclusion compounds (W histler and Daniel, 1985). It is thought that part o f the amylose in
the amorphous regions in starch granules is present, as araylose-iipid complexes but this has not been
conclusively proven (H izukuri, 1996).
Amylopectin is the major component o f starch granules. Follow ing the variation In amylose
content (see above), amylopectin content'in starch varies between 84 and 70%. Maize and potato contain
about 72 and 79% amylopectin, respectively (Swinkels, 1985). Certain plants (including maize) have been
bred to produce starches that contain 100% amylopectin. These are know , as “ waxy” starches.
Amylopectin molecules are branched to a much greater extent than amylose molecules. The
m ajority o f the glucose molecules are linked oc-1,4, but 4-5% are linked a-1,6 (Hoseney, 1.986). The
average chain length o f the a -1,4 branches is from 18 to 25 glucose residues (H izukuri, 1996). There has
been much research devoted to the elucidation o f the structure o f amylopectin (see Thompson, 2000, for a
more in-depth review o f current theories about amylopectin branching). Amylopectin is a large molecule
w ith a degree o f polymerization on the order o f 1 0 M 0 5, corresponding to a molecular weight in the order
o f iOM Q7 (Manners, 1989; Hizukuri, 1996). According to Hoseney (1986) it appears that at least p a rt i f
not 'A ll- o f the outer chains o f amylopectin molecules ocour as a double helix in starch granules.
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14
2.2.2 Starch Granules
Through the process o f photosynthesis, plants use sunlight to produce carbohydrates. Starch is the
most important form in which plants store these carbohydrates. Starch is synthesized in plastids, which are
called amyloplasts in the plant organs that specialize in production and storage o f starch (M artin and Smith,
1995). Microscopic examination o f starch granules shows a layered structure sometimes visible in the
intact granules, but more easily observed upon acid or enzymatic treatment. These layers have been
attributed to growth rings resulting from periodic fluctuation in the rate o f carbohydrate deposition. The
fluctuations may result on daily fluctuations in sunlight, but may also result from other fluctuations in
biosynthesis (M artin’ and Smith, 1995). Starch granules occur in various shapes and sizes, ranging in
diameter from 1 tolOO pm, depending on the plant source (Shi et al., 1991). Starch granule volumeaveraged median diameters fo r commercial samples o f common com, waxy maize and cross-linked waxy
maize are 14.9, 15.6,14.5 pm, respectively (Ziegler et al., 1993).
The radial growth structure o f the starch granules can be described as a spherocrystalline
arrangement (W histler and Daniel, 1985). As a result o f this order, the granules are birefringent. Birefringent
materials have two (or more) refractive indices for light polarized in different directions. Light transmitted
through a birefringent material is split into two orthogonally polarized components that propagate at
different speeds in the material (Serway, 1986). When birefringent samples are placed between cross­
polarized filters in polarized light microscopy, the combined orthogonal components o f light form an
interference pattern that is characteristic o f the sample. For starch granules, the interference pattern appears
as a Maltese cross, w ith its center at the point o f nucleation o f the granule (W histler and Daniel, 1985).
Finkelstein and Sarko (1972) measured the light scattering patterns o f single starch granules (resulting in
part from birefringence) and proposed that the patterns must result from radial arrangement o f helical
segments o f amylose and linear portions o f amylopectin.
Starch granules are composed o f both crystalline and amorphous regions. The crystalline regions
are attributed to the outer amylopectin chains. It is thought that these outer chains are arranged in double
helices that pack to form the crystalline regions (Sarko and Wu, 1978; Imberty et al., 1988; Imberty and
Perez, 1988). The m ajority o f the starch granule seems to be composed o f amorphous regions containing a
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heterogeneous m ixture o f amorphous amylose and branched amylopectin (Biliaderis, 1992; H izukuri,
1996), but the structure o f these regions is s till not clearly known (Eliasson and Gudmundsson, 1996).
Besides amylose and amylopectin, the starch granules contain other m inor constituents. One o f
those constituents, protein, is usually present at levels between 0.25% and 0.5% fo r cereals (maize, wheat,
waxy maize). For potato and tapioca, protein levels are about 0.06% and 0.1%, respectively (Swinkels,
1985). Lipids occur in cereal starches (maize, wheat, rice, and sorghum) at levels between 0.6% and 1%.
In potato and tapioca, the levels are 0.05% and 0.1%, respectively. Lysophospholipids are the predominant
lip id material in many cereal starches, but in maize and waxy maize the main lipids are free fatty acids
(Swinkels, 1985). Lipids are thought to be present in the amorphous regions, probably form ing amyloselip id complexes (Tester and Karkalas, 2004). Phosphorus is present in starch granules form ing phosphate
monoesters and phospholipids, and also as inorganic phosphate. Table 2.1 lists the phosphorus contents
reported by Kasemsuwan and Jane (1996) for maize, waxy maize, and potato. Ash content for maize and
waxy maize is about 0.1%, and for potato about 0.4%. Ash contains phosphorous residue from a ll the
forms mentioned above, as w ell as calcium, potassium, magnesium, and sodium (Swinkels, 1985).
Table 2.1 Phosphorus Content (% w/w) in Starches
Reported by Kasemsuwan and Jane (1996)
Starch
Maize
Waxy Maize
Potato
Phosphate Monoester
0.003
0.0012
0.086
Phospholipids
0.0097
not detectable
not detectable
Inorganic Phosphate
0.0013
0.0005
0.0048
2.2.3 Transformations of Starch During Cooking
As mentioned in the introduction to this chapter, starch has numerous applications in the food
industry. Many o f the applications depend on the transformations that occur to starch granules in aqueous
suspension both during and after cooking. The increase o f viscosity that occurs during heating o f a starch
suspension, and the formation o f a gel upon cooling some o f these cooked suspensions, are both fam iliar
phenomena. However, the fine details o f the mechanisms by which granules swell and break during
heating, and then form gels, pastes, and other useful structures, are s till under study. The follow ing
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16
paragraphs briefly summarize the phenomena that have been observed during the process o f gelatinization,
and some theories that try to explain these phenomena.
The semi-crystalline nature o f starch granules makes them insoluble in cold water (Eliasson and
Gudmundsson, 1996; Shi et al., 1991; W histler and Daniel, 1985). However, due to their amorphous
phases (Eliasson and Gudmundsson, 1996; Shi et al., 1991), dry starch granules w ill absorb up to 30% o f
their weight in water (Hoseney, 1986) and swell. The swelling causes the granule diameter to increase
anywhere from 9.1% fo r normal com starch to 22.7% fo r waxy maize (W histler and Daniel, 1985). Up to
this point, the water absorption and swelling are reversible (Hoseney, 1986; W histler and Daniel, 1985).
However, i f the starch suspensions are heated to a temperature (called the gelatinization temperature),
irreversible processes occur. W histler and Daniel (1985) describe the sequence o f events as follow s:
...as the temperature is increased, the starch molecules vibrate more vigorously, breaking
intermolecular bonds and allowing their hydrogen-bonding sites to engage more water
molecules. This penetration o f water, and the increased separation o f more and longer
segments o f starch chains, increases randomness in the general structure and decreases
the number and size o f crystalline regions.
As the water penetrates the starch granules and causes them to swell, amylose molecules and some
o f the amylopectin leach out (Eliasson and Gudmundsson, 1996; Cameron and Donald, 1993b; Hoseney,
1986). When granules uptake water and swell, their phase volume fraction increases, while, at the same
time, the concentration o f solubilized starch polymers (mostly amylose) in the interstitial aqueous phase
also increases. Both phenomena cause the viscosity o f the suspension to increase (Eliasson and
Gudmundsson, 1996). Varying the concentration o f starch, different viscosities can be obtained upon
heating. This ability to increase the viscosity o f aqueous suspensions makes starch an important ingredient
fo r foods such as puddings, pie fillings, soups, and gravies.
The natural variation in starch granule size and in amylose and amylopectin concentration can
cause individual granules to swell to different extents (K okini, 1998) and to begin gelatinization at different
temperatures (Hoseney, 1986). As a result o f these variations, a given sample o f starch gelatinizes over a
range o f temperatures. Additionally, differences among plant species and among different varieties o f the
same species make that range different for each specific starch. For example, Biliaderis et al. (1980) report
ranges o f gelatinization for commercial samples o f normal com starch, waxy maize starch and potato
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17
starch as 63-68°C, 64-70°C, and 64-67°C, respectively. The authors based these ranges on loss o f
birefringence, a method that w ill be discussed later.
Following gelatinization, a phenomenon called “ pasting” occurs. Pasting has been defined
(A tw e ll et al., 1988) as “ the phenomenon follow ing gelatinization in the dissolution o f starch. It involves
granular swelling, exudation o f molecular components from the granule, and eventually, total disruption o f
the granules.” As granular integrity is gradually lost during heating, the viscosity o f the dispersion
decreases and the concentration o f solubilized starch polymers continues to increase. Granule disruption
and solubilization o f starch polymers during pasting are accelerated w ith the application o f increased shear
stresses (e.g. w ith increasingly vigorous agitation) during cooking (Hoseney, 1986; Eliasson and
Gudmundsson, 1996).
When gelatinized starch dispersions are cooled, another phenomenon, retrogradation, can take
place. The definition proposed by A tw ell et al. (1988) includes as parts o f the process o f retrogradation the
re-association o f starch polymers, formation o f junction points that may extend into ordered regions, and
eventual appearance o f crystalline order under favorable conditions. Upon cooling, the behavior o f the
dispersion is greatly influenced by starch concentration. A t low concentration the amylose molecules
associate and precipitate, and at higher concentration the dispersion can form a gel (M itolo, 2006).
Researchers recognize that the proportion o f water in the starch suspension affects the
gelatinization o f the starch. However, there is widespread disagreement as to the mechanism by which
water influences the process. Some researchers, like W histler and Daniel, (1985) thought that it is not so
much the total quantity o f water that matters, but rather the water activity. Water activity, which is defined
as the ratio o f the water vapor pressure around the sample to the vapor pressure o f pure water at the same
temperature, plays an important role in the chemical and physical changes that occur during the cooking
and processing o f foods (O llivon, 1991). Its value can depend upon the salts, sugars, and other molecules
present in the food (O llivon, 1991; W histler and Daniel, 1985).
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18
Some researchers describe the phenomena o f starch gelatinization and retrogradation in terms
form erly reserved fo r the field o f polymer science. Slade and Levine (1989) compiled and cited extensive
amounts o f existing research to promote this food polymer science approach. According to this approach, a
starch granule can be modeled as a “ fringed m icelle” structure (Figure 2.4 shows a sim plified model o f
fringed micelle for a long linear polymer). In his chapter on polymer structures, Callister (1991) defines a
fringed micelle as a semi crystalline polymer consisting “ o f small crystalline regions (crystallites, or
micelles), each having a precise alignment, which are
crystalline regions
embedded w ith an amorphous m atrix composed o f
randomly oriented molecules” (Callister, 1991). As
starch granules are heated in water, the temperature
eventually surpasses the glass transition temperature (Tg)
o f the amorphous regions (Slade and Levine, 1989, 1996).
A t the glass transition temperature, a rigid, glassy, non­
crystalline material is converted into a rubbery material
(Callister, 1991). In the starch-water system, water
enables this transition to occur at normal cooking
amorphous regions
Figure 2.4 Fringed micelle structure o f a polymer
with crystalline and amorphous regions.
temperatures because it behaves as a plasticizer. A
plasticizer is a substance o f small molecular weight that is added to polymers to make them more fle x ib ility
and ductile while reducing their stiffness and brittleness. When the system has a low moisture level, the
effective Tg o f the amorphous regions is greater than the melting temperature o f the crystalline regions
(Tm). As water concentration is increased, the “ glassy” amorphous regions in the system are more
plasticized, and their effective Tg is reduced. When the temperature increases beyond this decreased Tg, the
amorphous regions transition to the rubbery state. The increased m obility o f the amorphous regions allows
water uptake and thermal expansion, and the crystalline regions melt at a Tmslightly above Tg (Slade and
Levine, 1989).
Tester and Morrison (1990a) observed granule swelling, birefringence, and thermal transitions
during gelatinization o f wheat starch w ith varying water contents. From their results they concluded that
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dissociation o f crystalline regions and beginning o f swelling occur in the same temperature range (4555°C). The authors proposed that the dissociation o f double helices formed by the external amylopectin
chains occurs at higher temperatures (55-60°C) than loss o f ciystallinity. Granular swelling was observed
up to 85°C indicating that there was s till some degree o f association among starch polym er molecules at
this temperature.
In a continuation o f their work (Tester and Morrison, 1990b), the same authors theorized that
starches w ith low gelatinization temperature have a lower proportion o f crystallinity, and less perfect
crystallites than starches w ith a higher gelatinization temperature. The difference in crystallinity is caused
by differences in the way the amylopectin chains in various starches form clusters o f double helices. The
researchers also found that when the amorphous regions were hydrolyzed by acid treatment (lintnerization),
the starch did not swell nearly as much as the native starch. From this they concluded that the whole
amylopectin molecule, and not only the outer chains involved in crystallite formation, is important fo r the
swelling process.
Cameron and Donald (1993b) studied the gelatinization o f wheat starch aqueous suspensions
using Small-Angle-X-Ray Scattering. They modeled the growth rings o f starch granules as stacks o f less
ordered regions (“ background material” ), w ith embedded alternating lamellae o f crystalline and amorphous
regions. The authors postulated that, for starch content o f 45% (w/w), gelatinization occurs by
destabilization o f granular order aided by water. Between 47 and 55°C, there is absorption o f water by the
background material (associated w ith radial granule swelling) Between 55 and 57°C there is absorption o f
water by the amorphous lamellae as the background material reaches a “ saturation level o f water
absorption.” They thought this phase is associated w ith complete loss o f birefringence, tangential swelling,
and dissociation o f double helices (as suggested by Morrison and Tester, 1990a, at a sim ilar temperature
range). Finally, between 57 and 61°C, there is rapid loss o f crystallinity, attributed to the acceleration in
the disruption o f crystalline lamellae as the amorphous lamellae are saturated w ith water. A t a higher
concentration o f starch (55% w/w), the authors hypothesized that water is absorbed in both the background
material and the amorphous lamellae, but that it does not reach a saturation level, and does not assist the
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disruption o f the crystalline lamellae. They thought that when loss in crystallinity fin a lly occurs at these
conditions (by 61°C) it occurs by a “ simple m elting process.”
2.2.4 Starch Modifications
Native starches cannot always meet the demands that modem food processes and consumers place
on starch-containing foods. For example, consumers expect cherry pie fillin g to be viscous and firm in
texture, but using a native starch in the baking process would result in the opposite effect. The acidic
environment would cause the starch to break down, thereby creating an unappealing, runny pie (Light,
1990). To avoid such problems w ith native starches, food scientists have developed ways to modify them.
Thus, “ m odified food starch” is an ingredient frequently listed on the label o f many food products. It may
be added to improve the products’ viscosity, adhesion, gelling ability and clarity, texture, freeze-thaw
stability, shelf-life, mouthfeel, capacity to either hold or inhibit moisture, and its resistance to harsh
processing conditions such as heat, shear, acid, and microwave radiation. Below some methods o f
m odifying starch w ill be briefly described, although a fu ll discussion o f this topic is beyond the scope o f
this text.
Plants have been bred to produce starches containing desired amounts o f amylose and
amylopectin. There are varieties o f com, barley, rice, and sorghum that are almost 100% amylopectin.
These are known as “ waxy” starches. Starches that contain very high levels o f amylose also exist. These
are known as amylotypes (Hoseney, 1986). They are generally more d iffic u lt to use in food applications
because they do not gelatinize readily at atmospheric pressure (Hoseney, 1986; Eliasson and
Gudmundsson, 1996). Researchers have developed lines o f high amylose starches that may be used alone
or in a blend w ith other starches in the confectionary industry (Light, 1990).
Chemically m odified starches are changed by breaking covalent bonds (e.g. the glycosidic
linkages between glucose molecules), or by form ing new covalent bonds (e.g. between two glucose chains,
or between a glucose molecule and a reactant) in the native starch (Eliasson and Gudmundsson, 1996).
Treatment w ith acid, the oldest method for chemical m odification o f starch, hydrolyzes the amorphous
regions without damaging the crystalline regions o f the granules (Hoseney, 1986). The resulting modified
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21
starch granules (called lintnerized or Naegeli starch) swell less and form less viscous dispersions upon
pasting (Hoseney, 1986). Acid-m odified starches are frequently used in the manufacture o f gum candies
(W histler and Daniel, 1985; Hoseney, 1986). Another method for hydrolyzing starch is based on enzyme
conversion (Eliasson & Gudmundsson, 1996).
O f particular interest to this study is the m odification method known as cross-linking. In the
cross-linking process, adjacent starch chains are covalently bonded at various locations w ithin the starch
granule (Hoseney, 1986; Tattiyakul and Rao, 2000; W histler and Daniel, 1985). There are two main
methods o f cross-linking: reaction w ith phosphoric acid to form a diester and reaction (usually w ith
epichlorohydrin) to form two ether bonds (Hoseney, 1986). The linking inhibits the swelling and
solubilization o f the starch granule, often resulting in an increase in gelatinization temperature range.
Cross-linking increases the ability o f a starch to withstand excess cooking, mechanical agitation, shearing,
and acidic conditions, and it yields a paste w ith “ shorter” texture (Hoseney, 1986; W histler and Daniel,
1985). Very highly cross-linked starches can be heated to much higher temperatures without swelling than
native starches, and some can even withstand autoclave sterilization (W histler and Daniel, 1985; Hoseney,
1986). Because o f their ability to withstand acid conditions, application o f shear, and high temperatures,
cross-linked starches are frequently used to increase the viscosity and stability o f baby food, fru it pie
fillings, salad dressings, and canned goods (Eliasson & Gudmundsson, 1996; Hoseney, 1986; Light, 1990;
W histler and Daniel, 1985).
Substitution, another important m odification, is introduction o f a new group into the starch
polymers. One example is the reaction o f starch w ith “ acid phosphate salts” (e.g., monosodium
orthophosphate, sodium trimetaphosphate, and sodim tripolyphosphate) to form starch phosphate
monoesters (W histler and Daniel, 1985; Solarek, 1986)). Because o f the large size and the charge on the
group added to the starch, the polymer chains tend to repel, rather than interact w ith, each other (Hoseney,
1986). As a result, these starches swell more, solubilize more readily, gelatinize at lower temperatures, and
resist retrogradation better than native starches (W histler and Daniel, 1985; Hoseney, 1986). These
qualities enable them to improve the shelf life and freeze-thaw stability o f frozen foods (W histler and
Daniel, 1985; Hoseney, 1986; Light, 1990, Zallie, 1988).
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2.2.5 Methods of Evaluating Gelatinization
Various methods have been developed to m onitor the gelatinization process. These methods
(some o f which w ill be brie fly discussed in the follow ing paragraphs) focus on measuring changes in
distinct phenomena that generally occur at different temperatures during gelatinization. The temperature
ranges that are recorded as the result o f these methods are often reported as the gelatinization temperature
range for a given starch, but they actually only represent the observed temperatures at which a specific
phenomenon occurred. In other words, no single phenomenon its e lf can represent the entire process o f
“ gelatinization.” Wanting to improve communication among researchers investigating gelatinization by
various methods, a committee undertook the task o f clarifying the terminology associated w ith starch
phenomena (A tw ell et al., 1988). According to the consensus that was reached at the Carbohydrate
D ivision Symposium o f the 1986 annual meeting o f the American Association o f Cereal Chemists (A tw ell
etal., 1988):
Starch gelatinization is the collapse (disruption) o f molecular orders w ithin the starch
granule manifested in irreversible changes in properties such as granular swelling, native
crystallite melting, loss o f birefringence, and starch solubilization. The point o f in itia l
gelatinization and the range over which it occurs is governed by starch concentration,
method o f observation, granule type, and heterogeneities w ithin the granule population
under observation.
The various techniques used to m onitor gelatinization measure the irreversible changes in
properties mentioned in the above definition. The techniques include light microscopy, polarized light
microscopy, scanning electron microscopy (SEM), x-ray diffraction, differential scanning calorimetry,
viscosity techniques, dye exclusion techniques, and electrical conductivity measurements. These
techniques w ill be briefly discussed in the succeeding paragraphs.
The swelling that starch granules undergo during gelatinization may be examined by several
microscopic techniques. Cunin et al. (1995) used light microscopy w ith two staining techniques that
targeted starch and protein. This combination o f procedures enabled them to inspect the changes that
occurred in wheat starch granules as pasta was cooked in water. Polarized light microscopy is a technique
that enables scientists to observe the birefringence o f the starch granules. As starch gelatinizes, the radial
order o f the native granules (and therefore their birefringence) is lost, and the characteristic “ Maltese cross”
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23
pattern disappears . This technique is useful for determining the “ onset” o f gelatinization, but it does not
reveal anything about the rest o f the process, since birefringence is lost before the granules complete the
swelling and pasting stages (Lineback and Wongsrikasem, 1980).
Light microscopy is, at best, lim ited in its resolution by the wavelength o f visible lig h t (0.4-0.7
pm, Ohanian, 1985). Scanning electron microscopy (SEM) is able to resolve much smaller features (in the
nanometer range) than light microscopy, and it has been used by a number o f researchers to observe the
structure o f starch granules before and after heating (see for example Lineback and Wongsrikasem, 1980;
Goebel et al., 1984; Umbach et al., 1990; Huang et al., 1990; and Cunin et al., 1995). Sample preparation
for SEM, however, can lead to artifacts, especially for high moisture specimens that need to be dried for
observation (see for example the discussion w ith reviewers in Goebel et al., 1984).
X-ray diffraction can also be used to examine the changes in the semi-crystalline structure o f the
starch granules during gelatinization. Intact, uncooked granules examined by x-ray diffraction show
evidence o f their partial crystallinity, but gelatinized starch shows no crystallinity (H izukuri, 1996).
In the technique known as differential scanning calorimetry (DSC), both a sample and a known
standard are heated separately, keeping them both at equal temperatures, even as the temperatures increase
linearly. A comparison between the electric power needed to heat the reference and the sample yields the
heat flo w to the sample, since the heat flow to the reference is known. The results are displayed as a
function o f temperature as a thermogram (A tw ell et al., 1988; Skoog and Leary, 1992,). The thermogram
obtained from heating starch in water exhibits a peak (called an endotherm, since gelatinization is an
endothermic transition) that corresponds to the process o f gelatinization. The data is analyzed to determine
both the onset temperature o f gelatinization and the temperature at which the maximum peak occurs. The
area under the peak indicates the enthalpy that must be supplied for the starch granules to progress from
their original ordered state to their more disordered, gelatinized state (Hoseney, 1986).
Viscosity techniques such as the viscoamylograph have also been used to evaluate gelatinization
by m onitoring the viscosity o f a starch suspension as it is heated in water. In this technique, a sample is
typically heated at a rate o f 1.5°C/min to a temperature o f about 95°C (Hoseney, 1986; Eliasson and
Gudmundsson, 1996). A fter being held at this high temperature for a certain amount o f time, the sample is
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then cooled at a rate o f 1.5°C/min. The viscoamylograph records the relative viscosity o f the system, and
plots the results as a function o f temperature and/or time (Hoseney, 1986; Eliasson and Gudmundsson,
1996).
Tester and M orrison (1990 a and b) developed a dye exclusion method to m onitor gelatinization
based on the fact that swollen starch granules w ill not absorb blue dextran dye. For their measurements,
they incubated starch suspensions at several temperatures. A t a set tim e, the suspensions were rapidly
cooled and blue dextran was added. Centrifugation was used to separate granular material from extragranular material. Then, the measured absorbance o f the supernatant provided the blue dextran
concentration. Knowing the concentration o f blue dextran, they were able to calculate the volume o f the
swollen granules. Thus, their dye exclusion method enabled them to m onitor granule volume during
gelatinization, using weight and absorbance measurements. They also calculated a derived quantity called
the “ swelling factor.” Tester and M orrison (1990a) compared their dye exclusion results w ith those from a
technique used to measure particle size, the Coulter method. In this method, particles suspended in an
electrolytic medium flow through a capillary tube w ith electrodes. As the particles pass between the
electrodes, they displace their own volume from the solution. This volume displacement is detected by a
sensor and recorded as a voltage pulse (Beckman Coulter, Inc., 2000). Since the voltage pulse is
proportional to the volume o f the particles, Tester and Morrison concluded that the Coulter method could
also be used to determine the start o f swelling (Tester and Morrison, 1990a).
Karapantsios’ group found that continuously measuring the electrical conductivity o f a starch
sample could be an effective means o f m onitoring gelatinization. More specifically, their research focused
on studying the flu id motion, heat transfer (Karapantsios et al., 2000), water dispersion kinetics
(Karapantsios et al., 2002), and mass transfer (Sakonidou et al., 2003) during gelatinization. Their
technique was based on the differences in the electrical properties o f the dispersed starch granules and the
continuous liquid phase. (These electrical properties w ill be discussed in greater detail in chapter 3). The
conductivity o f the system changes as gelatinization progresses because water diffuses from the bulk liquid
into the granules.
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The main disadvantage to techniques such as x-ray diffraction, small-angle x-ray scattering, SEM,
and dye exclusion methods is that process o f gelatinization must be interrupted before the analysis can be
performed. W hile some methods like DSC, viscometry, polarized light microscopy,and electrical
conductivity can be used to m onitor the gelatinization process continuously, they do not give direct
inform ation about the size o f the granules. It would be desirable to have a method to characterize
gelatinization in terms o f granule size w hile swelling is occurring, because swelling leads to increased
viscosity, and increased viscosity is usually the objective o f using starch as a food ingredient.
Non-intrusive and non-destructive methods o f particle size analysis based on laser diffraction exist
(ISO 1999, Ziegler et al., 1993). Laser scattering particle size analysis (LSPSA) is based on the fact that
particles w ith the same shape and size scatter light at the same characteristic angle (the scattering angle is
inversely proportional to particle size). It is possible, then, to obtain inform ation about a group o f many
particles that have the same shape but different sizes, by exposing them to a laser beam, and measuring the
intensity o f light refracted at different angles. To analyze the size o f starch granules, a starch suspension is
passed through a laser beam. When the beam hits the starch particles, the light scatters. The LSPSA
a
\
Figure 2.5 Schematic o f LSPSA apparatus: (a) laser beam; (b) individual rays o f light; (c) particles o f two
different sizes; (d) focusing lens; (e) detector with photosensitive rings.
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26
instruments have special lenses that focus the lig h t diffracted by particles o f the same size (w ith the same
angle o f diffraction) onto the same photosensitive ring in the detector (Figure 2.5). The pattern o f lig h t
intensity in the rings is fed to a computer, where it is mathematically transformed into the volum etric size
distribution for the starch. An important assumption that is made during this particle size analysis is that a ll
particles are spherical (ISO, 1999). Based on this assumption, the computer can use the volum etric data
obtained from the laser diffraction to compute the particle size distribution o f the starch granules.
The natural result o f the LSPSA method is a “ volume distribution.” The inform ation contained in
a volume distribution is different from the more usual distributions in which the number o f items is used to
calculate statistical parameters. Instead, the volume distribution uses the volume o f the items to calculate
the statistics. For example, the commonly used D [v,0.5], or median diameter o f the volume distribution, is
defined such that one h a lf o f the total volume o f particles has a smaller diameter, and one h a lf o f the
volume has a greater diameter. Results obtained w ith other methods such as microscopy lend themselves to
number distributions, so one most be very careful comparing results between different kinds o f distribution
(number, volume, and surface for example).
2.2.6 The Use of LSPSA During Conventional Heating
In a previous study (Ziegler et al. 1993), LSPSA was used to develop a dynamic method to
m onitor starch granule swelling during conventional heating o f starch suspensions. The method was based
on a measuring cell designed to produce adequate m ixing in a liquid suspension o f starch, elim inating the
non-uniform temperature distributions. W ith this arrangement, it was possible to preserve the dynamic
nature o f the cooking process and make the measurement in “ real time.” In other words, one important
benefit o f this method was that the process o f gelatinization did not have to be stopped in order to evaluate
its progress, unlike some other evaluation procedures.
The starch that was used was kindly supplied by the National Starch and Chemical Company
(Bridgewater, NJ). Three different varieties o f com starch were examined: a common com starch called
M elojel, a waxy maize starch called Amioca, and a cross-linked waxy maize called Cleargel. The
instrument used for the laser scattering and the particle size analysis was the Malvern MasterSizer, a laser
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27
diffraction particle size analyzer w ith a 300-mm range lens (M alvern Instruments Ltd., Malvern, England).
An accessory water-jacketed cell w ith a magnetic stirrer was also employed w ith the MasterSizer. The
starch solution samples placed in the cell had been prepared at two different concentrations: 7.5xl0"3%
(w /v) fo r the common and cross-linked starches; 4.5x10'3% (w /v) for the waxy starch. The cell was heated
by a computer-controlled water bath. During these experiments the temperature inside the cell was
controlled accurately and reproducibly. Two heating rates were chosen to facilitate comparison between
this method and the more traditional methods. One was the nominal rate o f 5°C/min (chosen for
comparison w ith DSC), and the other was 1.5°C/min (chosen fo r comparison w ith the Brabender
viscoamylograph). A series o f particle size distributions was collected during heating and the mean and
median diameters o f the particles were computed.
It was concluded that this method is a useful way to compare swelling data w ith thermal
transitions and changes in viscosity during starch heating. An adaptation o f this technique was used in this
current study to examine the swelling patterns during microwave heating. These patterns w ill be compared
w ith the results o f the experiments o f Ziegler et al. (1993) in section 2.5.
2.2.7 A Brief Summary of Microwave Heating Mechanisms
During conventional cooking, heat is generated usually by electrical heating or flame. The heat is
transferred to the surface o f food by conduction and radiation from heated surfaces, and by convection from
hot gases (in ovens and steaming) and liquids (in boiling and frying). The heat absorbed at the surface o f
the food is transferred throughout its volume by conduction fo r solid foods and also by convection for
liquid foods. Another effect o f conventional heating is the evaporation o f water from the surface o f the
food, w ith consequential browning and development o f characteristic color, aroma and texture. One
disadvantage o f conventional heating is that heat transfer to the inside o f foods is slow because the heat
conductivity o f foods is relatively low.
Heat generation w ithin food in microwave ovens, on the other hand, is generated by entirely
different mechanisms, which are s till not completely understood. Some theories w ill be discussed in much
greater detail in chapter 3, so a few main concepts w ill be b riefly summarized here. It is generally believed
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that two mechanisms come into play. The firs t and most important involves the dipolar rotation o f
molecules. The electric component o f microwaves produces a partial alignment o f polar molecules,
notably water. As the direction o f the electric fie ld in microwaves varies, the direction o f the molecules
changes. The proxim ity o f many molecules causes interactions that impede the radiation o f a ll the
absorbed energy. Therefore, part o f this energy is converted into random motion o f the molecules, or
thermal energy. The second mechanism o f heat production is the migration o f ions. Ions are permanently
charged and thus are accelerated in the presence o f an electric field. The movement o f ions in solution
produces many collisions between the ions and other molecules in the system. Then the energy absorbed
from the electric field by the ions is transformed into heat during these collisions.
The reason that microwave ovens are able to heat food faster than conventional ovens is that the
microwave energy can penetrate the interior o f the food, reducing the time required to reach the desired
temperature (Cumutte, 1980). This rapid heating rate, coupled w ith the fact that different food components
absorb the microwave energy differently, may result in less satisfactory products than conventional heating
(Goebel et al., 1984), as was briefly mentioned in chapter 1.
O f interest to this study is what occurs when aqueous suspensions o f starch are heated.
Conventional heating norm ally causes gelatinization to begin at the outer surface o f the starch product.
Conduction and convection then enable the process to proceed through to the interior, which usually results
in a uniform ly gelatinized product w ith sim ilarly swollen granules. Microwave heating can cause localized
areas o f relative high and low temperatures which vary in position for different product formulations,
product geometries, ovens, and even position w ithin a particular oven. The occurrence o f “ hot spots” and
“ cold spots” during microwave heating o f starch-containing foods often leads to non-uniform ly gelatinized
products w ith a wide variation in the extent o f swelling exhibited by the granules (Goebel et al., 1984;
Zylema et al., 1985). The next section w ill review some o f the literature dealing w ith the microwave
heating o f starch-containing products.
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2.2.8 Survey of Research on the Microwave Heating of Starch
In their review o f breads baked by
Conventional
Oven
conventional ovens and non-conventional
Thin crust
ovens (which included forced convection,
microwave, and electrical resistance ovens),
Y in and W alker (1995) outlined the baking
Thick Crust
Microwave
Cooking
stages used by those ovens (See Figure 2.6).
They explained that the heat transfer in bread
baked by conventional methods occurs from
Hybrid
Oven
Browning
the outside surface and moves inward. They
theorized that microwave heating is often
Figure 2.6 Schematic diagram o f the stages o f
baking and the types of ovens in which they occur
(adapted from Yin and Walker, 1995).
localized, depending in part on the geometry
o f the oven and the a b ility o f the food product
to absorb the radiation. Because the
microwave radiation raises internal temperature o f the bread rapidly, the first two stages (“ thin crust” and
“ thick crust” in Figure 2.6) are bypassed, so the microwave baking process essentially begins at stage 3,
“ cooking.” In addition, mass transfer o f water in microwave heating tends to occur from the center to the
outer surface, driving o ff more water from the bread dough than the conventional baking process. The
authors explained that this increased water loss could be caused by both the difference in temperature
distribution and the lack o f a crust to act as a moisture barrier.
Y in and W alker explained that these heat and mass transfer differences between conventional and
microwave ovens result in differences in the end products. The most obvious differences are that breads
baked in microwave ovens have no crust and they have a “ tougher, coarser, but less firm texture” than
bread baked in the conventional oven. Y in and W alker further speculated that microwave heating may not
allow time for enough enzymatic hydrolysis o f starch to maltose. Maltose is needed for yeast fermentation
and to decrease the viscosity o f the dough. The authors suggested that adding flours w ith high enzyme
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30
activity or starches that have been m odified to gelatinize at lower temperatures m ight be helpful in
im proving the quality o f microwave-baked breads.
A research group at the University o f Minnesota compared the effects o f microwave heating and
conventional heating methods on aqueous starch suspensions and starch-containing food products. In one
o f their studies (Goebel et al., 1984) they investigated the effect o f water concentration on the swelling o f
wheat starch granules heated. They prepared starch suspensions in beakers using four different starch:water
ratios (1:1,1:2,1:4, and 5:95, w/w). The suspensions were then heated to 75°C without agitation in an oven
capable o f heating using microwaves or convection. The researchers did not control the heating rate. The oven
was operated at three different settings: 1) “ convection mode” w ith oven temperature o f 177°C; 2) microwave
mode at low power; and 3) microwave mode at medium power. The temperature was measured at the center o f
the samples using a fiber optic probe. A fter the heating treatment, the researchers weighed the samples and
measured the shrinkage caused by moisture loss. Upon visually examining the samples, they identified various
degrees o f gelatinization within the samples, and classified these regions as “ gelled,” “ chalky” , or “ pasty.”
Then they examined the granules by light microscopy and SEM. As a result o f their observations, the
researchers developed a scheme that divided the gelatinization process into six stages. The definition o f
each stage was based on observable qualities in factors such as the size and shape o f the starch granules and
the presence o f birefringence.
The authors found that the heterogeneity in the microstructure o f the samples decreased w ith
increasing water content in the starch suspensions for both heating methods. For microwave heating, they
found that the time required to reach 75°C increased with water content and microwave power. From their
data, this increase with water content appears to be linear. When the oven was operated in convection mode,
heating time was longer and it did not have a linear dependency on water content. This time discrepancy was
not the only difference they noted between the convection and microwave modes o f the oven. W eight loss
and shrinkage were greatest fo r the samples that had been heated by convection. As expected, the samples
heated by convection had gelled regions closer to the surface o f the beaker. The samples heated by
microwaves had gelled regions closer to the center, probably caused by a “ hot spot” in that region. Fewer
distinct swelling stages were observed in the convection samples, and the stages that were observed were
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31
more advanced than those observed in the samples heated in the microwave oven. In other words, the
conventionally heated product appeared to be more structurally uniform .
The experiments o f Goebel et al. (1984) were intended to describe the differences between heating
methods and not to explain whether the differences were caused by the heating mechanisms, or by the
difference in time/temperature treatments. The authors also recognize that some settling may have occurred
during the experiments, and that this may have had an influence in the observed patterns o f gelatinization.
In order to determine the nature o f the differences between microwave and conventional methods, the
time/temperature treatment would have to be controlled, and a means to keep the starch granules suspended
would have to be devised. This present research sought to advance the understanding o f microwave
heating o f starch suspensions in part by perform ing experiments in which the time/temperature treatment is
controlled, and the starch granules are kept in suspension during heating.
The University o f Minnesota group continued their research w ith a series o f follow -up
experiments. Zylema et al. (1985), prepared dispersions o f wheat starch in water w ith four different
starch:water ratios, but this tim e they used test tubes rather than beakers so they could achieve a higher
heating rate. One set o f samples was heated by microwaves, while the other set was heated by conduction
in an o il bath. The temperature was measured at the center o f the dispersions. The o il bath temperatures
were adjusted so the time to reach the final target temperatures o f 65 and 85°C was the same as for the samples
heated by microwave irradiation. A fter heating, the samples were visually examined and regions were
designated as gelled, watery or chalky. Then the granules in the various regions were evaluated for
swelling by light and polarizing light microscopy, SEM, and DSC.
W ith regards to the role o f water in the gelatinization process, the researchers observed that
granule swelling became more uniform as water became less lim iting in both methods. The authors offer
three possible reasons why this increased uniform ity might occur for microwave heating: the additional
water 1) could increase the amount o f microwave energy absorbed; 2) could increase the convective and
conductive heat transfer; 3) could bring the water concentration throughout the sample above whatever
minimum would be required for the phase transition. The authors did not offer any explanation for the
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32
uniform ity in the case o f conductive heating, but it could be
inferred that the second and third possibilities mentioned
Microwave
65 C
85 C
Conduction
65 C
85 C
above are plausible.
chalk
Although the authors were careful to equalize the
times required fo r the temperature at the center o f the
samples to be equal, they noted that the time-temperature
profiles o f the samples heated by the two different methods
were not the same. In the early stages, conduction-heated
samples had a faster heating rate than microwave-heated
1:4
samples (w ith the exception o f the case w ith lim ited water).
A fter the temperature reached 60°C, however, the pattern
reversed, possibly in connection w ith the onset o f
gelatinization. A t this temperature, the heating rate for the
microwave-heated samples increased, surpassing the
heating rate for the conduction-heated samples. The
authors attribute this sudden increase in microwave heating
Figure 2.7 Test tube samples at various starchwater ratios exhibiting chalky, gelled, and watery
regions (adapted from Zylema et al., 1985).
rate to an increase in the absorption o f microwave energy
due to changes in the dielectric properties accompanying the transition from “ crystalline to less crystalline”
materials. The authors did not explain, however, how this process could increase microwave absorption.
Based on the SEM studies o f the morphology o f the granules and the development o f the matrix
between them, the authors identified five stages o f granule swelling. Zylema et al. (1985) found that the
distribution and range o f swollen granules were different for the two different heating methods. They attributed
this to differences in the heat and mass transfer occurring in the samples during the heating treatments. Like
Goebel et al. (1984), they did not find structures unique to either method. In the introduction to their paper, the
authors concede that it is not known whether the observed differences between microwave and conventional
heating are the result o f heat and mass transfer differences or i f they are the result o f interaction between
microwaves and molecules o f the food.
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33
Other researchers from the same group, Umbach et al. (1990), compared bagels that were baked (after
the in itial boiling step) using microwaves (300 Watts for 90 seconds) or forced-air convection heating (196°C
for 15 minutes). The authors chose the heating times for both methods so that the “ shape” o f the time versus
temperature curve would be similar and the final temperatures (as measured in the middle o f the dough) was
equal. However, no effort was made to make the heating times equal. The researchers found differences in
“ bagel structure” and starch swelling as observed by SEM as functions o f heating method and location w ithin
the bagel. The rate o f loss o f water was greater for microwave heating than for conventional heating, probably
because the microwave-irradiated bagels did not develop a dry crust to prevent moisture from escaping, unlike
the conventionally heated bagels, which lost water from their surface during baking causing a harder crust. The
total water loss was higher for conventional heating because the total heating time was longer.
Two years later, Umbach et al. (1992), reported the dielectric properties and the self-diffusion
coefficient o f water in mixtures o f starch, gluten, and water, using pulsed gradient spin-echo nuclear magnetic
resonance. The researchers had three important variables in their study: 1) water content o f the sample (54,
100, or 186% on a w/w dry basis); 2) sample treatment (unheated, heated in a conventional oven at 190°C for
25 minutes, or heated in a microwave oven at 700 W for 45 seconds); and 3) ratio o f starch to gluten (100:0,
80:20,50:50,20:80, and 0:100). The authors found differences in the dielectric properties o f the unheated,
microwave-heated, and conventionally heated samples that were dependent on moisture content and
composition. They observed that heating caused a decrease in the self-diffusion coefficient o f water, but there
was no great difference between the self-diffusion coefficient values for the samples heated by conventional
methods and for those heated by microwave methods.
The researchers from the University o f Minnesota mentioned in the above studies observed
differences between microwave and conventional heating o f starch-containing food systems in terms o f
moisture migration, gelatinization patterns, self-diffusion coefficient, and dielectric properties. These
differences might be related to differences in temperature distribution, heating rate, and heating time associated
w ith the different conventional and microwave heating methods. None o f the studies found any transformation
in starch that was unique to microwave heating.
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34
Sakonidou et al. (2003) compared the gelatinization o f starch in aqueous suspensions (2.5-20% solids
w /w) by conventional (submerged heating coil) and microwave heating methods. They attempted to equalize
the final temperature in their experiments, but did not equalize the heating rate. In order to prevent starch
granules from settling during microwave heating, they firs t preheated the suspensions to below their
gelatinization range (the authors specified T<60°C and the in itia l temperatures appears to be about 55°C in
their figures) in a glass beaker w ith a stirring rod on a heating plate. The authors thought this preheating
necessary because during prelim inary experiments they had found that starch settling during microwave
heating caused the appearance o f regions w ith different degrees o f gelatinization as observed by Goebel et
al. (1984) and Zylema et al. (1985). The preheated samples were then placed in the microwave oven
w ithout any agitation. The experiment ended when the suspension started to boil at the top o f the
container, even though a thermocouple indicated that the temperature at the center o f the vessel was s till
less than 100°C.
The samples that were heated conventionally were constantly stirred w ith an im peller, and thus
exhibited temperature uniform ity. (The temperature in a ll the samples was measured in two different
places: near the outside w all o f the vessel and at the center.) One noticeable difference between the two
heating methods was the presence o f small gas bubbles throughout the microwave heated sample, which
probably could not escape due to the lack o f agitation. The authors were using electrical conductivity to
m onitor gelatinization, and they acknowledged that the appearance o f the bubbles m ight have affected the
conductivity measurements.
The researchers evaluated “ the degree o f gelatinization” o f samples heated by both methods using
polarizing light microscopy and the application o f Congo red dye (they did not include details about their
method.) They observed that, while the conventionally-heated samples appeared to be fu lly gelatinized, the
microwave-heated samples did not gelatinize completely. The authors concluded that the fast heating rate
observed during microwave heating did not allow enough tim e for the water to fu lly disrupt the starch
granules, indicating that mass transfer could be a more important factor than heat transfer. Their
observations could also be explained by the presence o f “ cold spots” in their microwave-heated
experiments since there was no agitation.
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35
One important difference between conventional and microwave heating o f some products (e.g.,
puddings, gravies, sauces, and porridges) is that conventional methods often involve frequent or constant
agitation, whereas microwave heating involves very infrequent ( if any) agitation. This difference can lead
to cooked products w ith different properties. For example, Y iu et al., (1991) examined the difference
between microwave and conventional heating on regular and quick-cooking oats. The conventionallyheated samples were continuously stirred w ith a magnetic stirrer, and the microwave-heated samples were
stirred by hand at regular intervals. For both methods, the samples were heated for one or 20 minutes at
90-95°C after the samples had reached boiling temperature. The researchers found that, in general,
microwave-cooked samples were grainier and less viscous than the conventionally-cooked samples. They
also found that the conventionally-cooked samples had more solubilized starch and p-glucan than the
microwave-cooked samples, especially for the 20 m in cook time. The authors acknowledge that the
differences are probably caused by the difference in duration and intensity o f stirring during heating. Thus,
agitation can help to achieve uniform temperature distribution for both methods, but its intensity and
duration should be reasonably close between heating method in order to compare them under the same
conditions.
Huang et al. (1990) examined the effects o f both microwave and conventional heating on the
tissues and starch granules o f potatoes. They measured temperature in the center, side, and ends o f the
potatoes. A ll samples were studied by SEM microscopy, and evaluated fo r hardness and fo r weight loss
during heating. The researchers observed differences between the two heating methods in terms o f both
heating patterns and swelling patterns o f the starch granules. For conventional heating, the highest
temperatures and shortest times to reach the gelatinization temperature range were observed near the
outside o f the potatoes, and the lowest temperatures and longest times to reach the gelatinization
temperature range were near the centers. That pattern is expected, and it coincided w ith the observations o f
Goebel et al. (1984) and Zylema et al. (1985) for starch suspensions. For microwave heating, however,
Huang et al. (1990) found that temperature was relatively uniform throughout the potatoes. Therefore,
gelatinization occurred uniform ly in a ll the regions o f the potatoes. This uniform distribution o f gelatinized
starch is contrary to the findings o f Goebel’s group, who reported that gelatinization first occurred in the
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36
inner regions o f microwave-heated samples. This disparity illustrates the fact that microwave heating
patterns are far more d iffic u lt to predict that conventional heating patterns.
The mechanisms o f heat generation are different for microwave and conventional heating at the
molecular level. The nature o f this difference w ill be discusses in some depth in chapter 3 o f this thesis. In
addition to this difference at the molecular level, this b rie f review o f the available literature shows that, in
practice, the conditions prevailing during the application o f microwave and conventional heating methods
often d iffe r in a number o f aspects. Some o f the differing conditions include temperature distribution,
heating rate, pattern o f moisture migration, and rate o f water evaporation. Heat transfer during
conventional heating o f non-agitated systems, for example, usually occurs from the outside to the inside,
whereas microwave heating often produces “ hot spots” and “ cold spots.” The hot spots are often in the
geometric center o f a sample, but this is not always the case.
The literature reports differences in the gelatinization o f starch in aqueous suspensions and in
starch-containing products heated conventionally and using microwaves. Quality differences have been
also observed in baked goods, in baked potatoes, and in porridges. It is not known, however, whether these
differences are the result o f the different mechanisms o f heat generation at the molecular level (a so called
“ microwave effect” ), or i f they are the result o f the differences in conditions prevailing during the
application o f the heating methods. The quality o f microwave-heated products is often reported to be lower
than the quality o f conventionally-heated products. The speed and convenience o f microwave heating,
however, are desirable, and have created a market fo r “ microwaveable” products. A clear understanding o f
the differences between heating methods should facilitate improvement in the quality o f products to be
heated w ith microwaves.
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37
2.3 Objectives
The literature points to the need to equalize experimental conditions in order to have a meaningful
comparison between microwave and conventional heating o f starch suspensions. A means to keep the
suspensions mixed, which was not present in much o f the literature reviewed, could be used fo r both
heating methods to ensure uniform temperature distributions and to prevent the settling o f starch granules.
In order to match the time/temperature conditions during heating, a means would also have to be provided
to control the heating rate for both heating methods.
Another important consideration is what characteristic o f starch to study in order to compare
microwave and conventional heating. Since starch is often used to increase the viscosity o f a product upon
cooking, and since granule swelling is directly related to the increase in viscosity, it would be appropriate
to compare granule size behavior during microwave and conventional heating. As mentioned earlier in
section 2.2.5, the main disadvantage to many techniques to study gelatinization (like SEM and dye
exclusion methods) is that the process o f gelatinization must be interrupted before the analysis can be
performed. Some other methods (such as X-ray diffraction, polarized microscopy, and Coulter counter)
can only follow a sudden change in a property that indicates one particular stage during gelatinization.
Methods like DSC and viscometry do follow the gelatinization process continuously w ith m inim al
interference. They lend themselves to controlled heating rates, and they also maintain relatively uniform
temperatures throughout the samples under study. However, these two methods would be impractical for
measurements during microwave heating, since the available instrumentation was designed assuming the
use o f conventional heating.
In a previous study (Ziegler et al., 1993), a method was developed to measure the particle size
distribution o f starch granules during conventional heating. The main objective o f this present work was
the m odification o f this method to provide a means to obtain particle size information fo r starch granules
during microwave heating. Specifically the objectives o f this work were:
1)
To adapt the non-interfering laser-scattering method used in Ziegler et al. (1993) to measure
the particle size distribution o f starch granules in well-mixed, dilute aqueous suspensions
during microwave heating in-line, at a controlled heating rate.
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38
2)
To use the method to measure the particle size distribution o f common corn starch, waxy
maize starch, and cross-linked waxy maize starch in dilute aqueous suspensions during
microwave heating.
3)
To compare the results from the microwave experiments w ith those from Ziegler et al. (1993),
in order to investigate i f there are any differences in particle size behavior between the two
methods.
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39
2.4 Materials and Methods
2.4.1. Starches
Three starches, chosen for the sake o f comparison w ith the results o f Ziegler et al. (1993), were
obtained from the National Starch and Chemical Company o f Bridgewater, New Jersey. The names o f the
starches, some general inform ation about them, and the abbreviations by which they w ill be referred in the
rest o f the text are summarized in Table 2.2 below.
Table 2.2 Starches Used in Particle Size Analysis Experiments
Starch
Name
MELOJEL®
AMIOCA™
CLEARJEL®
Type o f Starch
T ypical
m oisture
content
common com
starch
waxy maize
waxy maize
T ypical L o t No. used
pH
in this study
L o t No. used in
Z iegler et al. (1993)
Abbre­
viation
11%
5
L.B.-5816
M .D. - 8660
CC
11%
5
A.C. - 4262
J.H. - 4406
WM
11%
6
B.C. - 4705
F.D. - 5924
CW M
2.4.2. Scanning Electron Microscopy (SEM)
Samples o f the starch granules were examined using SEM before the microwave heating
experiments. The surface o f metal stubs were covered w ith conductive adhesive and a small amount o f
starch was sprinkled on the surface. The samples were then sputter-coated w ith a 60-40% gold-palladium
alloy. A JMS5400 scanning electron microscope (JEOL, Peabody, M A ) was used to examine the samples.
The images were captured by an IM IX system (Integrated Microanalyzer for Imaging and X-Ray,
Microanalysis Version 8, Princeton, NJ).
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40
2.4.3 Heating and Flow System
Copper tubing
Plastic tubing
h i . nu
\~n
- k
Figure 2.8 Experimental set-up for PSA of starch granules during microwave heating: (a) microwave
oven cavity; (b) LSPSA cell; (c) peristaltic pump; (d) temperature measurement points; (e) ballast water.
A schematic diagram o f the experimental set up is shown in Figure 2.8. Figure 2.8a represents the
Tappan “ Space Saver” model 56-2277-10 (Tappan Appliances, Columbus, OH) w ith a measured power o f
605±17.5W (IM PI, 1989). The electric power supplied to the oven was conditioned using a Sola power
conditioner (EGS Electrical Group, Sola/Hevi-Duty, Skokie, IL 60077) to lim it the voltage fluctuations,
and a variable transformer (Powerstat type 3PN126DP124202, Superior Electric Company, B ristol, CT
06010) to control the voltage supplied to the oven at 115V. Throughout the experiments, the voltage was
monitored by a digital voltmeter.
Two holes, approximately 0.62 inches in diameter were drilled through the side o f the oven to
accommodate two pieces o f copper tubing that were used to carry the starch suspensions that were pumped
into and out o f the oven. Inside the oven, the copper tubing was connected to plastic tubing, that, in turn,
was connected to a 250-ml plastic vessel used to heat the starch suspensions (see Figure 2.9). The copperplastic connection was secured w ith plastic cable ties, and wrapped w ith tape and wet paper towels to avoid
excessive heating inside the oven. Outside the oven, copper tubing joined plastic tubing that connected the
heating vessel to the LSPSA system (see Figure 2.8).
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41
Figure 2.8b represents the
laser-scattering particle-size analysis
(LSPSA) cell. The starch
suspensions were continuously
pumped between it and the heating
vessel inside the oven cavity by a
Varistaltic pump (series A , Monostat,
New York, N Y , shown schematically
in Figure 2.8c). The speed o f the
pump was set at 2 (0.1 A in the pump
Figure 2.9 Inside view of Tappan microwave oven with heating
vessel, plastic tubing, and beaker with water.
current meter). This pumping speed
was chosen because it was the
minimum speed that would keep the granules in suspension though the duration o f the experiments.
The starch suspensions were heated from 30 to 80°C. A 1000-ml beaker w ith “ ballast” water was
placed inside the oven to provide extra thermal load and allow control o f the heating rate. The amount o f
ballast water (550 m l) was chosen to obtain a nominal heating rate o f 5°C/min. The experimental heating
rates were calculated by fittin g the time-temperature data w ithin the target temperature range (between 40
and 80°C) to a straight line. The heating rates ranged from 4.16 to 5.45°C/min, and the r2were 0.986 or
higher (see Table 2.3 and Figure 2.10). Three repetitions were performed for each starch.
The temperature was monitored at three
Table 2.3 Heating Rates for Starches
different locations (at the outlet o f the MasterSizer
Heating Rate*
Type o f
°C /m in
Starch
cc
5.29 ±0.14
WM
4.86 ±0.31
CWM
4.52 ± 0.32
* Mean based on 3 repetitions.
cell, at the outlet o f the pump, and inside the heating
vessel in the microwave oven, as shown in Figure
2.8d) using a fluoroptic thermometry system (Luxtron
750, Luxtron, Santa Clara, CA). The temperature probe inside the oven was threaded through a small hole
in the back oven w all, and then was introduced in the heating vessel through a rig id plastic fittin g w ith a
small rubber septum. The two temperature probes outside the oven were inserted in the plastic tubing
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42
(a) CC
80-
70-
o
60experiment 1
experiment 2
experiment 3
- 5.26 °C/min;r2 = 0.9920
5.17 °C/min;r2 = 0.9922
- 5.45 °C/min;r2 = 0.9921
50-
40-
100
200
300
400
Time (s)
500
600
700
(b) WM
80-
70o
60experiment 1
experiment 2
experiment 3
50-
- 4.60 °C/min;r2 = 0.9901
5.20 °C/min;r2 = 0.9947
- 4.78 °C/min;r2 = 0.9927
40-
100
200
300
400
Time (s)
500
600
700
(c) CWM
80-
70o
60experiment 1
experiment 2
experiment 3
- 4.16 °C/min; r2 = 0.9866
50-
40-
- 4.77 °C/min;r2 = 0.9860
- 4.62 °C/min;r2 = 0.9933
100
200
300
400
Time (s)
500
600
700
Figure 2.10 PSA heating curves for: a) CC; b) WM; c) CWM.
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43
through a plastic Y-connector (see Figure 2.11). One arm o f the Y-connector (f) was attached to a metal
barbed fittin g (d) w ith a short piece o f plastic tubing (f). A Swagelok fittin g and nut (c) (Swagelok
Company, Solon, OH) were screwed to the barbed fittin g . The fluoroptic probe was introduced through a
rubber septum (b) secured between the Swagelok fittin g and nut. The whole assembly was fille d w ith hotm elt plastic glue up to the Y-connector (e). A small hole was drilled in the hot m elt glue. The fluoroptic
probe was passed through the septum, and then through the hole in the hot-m elt glue before fin a lly being
threaded through the plastic tubing to the desired location. The purpose o f this assembly was to keep the
system sealed, preventing the leakage o f air or liquid, while m inim izing dead volume. The three measured
temperatures were typically w ithin 1.5°C o f each other (but never more than 3°C apart), and the highest
temperature was always in the heating vessel. The temperatures measured by the Luxtron instrument were
collected by a PC, via an RS232 cable. A program was written in QBasic to collect the three temperatures,
and to record them w ith a tim e stamp in a text file.
Figure 2.11 Close-up schematic view of Y connector: (a) Fluoroptic probe; (b)
Septum; (c) Swagelok fitting and nut; (d) Barbed fitting; (e) Hot melt plastic
glue with hole; (f) Plastic “Y” fitting; (g) Plastic tubing.
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44
2.4.4 Particle Size Analysis
Particle size distributions were measured continuously dining the heating experiments w ith a
Malvern MasterSizer w ith a 300 mm-range lens. The MasterSizer was used w ith a flow-through cell (a
difference from the previous experiments as described in Ziegler et al., 1993, that used a water-jacketed 15m l cell w ith a small magnetic stirrer). For the experiments, reverse osmosis-purified water was boiled
under vacuum to eliminate dissolved air. The de-aerated water was then carefully pumped from an
Erlenmeyer flask into the heating vessel, the PSA cell and the tubing. The fillin g process was done slowly,
avoiding the introduction o f bubbles into the system. A fter the MasterSizer was zeroed, enough starch was
added to the Erlenmeyer flask to take the obscuration o f the instrument to the lowest acceptable value (0.1).
The instrument was programmed to measure the particle-size-distribution every 20 seconds.
Diameters o f the median o f the volume distributions (D [v,0.5j) were obtained from the software provided
w ith the MasterSizer. The method reported by Ziegler, et al. (1993) was used to calculate the temperature
o f maximum swelling rate (Tmax) from a third degree polynomial regression (see section 2.4.5) on the
D[v,0.5] as a function o f temperature.
Ziegler et al. (1993) explained that temperature gradients in the measurement cell can produce
scattering at low angles. Since low-angle scattering is interpreted by the LSPSA instrument as large
particles, the scattering due to temperature differences would affect the accuracy o f the data collected. The
MasterSizer’s software enabled suppression o f the 10 rings corresponding to the spurious low-angle
scattering. Ziegler et al. (1993) observed that the elim ination o f this data did not change the results at the
angles o f interest for starch swelling measurements, so the suppression command was also employed for
the data from this current study.
2.4.5 Statistical Procedures
The median particle size diameter data was fitted to a third order polynomial o f the form
D = a,, + a,T + a2T2 + a3T3,
[2.1]
where the independent variable T represented the temperature. The regression was performed using
M A TLA B ® student version 6.0.042a, release 12 (The MathWorks, Inc., Natick, M A, 01760-2098). The
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45
maximum temperature was defined as the temperature at which the second derivative o f equation 2.1 is
zero. That is,
Tmax = T I d2D/dt2=o
[2.2a]
which is equivalent to saying
Tmax=
3^
[2.2b]
Analysis o f variance for Tmax and the in itia l and final median diameters was performed using the
one-way AN O VA command in M icrosoft Excel 2002 (M icrosoft Corporation, Redmond, W A). The Tukey
method for m ultiple comparisons w ith unequal sample size (Neter et al., 1990) was used to compare the
data from this study and the data in Ziegler et al., 1993.
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46
2.5 Results and Discussion
(a) CC
2.5.1 Starch Micrographs
Electron microscopy was used to obtain
direct images o f the intact starches before the
heating experiments. These images show general
morphological features and validate the LSPSA
in itia l measurements. Figures 2 .12a to c show
micrographs fo r common com starch (CC), waxy
maize starch (W M ), and cross-linked waxy maize
starch (CW M ) respectively. The granules for a ll
three starches are sim ilar in size and appearance.
They range in size from approximately 5 to 20
2.5.2
Particle Sizes
The follow ing discussion contains some
general observations about the particle size
distributions during heating. Figures 2.13 and
(c) CWM
2.15 are surface plots w ith axes o f temperature,
diameter, and frequency o f occurrence (as percent
o f the volume distribution) for CC and CWM.
The black curve in each o f these plots is D [v,0.5],
the median o f the volume size distribution.
Figures 2.14 and 2.16 are plots o f the particle size
distribution at three representative temperatures.
Some features in the plots were common to CC
Figure 2.12 SEM pictures of starches: a) CC; b) WM; c)
CWM.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
and CW M. The diameter distributions appear to be lognormal (i.e. log(D [v,0.5]) appears to be norm ally
distributed) w ith the exception o f the features described below.
A t low temperatures the size distributions fo r CC and CWM had a small fraction o f particles w ith
diameters o f less than about 3 pm, which could correspond to granular fragments. This fraction, also
observed in Ziegler et al, 1993, disappeared as temperatures increased near the gelatinization temperature
range. Another feature common to the starches was the development o f a “ shoulder” in the size
distributions to the le ft o f the main population. This shoulder (obvious in Figures 2.14 and 2.16) appeared
around the onset o f gelatinization, and persisted past 80°C. There are different possible interpretations fo r
this feature. One possibility is that the shoulder corresponds to a population o f granules that swelled at
higher temperatures, so those granules maintained their in itia l diameters even when the main population
had swollen. Another interpretation is that the shoulder corresponds to either swollen small granules, or to
fragments o f completely swollen granules. O f these, the latter two hypotheses seem more likely, since the
shoulder either was not present at low temperatures and appeared around the onset o f swelling (as in the
case o f CC), or it was smaller at low temperatures and developed w ith increasing temperature (as in the
case o f CW M). The last common feature is that the breadth o f the distributions increased w ith increasing
temperatures. This was most obvious as temperatures approached the temperature o f maximum swelling
rate (Tmax) when the height o f the distributions decreased, denoting a wider spread o f sizes. This probably
occurred because there were particles representing every swelling stage in the heated suspensions at that
point.
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Frequency (%)
48
Figure 2.13 PSA surface for CC.
Common Com
30°C
67°C
80°C
£10
.....[-
D[v,0.5] (urn)
Figure 2.14 Particle size distribution for CC at three representative temperatures.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Frequency (%)
49
Figure 2.15 PSA surface for CWM.
D[v,0.5] (urn)
Figure 2.16 Particle size distribution for C W M at three representative temperatures.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
2.5.3 Comparison with Conventional Heating
The software fo r the MasterSizer provided the mean diameters (D [4,3]) and the median diameters
(D [v,0.5]) o f the volume distribution o f granules. The D [v,0.5] is reported here as it was in Ziegler et al.
(1993), since this measure shows less scatter than D [4,3]. The data from three replicate experiments were
pooled, and means and standard deviations o f both the in itia l (DO and maximum (D max) diameters were
calculated. These results, along w ith those from Ziegler et al. (1993), are shown in Table 2.4. The last two
columns o f Table 2.4 contain calculated values for the maximum swelling power, which is defined as
Maximum swelling power =
D
3
( — j™5—)
[2.3]
Table 2.4 Comparison o f in itia l and maximum D [v,0.5] for microwave and conventional heating
Type
of
Starch
In itia l D iam eter (pm )
M icrow ave*
C onventional**
M axiimum
Swelling Power
Conven­
M icrow ave* C onventional** M icrow ave*
tio n a l**
36.55 ± 0.53 be* 33.32 ± 1.20 ab
14.38
11.2
41.70 ± 0.22 d
39.63 ± 1.62 cd
19.21
16.4
34.63 ± 0.25 ab 31.49 ±3.20 a
9.40
10.3
M axim um D iam eter (pm )
15.03 ± 0.12 abT 14.87 ± 0.38 ab
CC
WM
15.58 ± 0.37 ac 15.61 ± 0.24 ac
CW M 16.44 ±0.51 c
14.46 ± 0.07 b
* Mean based on 3 repetitions.
**D ata from Ziegler, et al. (1993). Means based on 3 repetitions, except CC that was based on 4 repetitions,
f Comparison o f Dj between starch types and heating treatments. Means followed by the same letter are not
significantly different (P<0.05).
%Comparison o f
between starch types and heating treatments. Means followed by the same letter are
not significantly different (P<0.05).
Analysis o f variance for in itia l and maximum D [v,0.5] for the microwave data showed that the
effect o f starch type was significant (P=0.01 for Dj, and P<0.0001 for Dmax). Ziegler et al. also reported
significant effect o f starch type fo r the conventional heating starches. The means and standard deviations
from both studies were used to estimate “ least significant differences” using the method o f Tukey (Neter et.
al., 1990). The means for both studies were compared between starch types and heating treatment as
shown in Table 2.4. The in itia l D [v,0.5] values were not significantly different between heating treatments
fo r CC and W M, but were significantly different for CW M (P<0.05). None o f the maximum D[v,0.5]
values were found to be significantly different (P<0.05). The maximum swelling powers are also reported
in Table 2.4 for completeness, and as expected, follow the trends in the values for D; and Dmax.
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51
Analysis o f the D [v,0.5] versus temperature curves can provide further insight on the sw elling
behavior o f starch. Figures 2.17-2.19 show the data for CC, W M, and CW M, and once again, there are
some common features among them. First, D [v,0.5] is constant or varies only slightly u n til the onset o f
swelling. Then after the onset o f swelling, D [v,0.5] increases suddenly, follow ing an sigm oidal curve.
These curves show an inflection point at Tmax (the temperature o f maximum swelling rate). Finally, the
curves reach a plateau at the end o f gelatinization. Depending on the kind o f starch, the particle size would
decrease, stay constant, or increases slightly w ith further heating time. Ziegler et al. (1993) observed the
same features for the conventional heating data.
The graphs in Figures 2.17-2.19 also show a second set o f curves. These are for third degree
polynomials obtained by regression on the value o f D [v,0.5] as a function o f temperature in the
gelatinization region. T ,^ was defined as the temperature o f maximum derivative o f these regression
curves as reported by Ziegler et al. (1993). For a ll o f the regressions, r2 > 0.99, except for one repetition o f
CW M, which had r2 = 0.9889. The regression lines and their derivative curves are overlaid on the data to
illustrate the procedure.
The plots in Figures 2.17-2.19 fo r CC, W M, and CW M show that the D [v,0.5] data as a function
o f time was repeatable for these starches. The values o f Tmax for the present experiments and for the data
reported by Ziegler et al., 1993 are summarized in Table 2.5. Analysis o f variance for the microwave data
showed a significant effect o f starch type (P=0.002) and Ziegler et al. (1993) also reported significant
differences between starches. The means and standard deviations for both studies were used to estimate
Table 2.5
Comparison o f T ,^ Values
“ least significant differences” as before. There was
no significant difference (P<0.05) between heating
Starch
CC
WM
CW M
T m„ ( ° C )
M icrow ave*
C onventional**
69.68 ± 0.83 a
69.51 ±1.44 a
68.36 ± 0.23 ab
69.01 ± 0.88 ab
67.01 ± 0.12 b
67.28 ± 1.09 ab
* r2> 0.99, except fo r one repetition o f CW M,
which had r2 = 0.9889; Means based on 3
repetitions.
** Data from Ziegler et al. (1993).
treatments for any o f the starches. This suggests
that, fo r the conditions in these experiments, there is
no important difference in swelling behavior
between microwave heating and conventional
heating.
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52
There are three sources o f potential differences between the previous experiments (Ziegler et. al,
1993) and the present experiments. The first one is the intrinsic lot-to-lot variation for the starch. This
variation (in terms o f particle size) is apparent for the D ; o f CW M (Table 2.4), but was not observed as
significant differences between heating treatments in any other case. The second potential difference is the
design o f the measuring cell. Ziegler et al. (1993) used a small-volume (15 m l) jacketed cell w ith a small
magnetic stirrer for their experiments. This cell was placed directly in the LSPSA instrument. In the
present study, heating was conducted in a 250-ml vessel inside a microwave oven, and the suspension was
circulated through a flow-through cell that was placed in the LSPSA instrument. I f there was a difference
in the shear stresses between the two treatments, it is expected that the shear in the present experiments
would be higher, since the suspensions were pumped and circulated through tubing. There was no
significant difference in the Dmax between the heating treatments, so the difference in shear stresses does
not appear to be important. The third potential difference is in the time-temperature treatment. In both sets
o f experiments the heating rate was controlled at a nominal 5°C/min between 40 and 80°C. The typical
time-temperature plot reported by Ziegler et al. (1993) had increasing heating rate between 30 and 40°C,
constant heating rate from 40°C to about 85-90°C, and then the rate visibly diminished. In the current
experiments heat loss from tubing and the LSPSA cell caused the heating rate to constantly decrease
through the heating treatment. The amount o f ballast water in the microwave oven was adjusted so that the
heating rate averaged about 5°C, but the heating rate was slightly higher that this value between 30 and 40°,
and slightly lower after 80°C. Straight lines were fitted to the data between 40 and 80°C, and the r2 values
for the regression were always close to one, meaning that, in the temperature range o f interest, the timetemperature curve was close to linear. The difference in shape o f the time-temperature curves between the
experiments o f Ziegler et al. (1993) and the present experiments was greatest at temperatures between 30
and 40°C. It is unlikely that, in this range o f temperatures, and in the corresponding duration (less than 2
minutes), the difference in the time-temperature treatment had an effect in the swelling phenomena
observed in the experiments.
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53
= 0 .9 9 8 5
r 2 = 0 .9 9 5 0
■ A -tL =
0 .9 9 5 1
620
_ .& ... a . .. ^
^
. . ■js.
. j j j . . oa.
Temperature (°C)
Figure 2.17 D[v,0.5] vs. T and calculation of Tmax for CC. Colored lines represent repetitions of experiments.
W at
r 2 = 0 .9 9 7 5
.& S L
-P-I5
o
Temperature (°C)
Figure 2.18 Dfv,0.51 vs. T and calculation of Tm„ for W M . Colored lines represent repetitions of experiments.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
!g 25
Q
3
8 20
f 15
o 10
Temperature (°C)
Figure 2.19 D[v,0.5] vs. T and calculation o f Tmax for CWM. Colored lines represent repetitions of experiments.
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55
2.6 Conclusions
This chapter focused on studying the swelling behavior o f three different starches in dilute
aqueous suspension as they were heated using microwaves. From the results o f this work, the follow ing
1.
Under the experimental conditions o f this work, there is no significant difference between
conventional and microwave heating o f common com starch, waxy maize starch, and crosslinked waxy maize starch in dilute aqueous suspension in terms o f maximum diameter,
temperature o f maximum swelling rate, and behavior o f D [v,0.5] as a function o f temperature.
2.
Laser Scattering Particle Size Analysis can be effectively used to study the swelling o f starch
granules in dilute aqueous suspensions, in-line during microwave heating. The well-m ixed
system used in this study allows uniform distribution o f temperature and starch granules, and
a controlled heating rate.
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56
CHAPTER 3
IN-LINE MEASUREMENT OF THE DIELECTRIC PROPERTIES
OF AQUEOUS STARCH SUSPENSIONS DURING MICROWAVE
HEATING
3.1
Introduction
3.1.1 Chapter Overview
Microwave heating is a complex process that is largely dependent on the foods being cooked.
Food characteristics such as composition, thickness, size, shape, density, heat capacity, thermal
conductivity, and dielectric properties a ll come into play. Since many o f those characteristics change w ith
temperature, they w ill vary throughout the heating process, making it more d iffic u lt fo r researchers to
determine the effect o f each individual characteristic. In addition, the dielectric properties, which are
perhaps the most influential food properties in microwave heating, vary w ith frequency. This variation
makes the analysis o f these properties more complex, but, fortunately, it can also be used to study some
molecular characteristics o f a food system.
The purpose o f this chapter was to both measure, analyze, and model the dielectric properties o f
aqueous dilute starch suspensions as functions o f temperature and frequency during microwave heating. In
order to do so, a technique was developed to measure the dielectric properties o f liquids in-line during
microwave heating and cooling. This technique was then used to measure the dielectric properties o f
aqueous starch suspensions as functions o f temperature and frequency during gelatinization. Some o f these
suspensions contained added sodium chloride to see how it would affect the dielectric properties o f starch
suspensions. The data was fitted to a theoretical mathematical model from the literature that had been
adjusted to account for the effects o f conductivity and two dielectric relaxations. The adjusted model used
here was called the Debye-Hasted model w ith Two Peaks, and it consisted o f fitted parameters that were
temperature-dependent
In order to better understand the work done in this chapter, a b rie f review o f electromagnetic
radiation and its interaction w ith molecules w ill follow this introduction. Then the literature relevant to the
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study w ill be reviewed. Section 3.2.1 w ill lis t the equations articulated by James Clerk M axw ell that
describe the behavior o f a ll electromagnetic radiation. These equations w ill then be translated into the
“ frequency domain” using a mathematical tool known as the “ complex method” in section 3.2.2. In section
3.2.3, two categories o f materials - conductors and insulators - w ill be described in terms o f their response
to electromagnetic radiation. A fter the concepts o f polarization and dielectric perm ittivity are defined in
section 3.2.4, dielectric relaxation w ill be described in section 3.2.5. Sections 3.2.6 and 3.2.7 w ill focus on
the models proposed by Peter Debye and by the brother-brother team o f Robert and Kenneth Cole to
predict the dielectric properties o f polar liquids. The dielectric properties o f a very important polar liquid,
water, w ill be b riefly outlined in section 3.2.8. Section 3.2.9 w ill expand upon the concept o f conductivity
to include frequency-dependent conductivity, which w ill be needed to discuss the w ork done by the B ritish
research team o f Hasted, Ritson, and Collie in section 3.2.10. Section 3.2.11 w ill summarize the heating
mechanisms occurring during microwave cooking, and section 3.2.12 w ill explain the mathematics behind
the penetration depth o f an electromagnetic wave im pinging upon a material. Various techniques used to
measure dielectric properties, and various factors that affect the dielectric properties o f foods, w ill be
surveyed in sections 3.2.13 and 3.2.14, respectively. The last topic o f the literature search, which w ill be
covered in section 3.2.15, w ill focus on how certain researchers modeled the dielectric property data they
had measured. A fter the literature review, the objectives o f this study w ill be listed (section 3.3), the
materials and methods used in the study w ill be outlined (section 3.4), the final results w ill be discussed
(section 3.5), and the conclusions w ill be summarized (section 3.6).
3.1.2 The Nature of Electromagnetic Radiation
The microwave radiation used to cook food in m illions o f household and industrial microwave
ovens is just a portion o f the entire spectrum o f electromagnetic radiation. This spectrum includes gamma
rays and x-rays on one end; ultraviolet, visible, and infrared light towards the middle; and microwaves,
radio waves, and television waves on the other end. Electromagnetic radiation, which is present a ll around
us, can be generated by both natural and man-made sources. The natural sources include radioactive
materials, as w ell as the sun, stars, and lightning. In fact, “ electromagnetic waves from the sun are the
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58
ultim ate source o f a ll the energy that keeps life on earth going” (Lehrman 1998). Man-made sources o f
electromagnetic radiation include everything from humble light bulbs to devastating nuclear explosions,
and other common sources like radio antennas, and, closer to the subject o f this study, magnetrons in
microwave ovens.
Great thinkers, philosophers and scientists have pondered the nature o f lig h t throughout history.
Today it is known that light, and in fact a ll electromagnetic radiation, has a dual particle-wave nature.
Classical physics describes electromagnetic radiation as waves, radiating outward from a source. They
travel through space (w ithout requiring a transmission medium) at the speed o f lig h t in a vacuum, 3x10s
m/s. This speed can be related to the wavelength and frequency o f the waves by the equation
v = Xf
[3.1]
where
v = speed o f propagation in the medium (m/s)
X = wavelength (m)
f = frequency o f oscillation (s'1, also known as hertz, abbreviated Hz).
Sometimes, v in equation 3.1 is replaced w ith c, the speed o f light in a vacuum and then the wavelength in
the vacuum is represented by Xq. By rearranging equation 3.1, it can be shown that the wavelength and
Table 3.1. Electromagnetic radiation, wavelengths, and sources
x-rays
10“ “ - 10'H
angstroms,
nanometers
micrometers,
nanometers
micrometers
i
Typical source
protons and neutrons w ithin a nucleus
changes in energy levels o f
atomic inner shell electrons
changes in energy levels o f
atomic outer shell electrons
changes in energy levels o f
atomic outer shell electrons
molecular vibrations and rotations
1
T
m illim eters
micrometers
ion5—10-1
centimeters
microwaves
klystrons and magnetrons
television and
meters
kilometers
antennas
Data from Cumutte, 1980; Shen and Kong,1987; Callister, 1991; Lehrman, 1998; and Ohanian, 1985;
Skoog and Leary, 1992; Serway, 1986; Inan and Inan, 1998; Yule, 1978.
0
infrared
4 x 10_/ —7 xH T7
visible light
4.
1
ultraviolet
Units used to
measure
wavelength
angstroms
c
1
Approximate
wavelength range
(meters)
10' 16- 10'10
Type o f
Electromagnetic
gamma rays
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59
frequency are inversely proportional to each other. Thus, radiation w ith an extremely short wavelength,
such as gamma rays ( 10"13m), have a very high frequency, whereas radiation w ith a very long wavelength,
such as radio waves (105m), have a very low frequency. Table 3.1 lists the various types o f
electromagnetic radiation and the units typically used fo r measuring each. There is considerable overlap
between many categories because the various types o f radiation are characterized not only based on their
wavelengths, but also on their methods o f generation and use (Ohanian, 1985).
The prefix “ micro” means “ short,” and as can be seen
from Table 3.1, microwaves are relatively short since their
wavelengths are typically measured in centimeters. Their
frequencies range between 300 MHz and 300 GHz, which
correspond to wavelengths between lm and 1mm. However, the
only frequencies permitted by the Federal Communications
Commission for cooking are 915 M Hz (wavelength o f 32.8 cm)
and 2450 M Hz (wavelength o f 12.2 cm) (Cumutte, 1980;
Schiffinann, 1986). Other frequencies o f microwave radiation are
Figure 3.1 Electromagnetic
used in telecommunications (cellular phones), in aircraft radar,
satellite dish antennas, and in instruments for analyzing molecular and atomic properties o f materials.
(Serway, 1986, p. 785).
Figure 3.2 Traveling plane wave.
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60
The waves o f electromagnetic radiation, produced
^
u
by moving electric charges, travel in three dimensions out
A
.....................................
from their source. As an illustration, consider an antenna
.
^
...............................
viewed from a large distance, as shown in a very sim plified
diagram in Figure 3.1. A t large distances, the waves travel
like a series o f concentric spheres. The shading o f colors
.t-Wvt..........
from dark to light indicates the fluctuating intensity o f the
wave as it travels away from its source. Shells o f equal
.....................
radius have equal intensities (amplitudes). Far enough
away from the source, however, the spherical wave fronts
o f a small area (as indicated by the area w ithin the large
Figure 3.3 Traveling pulse
rectangle on Figure 3.1) can be considered flat, parallel
planes. Thus, these are referred to as plane waves. Figure
3.2 illustrates a section o f a traveling plane wave. Once again, the shading o f colors from dark to light
indicates the fluctuating intensity o f the wave, this tim e as it travels from left to right in the z direction.
However, the intensity remains constant in a given x-y plane. Figure 3.3 shows a traveling pulse,
illustrated as a sigmoidal curve to show how the wave’s peaks and valleys (high and low intensities) change
in time. A wave that does not “ travel” is known as a standing wave. Such a wave pulsates, changing its
intensity at a particular position w ith time, although the locations o f peaks and fixed points where the
amplitude is zero remain the same. (See Figure 3.4) Standing waves can be formed by the superposition o f
two waves o f the same amplitude and frequency traveling in opposite directions (Ohanian, 1985). When
the maximum amplitudes o f the two waves coincide, their constructive interference forms the maxima o f
the standing wave, known as the antinodes. Sim ilarly, when the minimum amplitudes o f each traveling
wave coincide, their destructive interference forms minima called nodes. (When viewing the example o f a
standing wave in Figure 3.4, and any sim ilar representation o f an electromagnetic wave, it should be kept in
m ind that the representation o f a wave as a sine curve is not a picture or a real description o f the position o f
the wave. It is more correctly interpreted as a mathematical representation o f the intensity o f the wave
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Figure 3.4 Standing Wave (adapted from Ohanian, 1985).
along a one-dimensional line.) Standing waves in a microwave oven, in a much more complicated pattern
than that in Figure 3.4, are often responsible for the all-too-fam iliar phenomena o f hot and cold spots in
food. This is because the standing waves prevent uniform energy distribution, resulting in uneven heating
(Mudgett, 1986; IFT, 1989).
3.1.3 The Nature of Molecular Interactions and Electromagnetism
Although classical physics recognized the wave nature o f electromagnetic radiation, it did not
recognize the particle aspect o f the radiation’ s dual nature. It was not until the year 1900 that M ax Plank
recognized that his observations o f electromagnetic phenomena could be explained only i f the radiation
were emitted or absorbed in “ particles” or “ packets,” called photons that contained specific amounts o f
energy. (Atkins, 1990; Ohanian, 1985; Callister, 1991; Cumutte, 1980). In fact, he determined that the
energy o f a photon is completely specified by the frequency o f the radiation that produces it, according to
the formula
Ephoton = h f = h (\/X)
[3.2]
where h represents Planck’ s constant, 6.625 x 10'34 J s. This “ lim itation o f energy to discrete values is
called the quantization o f energy (from the Latin word quantum, amount),” (Atkins, 1990) and thus the new
branch o f physics that was bom from Plank’ s ideas became known as quantum physics. “ In general, the
higher the energies involved, the better the properties o f the radiation are described in terms o f particles
(photons) rather than waves” (Yule, 1978).
Nature only permits matter (including molecules, atoms, and even nuclei) to have certain energy
states. The lowest such state o f energy, which usually exists at room-temperature (Skoog and Leary, 1992),
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62
is called the ground state. A molecule exposed to
X X
Symmetric
i f that photon has the energy that the molecule needs to move it
Asymmetric
STRETCHING VIBRATIONS
to a higher energy state. Skoog and Leary (1992) represent the
interaction between the molecule (M ) and the photon as
M + Ephoton —> M *
X"
In-plane rocking
[3.3]
where M * represents the molecule in its excited state. The
In-plane scissoring
numerous energy states that a molecule possesses have been
categorized as three types: electronic, vibrational, and
rotational. These states are actually subdivided in such a way
that every electronic level has several vibrational states, and
Out-of-plane
wagging
Out-of-plane
wagging
BENDING VIBRATIONS
(Motion out o f page indicated by +
Motion into page indicated by - )
Figure 3.5 Schematic illustrations of
molecular stretching and bending
Leary, 1992).
every vibrational level has several rotational states. The energy
between the several rotational states is less that the energy
between the vibrational states (Skoog and Leary, 1992).
The different electronic energy states correspond to the
different energy levels o f the molecule’ s bonding electrons.
Photons o f ultraviolet and visible radiation have enough energy
to excite one or more bonding electrons to a more energetic state. When this occurs, the molecule’s nuclei
are subjected to various forces, which often induce some accompanying changes in vibrational and
rotational energy states (Atkins, 1990).
The vibrational energy states are associated w ith the vibrations o f
the bonds between atoms in a molecule. These bonds vibrate “ back and
5-
forth at a certain frequency, alternately stretching and compressing as i f there
were a spring connecting the atoms” (McMurray, 1988). Figure 3.5
8+
1
schematically illustrates several such vibrating motions, which are classified
as either stretching or bending vibrations.
Figure 3.6 Polar
molecule w ith dipole
from Mortimer, 1986).
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63
In order for a vibrational transition to occur, the molecule must have a dipole moment that changes
during the vibration (Ohanian, 1985). A dipole moment is a vector that points from the negative charge to
the positive charge in a molecule w ith an uneven charge distribution. (See the arrow in Figure 3.6.) A
permanent dipole moment occurs in a molecule formed by (at least) two different atoms that have different
attractions for the electrons that they share in a covalent bond. One atom w ill preferentially attract the
electrons, so the resulting electron distribution w ill be asymmetric. This unequal sharing o f the electrons is
known as a polar covalent bond, and the molecule formed in such a way is known as a polar molecule
(Atkins, 1990; M ortim er, 1986). In a polar molecule, the atom w ith the higher electronic density develops
a partial negative charge (given the symbol 5 - in Figure 3.6), while the atom w ith the lower electronic
density develops a partial positive charge (given the symbol 8+ in Figure 3.6). The asymmetric water
molecule shown in Figure 3.7 is a polar molecule that possesses a permanent
dipole moment because the excess o f negative charge from the two pairs o f
free electrons on the oxygen atom leaves the two hydrogen atoms w ith an
r
excess o f positive charge (Bouldoires, 1979; Ohanian, 1985). It is also
°
\
possible for non-polar molecules to experience a temporary dipole moment
Electron pairs
when its charges separate under the influence o f an applied electromagnetic
field. (Overall, however, in both cases the net charge on the molecule is zero.)
This separation o f opposite electric charges by a distance d is known as an
Figure 3.7 Water
from McMurray,
1988).
electric dipole. The value o f the dipole moment (given the symbol p<im) is
given by:
H dm
= Qd
where d is the distance vector pointing from the negative to the positive charge, and Q represents the
magnitude o f the charge. Dipole moments are usually reported in debye units, D (one debye is equal to
3.336 x 1(T30 coulomb-meter), named for Peter Debye (1884-1966), the Nobel-prize-winning physical
chemist (Atkins, 1990; Yule, 1978).
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[3.4]
64
The requirement for a vibrational transition is that the
dipole moment changes; it does not depend on whether that
dipole moment is permanent or temporary. Since homonuclear
diatomic molecules only experience bond stretching, which
does not affect dipole moments, those molecules do not
experience vibrational transitions. However, heteronuclear
diatomic molecules and polyatomic molecules do undergo
H
vibrational transitions because as their dipole changes, it can
interact w ith the oscillations o f the electromagnetic radiation, as
depicted in Figure 3.8 (Atkins, 1990; Skoog and Leary, 1992).
In the case o f the water molecule (which is o f interest to this
Figure 3.8 A nonpolar molecule can
undergo vibrational transitions if its
dipole moment changes during the
1990).
current study), oscillating electromagnetic radiation w ill push
and pull on the hydrogen and oxygen atoms, which already
vibrate at a natural frequency. When the frequency o f the
fie ld ’s oscillations is equal to the natural vibrational frequency
o f the molecule, a photon o f that radiation is absorbed by the molecule and its vibration increases in
intensity, like a spring stretching and compressing even more. (Cumutte, 1980,; M cMurray, 1998; Skoog
and Leary, 1992).
The third type o f excited state, transition to the rotational energy levels, involves the different
levels o f energy related to the rotations o f atoms around an axis w ithin a molecule. (Skoog and Leary,
1992). There are usually several rotational levels for each vibrational level in a molecule. These rotations
are the result o f the torque that a molecule experiences when the force o f an electromagnetic wave acts
perpendicularly to a bond w ithin the molecule (Cumutte, 1980).
Only polar molecules w ith permanent dipole moments can undergo rotational transitions. This is
because the dipole o f a polar molecule rotating around its center o f mass changes during the rotation,
allowing it to interact w ith the oscillations o f electromagnetic radiation, as depicted in Figure 3.9 (Atkins,
1990; Skoog and Leary, 1992). The dipole acts like a handle that the electromagnetic radiation uses to
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65
Figure 3.9 An electromagnetic field can cause the
permanent dipole moment (arrow) to oscillate, thereby
enabling the molecule to undergo rotational transitions.
“ stir” the molecule into oscillation (Atkins, 1990). These rotational transitions do not necessarily start in
the ground state; they can begin in other in itia l states (Atkins, 1990), and they often occur in conjunction
w ith vibrational transitions. As Atkins (1990) explains, “ A rotational change should be expected because
classically we can think o f the transition as leading to a sudden increase or decrease in the instantaneous
bond length. Just as ice-skaters rotate more rapidly when they bring their arms in, and more slowly when
they throw them out, so the molecular rotation is either accelerated or retarded.”
Photons from the infrared region o f the spectrum are not energetic enough to cause electronic
transitions, but they can induce a molecule to undergo vibrational transitions (Skoog and Leary, 1992).
Even less energy is required for rotational transitions; they can occur w ith radiation o f wavelengths greater
than 100 pm (Skoog and Leary, 1992), which includes the far infrared and microwave regions o f the
spectrum. The photons o f radiation that do not have the energy sufficient to excite the molecule to a higher
electronic, vibrational, and/or rotational state w ill be transmitted through the sample, while the others are
absorbed.
A fter this conceptual discussion o f the nature o f electromagnetic radiation, it is now time to turn to
a more mathematical explanation by introducing the equations that govern a ll electromagnetic radiation,
M axw ell’s Equations.
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66
3.2 Review of the Literature
3.2.1 Maxwell’s Equations
In 1864 the Scottish physicist James Clerk M axw ell (1831-1879) “ did for electromagnetics what
Newton had done two centuries earlier for mechanics” (Lehrman, 1998), by announcing that a change in an
electromagnetic field would propagate through space as a wave traveling at the velocity o f lig h t (Sttsskind,
1968a). An electromagnetic wave, as its name implies, is composed o f electric and magnetic fields that are
perpendicular to each other and to the wave’s direction o f propagation (Callister, 1991; Serway, 1986), as
shown in Figure 3.10.
M axwell expanded his wave theory
and published his two-volume
Treatise on Electricity and
Magnetism in 1873 (Ohanian, 1985;
Figure 3.10 Magnetic and electric field components o f an
electromagnetic wave, perpendicular to each other and to their
direction o f propagation.
SUsskind, 1968a). In this treatise,
he summarized in four equations the
relationships between electric and
magnetic flu x densities, electric and magnetic field strength, and the sources generating those fields. These
equations are the mathematical representation o f the fundamental laws that govern the behavior o f all
electromagnetic waves. In 1887, Heinrich Hertz provided experimental evidence to support M axw ell’s
predictions by generating the first a rtificia lly produced radio waves. He analyzed their properties and
realized they were sim ilar to light waves, the other known type o f electromagnetic radiation at that time
(Ohanian, 1985; Serway, 1986). The only difference was in their wavelength and frequency.
M axw ell’s four elegant equations are listed below (and are used in subsequent discussions) in the
differential formulation, w ith boldface type used to represent the vectors that are functions o f both time and
space. Because o f the importance o f M axw ell’ s equations to this current study, each o f them w ill be
discussed in the paragraphs that follow .
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67
SB
St
V xE
[3.5]
[3.6]
V xH
V • B
V• D
=
0
[3.7]
Pv
[3.8]
Where:
V is the differential “ del” operator in three Cartesian coordinates, defined mathematically as:
E = electric field strength (volts per meter), or electric field intensity, defined “ as the force per unit
charge on a test charge in the presence o f a second charge ...” (Hayt and Buck, 2001);
D = electric flu x density (coulombs per square meter); also called the electric displacement,
displacement density, or the displacement current vector;
H = magnetic field strength (amperes per meter);
B = magnetic flu x density (webers per square meter), also called the magnetic induction;
J = electric current density (ampere per square meter);
pV = electric charge density (coulombs per cubic meter);
x, y, and z are unit vectors in the x, y, and z directions, respectively.
Equations 3.5 and 3.6 are called either the “ curl equations” (because o f the del operator) or the
“ field equations” (because they refer to the magnetic and electric field vectors) (von Hippel, 1954). Hayt
and Buck (2001) describe curl as “ circulation per unit area.” In the Cartesian coordinate system, the
mathematical definitions o f the curl o f E and the curl o f H are:
[3.10]
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68
V x H = x(
5H:t
dy
dHy
dz
x
3x /
“ v
5x
gHx
3y
[3.11]
Note that the curl o f a vector is itse lf a vector. These operations can be w ritten more compactly and the
computations can be performed more easily i f the determinant format is used w ith the del operator
(Edminister, 1993; von Hippel, 1954):
c u rlH :
x
y
z
d
3x
d
dy
d
dz
Hx
Hy
Hz
[3.12]
V xH
Shen and Kong (1987) interpret the cross product o f
the del operator and a vector by referring to a small square in
the y-z plane, such as that shown in Figure 3.11. In this
figure, vector V
X
E has a component in the x-direction that is
coming out o f the page. (This is represented by the circle
w ith the dot in the middle.) To evaluate this component, the
right-hand rule must be used so that the thumb points in the
Figure 3.11 Definition of curl (adapted
from Shen and Kong, 1987).
direction o f the x-component o f V x E, as the fingers “ curl”
around the square. The curl o f E only in the x-direction can
be w ritten as
(V x E) • x =
dEz
dy
5Ey
dz
[3.13]
By referring to the letters labeling each side o f the square in Figure 3.11, and by applying the rules o f
partial differentiation, the terms on the right side o f equation 3.13 can be represented as:
SEZ
_
(Ez) on b - (Ez) on d
Ay
[3.14]
=
(Ey) on c - (Ey) on a
Az
[3.15]
dy
dEy
dz
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69
Substituting equations 3.14 and 3.15 into 3.13 yields
(V
X
E) • x
-
on a Ay + (E2) on b Az + (-Ey) on c Ay + (-Ez) on d Az
AyAz
[3.16]
One inherent assumption in the derivation o f equation 3.16 is that the area o f the square must be
infm itesim ally small, such that Ay and Az approach zero. The second assumption, according to Shen and
Kong (1987), is that the directional components o f the vector (Ey and Ez) must be “ constant on the
boundary o f the square.”
Equation 3.5 o f M axw ell’ s equations is a generalization o f Faraday’ s Law o f Induction. The
B ritish physicist and chemist Michael Faraday (1791-1867) first discovered this law after experimenting
w ith electric currents flow ing through wires wrapped around a doughnut-shaped piece o f iron (G uillen,
1995; Hayt and Buck, 2001; Ohanian, 1985; Shen and Kong, 1987). The unpretentious Faraday wanted
ordinary people to understand his findings, so in 1831 he simply stated his b rillia n t discovery in plain
English: “ Whenever a magnetic force increases or decreases; it produces electricity; the faster it increases
or decreases, the more electricity it produces.” (G uillen, 1995). M axw ell’s contributions to Faraday’s law
were tw o-fold: first, he realized that Faraday’ s findings were applicable also to space, without the need for
a wire (von Hippel, 1954); and secondly, he translated Faraday’ s law from English to mathematics, the
form al language o f science, resulting in equation 3.5 (Guillen, 1995). Equation 3.5 is the mathematical
declaration that a change in time o f the magnetic flu x density w ill indeed induce an electric field (Ohanian,
1985; von Hippel, 1954). The magnetic flu x density, also known as magnetic induction, is a measure o f the
internal field strength that an object experiences when it is subjected to an external magnetic field
(Callister, 1991). Another way to interpret equation 3.5 is that one can examine the electric fie ld ’ s change
in space to discover how the magnetic flu x varies w ith time. This interpretation w ill be used in chapter 4 to
model electromagnetic fields.
Equation 3.6 is M axwell’ s adaptation o f Ampere’s C ircuital law. The French physicist and
mathematician Andrd Marie Ampere (1775-1836) was the first to discover a magnetic field between two
wires carrying electric currents (Ohanian, 1985). He summarized the relationship between the currents and
the static field as
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70
V xH = J
[3.17]
The J term on the right side o f equation 3.17 represents a current density due to m oving charges such as
electrons or ions that can produce a magnetic fie ld (Edminister, 1993; Shen and Kong, 1987). However, as
Faraday discovered in his experiments in 1837 (Edminister, 1993), certain materials experience a
“ displacement” o f charge from one side to another when they are under the influence o f an electromagnetic
field (Barthel and Buchner, 1992; Edminister, 1993). I f that electromagnetic field alternates w ith tim e, the
displacement o f charge also changes w ith tim e, giving rise to what has been termed a “ displacement
current.” As w ill be explained further in section 3.2.9, displacement currents exist when imperfect
conductors carry conduction currents that vary w ith time. (Hayt and Buck, 2001). M axwell amended
Ampere’s law by adding the displacement current term to the right side o f the equal sign, resulting in
equation 3.6. This change made the law more general and valid for all circumstances, including magnetic
fields that are not constant and media that are not perfect conductors (Edminister, 1993; Ohanian, 1985,;
Shen and Kong, 1987; von Hippel, 1954). Equation 3.6 predicts that i f D should change in tim e, it w ill
induce a change in the magnetic field. Since D is proportional to the electric field, (as w ill be described
later in this section) a changing electric fie ld can also induce a magnetic field (Ohanian, 1985). The other
interpretation is that, given the change o f magnetic field in space, and the sources o f current in that space,
one can predict the change o f D in time.
The two field equations, therefore, describe the interaction o f the electric and magnetic fields with
each other and how they change w ith space and time. Ohanian (1985) explains how electromagnetic
radiation is the result o f m utually inducing electric and magnetic fields:
If, in itia lly , there exists an oscillating electric field, it w ill induce a magnetic field, and
this w ill induce a new electric field, and this w ill induce a new magnetic field, etc. Thus,
these fields can perpetuate each other. O f course, an oscillating charge or current is
needed to get the fields started, but after this initiation the fields continue on their own.
These self-supporting oscillations are electrom agnetic waves, either traveling waves or
standing waves.
I f the in itia l cause o f the electric field is removed, the field w ill start to decrease at its in itia l
location. This decrease w ill initiate subsequent changes in the other magnetic fields. To generate and
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71
maintain electromagnetic waves oscillating at a specific frequency, a source o f electric charges must
oscillate at that frequency (Cumutte, 1980). As Table 3.1 indicates, the typical oscillating sources fo r
microwaves are klystrons and magnetrons.
Scalar equations 3.7 and 3.8 are called the “ divergence equations,” and are Gauss’ magnetic and
electric laws, respectively. The dot product o f the V operator and a vector is called the divergence o f that
vector. Assuming that B and D have components in
a ll three rectangular coordinates,
V • B = <3Bx/5x + 5By/9y + dBJdz
[3.18]
V • D = dDJdx +
[3.19]
d D y /d y
+ dDJdz
Shen and Kong (1987) explain the physical
significance o f divergence by describing the flo w o f
a vector through a cube as pictured in Figure 3.12.
Figure 3.12 Definition of divergence
(adapted from Shen and Kong, 1987).
For ease o f illustration, the vector B is considered to
have only a component in the x direction, but Shen
and Kong (1987) indicate that the results obtained for this simple case resemble those that would be
obtained i f the vector B had components in a ll three directions. In this sim plified case, V • B = 3Bx/3x.
From the figure and the definition o f partial differentiation,
„
3BX
3x
_
{B x} at right - {B x} at left
Ax
[3.20]
R
_
{B x} at right AyAz- {B x} at leftAyAz
volume
[3.21]
The first term in the numerator o f equation 3.21 represents the flow o f vector B leaving the cube in
Figure 3.12 on the right side through an area o f dimensions AyAz. Sim ilarly, the second term in the
numerator represents the flow o f vector B entering the cube on the le ft side through an area o f dimensions
AyAz. In other words, the divergence o f B is equal to the net flow o f the vector from the cube divided by
the volume o f the cube. The inherent assumptions in this derivation o f equation 3.21 are that the vector B
is constant, and that the volume o f the cube is infinitesim ally small, such that Ax, Ay, and Az a ll tend
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72
toward zero. Based on this discussion, Gauss’ magnetic law (equation 3.7) indicates that the net flo w o f the
magnetic flu x density through a closed surface is zero (Ohanian, 1985). There can be no accumulation o f
magnetic flux; whatever enters the surface must leave it.
Gauss’ Law for electricity (equation 3.8) is based on the law o f the French physicist Charles
Augustin de Coulomb (1736-1806) that can be used to calculate the force between two charged particles
(Ohanian, 1985). In its divergence format, presented in equation 3.8, this law states that the net electric
flu x density through a closed surface is equal to the electric charge density. Thus, a positive value fo r the
divergence o f the electric flu x density in a region indicates that there is a source o f electric flu x in that
region. Conversely, a negative value for the divergence o f the electric flu x density indicates that there is
sink fo r electric charge density (Edminister, 1993).
Equations 3.5 through 3.8 are known as M axw ell’ s equations because M axwell generalized the
validity o f existing electric and magnetic laws to any kind o f medium. He also “ added the missing lin k
between the magnetic and the electric fields [equation 3.6], and thereby brought electromagnetic theory to
completion” (Ohanian, 1985). Referring once to M axw ell’ s four simply-stated equations, Hertz wrote
(Serway, 1986):
One cannot escape the feeling that these mathematical formulas have an independent
existence and an intelligence o f their own, that they are wiser than we are, wiser than
their discoverers, that we get more out o f them than we put into them.
Indeed, history has proven that the im plications o f these equations have been more enduring and more
extensive than M axwell ever suspected, since they have laid the foundation for much o f today’ s modem
technology. M axw ell’ s theory is also in line w ith Einstein’s special theory o f relativity.
However, M axw ell’ s equations alone are not sufficient to solve for all their independent variables
when one is solving electromagnetic problems. Two additional equations are needed. These equations are
provided by the constitutive relations (equations 3.22 and 3.23) which give inform ation about the media in
which the electromagnetic fields occur:
D = e'E
[3.22]
B = pH
[3.23]
where:
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73
s' = real dielectric perm ittivity (farads/meter, abbreviated F/m)
p = real perm eability (henries/meter, abbreviated H/m)
Equation 3.22 states that the electric flu x is proportional to the electric field. The “ constant” o f
proportionality is s', the real dielectric perm ittivity. This quantity relates the interaction o f an object w ith
an electric field (Hewlett-Packard, 1992). However, as w ill be shown later, the p erm ittivity is not a true
constant since it depends on many factors, including the frequency o f the applied electric fie ld oscillation,
the temperature, the pressure, and the chemical composition, orientation, physical and molecular structure
o f the material (Hewlett-Packard, 1992; Nelson, 1981; Shen & Kong, 1987). In general, the perm ittivity o f
a material is higher that the perm ittivity o f a vacuum (also known as the perm ittivity o f free space), which
is denoted by the symbol s0, and has the value eD= 8.85 x 10' 12 F/m (rounded to two decimal places). The
ratio o f a material’ s perm ittivity to the perm ittivity o f free space is known as the relative dielectric constant,
or relative dielectric perm ittivity,e/. That is,
s / = e'/e„
[3.24]
Equation 3.23 states that the magnetic flu x density (magnetic induction) o f an object is
proportional to the external magnetic field applied to that object. This proportionality “ constant” is p, the
real permeability. Like s', p is a function o f many variables. However, a complete discussion o f p is
beyond the scope o f this paper. The focus here is the interaction o f microwaves w ith food components,
which, due to their negligible content o f metals such as iron, cobalt, and nickel, experience little or no
magnetic effects (Hewlett-Packard, 1992; IFT,1989). Thus, the permeability o f food materials is, in
general, equal to the permeability o f a vacuum, p = p0 = 1.26 x 10'6 H/m (rounded to two decimal places).
The versions o f M axw ell’ s equations listed in this section were given in the time domain. The
next section w ill present them from another perspective used frequently in modeling electromagnetic
problems - the frequency domain.
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74
3.2.2 Maxwell’s Equations in the Frequency Domain
The electromagnetic oscillations produced in a microwave oven have a periodic wave nature, as
described in section 3.1.2. The French mathematician Jean Fourier (1768-1830) demonstrated that any
periodic phenomenon can be mathematically modeled as the sum o f sine or cosine terms (Skoog and Leary,
1992). Thus, a general mathematical form ula to represent an electromagnetic wave as a complex function
o f time in polar form is:
w (t) = A[cos (cot + (p) + j sin (cot + <p)]
[3.25]
where
j = (—1) 1/2, the so-called “ imaginary” number;
cp = phase angle
(o = angular frequency in radians per second (abbreviated rad/s), calculated from frequency by the
equation:
co = 2 n f
[3.26]
Electromagnetic oscillations are completely governed by M axw ell’s equations (equations 3.5-3.8),
which elegantly and concisely summarize the time-dependent relationships between electric and magnetic
phenomena. However, the second-order differential equations that are derived from M axw ell’ s equations
are d iffic u lt to solve (Serway, 1986). In fact, Yee (1966) says, “ Solutions to the time-dependent M axw ell’s
equations in general form are unknown except fo r a few special cases.” Thus, scientists and engineers
studying oscillatory phenomena must use sim plified versions o f these equations. They customarily
transform problems like equation 3.25 from the “ time domain” to the “ frequency domain.” This
transformation (called the “ complex method” by Kreyszig, 1993) allows the sim plification o f differential
equations into a relatively easy-to-solve exponential form. The basis o f the transformation used in the
complex method is Euler’s identity:
e*6 = cos 0 + j sin 0
[3.27]
By this identity, equation 3.25 can be alternatively w ritten as
f(t) = Aei(“ t+'I,) = A e iv ei“ t
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[3.28]
75
Equation 3.25 represents a complex function in polar notation, and equation 3.28 is that same
complex function in exponential form. Any complex number w ritten in polar notation can be converted to
its complex counterpart in the Cartesian coordinate system, which has the general form:
w = x + jy
[3.29]
In equation 3.29, x is the “ real” part o f the complex number and y is the “ imaginary” part. The real
component o f equation 3.25 is
x = A cos((ot + <p).
[3.30]
An equivalent manner o f representing equation 3.30 would be to write equation 3.25 w ith the abbreviation
“ Re” for “ real,” as in
x = R e{A cos(cot+cp) + A j sin(cot+cp)}
[3.31]
Employing Euler’ s identity (equation 3.27), equation 3.31 becomes
x = Re{ A eiq>ei“ t }
[3.32]
The foregoing b rie f explanation o f how complex numbers can be readily converted from one form
to another illustrates some o f the steps o f the complex method used for sim plifying M axw ell’ s equations.
In brief, the complex method consists in finding a complex number o f the form A eiq>e*®* (as in equation
3.32) containing a real part that is the periodic function o f interest (given as A cos((ot+(|>) in equation 3.30).
Then the problem is solved in terms o f exponential expressions which are manipulated more easily than
cosine expressions. Once a solution is found, it can be converted back into the corresponding “ real”
solution in the tim e domain by taking the real part o f the complex solution. In equation 3.32, a ll the time
dependency is contained in the factor e*“ '. I f there is only a single frequency to consider in a particular
system, this term w ill be the same forall vectors, and therefore contains no extra information. Thus, all
needed information is contained in the factor A e’9, which is called a “ phasor.” This notation can befurther
sim plified by w riting A = A e’9 (the phasor is indicated by the different italicized font). For a vector, the
notation becomes A. It can be said, then, that two equivalent representations o f a time harmonic field are:
A cos(cot + cp) *-*■ A = A e*9
When such an equivalence is used to transform a differential equation, the problem is said to be
transformed from the time domain to the frequency domain. Figure 3.13 gives a three-dimensional
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[3.33]
76
Amplitude
(power) 1
Frequency domain
measurements
Time domain
measurements
Figure 3.13 Three-dimensional representation of relationship between time,
frequency, and amplitude (adapted from Agilent Technologies’ net seminar, 2000).
schematic representation o f the relationships between time, frequency, and amplitude. The signal in tim e
domain is a superposition o f three single-frequency signals. The frequency domain in this sim plified
representation shows only the magnitudes o f the three signals. In order to completely define the original
signal, the phase angle would also have to be specified.
I f it is assumed that the time-domain oscillations o f the electromagnetic fields, flu x densities, and
currents are cosine functions o f a single angular frequency © (Shen & Kong, 1987), the complex method
can be used to re-write M axw ell’s time-harmonic differential equations by replacing the time derivative
(9/9t) w ith the quantity j© . The end result is a set o f complex equations depending only on frequency, not
on time.
Upon performing the complex transformation on equations 3.5- 3.8, the follow ing frequencydomain versions o f M axwell’s equations (again using the notation in Shen and Kong, 1987) are obtained:
V x £ = - j© fl
[3.34]
V x H = j© D + J 0
[3.35]
V«S =0
[3.36]
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77
V • D = pv
[3.37]
The constitutive relations (equations 3.22-3.23) also can be transformed to complex functions:
D = e*E
[3.38]
B = p*H
[3.39]
where e* and p* are the complex perm ittivity and permeability, respectively. Equation 3.38 w ill be vita l to
this study, and the complex perm ittivity w ill be defined in section 3.2.5. Its relevance to the microwave
heating o f food w ill be described in much greater detail throughout the rest o f this chapter. On the other
hand, the complex permeability w ill not be discussed in greater detail since the permeability o f food
materials is (as mentioned in the previous section) generally equal to the permeability o f a vacuum, p0.
M axw ell’s equations govern the way electromagnetic radiation interacts w ith various materials.
The next section w ill discuss how certain materials interact w ith electromagnetic radiation and, based on
those interactions, which types o f materials are used in the construction o f the various parts o f a microwave
oven.
3.2.3 Materials in a Microwave: Conductors and Insulators
A ll matter interacts w ith electromagnetic radiation by absorbing, reflecting, or transmitting it, as
illustrated in Figure 3.14. The material property that accounts for the difference in behaviors o f materials
is in the presence o f an electric field is the dielectric perm ittivity. This quantity was seen in constitutive
equation 3.22 that related the electric field intensity E to the flu x density, D. Based on how materials
interact w ith the radiation, they can be classified into two general categories: conductors and insulators.
Reflected
transport electric charge (Serway, 1986), such as
Transmitted
electrolytic solutions or metals. In an electrolytic
solution, the charge carriers are ions drifting
Absorbed
through the solution (Atkins, 1990). In metallic
may be either reflected, transmitted, or
absorbed by a material, (adapted from
Engelder & Buffler, 1991).
conductors, electrons are the charge carriers. They
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78
move “ throughout a region o f immobile positive ions which form a crystalline array giving the conductor
its solid properties.” (Hayt and Buck, 2001). The atoms o f m etallic conductors are packed closely together,
resulting in a large number o f electrons and available energy levels. Several energy levels that are located
near each other are said to form a band. (Hayt and Buck, 2001). The electrons w ith the greatest amount o f
energy are located in the valence band, so they are given the name “ valence electrons.” They are also
frequently referred to as “ free” or “ conduction” electrons (Hayt and Buck, 2001; Serway, 1986). Just about
every atom in the metal donates a free electron (Edminister, 1993; von Hippel, 1954). These are the only
electrons that are free to move, and some travel w ith kinetic energies in the range o f several electron volts
(von Hippel, 1954). In the absence o f an electric field, the free electrons move in random directions,
bumping into the atoms o f the metal in much the same way as the molecules in a gas would bump into the
walls o f a vessel that held it. (Callister, 1991; Edminister, 1993; Hayt and Buck, 2001; Serway, 1986).
Since approximately the same number o f electrons travel in each direction during this random movement,
they cancel each other out; therefore there is no net flow o f charge, or, in other words, no current. (Serway,
1986).
The valence electrons behave somewhat differently in the presence o f an electric field. They can
Empty
conduction
band
A
Energy
Partially
filled
band
Energy gap
Pilled
valence
band
Filled
valence
band
(a) Conductor
(b) Insulator
Figure 3.15 Illustration of valence and conduction bands at 0 K:
a) conductors and b) insulators, (adapted from Hayt and Buck, 2001; and von
Hippel, 1954).
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79
absorb photons o f energy from the fie ld that w ill increase their own kinetic energy and “ boost” them into a
higher groupo f energy levels known as the conduction band (Hayt and Buck, 2001), illustrated in Figure
3.15a. This flow o f electrons (which moves in the direction opposite to the direction o f the electric fie ld ) is
a conduction current. The density o f this conduction current, J^nd, measured in units o f amps per square
meter (A/m 2), is proportional to the applied electric field, E, measured in units o f volts per meter (V /m ),
according to what is known as the point form o f Ohm’ s Law:
Jcond — ^chargesF
[3.40]
The proportionality constant, a Charges> is known as the conductivity o f the medium, which is
measured in siemens per meter (S/m) or mhos per meter (fi'V m ). The subscript “ charges” is used here to
indicate that this is a true conductivity due to m oving charges (as opposed to the one that w ill be discussed
in section 3.2.9). Conductivity is a function o f temperature and the properties o f the crystalline structure o f
the material. The conductivity takes into account the number o f conducting electrons per unit volume and
their m obility throughout the volume o f the material (Edminister, 1993; Hayt and Buck, 2001). The inverse
o f the conductivity is known as the resistivity,
resistivity = l/CTcharges
[3.41]
which can be roughly thought o f as a measure o f the d ifficu lty the electrons have in moving through the
metal due to their frequent collisions w ith atoms.
In the idealized case in which the electrons do not collide w ith atoms (i.e., the resistivity o f the
metal is zero, or, in other words, the conductivity o f the metal is infinite), the metal is called a “ perfect”
electric conductor (PEC). Such a thing does not tru ly exist, but a b rie f discussion o f the physics o f perfect
conductors from a theoretical perspective is useful fo r understanding this study. When an electromagnetic
wave impinges upon the surface o f a PEC, its oscillating electric field causes the electrons in the conductor
to oscillate accordingly. The direction o f oscillation o f the electrons is such that it tends to cancel the
electric field. (The electric field o f a PEC is completely cancelled and the wave does not penetrate). The
result o f the electron movement is an oscillating current that acts as a source for a reflected wave. The net
effect is that a wave is reflected w ith no loss o f power (Bloom field, 2001; Cumutte, 1980; Hayt and Buck,
2001; Ohanian, 1985; Shen & Kong, 1987).
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80
Perfect conductors do not exist, but very good ones do.
Some examples are gold, copper, and aluminum, and the values o f
their conductivities are listed in Table 3.2 (Inan and Inan, 1998;
Serway, 1986; Shen and Kong, 1987). In good conductors, the
electrons are not completely free to move. As a consequence, an
incident electromagnetic wave is not completely cancelled at the
Table 3.2 Conductivities at 20 °C
(data from Inan and Inan, 1998).
M aterial
C onductivity
S/m
aluminum
3.82 x 10'
gold
4 .1 0 x 1 0 '
copper (annealed)
5.80 x 10'
glass
~ 10' “
polystyrene
~ 10'16
porcelain
~ 10'14
surface, and penetrates a small depth into the conductor w ith a
rapid exponential decrease o f power. Most o f the wave, however, is reflected as in the PEC.
The electrons in many non-metallic materials cannot be excited into the conduction band by the
application o f an external field. Such materials are called insulators, and are defined as materials through
which electric charge does not readily flow (Ohanian, 1985; Serway, 1986). Figure 3.10b shows the reason
fo r this failure to carry electric charge: a large energy gap between the valence and conduction bands. The
electrons can only bridge this gap by absorbing photons o f a large amount o f energy, which they norm ally
cannot absorb from an applied electric field (Hayt and Buck, 2001). Instead, the insulators slightly reflect,
but prim arily transmit the electromagnetic waves, just as glass reflects some light, but transmits the rest
(Tinga, 1970). Insulators have extremely low conductivities, as shown by the values fo r glass, polystyrene,
and porcelain in Table 3.2. For this reason, these materials, as w ell as other ceramics1, plastics, and paper
can be used to hold food that is cooked in the microwave oven.
A sub-category o f insulators is dielectrics. Dielectrics are insulator materials w ith electrons that
are more confined than those in a conductor. (The exception is a dielectric material that contains ions. In
such a case, the dielectric does have some conductivity, although it is less than the conductivity o f metals.)
D ielectric materials either possess or experience a separation o f positive and negative charge. In some
cases, this separation is permanent due to the polar structure o f the molecules. In other cases, an applied
electric field causes the electrons o f the dielectric to shift slightly, causing a separation o f positive and
negative charges. These positive and negative charges (which can either be protons and electrons or partial
1 For a discussion about some ceramics that do indeed absorb microwave radiation, see Roy et al.,
1985, and Komareni and Roy, 1986.
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81
charges) cannot move freely enough to produce a flo w o f charge and conduct electricity (Isaacs, et al.,
1999), and they do not completely cancel the effect o f an applied electric field. Instead, they sim ply reduce
the strength o f the applied field, as the dielectric essentially stores some o f the fie ld ’ s energy (C allister,
1991; Hewlett-Packard, 1992; Ohanian, 1985). It is this storage ability that characterizes dielectrics (Hayt
and Buck, 2001). Perfect dielectrics release a ll their stored energy when the fie ld is released, but in reality,
perfect dielectrics, like perfect conductors, do not exist. What actually happens is that some o f the
dielectrics’ stored energy is lost as heat, so they cannot release all o f the energy when the electromagnetic
field is removed. This gives rise to the term “ lossy” dielectric. It is worth noting that the properties o f
materials are strongly dependent on frequency, so it is possible that a certain material can be “ lossy” at one
frequency, but not at a higher or lower frequency.
In order to give an
example o f how the
Power Supply
AC Power
different materials
described may be usefully
Power Generator
(Magnetron)
Waveguide
Power
Feed
System
Oven
Cavity
Figure 3.16 Basic block diagram of microwave oven power system
applied, the operation o f a
microwave oven w ill be
discussed. A complete
description o f this complex
device is beyond the scope
o f this work, but a sim plified description w ill be attempted. Figure 3.16 shows a schematic diagram o f a
microwave oven. The first block indicates the power supply (which is composed o f capacitors,
transformers, diodes, and rectifiers) that converts the 120 V alternating electrical current from a normal
w all outlet into high voltages that are delivered to a power generator as a direct current (Bloom field, 2001;
Decareau, 1992, Gerling, 1987). The power generator, which actually produces the microwaves, is usually
a vacuum tube called a magnetron. In the top and the bottom o f the magnetron, two permanent magnets
produce a magnetic field parallel to the central axis o f the magnetron. Inside the magnetron is a filament
surrounded by resonant cavities separated by metal walls arranged like the spokes o f a wheel (Bloom field,
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82
2001; McConnell, 1999). When the filam ent receives enough power from the power supply, it heats up and
emits electrons that are in itia lly attracted to the positively charged walls o f the resonant cavities. However,
the magnetic field present in the magnetron alters the flo w o f these electrons, causing them to curve in their
path. As more charges are added, they continue to curve in a rotating pattern, form ing a sort o f “ cloud” o f
electrons. When the cloud touches a w all o f one o f the resonant cavities, it becomes negatively charged
and repels the cloud to another positively-charged w all. The time it takes for this to occur corresponds to
the frequency o f the magnetron, either 2.54 GHz or 915 M Hz for a microwave oven. As the charges
resonate w ithin the magnetron, a small antenna attached to one o f the cavity walls radiates these
oscillations (microwaves), into the waveguide (Bloom field, 2001; Decareau, 1992; Gerling, 1987; Ohanian,
1985).
The waveguide is a hollow metal tube, often constructed o f copper (Ohanian, 1985), that
ultim ately channels the microwaves into the oven cavity. First, however, the microwaves, which behave as
traveling waves inside the waveguide, may pass through a power feed system which helps to achieve
relatively uniform heating in the oven cavity. Such a power feed system may have an apparatus called a
mode stirrer that rotates like a fan. When the microwaves h it the mode stirrer, they are reflected and
distributed more evenly into the oven cavity (Decareau, 1992).
Inside the oven cavity the microwaves form a complex pattern o f standing and traveling waves. I f
it is also assumed that the metallic cavity walls are perfect conductors, a wave w ill reflect when it hits the
interface between the perfect conductor and the air (which acts as an insulator). The microwaves reflecting
o ff the w all can then impinge the food placed in the microwave oven for heating. The food is usually
contained in a vessel made o f an insulator material such as ceramic, paper, or plastic, that does not exhibit
either a permanent or induced separation o f charges at microwave frequencies. These materials are used
because they transmit most o f the microwave energy, reflecting a small part, and absorbing even a smaller
part. Because food is composed prim arily o f the polar molecule water, it behaves as a lossy dielectric
material, w ith the process o f polarization playing an essential role in microwave heating. The next section
examines polarization in more detail, and the mechanism o f microwave heating w ill be discussed further in
section 3.2.11.
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83
3.2.4 Polarization and Dielectric Permittivity
It was mentioned in the previous section that dielectric materials reduce the intensity o f the net
electric field. This reduction is possible due to a phenomenon known as polarization. The purpose o f this
section is to b riefly define five different mechanisms o f polarization (orientational, atomic, electronic, ionic
and interfacial), as w ell as to mathematically illustrate the relationships among polarization, electric flu x
density, electric fie ld intensity, dielectric perm ittivity, and dielectric loss.
In the absence o f an electric field, the dipoles in polar molecules are randomly positioned (as
shown in Figure 3.17a) so that the net vector sum o f their dipole moments is zero (Daniel, 1967; Geyer,
1990,; Hayt and Buck, 2001; Hewlett-Packard, 1992). However, i f an electric field is applied, the dipoles
tend to align w ith the field, rotating as illustrated in Figure 3.17b. The angle o f rotation o f each dipole is
very small (Grant et al., 1978), and the alignment is only partial, but the end result is that the equilibrium
average dipole moment o f the entire system points in the direction o f the applied electric field. This
produces an opposing internal electric fie ld that is significant enough to reduce the electric field in the
volume occupied by the molecules (Grant, et al., 1978; Serway, 1986; Tinga, 19701). Such behavior is
known as orientation, or dipole polarization.
A p p lied Electrom agnetic Field
*4
«*X
Figure 3.17 Orientation of polar molecules: (a) random orientation in the absence of an electric
field; (b) orientational polarization under the influence of an applied electric field (adapted from
Atkins, 1990; Callister, 1985; and Mortimer, 1986).
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84
The time required fo r orientation polarization to occur depends largely on the temperature and
viscosity o f the system, as w ell as on the size o f the
polar molecules. These factors are influential
because the dipoles’ rotation is opposed by both
thermal and frictional forces. The natural thermal
(Brownian) motion o f the molecules opposes the
alignment process, driving them to assume random
positions in the medium. Since this thermal motion
increases w ith increasing temperature, at
for orientational polarization
(Data from Smyth, 1955)
Realignm ent C orresponding
Substance
Tim e
(seconds)
gas
10'12
infrared
small molecule
in low
10‘u or lO' 10
microwave
viscosity flu id
large molecule
or viscous
IQ-6
flu id
solid
orientation often not observable
sufficiently high temperatures the dipoles’ rotation
is so hindered that the orientational polarization is effectively zero. The frictional forces are related to the
size o f the molecule and the viscosity o f the medium. Large molecules in a viscous medium require more
time to align than small molecules in a gaseous medium. In other words, a higher force, or lower
frequency, is required to torque those large molecules. Table 3.3 lists the general tim e ranges and
corresponding frequencies o f radiation required to cause orientation polarization in the given substances.
Note that these values are just general trends, and that in actuality “ The high internal frictional resistance o f
very viscous liquids, glasses, and solids may lengthen the tim e required for the polarization process to
seconds, minutes, or longer, so that it may not make its e lf evident under the conditions o f observation”
(Smyth, 1955).
Orientational polarization can only occur for polar molecules that possess a permanent dipole
moment that can be torqued by an electric field. However, an applied electric field can induce polarization
even in nonpolar molecules by causing positively and negatively charged particles to “ shift in opposite
directions against their mutual attraction and produce a dipole which is aligned w ith the electric field”
(Hayt and Buck, 2001). This is known as induced polarization, which acts more quickly than and
independently o f orientational polarization.
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One type o f induced polarization is
atomic polarization. This occurs in molecules
(a)
(b)
Figure 3.18 Atomic Polarization
(a) Molecule with atoms that don’t share electrons equally
in the absence of an electric field;
(b) Atomic polarization under the influence of an electric
field (adapted from von Hippel, 1954, p. 96).
or solutions in which the different atoms do
not share the electrons equally (i.e., polar
molecules) (Geyer, 1990,; von H ippel, 1954).
An electric fie ld then displaces the charged
atoms or groups o f charged atoms from their
equilibrium positions, as illustrated in Figure 3.18 Geyer, 1990,; Smyth, 1955; von Hippel, 1954). This
usually requires 10"12- 10' 14 seconds to occur, corresponding to the infrared region o f the spectrum (Smyth
1955).
A second type o f induced polarization, electronic polarization (Figure 3.19), occurs when an
electric field induces a temporary dipole in an atom. The field produces a separation o f charge by pulling
the negatively charged electron cloud and the positively charged protons in opposite directions (Callister,
1991; Geyer, 1990; McConnell, 1999; Smyth, 1955; von Hippel, 1954). Because the electrons have such a
low mass they can move quickly, so the process o f electronic polarization requires only about 10"15
seconds. Thus, it can occur in a ll dielectrics subjected to an electromagnetic field oscillating at
approximately the frequency o f ultraviolet lig h t (Callister, 1991; McConnell, 1999; Smyth, 1955).
Another type o f polarization, known as ionic polarization (Figure 3.20), occurs only in ionic solid
materials (Callister, 1991; Smyth, 1955). The influence o f an electromagnetic field causes the positions o f
the positive and negative ions to shift (Callister, 1991; Smyth,
1955). Ionic polarization is usually larger than atomic
____
(
j
polarization, but it takes about the same amount o f tim e to occur
(approximately 10"12 seconds), so it also requires infrared radiation
(Smyth, 1955).
The final type o f polarization to be discussed here is
interfacial polarization, also known as Maxwell-W agner (Geyer,
1990) or space charge (von Hippel, 1954) polarization (Figure
\
------------
^
^^
Figure 3.19 Electronic Polarization
(a) Electron cloud symmetrically
surrounding the positive nucleus in the
absence of an electric field.
(b) Asymmetrical distortion of the
electron cloud in the presence of an
1991; and von Hippel, 1954).
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86
0
0
0
0
0
(a)
(b)
Figure 3.20 Ionic Polarization
(a) Ions in the absence of an electric field.
(b) Ionic polarization due to application of an electric field (Adapted from Callister, 1991).
3.21). It occurs in heterogeneous materials that contain m obile charge carriers and at least two phases w ith
different dielectric perm ittivities and conductivities (McConnell, 1999; Smyth, 1955,; von Hippel, 1954).
Under the influence o f an electric field, the charge carriers begin to move, but they are hindered in their
motion (e.g. by an interface). Thus, the charge carriers tend to collect at interfacial boundaries, and this
build-up o f charges polarizes the material (Geyer, 1990; McConnell, 1999; Smyth, 1955,; von Hippel,
1954). The frequency at which this polarization occurs depends on the speed o f the charge carriers (Smyth,
1955; von Hippel, 1954).
A ll o f these types o f polarization involve the separation o f opposite charges that results in either a
temporary or a permanent dipole moment.
Usually more than one type o f polarization
occurs in a dielectric material (Daniel, 1967;
Geyer, 1990). The total polarization o f a
material subjected to an electromagnetic field is
(a)
(b)
a vector (given the symbol P) that points in the
Figure 3.21 Interfacial Polarization
(a) Charge carriers at an interface in the absence of an
direction o f the applied field. The vector P
electric field.
(b) Charge carriers collecting at interface in the presence
of an electric field (Adapted from von Hippel, 1954).
actually represents the summation o f the
polarization o f a ll the molecules present in the
material (Callister, 1985; Edminister, 1993; von Hippel, 1954). It is defined as the dipole moment per unit
volume (Edminister, 1993; Grant et al., 1978; Hayt and B u ck, 2001; Kaatze, 1997), which is represented
mathematically as:
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87
P=
nAu
lim
(i/M r) I p dmi
Air
0
i= l
[3.42]
where
Air = volume
n = number o f dipoles per unit volume.
Von Hippel (1954) reduces equation 3.42 to an average dipole moment, which he then m ultiplies
by the number o f polar molecules involved. Needless to say, as it becomes necessary to mathematically
account for great numbers o f molecules in a material w ith different individual dipole moments,
computations can become extremely complicated. Much effort has been expended both experimentally as
w ell as theoretically in trying to relate the material properties o f the molecules in a dielectric to the
polarization vector, and subsequently to the local and total electric fields. Such a discussion is beyond the
scope o f this thesis, but interested readers can read more o f how the subject is treated in von H ippel (1954)
and Hayt and Buck (2001).
In a discussion o f polarization, the example o f a parallel plate capacitor is frequently used. In a
capacitor, the total charge that can be accumulated between the two plates can be increased by the
introduction o f a dielectric material (i.e., a material in which a dipole exists or can be induced). The
polarization acts as a mechanism by which the effective charge in the plates is decreased, or “ bound” . Von
Hippel (1954) accounted for the charge in the capacitor as a total charge, that is composed o f both the
bound charge and the “ free” charge (the effective charge). The electric flu x density D then corresponds to
the total charge, the product e0E corresponds to the free charge and P corresponds to the total charge. This
charge balance can be written as (von Hippel, 1954, Callister, 1991; Edminister, 1993)
D = e0E + P
[3.43]
Assuming that the dielectric material in question is isotropic, both the electric field vector and the
polarization vector are parallel and linearly related by the follow ing equation (Edminister, 1993; Hayt and
Buck, 2001).
P = XeSoE
where & is a quantity known as the electric susceptibility, defined by von Hippel (1954) as:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[3.44]
88
_
bound charge density
free charge density
^
^
Substituting equation 3.44 into equation 3.43 and factoring yields
D = s0E (1 + Xe)
[3.46]
The quantity in parentheses in equation 3.46 is actually the relative dielectric constant. That is,
£r = Xe+l
[3.47]
D = EoErE
[3.48]
such that equation 3.46 can be re-written as
which was expressed in section 3.2.1 as constitutive relationship 3.22. The beauty o f equation 3.22 is that
the effect o f polarization is completely taken into account by the real dielectric perm ittivity o f the dielectric
material, so it is not necessary to calculate the polarization and use equation 3.43 in order to determine the
flu x density.
This section has described how electric dipoles tend to align themselves w ith an applied electric
field. However, what happens when the field is removed or oscillates between positive and negative
values, as in the case o f a microwave oven? The next section w ill attempt to answer that question.
3.2.5 Dielectric relaxation
When an applied electric field causes orientational polarization by torquing the permanent dipoles
in a material, it reorders the molecules and enables them to increase their own potential energy by taking
some o f the energy from the field (see for example, Grant, 1978; Tinga, 1970; von H ippel,1954). This is
analogous to the increase in the potential energy o f a spring when it is compressed (Edminister, 1993; Hayt
and Buck, 2001). When the spring is released, a restoring force that follows Hooke’ s law enables the
spring to return to its equilibrium position. In the restoration process, the spring’s potential energy is
converted into kinetic energy. Sim ilarly, the restoring force on the dipoles is the thermal motion o f the
molecules. When the electromagnetic field ceases, this thermal motion acts to return the dipoles to their
former random, “ relaxed” orientations (as shown before in Figure 3.17a), and polarization decays
exponentially w ith time (Smyth, 1955). This process is known as dielectric relaxation. The relaxation
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89
time, given the symbol t, is defined as the time required fo r the polarization to be reduced to 1/e (where e is
the Euler number, so 1/e is approximately 36.8%) o f its maximum level (Callister, 1991, Decareau, 1992;
Hewlett-Packard, 1992; Pethig, 1992, Smyth, 1955; Tinga, 1970; von Hippel, 1954). The reciprocal o f the
relaxation time is the critical angular frequency. That is,
t
= 1/COcn, =
1/(2tc fcrit)
[3.49]
where tocrit is the critical angular frequency and fcrit is the critical frequency.
As they return to randomness, the dipoles decrease their potential energy by releasing the energy
that they had taken from the field. In some cases, a ll the energy can be reemitted as electromagnetic
energy. More often than not, however, only part o f the energy is reemitted in the same form it was
absorbed. The rest is converted into heat and dissipated as the dipoles move and jostle their neighbors
(Daniel, 1967; Tinga, 1970). The property that indicates how much o f the stored energy can be restored is
the dielectric perm ittivity, s', and the property that indicates how much energy is “ lost” through dissipation
as heat is the dielectric loss, e" (Engelder and B uffler, 1991; Nelson, 1981; Hewlett-Packard, 1992; Tinga
1970). Both the dielectric perm ittivity and the dielectric loss are influenced by such factors as the object’s
chemical composition, the temperature, the pressure, and the frequency o f the applied electric field
oscillation (Hewlett-Packard, 1992; Nelson, 1981; Shen & Kong, 1987), as w ill be discussed in greater
detail later.
When the applied electromagnetic field is time-harmonic, its intensity oscillates and its direction
fluctuates w ith time. The terms associated w ith the dielectric perm ittivity result in a sinusoidal wave w ith
constant amplitude, and the terms associated w ith the loss factor result in an exponential decay o f the
amplitude o f the wave. In the case o f a 2.45-GHz microwave oven, the field oscillates 2.45 b illio n times
each second. I f the material subjected to this field is contains polar molecules that are free to rotate (as in a
gas or a liquid), the permanent dipoles w ill continually tend to reorient themselves, alternating between
partially aligning w ith the field and relaxing to their random orientations when the fie ld ’s intensity cycles
to zero. The dipoles are able to follow a field oscillating at low frequencies because there is enough time in
each cycle o f oscillation for the dipoles to rotate and re-align. This leads to a significant contribution to
orientational polarization, which, in turn, leads to a higher value o f dielectric perm ittivity. However, the
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90
dipoles are unable to “ keep up” w ith an applied fie ld oscillating faster than what is known as their
relaxation frequency (Callister, 1991) because there is not enough tim e fo r the dipoles to rotate and
completely align before the direction o f the fie ld reverses. When this happens, equilibrium is not achieved,
the orientation polarization is out o f phase w ith the alternating electromagnetic field; its amplitude
dampens, decays exponentially to zero, and no longer contributes to the dielectric perm ittivity (Atkins,
1990; Barthel & Buchner, 1992; Callister, 1985; Pethig, 1992; Smyth 1955, Tinga, 1970). As a result, the
value o f the dielectric perm ittivity drops sharply as the frequency increases, a phenomenon known as
anomalous dispersion (Debye, 1929; Skoog and Leary, 1992; Smyth, 1955).
Ionic
Interfacial
O m o
o
Orientational
Atomic polarization
Electronic
polarization
polarization
Ionic
l.E+06
l.E+09
Frequency (Hz)
l.E+12
l.E+15
Permittivity - - - - Loss
Figure 3.22 Graph of dielectric permittivity and loss as (unctions of frequency, with dominating processes
indicated. (Adapted from Callister, 1991; Geyer, 1990; and Hewlett-Packard, 1992.)
Figure 3.22 shows the various types o f polarization that are likely to occur at certain frequency
ranges and contribute to energy absorption and storage in many materials. It also sketches the variation o f
both dielectric perm ittivity and dielectric loss w ith frequency. The anomalous dispersion described above
is evident in the middle o f the graph (at the frequencies corresponding to orientational polarization), where
there is a dramatic drop in perm ittivity that coincides w ith a peak in the loss curve. Both o f these are the
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91
signature features o f dielectric relaxation. The frequency at which the perm ittivity curve exhibits an
inflection point, and the loss exhibits a peak, is the critical frequency, fcrit, (Barthel and Buchner, 1992, p.
264). The orientational polarization is dampened and it eventually disappears as the frequency increases,
but then ionic, atomic, or electronic polarization may dominate. These three polarizations, however, do not
exhibit a relaxation process, but rather behave like a resonant system, “ the state o f a harmonic oscillator
when being driven at its preferred frequency” (Geyer, 1990), w ith a dampening effect. These responses are
shown schematically in Figure 3.22 for atomic and electronic polarization. The ionic polarization would
occur in the appropriate material at a sim ilar frequency as the atomic polarization, but it is not specifically
shown in the graph. When the orientation polarization ceases to be the dominating effect, the perm ittivity
reaches a plateau. The resonant effects show as a peak, follow ed by a sharp drop in perm ittivity, before its
value settles at another plateau slightly lower than the in itia l one. Electronic polarization is usually the last
polarization mechanism to be lost (Atkins, 1990). A t very low frequencies, another mechanism contributes
to polarization and causes the low-frequency end o f Figure 3.22 to have a high value o f perm ittivity and a
loss value that decreases as frequency increases. This mechanism is ionic conductivity (which w ill be
discussed in greater detail in sections 3.2.9 and 3.2.10).
I f the dielectric loss has a value o f zero, all the energy absorbed from the first h a lf o f the applied
electromagnetic fie ld ’ s oscillating cycle is recovered during the second h a lf o f the cycle, so there is no net
energy absorption (Grant, 1978). However, when the applied external electric field increases to such an
extent that polarization mechanisms cannot “ keep up,” there is a difference in phase angle (5) between E
and D (Grant, 1978). In this situation, the real dielectric perm ittivity in equation 3.22 is no longer
sufficient to characterize the system. Instead, the quantity known as the complex perm ittivity, e*, must be
used (Barthel & Buchner, 1992). The complex perm ittivity is the complex sum o f the “ real” dielectric
perm ittivity term and the “ imaginary” dielectric loss term, both o f which are functions o f the angular
frequency o f the applied field (Barthel and Buchner, 1992; Debye, 1913; Grant, 1978; Hayt and Buck,
2001; Smyth, 1955):
£ * (© ) = e '(© ) —je " ( to )
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[3.50 ]
92
Although they are frequently referred to as separate quantities, Von Hippel (1954) reminds us that e' and s "
.. are conjugate functions, and therefore not entirely independent o f each other. Physically speaking, the
mechanisms o f energy storage and energy dissipation are two aspects o f the same phenomenon; hence i f
one o f them is given over the whole frequency spectrum, the other one is prescribed.”
The ratio o f the loss factor to the perm ittivity represents the ratio o f energy lost to energy stored in
each cycle. It is the tangent o f the phase difference between D and E, known as the loss tangent (Geyer,
1990; Hewlett-Packard, 1992; Marsh and Wetton, 1995):
tan 5 =
loss current
charging current
[3.51]
The loss tangent can be represented on a vector diagram o f complex
perm ittivity. In such a diagram (see Figure 3.23), the “ real” component,
e', is drawn on the horizontal axis, and the “ imaginary” component, s",
is drawn on the vertical axis (Hewlett-Packard, 1992). The complex
perm ittivity its e lf is the vector sum, which forms the angle 8 w ith the
perm ittivity.
8
Figure. 3.23 Loss tangent diagram
What does this discussion o f events at the microscopic level
have to do w ith the cooking o f food in the microwave? The connection lies in the fact that most food
materials are generally considered to be “ lossy” at microwave frequencies. (Hewlett-Packard, n.d.; Shen &
Kong, 1987). A lossy material is one that has a high e " value, so it w ill absorb energy from a microwave
fie ld and heat quickly. Conversely, a material w ith a low e" value w ill transmit, rather than absorb, the
fie ld ’s power (Engelder and Buffler, 1991). For this reason, many researchers have focused on trying to
better understand a material’s dielectric properties and model them. One o f the first to do so was Peter
Debye, whose work w ill be briefly described in the next section.
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93
3.2.6 The Debye Model
The theory o f dielectric relaxation due to dipole orientation was developed by Peter Debye
(Debye, 1913, 1929). Debye modeled polar molecules as spheres rotating in a homogeneous, non-polar,
viscous liquid. Daniels (1967) further clarifies this model by pointing out several o f the assumptions upon
which it is based. Those assumptions include the sim plifications that dipole electrostatic interaction is
negligible, that no external forces influence the position o f the dipoles, and that the change in position and
orientation o f the dipoles is due only to thermal motion.
According to Debye’s model, the relaxation time for a polar molecule can be calculated according
to the equation
x = 47tr|a3/(kT )
[3.52]
where
a is the radius o f the sphere (m),
k is the Boltzmann constant, 1.38 x 10"23J/K,
T is the temperature (K ),
r| is the coefficient o f microscopic internal friction in poise (abbreviated p), although frequently
(and usually unsuccessfully, according to Smyth, 1955), the measured value for macroscopic viscosity is
used.
Debye also derived an equations to calculate the value o f the relative dielectric perm ittivity and
loss as functions o f the frequency. Although there are several different versions o f these equations (Marsh
and Wetton, 1995; Smyth, 1955), the ones presented below are in a format that is frequently used in the
literature:
Sr' = e=o + [(es- £«,)/( 1 + 002t 2)]
E r"
=
[(E s
-
Eco) oot ] / ( 1
+
co V )
[3.53]
[3.54]
These equations can also be combined into one for the relative complex perm ittivity:
Er* =E„ + (es- e „)/(l +j(OT)
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[3.55]
94
In equations 3.53-3.55, es is the static perm ittivity, which represents the value o f the relative
perm ittivity at low frequencies o f the electric fie ld when there is equilibrium between the dielectric and the
external field (Smyth, 1955). A t this point, the dipoles s till have sufficient tim e to “ keep up” w ith the
oscillations o f the field by rotating and re-aligning themselves. In some literature (C ollie, et al., 1947;
Debye, 1913; Hasted, et al., 1948), this is designated as s0 (which is easily confused w ith the optical
perm ittivity discussed below and/or w ith the perm ittivity o f a vacuum), so that notation w ill not be used
here.
The term e„ represents the infinite perm ittivity, which is also frequently called the optical
dielectric constant (Smyth, 1955); it refers to the value o f the perm ittivity at high frequencies. As Debye
(1913) explains, “ The symbol oo in ew is to be understood to mean that this quantity refers to oscillations for
which the effect o f the dipoles has vanished and not to oscillations which are in fact in fin ite ly fast.” The
infinite perm ittivity, therefore, only includes contributions from the molecules’ vibrations, from
intramolecular processes, and from induced polarization (Hasted, 1972).
The difference es - e«, is known as either the amplitude o f the relaxation process (P izzitutti and
Bruni, 2001) or the relaxation strength (Mashimo et al., 1987, 1992). Because the ss and emterms in
Debye’ s equations are properties o f the molecules in the sample being studied and are independent o f the
electromagnetic field, theoretically
their values should be predictable
70
based on molecular theories (Ellison,
60
50
etal.,1996). Often, however, the
40
predicted values vary significantly
20
from measured values, although the
model does hold rather w ell fo r water.
1.00E+06
1.00E+08
1.00E+10
1.00E+12
1.00E+14
1.00E+16
Figure 3.24 is sim ilar to
Frequency (Hz)
Permittivity
Loss
Figure 3.22, except that it only shows
dielectric relaxation, w ith its
Figure 3.24 Relative dielectric permittivity and loss curves showing
^s) fcrit. and E®.
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95
characteristic sigmoidal shape o f the perm ittivity curve (also known as the dispersion curve) and the peak
in the loss curve (also known as the absorption curve) (Marsh and Wetton, 1995). The plateau in the
perm ittivity curve at low frequencies represents the static perm ittivity. In itia lly the dielectric loss increases
w ith the frequency o f oscillation. However, when the frequency o f the fie ld increases and the dipoles’
rotation lags behind the oscillation o f the field, energy storage ( e ' ) decreases sharply while energy loss (s ")
increases. A t the critical frequency (fcnt), the loss is at a maximum and the perm ittivity curve has an
inflection point. The relaxation time can be calculated according to equation 3.49.
Eventually, as the frequency surpasses the relaxation frequency, the dipoles are less able to
respond to the changing field, so both e' and e ' ' decrease dramatically as orientation polarization is lost
(Hewlett-Packard, 1992). The perm ittivity curve reaches a plateau at the value o f the infinite perm ittivity.
The Debye equations can be rearranged to express the perm ittivity as a function o f the product o f
the frequency and the loss. The resulting equation is
s / = co£r" ( - x ) + e s
Thus, using experimental data to plot a graph o f e /
vs
[3.56]
cosr" can be a useful method fo r determining the
parameters in Debye’s model. Such a plot yields a slope o f -x and an intercept o f e s (Figure 3.25). Haynes
and Locke (1995) used this regression method in their study o f the perm ittivities o f wheat.
According to Ellison, et al. (1996), Debye’s model “ is essentially the only one which starts from
clearly defined hypotheses and deduces in a rigorous way a functional dependence” o f a material’s complex
perm ittivity on the properties o f the material and
on the frequency o f the applied electric field.
This model can be quite useful for media that
experience only one dielectric relaxation and
-T
negligible losses due to conduction. Such
CDS,
Figure 3.25 A plot of e' versus toe" yields a
slope of -t.
materials are known as Debye materials, and
many liquids fit this category (Ellison, et al.,
1996; Geyer, 1990). The model works w ell for
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96
water at certain temperatures and frequencies (as w ill be b riefly discussed in section 3.2.8), but under other
conditions, water’s behavior deviates from that predicted by the Debye equations. Daniel (1967)
acknowledges that this is because “ No exact theory o f the liq u id state has yet been developed, even fo r the
equilibrium properties o f simple liquids.” The reality is that most systems exhibit broad dispersion curves
and m ultiple relaxation peaks due to the com plexity o f the various polarization mechanisms (Geyer, 1990;
Grant, et al., 1978,; Marsh and Wetton, 1995). Daniel further (1967) summarizes the com plexity o f the
situation by saying:
The task o f dielectric theory is d iffic u lt not so much because permanent dipoles cannot
always be identified, but m ainly because they influence one another m utually; a dipole is
not only subject to the influence o f a fie ld but also has a field o f its own. The mutualness
o f the influence o f dipoles, permanent or otherwise, on one another makes the response o f
the assembly a cooperative phenomenon and causes it to depend on the size and shape o f
the assembly. The effective field acting on a dipole is in general not the externally
applied field E but is augmented by a contribution caused by cooperation.
For this reason, other empirical models to predict dielectric properties o f materials have been developed,
including the Cole-Cole and Hasted models that w ill be examined in greater detail in sections 3.2.7 and
3.2.10, respectively.
3.2.7 The Cole-Cole Model
Figures 3.22 and 3.24 are both graphs o f permittivity-loss-frequency data. Another type o f graph
is known as an Argand diagram, in which (like in the loss tangent diagram o f Figure 3.23) the imaginary
term (er" ) is plotted against the real term (e /). However, the difference is that an Argand diagram is based
on the fact that the Debye equations can be rearranged into
an equation for a circle (Cole and Cole, 1941; Daniel, 1967;
Smyth, 1955):
[er' - (es - s„)/2] ' + er" 2 = [(es - e„)/2]
[3.57]
When dealing w ith actual data, though, all values must be
positive, so the graph actually is a semicircle. Its center is
Figure 3.26 Argand diagram of
substance behaving according to
Cole and Cole, 1941).
located at (sM+ss)/2, and its intercepts at £«, and es on the
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97
real (e /) axis in Figure 3.26. Although © does not actually appear on the graph, each point on the curve
actually represents a single frequency, spanning the range from zero to infinity (Cole and Cole, 1941; Daniel,
1967). An Argand diagram like this serves as a convenient test to determine i f a system follow s the Debye
model. That is, i f the resulting graph is a semi-circle when measured values o f dielectric loss are plotted
against values o f perm ittivity measured at the same frequency, the system obeys Debye’ s model and has a
single relaxation time.
In 1941 Kenneth S. Cole and Robert H. Cole published a paper after constructing such graphs from
their data on various polar liquids (such as water, glycerine, various alcohols and glycols) and solids (such as
ice and organic crystals). These researchers found that the graphs were not perfect semicircles. Instead, they
all could be approximated by circular arcs, whose centers lay below the e' axis. Upon observing this, Cole and
Cole questioned the general validity o f Debye’s theory o f the complex dielectric constant for dispersion in all
media. They pointed out that Debye had originally derived his equations only fo r polar gases and polar liquids
in dilute solutions, but that it was frequently used for pure polar liquids. However, they realized that Debye’s
equations 3.53-3.55 did not adequately describe the frequency dependence o f the real and imaginary parts o f
the o f the dielectric constant for liquid and solid dielectrics. Thus, they sought to derive a more appropriate
equation to predict the dielectric constant by carefully examining the experimental data from several
researchers who had measured the dielectric constants for various materials.
Needing an indication o f the deviation o f the experimental results from the Debye predictions, Cole
„
er
and Cole examined the arc-shaped Argrand diagrams they had
it
produced from their data. They drew a radius from the point s0to
the center o f each o f the arcs which lay below the e/ axis, as in
e/
Figure 3.27. This radius defined an angle w ith the e ' axis, and the
>
measurement o f this angle in every case was a fractional m ultiple o f
Figure 3.27 Argand diagram o f
substance that does not obey
Cole and Cole, 1941).
90 , which they denoted as cx7i/2. From this they derived a new
expression for the complex perm ittivity that is similar to Debye’s,
but that contains the additional parameter a:
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98
S f*
£oo
[3.58]
[l+ y w i0^
a)]
The parameter a, which has an empirical value that can vary between 0 and 1, is a measure o f how broad
the relaxation peak is, which, in turn, indicates the range o f relaxation times (Cole and Cole, 1941; Grant, et
al., 1978; Hasted, 1972a; Smyth,1955; Von Hippel, 1954). Hasted (1972a) points out that the value o f a
does depend on temperature, and that “ Its interpretation is s till open to question.” Note that the Cole-Cole
equation in 3.58 reduces to the Debye expression 3.55 when a = 0.
Equation 3.58 can be separated into the real (s') and the imaginary (s ") parts o f the complex
perm ittivity, which are represented as:
(Ss-s«> )[1+(QTq)‘ ° (sin a7t/2)1
[1+2(oot0) (1 a) sin arc/2 +(cot0)2(I a ) ~
[3.59]
fe - So Xcotp) 1 a (cos an/2)
[1+2(( ot0) (1 a) (sin an/2) + (g)t0)2(1 a)]
[3.60]
Cole and Cole (1941) were able to experimentally determine the value o f a through a procedure
that involved drawing vectors in the complex plane o f their Argand diagrams, plotting the logarithm o f the
vectors’ ratio against frequency, and determining the slope o f the curve. A three-dimensional version that
%
Frequency (Hz)
80 io 15
Figure 3.28 Three-dimensional plot depicting e/ and e/' as functions of
frequency.
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99
depicts both e/ and e /' as functions o f frequency is shown in Figure 3.28.
The Cole brothers were quite successful in using their new equation w ith a to analytically
represent many o f the deviations from Debye’ s equations. Their empirical function can also be used
successfully to describe the dielectric behavior o f water, for which a is typically 0.013 (Ellison, et al.,
1996). More w ill be said about the dielectric properties o f water in the next section.
3.2.8 The Dielectric Behavior of Pure Water
Much has been written about the dielectric properties o f water, and some o f the basics are b riefly
discussed below. A fu ll discussion o f the topic is beyond the scope o f this text, but interested readers can
consult the bibliography for other possible sources o f more detailed information.
A dielectric spectrum showing general behavior o f the relative dielectric perm ittivity and loss for
water at various frequencies and temperatures is shown in Figure 3.29. This dielectric spectrum was
100
80°C
AAA
70°C
***
60°C
50°C
40°C
30°C
ooo
ffeq. range studied
915 and 2450 MHz
90
£r'
80
70
60
50
40
30
20
£r
10
105
106
107
108
109
10 ’ °
10"
1012
1013
frequency (Hz)
Figure 3.29 Dielectric spectrum of water showing orientational polarization.
Generated from Cole-Cole parameters given by Hasted (1972).
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100
generated w ith data from Hasted (1972b), who has reported parameters for the Cole-Cole equation for
water at different temperatures. Note that Figure 3.29 only shows the effect o f orientational polarization.
(The other polarization effects, even though they do occur in water, do not appear in this graph. For more
complete spectra, see Ellison et al., 1996). As Figure 3.29 indicates, the variation o f s / w ith frequency and
temperature produces a sigmoidal shape in these curves. A t low frequencies, water’ s static perm ittivity
decreases as temperature increases. A t about 1010 Hz, the perm ittivity curves fo r each temperature “ cross
over,” so the behavior reverses, showing an increase in relative dielectric perm ittivity as temperature
increases. This trend continues u ntil a frequency o f approximately 4 x 1011 Hz, when the relative
perm ittivity fo r a ll temperatures levels out to the value o f e*,. The nature o f the relationship between es and
temperature has been described in different forms, among them exponential decay and polynom ial, as
mentioned by Kaatze (1997) and Grant et al. (1978), respectively, in their reviews o f the available
literature.
Water’s relative dielectric loss, as shown in Figure 3.29, asymptotically approaches zero at both
very low and very high frequencies. In between those extremes, it shows a maximum peak, w ith a critical
frequency that shifts towards higher frequencies as temperature increases. That shift to higher frequencies
results in a an almost exponential decrease in relaxation times w ith increasing temperature at microwave
frequencies (Grant, et al., 1978; Hasted, 1972b). The height o f each peak is a function o f the amount (es &*,). Since es decreases significantly w ith temperature but £*, does not change much, their difference (and,
therefore, the height o f the peak) decreases w ith temperature.
Many researchers have examined water’s dielectric behavior at various frequencies, but at lim ited
temperatures, such as those between 20 and 30 °C. A t 25 °C, water’ s spectrum displays a principal
absorption band (indicated by a steep drop in perm ittivity and a peak in the loss value) centered around 1819 GHz that is considered to be due to the dielectric relaxation (Ellison, et al., 1996; Hasted, 1972b;
Kaatze, 1997). According to Hasted (1972b), this principal relaxation “ accounts fo r more than nine-tenths
o f the static dielectric constant.” In the high-frequency infrared range (30 x 1012 Hz - 150 x 1012 Hz), the
dielectric spectrum shows evidence o f atomic resonance processes occurring (Ellison, et al., 1996).
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101
However, Ellison et al. (1996) acknowledge that there is uncertainty over the meaning and origin o f other
absorption bands that appear in the middle o f the graph:
In the intermediate range, 100 GHz - 30 THz, doubt and confusion reign. There are
certainly some absorption bands. The precise number and their origin is s till subject to
controversy. It would seem reasonable that there are inter-molecular, molecular, and
atomic processes at work, but the lack o f experimental data makes it impossible to decide
which are the incorrect interpretations.
How closely the Debye equations predict the actual dielectric properties o f water depends largely
on the temperature and frequency range studied. Kaatze (1997) found that in the ambient temperature
range (20-30 °C), the Debye model was satisfactory fo r predicting the dielectric behavior o f water “ w ithin
the lim its o f experimental error” at frequencies below 300 GHz. A t higher frequencies, however, water’ s
behavior deviates from that predicted by the Debye equations. Ellison’s group (Ellison, et al., 1996) made
a more conservative assessment o f the effectiveness o f the Debye model than did Kaatze, stating that the
Debye model seemed to be appropriate only up to 100 GHz. In some cases, however, even that lim ited
frequency range s till seemed to be too broad, so they divided it up into smaller frequency intervals and
applied a separate Debye equation to each interval.
Grant et al. (1978) thought that the Debye model was “ a good first-order approximation,” but they
believed the Cole-Cole model offered a better representation o f the data for water. Hasted (1972b), using
his own data (see Collie et al., 1948) as w ell as the data o f some other authors, performed non-linear
regression to fit the Cole-Cole parameters for water at different temperatures. These parameters were used
in this study to produce Figure 3.29, and they were used in an algorithm (that w ill be discussed later) to
reduce systematic errors in the data.
Recognizing the complexity o f the modeling situation, Ellison et al. (1996) acknowledged that
much literature is “ devoted to the question o f whether the best fit to the experimental data is obtained by
one Debye function, a Cole-Cole function w ith ‘sm all’ a, or by a sum o f two or more Debye functions.
There is no clear loser in these minor skirmishes.” One other model that is often applied to electrolytic
solutions is the Hasted model. However, before that is discussed in section 3.2.10, the concept o f total
system frequency-dependent conductivity w ill be first addressed.
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102
3.2.9 Total System Frequency-Dependent Conductivity
As mentioned in section 3.2.3, in an electrolytic solution, the charge carriers are ions d riftin g
along in solution. As a result, ionic conductivity is directly dependent on the m obility o f the ions. In turn,
that m obility (just like the rotation o f the dipoles) is dependent upon both their size and the motion o f the
molecules in the solvent (Debye and Falkenhagen, 1928b). “ Size” does not necessarily refer only to an
ion’s actual physical radius, but rather the hydrodynamic radius. This quantity takes into account the fact
that an ion in an electrolytic solution is surrounded by solvent molecules and other ions that move w ith it as
a bigger sphere. This becomes particularly important fo r small ions which are solvated to a greater degree
than big ions (Atkins, 1990).
When analyzing electrolytic solutions that contain both positive and negative charges, it is
customary to focus on a quantity known as the current density, J, rather than focusing simply on the current
(Edminister, 1993). There are three types o f current densities that are relevant to this current study:
convection current density, conduction current density, and displacement current density (Hayt and Buck,
2001).
Convection current density can be thought o f as
“ a cloud o f charged particles in motion” passing a
surface, as depicted in Figure 3.30 (Edminister, 1993).
The product o f the cloud’s velocity and its charge density
Figure 3.30 Convection current density: the
product o f the charge density and velocity of the
cloud o f particles (adapted from Edminister,
id d ^
_ A, 2
1993).
o f A /m :
yalue ofthe convection current densi^ in units
J
Jconv — Pv^
[3.61]
where
pv = charge density (charge per unit volume measured in C/m3), and
v = d rift velocity (m/s)
Conduction current density, as its name implies, is found in conductors. It refers to the passing o f
charges (either ions in solution or electrons in metal) through a surface under the influence o f an
electromagnetic field, as depicted in Figure 3.31 (Edminister, 1993; Hayt and Buck, 2001). This is the type
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103
o f conductivity discussed in section 3.2.3. Just like
convection current density, conduction current density
may be defined as the product o f the charge density and
velocity o f the m oving charges. However, in this case,
the velocity o f the electrons is directly proportional to the
strength o f the electric field, and the proportionality
Figure 3.31 Conduction current
1993).
“ constant” is known as the m obility o f the charges,
^charges, w ith units o f m2/(Vs):
V
—p. charges f
[3.62]
Equation 3.62 can be used to relate the conduction charge density directly to the electric fie ld by employing
the conductivity term, crcharge, which is the product o f the m obility and the charge density. That is,
^charge — Pv
[3.63]
charge
Rearranging equation 3.63 to solve for pv, then substituting it and equation 3.62 into equation 3.61 for
current density results in the now -fam iliar point form o f Ohm’ s Law (equation 3.40) for conduction current
(Edminister, 1993; Hayt and Buck, 2001):
Jcond
^charges®*
The J term in Ampere’ s original law (equation 3.17) includes a conduction current symbolized by
Jcond, and a source current (which can actually either be a conduction or a convection current) symbolized
by J0 (Shen and Kong, 1987). That is,
j = Jo "*■Jcond
[3.64]
However, there is s till a third type o f current density, the displacement current density. M axwell
generalized Ampere’ s law by adding a term for the displacement current density(3D /3t) to the right side o f
equation 3.17, resulting in the version o f the law given in equation 3.6. Thisdisplacement current density
is found whenever an imperfect conductor carries a conduction current that changes w ith time (Hayt and
Buck, 2001). “ Imperfect conductors” includes dielectric materials, even though dielectrics are technically
classified as insulators. Perfect insulators have a conductivity value o f zero (Edminister, 1993)), but in
reality, most materials that are considered to be dielectrics have a conductivity greater than zero, which
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104
means they can carry a conduction current. D ielectric materials exposed to time-dependent
electromagnetic radiation also carry a displacement current because o f the polarization (Barthel and
Buchner, 1992). However, the displacement current density is not a true current and it cannot be measured
directly; only the sum o f it and the conduction current can be experimentally determined (Barthel and
Buchner, 1992; Hayt and Buck, 2001; Ohanian, 1985). Substituting equation 3.64 into equation 3.6 yields
V x H = J„
Jcond + dD/dt
[3.65]
By substituting equation 3.22 into equation 3.65 the displacement current density can be related to the
dielectric constant (Debye and Falkenhagen, 1928a)
V x H = J0 + Jcond + S/St (s'E )
[3.66]
Converting equation 3.66 from the time domain to the frequency domain w ith the factor ei<Dt, as described in
section 3.2.2
(Edminister, 1993;Shen and
Kong, 1987) yields:
V
X
H = Jo+ Jcond+ jcos'E
[3.67]
A t this point in the derivation it would be natural to substitute the point form o f Ohm’ s law
(equation 3.40) into equation 3.67 for the J COnd term. However, this is where dielectric measurements get
more complicated. In the words o f Geyer (1990):
It is often stated that it is artificia l to make distinctions between ohmic carrier transport
phenomena and dielectric loss characteristic o f a material when the material is placed in a
tim e-varying electric field. Actual dielectric measurements are indifferent to the
underlying physical processes.
In other words, a measured value o f the dielectric loss o f a material represents not only true ionic
conduction, but also other energy-consuming processes that are not fu lly understood. The uncertainty in
the latter has caused general confusion in the literature w ith regards to the conduction term in equation
3.67. Most writers have chosen to sim plify the problem by “ lumping” all energy-consuming processes
together. Von Hippel (1954) summarized this approach by stating, “ the conductance term need not stem
from a m igration o f charge carriers, but can represent any other energy-consuming process.”
The two energy-consuming processes addressed in this study are the true conductive loss
component and the dipole loss component, both o f which are functions o f frequency and temperature. The
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105
true conductive loss is that due to the movement o f the charge carriers in the system (as represented by the
previously-discussed conductivity, ocharge)> and the dipole loss is the amount o f energy that the system loses
during dipole relaxation (Grant, 1978; Hayt and Buck, 2001; Pethig, 1992; Smyth,; Swami and M udgett,
1981; von Hippel, 1954). The term that w ill be used to take into account both o f these loss mechanisms
throughout the rest o f this study is the “ total frequency-dependent” (Grant et al., 1978) conductivity term,
o, as opposed to acharge which is conductivity due to moving charges. The total frequency-dependent
conductivity includes “ a ll dissipative effects,” (Von Hippel, 1954), such as the frequency-independent
ohmic conductivity from moving charges as w ell as energy losses from the frequency-dependent relaxation
ofthe dipoles (Grant, 1978; Von Hippel, 1954). Mathematically speaking, the total frequency-dependent
conductivity as used in this study is written as the follow ing sum:
^
— ^charge
^ d ip o le
[3.68]
A t this point, the charge conductivity term in Ohm’ s law (equation 3.40) can be replaced by the
total frequency-dependent conductivity term. Then, assuming that the material under study is isotropic
(that is, it has the same electrical conductivity in a ll directions), the revised form o f Ohm’ s law can be
substituted for the conduction current density in equation 3.67 to obtain
V x H = J 0 + o E + jcos'E
[3.69]
Factoring out E and the “ imaginary” terms j© leaves
V x H = j© E (e' - ja /o ) + J0
[3.70]
(which is essentially a mathematical construct) that was present in the curl equation 3.6 has disappeared
completely from equation 3.70. Secondly, the term in parentheses in equation 3.70 is very sim ilar to the
definition o f complex perm ittivity in equation 3.50. A closer comparison o f the real and imaginary terms
yields a new expression to define dielectric loss:
e" = o/(o
or, for the more commonly used relative dielectric loss:
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[3.71]
106
£," = cr/(ct)£0)
[3.72]
It is worth keeping in mind that the dielectric loss given in equation 3.71, e", takes into account losses in a
medium related to both charge carriers and dipole processes (Geyer,1990).
3.2.10. The Hasted Model
One o f the first group o f researchers to acknowledge the effect o f ions on the dielectric properties
o f solutions was the team o f Hasted, Collie, and Ritson in Oxford, England. Their first objective (Collie,et al.,
1948) was to investigate the dielectric properties o f water and heavy water. Knowing that these properties were
very dependent on wavelength and temperature, and that there was no standard method for measuring them,
Collie, Hasted, and Ritson employed high-frequency techniques that had been perfected during W orld War II.
They used various resonators and waveguides, to obtain their data. Because o f all the work that was involved,
they were only able to determine the dielectric constant and relative dielectric loss o f water and heavy water at
3 different wavelengths: 10 cm, 3 cm, and 1.25 cm. However, they did this for a significant range o f
temperatures: 0-75°C for water, and 5-60°C for heavy water. They observed that both liquids exhibited
Debye behavior w ith a single relaxation time (Collie,et al., 1948).
Their second objective (Hasted, et. al., 1948) was to determine the dielectric constants and relative
dielectric loss o f aqueous ionic solutions at those same three wavelengths. They knew that the relative loss
values that they would measure would contain contributions from both the dielectric and ionic processes.
Wanting to determine the portion o f the total loss that was due only to dielectric processes, they employed a
correction factor that they could use to compensate for the loss contribution o f the ions. This correction in the
notation they used in Collie, et al. (1948) was:
£r apparent
^ 4jtOesu/(0
[3.73]
where
e /' apparent= the observed (measured) value o f the relative dielectric loss for the ionic solution;
e /' = the “ true” value o f the relative dielectric loss o f the solution due only to dielectric processes
excluding any contribution from the conductivity o f the ions;
oesu = the conductivity o f the ions in electrostatic units.
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107
Electrostatic units, von Hippel (1954) explains, were used in “ earlier times,” when the study o f
electromagnetism was first beginning. The concepts o f charge and current were new quantities that could
not be completely described by the standard units o f length, mass, and tim e that had previously been
sufficient fo r the study o f mechanics. Therefore, many scientists
Table 3.4 Factors used to convert
from mks units to
electrostatic units
(Source: von Hippel, 1954)
Ocsu
~ CTmlis * 9 x 1 0 ______________
£oesu
~ Co mks* 3671 X 1 0
Cr esu
Cr esu
mks * 3671 X 10
Sf mks
*
3671
X
chose to circumvent the problem by “ prescribing that either the
dielectric constant (electrostatic system) or the permeability
(electromagnetic system) be a plain number.” Fortunately, von
Hippel (1954, p.22) has published a table o f conversion factors that
10________
can be used to convert various quantities from our more customary
mks units (which now include units o f coulombs and amps to quantify charges and currents) to the
electrostatic units used by Hasted, Collie, and Ritson. The conversion factors pertinent to this study are
given in Table 3.4.
The two epsilon terms in equation 3.73 represent relative losses, meaning that they are the
quotients o f the actual loss values divided by by the value o f s0, analogous to equation 3.24. Therefore, the
conversion factors that we would apply to the numerator and the denominator o f those terms would actually
cancel out. However, the correction term in equation 3.73 does have to be addressed. In the electrostatic
system o f units, e0esu = 1, so it is understood to be in the correction term even though it is not written there.
To avoid some confusion, the correction term could be rewritten so that it contains e0esu and then converted
as follows:
4 n g esu
® Co esu
=
4710^3 ■9
X
109
0) Co mks * 36 71 X 10
=
g mks
„
tOG0 mks
Therefore, another way o f w riting equation 3.73 is:
Gr apparent —
Omks'( 0 ) £ 0 mks)
[3.75]
Once they had determined the relative dielectric loss, Hasted, et al. (1948) calculated the
wavelength corresponding to the relaxation time, and the static dielectric constant, both o f which are
parameters used in the Debye equations. Comparing these calculated parameters w ith those o f pure water
from their earlier work, they determined that the static dielectric constant o f water could be also corrected
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108
to compensate fo r the ions. Their resulting equation that predicts the depression o f the dielectric constant o f
water for electrolytic solutions with concentrations less than 2 M is:
Es solution
Es watcr^ 2 5c
[3.76]
where
sswater = the static dielectric constant o f water
c = the concentration o f the electrolyte (m ol/liter)
8 = the average hydration number, which represents the average number o f water molecules
attached to the dissolved ions. The value o f 8 is found from the equation
8 = (8+ + 8" )/2
[3.77]
where S+ and 8" are the hydration numbers for the positive and negative ions, respectively. These values
are listed in a table included in Hasted, et al. (1948). The value o f 8+ for the Na+ ion is -8, and the 8- value
for the C l' ion is -3,so according to equation 3.77, 8 fo r a sodium chloride solution is -5.5.In general, the
average hydration number for an electrolytic solution w ill be a negative number between -5 and -22.
The fact that the addition o f ions depressed the dielectric constant seemed to indicate that the ions
have the ability to hinder the rotation o f the solvent molecules. The researchers theorized that the water
molecules that each ion was able to attract to itself formed layers known as the hydration sheath. The formation
o f the hydration sheath led to a decrease in the number o f water molecules that could be polarized by the
electric field, resulting in the decrease o f the dielectric constant. The water molecules comprising the sheath
have dielectric properties that differ from those o f water molecules further away from the ion. Much later,
Hasted (1972a) further described the hydration sheath as
essentially a dielectric saturation phenomenon; the strong electric fields in the
neighbourhood o f the ions produce a non-linear polarization, which renders the local
water molecules ineffective as regards orientation in the applied field. It is possible to
make estimates o f the extent o f hydration, or “ hydration number,” o f water molecules
considered to be ‘bound irrotationally’ to the average ion; these estimates are in
reasonable agreement w ith hydration numbers estimated on the basis o f activity
coefficients, entropies, m obilities, and viscosities. The hydration number must be
distinguished from the number o f water molecules actually adjacent to the ion in the first
or second layers o f hydration (the hydration sheath); it does not follow that a ll o f these
molecules can be considered to be attached to the ion as it moves in the solution.
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109
There has been some confusion in the literature (see for example: Mudgett, 1974; Mudgett et al.,
1974a; Kudra et al., 1992) on whether the correction is to e / or to 8s o f water. However, close examination
o f Hasted et al. (1948) paper shows that the correction is actually to ss, the static dielectric constant o f
water, and Mudgett (1982; 1986) seems to agree w ith this observation. Thus, the term for the static
perm ittivity in equations 3.53-3.55 is replaced w ith the sum on the right side o f equation 3.76 to yield the
follow ing Debye-Hasted equations to predict the relative dielectric perm ittivity and loss for ionic solutions:
i
Sr ionic solution
tt
Sr tonic solution
_
— (^s water
#
_
Sr ionic solution
2 8 C—Sqq
i , „2 2
1+ CO T
Ss water
2
5c
J+ ^
(Ss water
Sqq water)
Sqqwater)
t , • „
1 + JC0X
^co water
.
+
[J./oJ
^charge
^
_i_
n
J^charge
water
COS0
7 0 *1
r« q a .
|y .o U J
where:
Sonwater= infmite-frequency (optical) dielectric constant o f water
sswater= relative static dielectric constant o f water
Both the dipole and the ionic components o f the loss factor in equation 3.79 vary w ith temperature
and frequency, albeit by different mechanisms, since the dipole loss depends on the rotation o f the water
molecules and the ionic loss depends on the m igration o f ions (Mudgett, 1986). The infinite dielectric
constant, however, does not vary much w ith temperature. Based on their measurements o f the dielectric
properties o f water, Hasted’ s team had determined that the value o f £«, is 5.5 ± 1 (C ollie et al., 1948).
Mudgett’ s group (Mudgett, 1974; Mudgett, et al., 1974a) used the Debye-Hasted model to predict
the dielectric constant and loss o f nonfat m ilk. The researchers measured both the dielectric properties and
the conductivities o f their m ilk solutions at frequencies o f 300, 1000, and 3000 MHz, and at tempertures o f
25°, 35°, 45°, and 55°C. Their predicted results were, in general, w ithin 5-15% o f the results from their
measurements, and the researchers were able to improve the agreement o f the results when corrections
were made for binding and exclusion effects. The group (Mudgett and Westphal, 1986) later used the
Debye-Hasted model to predict the dielectric properties o f an aqueous cation exchange resin, and Mudgett
claimed (Mudgett, 1986) that the model would also be useful for estimating the dielectric properties o f
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110
aqueous fluids such apple juice that have low content o f suspended solids. When Kudra et al. (1992) used
their version o f the Debye-Hasted model to predict the dielectric properties o f m ilk at 20 C and 2.45GHz,
they found the predictions differed “ substantially” from the data. When trying to account fo r the
discrepancies, Kudra’ s team referred to Mudgett’ s (Mudgett et al., 1974) need for correction factors to
improve the results o f the model, and they further suggested that perhaps a ll the components o f m ilk
together may have a synergistic effect on the dielectric properties o f the aqueous solution.
3.2.11 Heat Transfer in a Microwave Oven
In the previous section it was shown how electrolytes can change the dielectric constant and loss
o f an ionic solution from those o f pure water. Foods that contain high concentrations o f electrolytes (such
as salty foods) tend to have high dielectric loss values due to the “ loss” o f energy, not only through
dielectric rotation, but also through imperfect conduction. Such foods are therefore termed “ lossy”
(Decareau, 1992). Although the term may sound like it has a negative connotation, lossy foods are actually
desirable from the point o f view o f microwave processing because they heat quickly in the microwave oven
(Decareau, 1992). This section w ill focus on a basic explanation o f how microwaves heat food, although a
complete treatment o f the subject is beyond the scope o f this work.
3.2.1 l.a
The “Molecular Friction” Effect
Most foods are composed o f water and other polar materials (Grant, et al., 1978). When the food
is subjected to microwave radiation, the polar molecules tend to align themselves in response to the
alternating field, yet they barely begin to move one way before they are forced to change directions. The
process through which this oscillatory movement produces heat has been called “ molecular friction.” In
non-technical terms, as the molecules oscillate, they “ rub” against each other, thereby extracting kinetic
energy from the microwave field and converting it via “ friction” to thermal energy. This “ molecular
friction” effect has been thought to be the primary mechanism by which food is heated in the microwave
oven (Bloom field, 2001; Cumutte, 1980; Decareau, 1992; Goebel, et al., 1984; Mudgett, 1986; Zallie,
1988). Bloom field (1997-2000) offers an analogy to illustrate the concept o f molecular friction:
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Ill
Visualize a boat riding on a passing wave—the boat begins bobbing up and down as the
wave arrives but it stops bobbing as the wave departs. Overall, the boat doesn't absorb
any energy from the wave. However, i f the boat rubs against a dock as it bobs up and
down, it w ill converts [sic.] some o f the wave's energy into thermal energy and the wave
w ill have permanently transferred some o f its energy to the boat and dock.
Two researchers, Vala and Szcezepanski (1997), wanted to know “ whether the application o f
microwave energy to a sample merely heats it therm ally” or i f there was another mechanism o f heat
transfer. They sought to answer the question, “ Is there a microwave effect?” In order to find out, they used
infrared spectroscopy to investigate the behavior o f gaseous water at the molecular level. Infrared
spectroscopy is an analytical technique that involves exposing a sample to various wavelengths o f infrared
radiation and then observing which wavelengths are absorbed and which are transmitted (M cM urray,
1988). The absorption spectrum is usually plotted on a graph o f wavelength versus the amount o f radiation
transmitted. Since the absorption o f infrared radiation can result in a change in the vibrational energy
levels o f a molecule, the locations o f the lines on an infrared spectrum indicate the types o f bonds present
in a sample (McMurray, 1988; Skoog and Leary, 1992). The intensities o f the lines on the spectrum
indicate the number o f molecules in the various vibrational and/or rotational energy levels (Atkins, 1990).
Because they possess several rotational states for each vibrational energy level, gases display closely
spaced lines on their infrared spectra. In contrast, liquids and solids, which have restricted rotation, do not
show discrete lines. Instead they display rather broad peaks from their vibrations (Skoog and Leary, 1992).
The variety o f qualitative and quantitative inform ation that infrared spectroscopy reveals makes it a
valuable tool for analyzing and identifying the ingredients in unknown substances.
Since the water molecules that are the main component in foods are polar and possess a permanent
dipole moment, they can undergo both rotational and vibrational transitions when they are excited by
photons o f energy. In addition, since the water molecules tend to form hydrogen bonds w ith each other,
any given volume o f liquid water w ill have a tremendous number o f allowable energy levels, some o f
which w ill correspond to microwave frequencies (Cumutte, 1980; Vala and Szczepanski, 1997). However,
the molecules do not remain in the excited state for long. An electronically excited species lasts for only
about 10"8 to 10‘9 seconds, and a vibrationally excited state lasts for about 10‘ 12 seconds or less (Skoog and
Leary, 1992). There are several relaxation mechanisms by which these species can return to their ground
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112
states. Electronically excited species frequently em it their excess excitation energy as heat, as represented
by expression 3.80:
M * - * M + heat
[3.81a]
However, usually the amount o f heat released through such a relaxation is “ not detectable” (Skoog and
Leary, 1992). Another mechanism, known as radiative decay or relaxation, occurs when the molecule
sim ply emits a photon o f energy, thereby returning its e lf to a lower energy state:
M * -> M + Ephoton
[3.81b]
In the case given by expression 3.81, the temperature o f the system does not change. A third mechanism is
nonradiative decay or relaxation. This can be a several step process that converts the internal excitation
energy o f an individual molecule into the vibrational, rotational, and translational energy o f neighboring
molecules. During this redistribution o f energy, one molecule exerts forces on and collides w ith its
neighbors. Kinetic energy is exchanged during the collisions, and the temperature o f the system is slightly
increased (Atkins, 1990; Cumutte, 1980; Skoog and Leary, 1992). Because this process increases the
entropy o f the system, it is irreversible. In other words, the electromagnetic energy obtained from the field
is not “ reradiated” (Cumutte, 1980). This frequently occurs w ith molecules in solution (Skoog and Leary,
1992).
When Vala and Szcezepanski (1997) compared the infrared spectra o f water vapor heated
therm ally w ith that o f water vapor subjected to microwave radiation, they did find a difference, and
declared that “ the microwave energy is not simply heating the water vapor.” What they defined as “ the
microwave effect” was the change in the water molecules’ angular momentum due to the transitions that
the microwaves induced between rotational levels in the ground vibrational ground state. In addition, these
authors believed that collisions occurred among molecules, especially at higher pressures, when the water
vapor would condense into liquid water. This would result in an exchange o f energy w ith neighboring
molecules that would, in the authors’ words, be “ closely equivalent to heating water therm ally.”
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113
3 .2 .1 1 .b
The Role of Ionic Conduction in Microwave Heating
Ions also play an important role in microwave heating through the mechanism o f ionic conduction
(B uffler and Sanford, 1991; Decareau, 1992; IFT, 1989). Since the ions dissolved in solution are charged,
they too are influenced by an alternating electromagnetic field. Like the dipoles, they firs t move in one
direction, then in the other as the fie ld oscillates. In the process o f moving, they collide and disrupt
hydrogen bonds in water, liberating energy in the form o f heat (B uffler and Sanford, 1991; IFT, 1989).
The mechanism fo r this “ liberation o f energy” is sim ilar to that o f dipolar rotation.
3.2.11.C
Diffusion of Heat
Once the energy from the electromagnetic radiation has been converted into thermal energy and
transferred to the bulk o f the food, it diffuses. Arpaci (1991) defines diffusion as the mechanism by which
“ heat is transferred through a medium or from one to another o f two media in contact, i f there exists a
nonuniform temperature distribution in the medium or between the two media.” One mechanism o f
diffusion is known as conduction. In this case, the heat transfer occurs as an exchange o f kinetic energy
when energetic molecules in a high-temperature region o f a medium collide w ith less energetic molecules
in a lower-temperature region o f a medium (Arpaci, 1991; Serway, 1986). A second method o f diffusion is
convection, which is defined as the “ motion o f the medium which facilitates heat transfer” (Arpaci, 1991).
Convection is especially influential fo r the heat transfer in liquids that have been heated by microwaves
(Anantheswaran and Liu, 1994).
B uffler and Stanford (1991) point out that the material properties o f the food being studied affect
the rates at which the process o f conduction and convection occur. One such property is the thermal
conductivity, which Callister (1991) defines as “ the proportionality constant between the heat flu x and the
temperature gradient” fo r equations involving steady-state heat flow . A second important property is heat
capacity, the amount o f heat required to produce a unit temperature increase in a unit o f mass o f a
substance. Viscosity and density also come into play because they affect how freely the heat w ill diffuse
through a food system (Anantheswaran and Liu, 1994; Stanford & Buffler, 1991).
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114
3.2.12 Penetration Depth of Electromagnetic Radiation
As mentioned in section 3.2.3, when an electromagnetic wave encounters an interface between
two materials w ith different dielectric properties (such as air and food), some radiation w ill be reflected,
while the rest is transmitted into the second material (Barringer, et al., 1995; Skoog and Leary, 1992). This
section focuses on how far microwave radiation penetrates into the food. The mathematical results for
homogeneous, isotropic dielectric materials are presented here, but the complete derivation o f the results is
in Appendix A.
The second order differential equation for
the vector E that results from M axw ell’ s equations is
V2E + ro2po e* £ = 0
[3.82]
(A sim ilar equation can be derived for H; see
Appendix A .) The solution to equation 3.82 for the
sim plified case o f an £ fie ld that is a function only o f
z and is parallel to the x-axis (as shown schematically
Figure 3.32 Electric field vector as a function
of z, parallel to the x-axis (adapted from Hayt
and Buck, 2001).
in Figure 3.32) is:
E = x£x
[3.83a]
£ = x E 0e'jk*z
[3.83b]
where
Ex is the magnitude o f the electric field in the x-direction,
E0 is the intial amplitude o f the wave at z = 0 and t = 0 (Hayt and Buck, 2001), and
k * is the complex propagation constant (which really is not constant), defined by Shen and Kong
(1987) as:
k* = co(p0e*)'/j
[3.84]
k* = kR- jk i
[3.85]
where kR and ki represent the real and imaginary parts, respectively, o f the complex propagation constant.
I f the right-hand-side o f equation 3.85 is substituted into equation 3.83b, and the expression is converted
back to the time domain, the resulting “ real” part is
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115
Ex (z,t) = E0 e‘kiz cos (cot - kRz)
[3.86]
The negative sign in front o f lq (w hich has a positive value in its usage here) in equation 3.86
indicates that the wave w ill decay exponentially as it penetrates deeper (w ith increasing z) into a m aterial.
For this reason, lq is known as the attenuation factor (Hayt and Buck, 2001). The original amplitude o f the
wave at the surface o f the material (at z = 0) is E0. When the wave penetrates a distance z into the m aterial
such that the amplitude o f the wave has decayed to 1/e (approximately 36.8%) o f its original E0 value, it is
said that the wave has reached its penetration depth, dp, defined as (Shen & Kong, 1987, p. 53):
dp= 1/k,
[3.87]
In order to calculate the penetration depth, kr must first be determined. Substituting equation 3.50
into equation 3.84 yields:
k * = ffl[p 0(s' —je " ) ]1/2
[3.88]
Factoring out e' leaves
, r
c” t
1/2
k * = o)(p0s') 12 [
[3.89]
Substituting in equation 3.71 yields the equation for k * in the notation o f Shen and Kong (1987)
[3.90]
Equations 3.89 and 3.90 are both used in the literature. B y comparing the coefficients o f the j terms in
both, and keeping in mind the definition o f relative dielectric property and equation 3.51, the follow ing
terms can be equated:
[3.91]
Equation 3.91 is important to keep in mind when reading the literature because its terms are frequently used
interchangeably, according to various authors.
A fter some manipulations (described in Appendix A ), the imaginary part o f equation 3.90 is found
to be
k! = ra(n0s ')1/2{ l + [CT/(co£')]2} 1/4{sin (14 tan 1 [c/(cos')]}
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[3.92]
116
Equation 3.92 is the general equation fo r kx. Its reciprocal (according to equation 3.87) is the penetration
depth.
(o(p0e') 1/2 {1 + [a/(cos')]2} 1/4{sin ('A tan 1 [cr/(ooe')]}
It is clear from equation 3.93 that the penetration depth depends not only on the perm ittivity, but
also on the conductivity o f the material, w ith that conductivity (as described in section 3.2.9 and defined in
equation 3.68) including both ionic and dipole conductivity. The penetration depth is also very much
frequency-dependent. Generally speaking, power penetration in a food is about 2-3 times greater in a
microwave operating at 915 M Hz than in a microwave operating at 2,450 M Hz (IFT, 1989). Frequently,
different notations and versions o f equation 3.93 are used in the literature (Hayt and Buck, 2001; Mudgett,
1982, 1986; Umbach, et al., 1992; von Hippel, 1954). Appendix B shows that these other versions are in
fact equivalent to equation 3.93. This topic o f penetration depth w ill come up again in the next section w ith
a discussion o f the results o f some researchers who investigated the effect o f salt on penetration depth.
3.2.13 How Dielectric Properties Are Measured in Literature
There are several different methods for measuring dielectric properties reported in the literature,
and they can be loosely classified according to the type o f cell or holder used to hold the sample while its
properties are being measured, the signal that is measured, and the domain (frequency or tim e) in which
measurements are made. Cell types and holders include various types o f coaxial probes, transmission wave
guides or coaxial transmission lines, resonant cavities, free-standing samples between antennas, and
parallel plates. The signal that is measured can be either the signal that is transmitted or reflected after the
sample is impinged w ith an incident test wave. The domain in which the measurements are made has
traditionally been the frequency domain (Grant, 1978), but making measurements in the tim e domain has
become increasingly popular. The characteristics o f available methods often overlap these classifications,
and researchers choose a method based on such factors as accuracy, cost, convenience, frequency range
desired, and the geometry and physical state o f the substance to be tested. Below is a b rie f summary o f
some ofth e most frequently used techniques for measuring dielectric properties.
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117
Reflection
~~N
f
s *r
►
Reflection
liquids
(a)
solids
(b)
(c)
(d)
Figure 3.33 Coaxial Probe Method, (a) Probe used with liquids; (b) probe used with solids; (c) fields and reflection
at the probe surface; (d) computer that analyzes data and calculates the complex permittivity (adapted from Hewlett
Packard, 1992).
When the signal reflected from the test sample is to be monitored, an open-ended coaxial probe is
often used to transmit the waves from their source. The probe is a cut-off section o f coaxial transmission
line that can be used to measure the dielectric properties o f both liquids and solids, as illustrated in Figure
3.33a and b (Engelder and Buffler, 199; Hewlett-Packard, 1992; Hewlett-Packard, n.d). Figure 3.33c
shows the immersed end o f the probe w ith schematic curving lines represent the changing electromagnetic
fields. A detector measures the reflected signal, and computer w ith special software converts the
measurement data into the relative complex perm ittivity e*r, as shown in Figure 3.33d. The coaxial probe
method is popular due to its convenience, its good experimental repeatability, its wide frequency range, and
its non-destructiveness, although one source o f error can be air space between the probe and a solid sample
surface (Engelder & B uffler, 1991; Haynes and Locke, 1995; Hewlett-Packard, n.d.). Some researchers
who have used a dielectric probe include Barringer et al., 1995; Boughriet et al., 1999; Haynes and Locke,
1995; Ndife et. al, 1998; Umbach et al., 1992; Zheng et al. 1998.
I f signal transmission is to be monitored, a transmission line, rather than a coaxial probe, is used.
The transmission line (shown in Figure 3.34a) is usually a section o f a waveguide or coaxial line that
actually contains the material that is to be tested (Engelder & Buffler, 1991; Hewlett-Packard, 1992). The
method is based on the principle that a dielectric object inside the line changes the propagation o f the wave
in the line, and it can be used at any frequency for which the proper equipment is obtainable (ASTM,
1976). The detector used in this technique measures not only the reflection from the material being tested,
but also the transmission through the material. The computer and software then use this reflection and
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118
Transmission line with
material to be tested inside
Transmission
Reflection
/■
s*.
Transmission
f
\
Reflection
(a)
(b)
Figure 3.34 Transmission line method, (a) Transmission line with material to be tested inside; (b) Computer that
analyzes data and calculates the complex permittivity (adapted from Hewlett-Packard, 1992).
transmission data to calculate the relative complex perm ittivity, as illustrated in Figure 3.34b (Engelder &
Buffler, 1991,; Hewlett-Packard, 1992). This method is “ more accurate and sensitive” (Engelder &
B uffler, 1991) than the coaxial probe technique, but it does have some disadvantages. One is that the
method can only be used w ith solid, isotropic samples. A second disadvantage is that each sample must be
carefully cut to specific tolerances so that it perfectly fits in the cross-section o f the transmission line.
Otherwise, air gaps around the sample w ill introduce measurement errors (ASTM , 1976; Engelder &
Buffler, 1991; Hewlett-Packard, 1992).
Two other methods for measuring dielectric properties use a resonant cavity. A resonant cavity is
defined by ASTM (1976) as “ an enclosure w ith conducting walls which w ill support electromagnetic
resonance o f various specific modes dependent on the cavity geometry and dimensions, and on the integral
number o f h a lf waves and their directions o f propagation as terminated by the cavity walls.” In practice, a
resonant cavity can simply be a section o f waveguide that has been closed on both ends by m etallic plates
(ASTM , 1976; HP, n.d.). Constructive interference from reflected waves creates standing waves in the
cavity, causing it to resonate at its own frequency (McConnell, 1999). Each cavity can also be classified by
its “ quality,” or “ Q” factor, which is the inverse o f tan 8 given in equation 3.51 (ASTM , 1976; HewlettPackard, n.d.; Hewlett-Packard, 1992). When a dielectric material is placed inside the cavity, both the
cavity’ s frequency and Q are lowered (ASTM , 1976). These changes are used as the basis for two
measurement methods that w ill be described below: the resonant cavity perturbation method and the
resonanat cavity method for specimen o f reproducible geometric shape (ASTM, 1976). Some researchers
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119
that have used a resonant cavity in their w ork include (Bengtsson and Risman, 1971; Johri et al, 1991;
Kudra et al., 1992; M ille r et al., 1991; Risman and Bengtsson, 1971; Ryynanen et al., 1996; Tong et al,
1994; Tsoubeli et al., 1995.
The resonant cavity perturbation method can only be used w ith small, uniform ly shaped,
homogeneous samples that are “ placed symmetrically in a region o f maximum electric fie ld ” (ASTM ,
1976) inside the cavity. The complex dielectric perm ittivity is calculated based on measurements o f the
cavity’ s frequency and Q factor before and after the dielectric sample is introduced, the dimensions o f the
cavity, and the dimensions o f the sample (ASTM , 1976; HP, 1992; HP, n.d.). Two advantages o f this
method are its accuracy, sensitivity to low dielectric losses, and high-temperature tolerance (ASTM , 1976;
Bengtsson and Risman, 1971; Engelder and B uffler, 1991; Hewlett-Packard, n.d.; Risman and Bengtsson,
1971; and Tong, et al., 1994). It has an advantage over the transmission line method in that the sample to
be tested does not have to be precision-cut to fit inside the apparatus, although its dimensions do need to be
accurately measured (ASTM , 1976). A disadvantage is that the frequency at which the complex dielectric
perm ittivity can be determined is lim ited to the resonant frequencies o f the cavities (ASTM , 1976).
The resonant cavity method for specimen o f reproducible geometric shape can test larger samples
that takes up more volume inside the cavity. However, before measurements o f the test sample are made,
standard samples o f known perm ittivity are first used to calibrate the cavity. This calibration procedure
results in a curve or a mathematical expression relating the perm ittivity o f a material to resonant frequency.
As the name o f this method implies, the sample to be tested must be o f the exact same size and shape as the
calibrating standards, and it must be placed in the same location inside the cavity. The resonant frequency
o f the cavity containing the sample is then measured and the calibration curve or expression is used to find
the corresponding value o f the sample’s real perm ittivity. This information, along w ith the observed Q
factors, is then used to calculate the sample’ s dielectric loss factor. The main advantage o f this method is
that it can be used for large samples o f any shape, as long as standards o f the same shape can be found. In
addition, “ Because the cavity dimensions do not contribute to the calculation o f results, considerable
freedom is allowed on tolerance o f dimensions and geometry o f the resonant cavity.” The main
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disadvantage o f this method is that standard substances w ith well-known dielectric properties can be
d iffic u lt to find.
The free-space method uses two antennas to m onitor dielectric properties. This method is useful
for large, fla t materials or materials that are non-uniform, but it is not usually used for testing the dielectric
properties o f food. The material to be tested is sim ply placed between two antennae (Hewlett-Packard,
n.d.). One antenna directs microwave radiation at the material to be tested, and the other returns the
transmitted signal to an instrument that analyzes it. This data can then be used to calculate the relative
perm ittivity. The free-space technique has the advantages that it is non-destructive and can be used w ith
materials at high-temperatures. Its disadvantages include difficulties in measuring the perm ittivities o f
low-loss materials and errors resulting from sample size (Hewlett-Packard, n.d.).
One instrument that is frequently used in conjunction w ith a ll o f the afore-mentioned techniques
(usually in the radio and/or microwave range o f frequencies) is the network analyzer (Hewlett-Packard,
n.d.). The network analyzer not only
Incident wave Reflected wave
detector
detector
Source
Transmitted
wave detector
(^ )
generates the test wave that impinges the
test material, but also detects the resultant
Test
Material
reflected and/or transmitted waves. The
6
Figure 3.35 Network Analyzer Block Diagram (adapted
from HP Electronic Materials Measurement Seminar and
Application Note 1217-1).
instrument is usually connected to a
computer loaded w ith special software,
and the data it collects can be converted
properties. Figure 3.35 shows a basic block diagram o f a network analyzer. The source generates a
microwave signal at the frequency o f interest which is then transmitted to the material being tested. As the
wave interacts w ith the material, its signal changes, as some o f it is reflected and some o f it is transmitted.
The receiver detects the difference between the in itia l incident signal and the reflected/transmitted signals.
The software on the computer uses the transmission and/or reflection data, as w ell as information about the
physical geometry o f the material being tested, to determine the material’ s relative complex perm ittivity as
a function o f frequency (Engelder & Buffler, 1991; Hewlett-Packard, 1992; Hewlett-Packard, n.d.).
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Another method o f measuring dielectric properties requires parallel-plate electrodes or a parallelplate type o f cell that may be used in conjunction w ith an impedance analyzer or an LCR meter. The
material to be tested is placed between the two parallel electrodes, form ing a capacitor, and the capacitance
(which is proportional to the dielectric perm ittivity) o f the system is measured. This method is often used
fo r measuring the dielectric properties o f thin sheets o f materials and liquids (Hewlett-Packard, n.d.).
Researchers who used parallel plates include Bohidar et al., 1998 (parallel plates w ith LCR meter); Hagura,
et al., 1997 (two parallel electrode plates w ith capacitance meter); Kumagai et al., 2000 (parallel plate cell
w ith LCR meter); Laaksonen and Roos, 2001 (dielectric analyzer w ith parallel plate sensors); Noel et al.,
1995 and 2000 (parallel-plate cell w ith RLC Digibridge). Piyasena et al. (2003) used an LCR meter
connected to a liquid test cell.
As mentioned earlier, dielectric measurements have traditionally been performed in the frequency
domain. In such experiments, a source generates an oscillating incident wave at a specific frequency that
strikes the sample in a holder. The dielectric properties o f the material at that specific frequency are
measured. The source then changes the frequency, and another measurement is taken. This procedure is
repeated until data is collected at all the frequencies o f interest. In the case o f the network analyzer, two
different types o f techniques that can be employed to measure dielectric properties: broadband techniques
and resonant techniques. The broadband technique can measure properties over a wide range o f
frequencies. This method was used by Tran and Stuchly (1987) to measure the dielectric properties o f meat
between 100 and 2500MHz. The resonant technique can make measurements only at isolated frequencies,
but it yields more accurate loss measurements. Marsh and Wetton (1995) indicate that frequency-domain
methods are useful when measuring the dielectric properties o f food because they are not as sensitive to the
conductivities o f salts frequently found in food. However, these frequency-domain measurements are
lim ited in terms o f the frequency ranges that they can cover (Grant, 1978; Barthel, et al., 1990).
Dielectric measurements can also be made in the tim e domain using a method known as timedomain reflectometry. As its name implies, this method involves measuring, as a function o f time, the
signal reflected from a test sample after a stimulus (a b rie f voltage pulse) is applied to it via a transmission
line. Comparison o f the incident and reflected responses, mathematically converted back to the frequency
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domain by Fourier transforms, can yield the entire complex dielectric spectrum fo r the sample (Grant,
1978; Kaatze, 1997; M iura, et al., 1994). Researchers who used time domain reflectom etry include
Mashimo et al., 1987 and 1992; M iura et al., 1994,2003; Sudo et al., 2002; and Sun et al., 2004). Kent
(1990) points out that a disadvantage o f this method is the time delay between the measurement o f the
incident and reflected signals. Even i f very small errors (on the order o f picoseconds) in time-referencing
occur, large errors appear when the phase o f the reflected signal is computed. This problem worsens as the
frequency increases. Therefore, Kent’ s (1990) research employed a method known as time domain
spectroscopy, which is based on transmission rather than on reflection. In this method, errors in phase are
not so dependent on frequency, so measurements can be made more accurately up to 10 GHz (Kent, 1990).
It is worth mentioning that Kaatze (1997) warns that neither time-domain nor frequency-domain methods
may be completely accurate at low frequencies fo r liquid test samples that have high conductivities because
ohmic currents dominate at low frequencies.
3.2.14 Survey of the Dielectric Properties of Foods
Most foods are composed o f several different substances, and the dielectric properties o f these
complex food systems have not yet been completely analyzed and understood (Engelder and B uffler, 1991,
p. 8; Umbach, et al., 1992). Nevertheless, in this section some o f the ways in which certain food
components typically influence the dielectric properties o f a food system w ill be discussed. Water and salts
are the food components that most affect dielectric properties. The other major food components (fats,
proteins, and carbohydrates) have lower dielectric properties, but their effect is s till significant, especially
when they interact w ith water and salt (Tsoubelli et al., 1995; Haynes and Locke, 1995; Wei et al., 1994).
The complexity o f the study o f dielectric properties is increased by the fact that physical properties such as
density, the amount o f air in a substance, temperature, and glass transition phenomena also play an
important role, as do other factors such as chemical m odification (in the case o f starches) and
stoichiometric charges (in the case o f aqueous solutions). It is often d iffic u lt to determine which factor is
more dominant and prim arily responsible for the observed dielectric behavior o f a food system, especially
when a ll dielectric properties vary w ith the frequency o f measurement. Many researchers have focused on
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particular aspects o f the complex challenge o f measuring, modeling, and understanding the dielectric
behavior o f foods, and some o f their observations and conclusions w ill be b riefly reviewed in this section.
Both water content and temperature exert an overwhelming influence on the dielectric properties
o f foods, and the exact contribution o f each is often d iffic u lt to ascertain. Much o f the research
summarized below w ill list both as variables. In general, however, both the dielectric constant and the
dielectric loss o f foods tend to increase w ith increased moisture contents at microwave frequencies,
although Tinga (1970) reports that the loss can actually level o ff or even decrease at certain moisture
contents. Temperature affects dielectric properties in different ways. First, it affects the kinetic energy o f
the molecules, and therefore their ability to respond to the changing electric field. Second, phase changes
associated w ith temperature variations also affect molecular m obility and response to the electric field.
3.2.14.a Fats
Because o f their non-polar nature, fats, tend to have low values for both dielectric perm ittivity and
loss (Kent, et al., 1992; Bengtsson and Risman, 1971; Zallie, 1988). As a result, they do not readily
experience the internal generation from the “ molecular friction effect” that forms the basis o f the
microwave heating o f foods. However, since fats have a low specific heat, they require less energy to
produce an increase in temperature, so they do tend to heat faster than water (Bengtsson and Risman, 1971;
Zallie, 1988). This phenomenon can have a significant effect in foods that combine fats, starch, and water.
I f such a food is heated in the microwave, the fat w ill tend to heat faster than water and thereby, through
conduction, aid in uniform heating and swelling o f the starch (Zallie, 1988). In contrast, i f the food is
heated by conventional methods, the fats probably w ill inhibit starch gelatinization by covering the
granules w ith a protective barrier that keeps out the moisture. Another possibility is that some fats may
penetrate a swelling starch granule and interact w ith it on a molecular level, thereby changing its texture
(Zallie, 1988).
Bengtsson and Risman (1971) measured the dielectric properties o f fats and oils at a single
frequency (about 2.8 GHz). They noted that when they increased the temperature from -10°C to 60°C the
perm ittivity did not change very much, but that the loss did increase. This behavior was particularly
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evident in “ hard” fats, such as lard and tallow , that experienced a phase change as the temperature was
increased. The authors suggested that the loss value o f these fats is lower at lower temperatures because
the solid state inhibits the molecules from m oving freely.
Hagura et al. (1997) monitored the changes in the capacitance (which is directly proportional to
the dielectric constant) o f cocoa butter at 500 Hz, 5 kHz, and 50 kHz at a constant temperature o f 10°C
over a period o f 23 days. Higher capacitance values were recorded at the lower frequencies, but the
capacitance values at all three frequencies decreased over the entire time period. The researchers observed
changes in the capacitance values when the cocoa butter underwent a polymorphic transformation to
another crystalline form. They suggested that this method could be used as a non-destructive and
continuous method fo r m onitoring polymorphic changes, especially for chocolate and other foods w ith a
crystalline structure.
3.2.14.b Proteins
Some researchers have reported on efforts to study the interactions between protein and water
molecules by measuring dielectric properties. W ei et al., (1994) presented the results o f their own
experiments, and Pethig (1992) conducted a substantial review o f the available literature. The complete
findings o f both o f these studies are beyond the scope o f this text, but the authors’ main idea is that water
molecules are attracted to and form layers o f a hydration shell around a protein molecule in solution. The
authors conjectured that some water molecules make up part o f the protein’s structure, whereas other water
molecules are so tightly “ bound” that their relaxation times are the same or greater than the protein
molecule’ s relaxation time. Pethig (1992) explains that in the presence o f an electric field, hydrogen bonds
are broken and re-formed as the dipoles o f the water molecules attempt to orient themselves. In addition,
the polarizability o f the system is affected since proteins may contain not only polar side-groups that
undergo their own orientation, but also ionized side-groups that can transfer protons, and polypeptides that
can vibrate. These findings are consistent w ith those o f Mashimo and M iura’s group (Mashimo et al.,
1987: M iura et al., 1994; M iura et al., 2003), who examined numerous proteins in their time domain
reflectometry studies o f biological materials. This group reported that other relaxation events seem to
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occur, as evidenced by their observation o f different relaxation peaks at frequencies lower than the main
relaxation peak o f water. Some excerpts o f their w ork w ill be discussed in greater detail later.
Kent (1990) measured the dielectric properties o f herring fish at 22°C and at frequencies between
0.1 and 10 GHz. The behavior o f the dielectric perm ittivity could be directly correlated to the water
content in the fish. However, the perm ittivity decreased gradually between 100 M H z and 10 GHz,
indicating a broad dielectric relaxation. A fter considering the broadness o f the dispersion and comparing
the measured perm ittivity values w ith those o f pure water, the author speculated that proteins and other
compounds hindered the rotation o f some o f the water dipoles in the fish. Kent also attributed the dielectric
losses that were observed below 1 GHz to the conductivity o f the fish’s muscle rather than to water content.
Pea puree is a food that contains a relatively high amount o f protein, but it has an even higher
concentration o f w a te r- 80% in the samples tested by Tong, et al. (1994). The researchers reported that
the dielectric constant decreased as the temperature increased at both 915 M Hz and 2450 M Hz. However,
the loss behaved differently at the two frequencies. A t 915 MHz, the loss simply increased as the
temperature increased from 25 to 125°C. On the other hand, at 2450 M Hz the loss in itia lly decreased
between 25 and 75°C, then it slowly increased as the temperature continued to rise. The authors reasoned
that at this higher frequency, the decrease in dipolar losses w ith increasing temperature was balanced by an
increase o f ionic losses resulting in relatively small changes in loss factor w ith temperature.
3.2.14.C Carbohydrates and Starches
The group working w ith Nelson from the USDA - Agricultural Research Service studied the
effects o f different variables (such as water content, temperature, and frequency) on various starches and
carbohydrates. To begin with, Nelson (1981) reviewed literature from several researchers regarding the
factors affecting the dielectric properties o f bulk cereal grains. In spite o f the wide variation in grains and
study conditions, he was able to make some generalities. He found that the parameter that most influenced
the dielectric properties was water content, and that the most predictable property was the dielectric
constant. When the frequency was held constant, the dielectric constant tended to increase as the water
content increased. This pattern was so predictable that the behavior o f the dielectric constant has actually
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been used to m onitor the moisture content in grains (1981). The dielectric constant also increased almost
linearly w ith increasing temperature, but decreased w ith increasing frequency. The behavior o f the relative
dielectric loss factor and loss tangent were much less predictable than the dielectric constant since their
values could increase or decrease depending on a combination o f temperature and frequency. A fte r water,
frequency was the next most influential factor affecting the dielectric properties o f grains. A third
influential parameter was bulk density, which is related to the amount o f air in the sample. The amount o f
air present in a food material w ill greatly affect the dielectric properties since the dielectric properties o f air
are virtually those o f free space. W ith a ll else being constant, Nelson found that the dielectric properties o f
a given sample o f grain generally seemed to vary linearly w ith bulk density. However, he was unable to
make further generalizations due to the fact that the bulk density o f a grain sample can vary due to factors
such as kernel size, kernel density, chemical composition, variations in grains, contaminants, and
measurement techniques used by various laboratories.
In a subsequent study by the same group, Lawrence et al. (1990) measured the dielectric properties
o f Stacy soft red winter wheat at moisture contents between 8.2 and 23.4% on a wet basis; temperatures
between 0-50°C; and frequencies between 0.1 and 100 MHz. Under these conditions, both the dielectric
constant and loss increased w ith moisture content. They both also tended to increase w ith temperature
when the frequency and moisture content were held constant. Two years later, the group (Prakash et al.,
1992) published the results o f their investigation o f not only the effects o f moisture content and temperature
on the dielectric properties, but also the influence o f stoichiometric charge for potato starch, locust bean
gum, carrageenan, carboxymethylcellulose, and gum arabic. They found that both dielectric perm ittivity
and loss seemed to increase w ith increasing moisture level and temperature, but they decreased w ith
increasing charge. The authors suggested that as the amount o f charge increases, more water becomes
“ bound to the charged groups” and less able to respond to the applied electric field. Therefore, both the
dielectric constant and loss decrease. The fact that the effect o f charge was not observed in much drier
samples lends support to this theory.
RyynSnen et al. (1996) studied the effects o f water content on the dielectric properties o f both
gelatinized and non-gelatinized com, waxy com, wheat, and potato starches. They took the dielectric
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measurements o f their starch suspensions (which ranged between 5 and 30% w/w starch) at 2.75 GHz and
at temperatures o f 3,20, 60, 80, and 94°C. The researchers found that the dielectric properties o f the
different types o f starch suspensions at the same conditions o f temperature and water content did not d iffe r
significantly from each other. However, the perm ittivity and loss fo r a given starch type did decrease as the
temperature and the starch concentration increased (i.e., water content decreased). The water content
exerted the most influence, and the researchers remarked that behavior o f the starches’ dielectric properties
was actually very sim ilar to that o f water. The authors reported some slight differences between the
dielectric properties o f the gelatinized and the nongelatinized starches: the nongelatinized starches had
perm ittivity values that were 2-3% higher, but loss values that were up to 5% lower than those o f the
gelatinized starches. The researchers suspected these differences were simply due to settling problems in
the nongelatinized starches that led to uneven gelatinization when the samples were heated to about 60°C
during the measurement process.
Ndife, et al. (1998) measured the dielectric properties o f tapioca, com, wheat, rice, waxy maize, and
amylomaize starches in both granular and suspended forms at 2450 MHz. The suspensions were in the
starch:water ratios o f 1:1,1:1.5, and 1:2 w/w, and measurements were made every 5°C between 30 and 95°C.
Not only did they find that both the dielectric constant and loss factor increased as the amount o f water
increased, but also that the temperature dependence o f those dielectric properties increased as moisture content
increased. The authors made two other interesting observations about their data. The first was that the effect o f
temperature was different for the starch in the low-moisture granular form than it was for the starch in the highmoisture suspensions. As temperature increased, the dielectric properties increased for the grains but decreased
fo r the suspensions. The second observation was that the dielectric loss factors o f suspensions o f com wheat
and rice were statistically different from those o f suspensions o f waxy, amylomaize, and tapioca. Because their
loss factors were generally higher than those for the other starches, the authors hypothesized that wheat, rice,
and com starches would heat “ and gelatinize faster than the other starches in the microwave oven.”
The group at the University o f Minnesota also has done extensive research on the dielectric
properties o f starches. M ille r et al. (1991) studied a variety o f modified and unmodified starches in an
attempt to determine how the added functional groups (which included positive or negative ionic charges,
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polar, and hydrophilic groups) affected the dielectric and thermal transition properties o f starch-water
suspensions. They prepared the starches in 1:1 and 1:2 starch:water ratios (w /w ), and monitored the
dielectric properties o f each starch system as it was heated at 2450 M Hz from 30-90°C. The authors
reported that the dielectric constant values for both the normal and waxy com starches remained constant
throughout the entire temperature range. However, their plots seem to show some in itia l decrease o f s /
followed by a change in slope around gelatinization temperatures (between 60 and 70°C). The researchers
noted that the loss factor did decrease slightly w ith increasing temperature. They also commented that, in
general, the dielectric properties for those starches did not change in the gelatinization range in spite o f the
aforementioned change in slope shown in their plots and the decrease in e /'.
In the case o f the commercially prepared waxy com substituted w ith quaternary ammonium or
phosphate, the authors reported that the loss and absorptivity in itia lly decreased, then increased between
about 50-65°C, and leveled o ff and remained fa irly constant in the temperature range o f approximately 7090°C. Since a ll salts used in the m odification processes had been washed from the starches before the
experiments began, the authors suspected that the increase in the relative dielectric loss was due to the
m odification itself, which added an ionic charge. The researchers hypothesized that the type o f
m odification (including the degree o f substitution) affected the dielectric properties o f the m odified
starches more so than the process o f m odification or the amount o f water involved, at least at the
concentrations they studied here. Since all the m odified starches underwent thermal transitions and
swelling at lower temperatures than did the non-modified starches, they further concluded that “ Dielectric
behavior does not always relate to differences in thermal behavior.”
The follow ing year, the same group (Umbach et al., 1992) examined the dielectric properties and
the diffusion o f water molecules in mixtures o f starch, gluten, and water in an effort to better understand
how water moves and interacts w ith starch and gluten during the baking o f starch-based foods. The three
important variables in their study were: 1) water content o f the sample (54, 100, or 186%, w /w dry basis);
2) heat treatment (unheated; heated in a conventional oven at 190 °C for 25 minutes; or heated in a
microwave oven at 700 W fo r 45 seconds); and 3) ratio o f starch to gluten (100:0; 80:20; 50:50; 20:80; or
0:100). They measured the dielectric perm ittivity and loss (at 2450 M Hz and 22°C) o f some o f the samples
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before heating, whereas they heated the rest o f the samples firs t before measuring the dielectric properties.
N ot surprisingly, the dielectric properties that they measured for the dry starch and gluten powders
indicated that those materials did not significantly interact w ith the electromagnetic radiation. The
researchers found that the amount o f water in the starch-gluten mixtures had a greater effect on dielectric
constant (which tended to increase w ith increasing water content) than on the loss. The addition o f more
water to the mixtures, they commented, apparently enhanced the ability o f the samples to rotate in response
to the electromagnetic field. Both the dielectric constant and loss o f the heated samples were lower than
the values for the unheated samples.
Haynes and Locke (1995) also examined systems o f wheat starch, gluten, and water, as w ell as
commercial cracker dough. They sought to determine the dielectric properties as functions o f frequency
(between 0.2 and 20 GHz) and water content. They found that the dielectric loss factor o f cracker dough,
as w ell as the optical and static dielectric constants for a ll samples o f cracker dough, wheat starch, and
gluten generally increased w ith increasing moisture content. Concomitantly, the relaxation time decreased.
They attributed this to water’ s ability to act as a plasticizer which enables the water molecules to move
more easily. Furthermore, a direct result o f this increased m obility was the decrease in relaxation tim e that
accompanied the increase in water content for the samples o f cracker dough and wheat starch. The
researchers observed a deviation from Debye’ s model caused by the presence o f salts in the cracker dough,
and they admitted that the relaxation time studies for the gluten mixtures were “ inconclusive” and required
more research because gluten undergoes a physical change at moisture contents over 30%.
Mudgett, et al. (1974b) studied the dielectric properties o f alcohol-water mixtures at 3 GHz and
25°C, and they suspected that the conclusion made from these findings could apply to other food systems
containing sugars or carbohydrates. Their conclusion was that dissolved hydroxyl substituents influence
the dielectric properties o f a mixture by forming hydrogen bonds that “ stabilize” the structure o f the liquid.
This changes the value o f the relaxation time from what it was for the pure components o f the system, and
leads to “ synergistic loss effects at microwave frequencies.” Lending support to this latter theory, Sudo et
al. (2002) observed that the broadness o f the loss peak in the various alcohol-water mixtures that they
studied was related to both the water content and the number o f carbon atoms in the alcohols.
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3.2.14.d Influence o f Salts on Dielectric Properties of Foods
Ions can significantly influence the dielectric properties o f solutions due to their interaction w ith
both the changing electromagnetic fields and the solvent (Hasted et al., 1948; Mudgett, 1986). The team o f
Barthel and Buchner (Barthel et al., 1990b; Barthel and Buchner, 1992) classified the ions’ influence as
three different effects: the volume effect, irrotational bonding, and kinetic depolarization. The volume
effect refers to the dilution o f the polar solvent by the ionic solute. In other words, when an electrolyte is
added to an aqueous solution, there are, in essence, fewer water molecules available, so fewer dipoles can
move in the presence o f an electric field. This may be particularly important for salts w ith large ions
(Barthel et al., 1990b; Barthel and Buchner, 1992; Hasted et al., 1948). Irrotational bonding refers to the
fact that the electric field around an ion in solution actually polarizes the nearby solvent molecules,
changing their ability to rotate in response to a changing external electromagnetic field. The third effect,
kinetic depolarization, actually depends upon the other two effects. It refers to the way in which an ion
responds to the external electromagnetic field by moving in a direction opposed to the alignment o f the
solvent dipoles (Barthel et al., 1990b; Barthel and Buchner, 1992). This movement results in “ dielectric
friction” and the breaking o f hydrogen bonds w ith water. As a result, the ions’ m obility is impeded, the
solvent’s perm ittivity is decreased, and heat generation is increased (Barthel and Buchner, 1992; IFT, 1989;
Mudgett, 1982).
The consequences o f these three different phenomena are observed as changes in the values o f the
dielectric parameters o f electrolytic solution. To begin with, as mentioned already in section 3.2.10, one o f
the consequences o f the addition o f ions is the reduction o f the static dielectric perm ittivity. This
depression depends on the concentration, and therefore on the conductivity, o f the electrolyte (Barthel and
Buchner, 1999; Boughriet, et al. 1992; Hasted, 1972a; Hasted, et al., 1948; Lestrade, 1975; Mudgett, 1974;
Mudgett, et al., 1971; Swami and Mudgett, 1981; W ei, et al., 1992). Depression o f the static dielectric
perm ittivity would obviously decrease the relative dielectric perm ittivity, and this effect is frequencydependent. In fact, Boughriet et al. (1999) found that the dielectric constant o f saline solutions could
change by as much as 50% as the frequency varied between 400 M Hz to 20 GHz.
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Section 3.2.9 introduced the concept o f dielectric loss having both a dipolar component and an
conductive component. A second consequence o f adding electrolytes to a solution is that they decrease the
dipolar component, but increase the conductive component o f the dielectric loss o f the solution. In the
literature the decrease in the dipolar component o f the loss is often attributed to the impact o f “ bound
water” on the rotation o f the dipoles, whereas the increase in conductive loss is attributed to the
electrophoretic movement o f the ions (Hasted, et al., 1948; Mudgett, 1974, 1986; Swami and Mudgett,
1981). The effect on the overall loss term for the system depends largely upon the frequency o f the
microwave radiation and the temperature (Boughriet et al., 1999; Calay et al., 1995; Swami and Mudgett,
1981; Piyasena et al., 2003). Calay, et al. (1995) found that the total dielectric loss at 915 M Hz increased
w ith increasing temperature because the conduction losses dominated. A t a higher frequency (2450 M H z),
they report that the total dielectric loss in itia lly decreased w ith increasing temperature follow ing the trend
o f the dipolar loss. A t higher temperatures, however, the conductive losses dominated, and the total
dielectric loss then increased.
A third consequence o f electrolytes is a change in the relaxation time, but the extent o f that change
varies fo r different solvents and electrolytes. W ei et al. (1992) found that solutions o f “ moderate”
concentration o f LiC l, RbCl, and CsCl s till exhibited Debye behavior as their pure solvents would,
although their relaxation times decreased linearly as the concentration increased at 25 °C and at frequencies
between 45 M H z and 20 GHz. However, the researchers observed that the relaxation tim e actually reached
a minimum and then began to increase when the dielectric behavior deviated from the Debye model at
higher concentrations o f L iC l (greater than 5 M ). Like Wei, Hasted (1972a) reported that the relaxation
tim e o f dilute electrolytic solutions decreased as concentration increased, reaching a minimum at a
concentration o f about 5M, or at even lower concentrations in cases o f electrolytes o f small ions. When the
concentration was then increased further, the relaxation tim e actually increased, surpassing that o f pure
water. In contrast to W ei’ s work, Johri et al., (1991) did not find a linear relationship between relaxation
tim e and concentration when they studied room-temperature solutions o f three different concentrations o f
NaCl and CuS04 solutions at frequencies o f 9GHz, 20.9 GHz, and 29.967 GHz. These authors admitted
that “ a clear pattern does not emerge from the results,” and suggested that an increase in the ionic
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concentration m ight lead to “ some shielding o f the interior o f the samples” . The authors recommended that
the problem be examined more thoroughly by experimenting w ith more frequencies and concentrations.
The fourth consequence o f electrolytes is the reduction o f penetration depth. This effect is
actually dependent on the other three, since it is a direct result o f the increase in conductive losses and the
relationship between penetration depth and conductivity given in equation 3.93. As a result, salty foods
that are cooked in a microwave often experience rapid heating on their surfaces and comers. For example,
Dealler et al. (1992) found that the heating at the core o f a sample o f mashed potatoes decreased after salt
was added. In their experiments w ith pre-gelatinized com starch, Piyasena et al. (2003) found that the
salted suspension exhibited penetration depths that were a fu ll order o f magnitude lower than those o f the
non-salted samples. These researchers also found that penetration depth had an inverse relationship not
only w ith frequency (which agrees w ith equation 3.93), but also w ith temperature. Lentz (1980) observed
that the addition o f salt to water caused an increase in heating on the edges o f the load during microwave
heating. This, in turn, led to an increase in the rate o f evaporation o f the saline solution, even though its
average temperature was less than that o f the pure water load. As microwave heating continued, the edges
o f the saline solution began to boil “ long before” the solution’s average temperature climbed to the boiling
point. When Zheng et al. (1998) found that marinated seafood samples exhibited a greater difference in
temperature from their surface to their center than did their non-marinated counterparts, they attributed this
to the salts and spices in the marinade. Van Remmen et al. (1996) found that the centers o f spherical and
cylindrical samples o f agar gel w ith salt remained much colder than their outer layers after microwave
heating, in marked contrast to the pronounced center heating the researchers had observed in samples
without salt. Anantheswaran and Liu (1994) reported that the center o f an electrolyte solution was even
more likely to be cold when the solution’ s viscosity was high enough to hinder convective heat transfer.
Because the microwaves are preferentially absorbed by salty food compounds and often do not reach the
center o f the food, developers o f prepared foods w ith salty sauces or food components must carefully
consider food safety and consumer convenience (Anantheswaran and Liu, 1994, p. 122; B uffler and
Stanford, 1991; Casasnovas and Anantheswaran, 1997; Dealler, et al., 1992; Schiffmann, 1986).
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Taking a unique approach to the interaction between starch and electrolytes, Oosten (1982)
proposed that starch can be seen as a weak ion exchanger. In his model, starch in aqueous suspension is a
weak acid that is partially dissociated (Starch-OH dissociates to Starch-O" + H+). A t equilibrium , this
dissociation would cause a gradient o f concentration o f H + and a migration o f these H+ ions to the bulk
water in the suspension. The net effect, according to the author, is a partially negatively charged starch
granule, w ith a partially positively charged water, which establishes a potential. Oosten further explains
that sodium ions can displace hydrogen from the starch form ing “ starch alcoholates” (Starch-0"Na+). Since
these starch-sodium salts are better dissociated than the weak acid, the author hypothesizes that the addition
o f sodium chloride to a starch suspension w ill increase the potential to some extent. The in itia l objective o f
this hypothesis was to explain the increase in gelatinization temperature by the addition o f certain salts
(including sodium chloride) to alkaline starch suspensions. However, from the point o f view o f dielectric
properties, it is interesting to keep in mind the possible effect that starch may have on the m obility o f the
Na+ ion.
The dielectric behavior o f proteins can also be affected by salts. Bohidar, et al (1997) studied the
dielectric behavior o f gelatin solutions and gels (both w ith and without sodium chloride) in the temperature
range o f 20-60°C and the frequency range o f 20 kHz and 10 MHz. They found that the dielectric constant
o f the samples w ith sodium chloride were greater than those without it. The researchers observed that “ all
the significant dispersion behavior” occurred at frequencies below 100 kHz. A t 10 kHz, they found that the
perm ittivity at a given frequency increased sharply at the temperature o f gelation. This jum p was most
pronounced in samples containing sodium chloride, and the magnitude o f the jum p increased as the
concentration o f protein increased. The authors hypothesized that the change in the dielectric constant at
the gelation temperature was related to the amount o f water contained in the gel network. They suspected
that the dielectric constant’ s higher-than-normal value had been inflated by a significant contribution from
interfacial polarization “ either due to electrochemical reaction or due to ions getting trapped at some
interface w ithin the bulk.” They also reasoned that the unexpected measurements that they found at 100
kHz were not truly representative o f the dielectric behavior o f the bulk o f the sample, but rather were more
indicative o f ionic m obility.
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Proteins can significantly affect the dielectric behavior o f carbohydrate mixtures. Tsoubeli et al.
(1995), contrasted the different effects o f calcium caseinate and whey protein isolate (W PI) on the
dielectric properties o f their wheat starch-water mixtures (at a concentration o f 30% moisture). They found
that the mixtures experienced an increase in e' when whey protein isolate (W PI) was added to the system,
and the e' value continued to increase as the temperature increased. They reported that the dielectric loss
increased w ith the addition o f caseinate, but decreased w ith the addition o f W PI. The effect o f temperature
on the dielectric loss o f the two protein fractions was reversed. During heating, the dielectric loss o f the
caseinate decreased, but it increased for W PI. They concluded that the W PI increases the “ energy coupling
o f starch” as w ell as its dielectric loss and “ its ability to release energy as frictional heat.” They also
commented that the behavior o f the WPI was sim ilar to the behavior their group had observed (M ille r et al.,
1991) for com starch substituted with quartenary ammonium.
Gluten, the protein in wheat, has also been found to affect the dielectric properties o f carbohydrate
systems. Umbach et al., 1992, who measured the dielectric properties o f starch-gluten-water systems,
found that the dielectric constant tended to decrease as the amount o f gluten in the system increased. They
also found that the water in the samples that were heated in the microwave (as opposed to the samples that
were heated by conventional means) interacted more w ith the gluten than w ith the gelatinized starch.
These high-gluten samples had lower values o f both the dielectric constant and loss factor than the
corresponding high-starch samples at greater moisture contents. In related research, Haynes and Locke
(1995) observed that increased water content caused the static perm ittivity for gluten-water suspensions to
increase but relaxation time to decrease except at high (39%) concentrations o f water. They recognized
that the relaxation time for the samples w ith high moisture contents deserved more investigation because
the material properties o f gluten change: “ Below 30% moisture, gluten is a crumbly powder, and above
30% moisture, gluten becomes a cohesive rubber substance presenting a handling problem.”
3.2.I4e. Mixtures and Solutions
Engelder and B uffler (1991) declared, “ Foods are often complex mixtures o f many substances.
However, the composite dielectric response may not be the simple linearly-weighted sum o f the individual
responses.” Mudgett’s group (Mudgett et al., 1971; Mudgett, 1974) found this to be quite true in their
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experiments w ith m ilk. They used salts to chemically synthesize aqueous m ilk solutions at various
concentrations, replacing the ash, protein, and carbohydrate components o f real nonfat m ilk w ith sodium
chloride, sodium caseinate, and lactose, respectively. The authors then measured the relative dielectric
losses o f each o f these components at 3 GHz and 25°C, and calculated their weighted average in an effort
to predict the value o f the relative dielectric loss o f an aqueous solution o f nonfat dried m ilk. They found
that this predicted value was much higher than which was actually measured for both their synthetic m ilk
solution and for the real m ilk. Further investigation showed that some o f the m ilk salts did not dissociate at
a ll because o f saturation conditions in the solvent. This, obviously, would lower the conductivity and loss
o f the solution. In addition, the authors suspected that only some o f the dissociated salts actually
contributed to the conductivity and the loss, and they hypothesized two possible reasons fo r this. The first
was that some o f the dissociated salts may have become bound by other constituents in the m ilk, possibly
the proteins. The second was that, at high ionic concentrations, the non-bound ions w ith their surrounding
hydration sheath are close to other ions w ith opposite charge, and could form complexes. The researchers
theorized that these complexes may have a role in lowering the dielectric loss. A fter quantifying the
influence o f these interactions, the authors were able to accurately predict the relative dielectric loss o f the
m ilk.
In another study o f m ilk, Kudra et al. (1991) focused on the heating characteristics o f m ilk
components (fat, lactose, and protein) using their specially designed microwave pasteurization system.
They found that protein was the component that contributed the most to heating, causing a temperature rise
o f approximately 0.45°C for each percent o f protein. Although they do not report dielectric measurements,
their results im ply that the dielectric loss o f m ilk is strongly related to its protein content. To explain this,
they point out that m ilk proteins are charged and are in closely associated w ith calcium ions and other
minerals. In a subsequent study (Kudra, et al., 1992), these researchers actually measured the perm ittivities
and loss factors o f m ilk and o f various solutions o f m ilk’ s constituent components at 20 C and 2.45GHz.
The quantitative results showed that the perm ittivity decreased and the loss increased as the protein
concentration increased.
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A class o f food mixtures that deserves special mention is emulsions. Mudgett’s group (Mudgett,
1974, Mudgett, et al., 1974b) used a theoretical model as w ell as experimental standing wave
measurements to calculate the dielectric properties o f various concentrations o f olive oil-water emulsions at
25°C and 3 GHz. The results o f both methods were in reasonably good agreement. In each case, both the
dielectric constant and loss increased as the concentration o f water increased. The results from the standing
wave measurements were slightly lower than those predicted by the theoretical model. This led the
researchers to theorize that “ The apparent effect o f suspended colloids is the volumetric exclusion o f
dielectrically active water from aqueous mixtures by colloidal materials o f low dielectric activity, thus
depressing dielectric constant and loss.”
Bouldoires, 1979, examined the dielectric constant as a function o f time at 25°C for the bases used
in instant drinks. The results showed that the dielectric properties o f two samples w ith the same water
concentration (one a simple water-glucose m ixture; the other an emulsion o f water, glucose, and aroma
oils) behaved sim ilarly. They decreased linearly w ith time due to the crystallization o f glucose
monohydrate, and then flattened out at a value which was about the value o f the dielectric constant o f
anhydrous glucose. However, the e' o f the sample w ith the aroma oils was slightly higher at the beginning
and end o f the experiments.
The group from the University o f Minnesota studied the microwave power absorption o f
emulsions and layered systems composed o f various concentrations o f o il and water, some also containing
sodium chloride (Barringer et al., 1995). The samples were heated in a microwave oven (w ith a power
output o f 600 W) to 55°C. They found that dielectric properties played a role in increasing the heating rate
in samples o f certain sizes because o f the constructive interference that was set up in the microwave oven.
In addition, they observed that the heating rates o f emulsions w ith the same composition but different
dispersed phases were different. More specifically, they reported that the heating rate for the emulsion o f
50% oil-in-w ater was 13% higher than the heating rate fo r the emulsion o f 50% w ater-in-oil. The
researchers attributed this to the way the electric field w ithin droplets o f o il is actually enhanced when
water is the continuous phase, whereas the electric field w ithin droplets o f water is reduced by the shielding
effect o f o il in the continuous phase. This m ight also explain why the oil-in-water emulsion heated much
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more quickly than a layered system w ith the same concentration o f o il and water, whereas the w ater-in-oil
emulsion had a heating rate that was sim ilar to its corresponding layered sample. Since the heating rate
increased only slightly when the researchers added sodium chloride to the emulsions, they concluded that
the number o f interfaces has a greater affect on heating rate than simply adding a high-loss component to
one phase.
3.2.15 How Some Researchers Have Modeled Dielectric Properties
The previous section discussed some o f the trends in dielectric properties that authors reported in
the literature. This section w ill briefly describe some o f the attempts that certain researchers made to
model the dielectric properties that they studied. Some authors were able to use a Debye or Cole-Cole
model, whereas others developed their own empirical models based on the data they collected. S till other
authors encountered more complicated cases in which the systems they studied displayed more than one
relaxation peak, and often those m ultiple peaks were asymmetrical and broader than those predicted by the
Debye model (Grant et al., 1978; Marsh and Wetton, 1995; Einfeldt, et al., 2001). These cases were
usually addressed by summing up several Debye or Cole-Cole expressions.
Calay, et al. (1995) searched the literature for data relating dielectric properties o f foods to
frequency, temperature, and composition. The group performed least squares regression analysis on their
data and developed empirical equations to predict the dielectric constant and dielectric loss factor for the
foods as functions o f moisture, salt, and fat content. Most o f their equations predicted values o f dielectric
properties that agreed w ith the measured values w ithin the lim its o f experimental uncertainty. However,
they found some discrepancies between the predicted and measured values because the compositional data
in the literature was incomplete and had to be assumed in some cases.
Tran and Stuchly (1987) measured the dielectric properties o f raw beef, chicken, and salmon at
temperatures between 1.22° and 64°C and frequencies between 100 and 2500 M Hz. Instead o f least
squares regression, they used a cubic spline algorithm to model the relative dielectric perm ittivity and loss
as functions o f frequency at certain temperatures. They reported that their results agreed “ reasonably with
other published results.”
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Boughriet et al. (1999) measured the real and imaginary parts o f the relative dielectric p e rm ittivity
o f methanol, 2-propanol, and butanol. Using this data (which they assumed follow ed the Cole-Cole
model), they calculated the Cole-Cole parameters 8s, e*,, t, and a. They found that their results were in
good agreement w ith literature values. The researchers next measured the relative dielectric perm ittivity
and loss factor o f saline solutions ranging from 0.2 to 3.5% NaCl. They fit this data to a Cole-Cole model
w ith a=0 (which is equivalent to the Debye model), but w ith an added term (-jo/(co£0) to correct for
conductivity. Once again, their calculated parameter values agreed w ell w ith the values in the literature.
The data for the relative dielectric perm ittivity in this group’s work show a departure from the Debye
model at NaCl concentrations o f 2% and higher. They attributed this departure to electrode polarization,
but this phenomenon was not modeled by the authors.
Electrode polarization has been reported by various authors; it is the topic o f much research
because it is not a well understood phenomenon. According to Adamec and Calderwood (1989) it is basically
caused by a difference in transfer rates between charges in the bulk liquid and charges at the interface o f the
electrodes that leads to a buildup o f charges near the electrodes. It particularly comes into play at low
frequencies (Feldman, et al., 2001).
Pizzitutti and Bruni (2001) attempted to correct dielectric properties obtained w ith a parallel plate
cell by subtracting the effect o f sample heterogeneities and electrode polarization. They modeled the effect
o f heterogeneities (a Maxwell-Wagner effect) as an additional dielectric relaxation, and the electrode
polarization as an equivalent circuit element behaving like a mixed resistive-capacitive element. Using
measurements for ice as a test case, the author used their corrections (adding also a -ja/(coe0) conductivity
term) and obtained satisfactory agreement w ith literature data.
W ei et al. (1992) also used a Cole-Cole model that contained a correction term for ionic
conductance to fit their data on the complex dielectric properties o f aqueous solutions o f CsCl and RbCl.
The authors then extracted from that model the parameters o f static and optical perm ittivities, relaxation
time, and conductivity o f the solutions at various concentrations.
Piyasena’ s group (Piyasena et al., 2003) performed statistical analysis on starch suspensions with
and w ithout salt to assess the effects o f temperature, starch concentration, salt concentration, and frequency
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on the dielectric properties. The authors then used m ultiple regression to develop sets o f predictive
equations to calculate the logarithms o f the relative dielectric perm ittivity, relative dielectric loss, and
penetration depth as functions o f these variables. Each set contained one equation fo r samples w ithout salt
and one equation for the samples w ith salt. When they compared the predictions o f their models w ith the
Debye-Hasted models (Hasted et al., 1948, Mudgett, 1982), these authors thought that the Debye-Hasted
model m ight be “ more appropriate for predicting the loss factor at low temperatures as opposed to the
dielectric constant for salt-enriched solutions.” They also suggested that the Debye-Hasted model might
need a correction factor to calculate the dielectric constant more accurately for the starch-salt suspensions,
sim ilar to the correction factor Mudgett et al. (1974) used to improve the predictions fo r m ilk.
Nelson (1981) reprinted equations that he had developed in an earlier study to predict the
dielectric constant o f shelled, hybrid yellow-dent fie ld com. He gave three different equations, valid at 20,
300, and 2450 MHz, respectively. Each equation calculated the dielectric constant o f the com as a function
o f moisture content, bulk density, and temperature. In a later paper, (Nelson, 1987) Nelson developed
mathematical models to predict the dielectric constants o f other cereal grains and soybeans as functions o f
frequency, moisture, and bulk density at 24°C using two empirical equations: one that calculated the cube
root o f the dielectric constant, and one that calculated its square root. The parameters in each equation
varied according to the particular grain and the particular equation being used, and the author listed them
a ll in a table. Both models seemed to be equally reliable, w ith an average accuracy o f 3-4.5% for their
range o f validity. That range was (in most cases) frequencies between 20 M Hz and 2450 M Hz and
moisture contents between 8% and 25% (on a wet basis). In spite o f this accuracy, Nelson (1981, 1987)
advised that i f researchers want to know very accurate values o f dielectric properties for a particular
substance, they should directly measure those properties under the conditions o f interest.
A few years later, Nelson’s group (Prakash et al., 1992) developed empirical models to predict the
dielectric properties for five hydrocolloids (potato starch, locust bean gum, gum arabic, carrageenan, and
carboxymethyl cellulose) in the temperature range o f 20-100°C and at 2.45 GHz. The authors performed
stepwise m ultiple regression on their dielectric data to obtain equations for the dielectric constant and loss
factor that were functions o f stoichiometric charge-to-mass ratio, temperature, and moisture content, w ith
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the latter being the most influential variable. The predictions made by these equations were close to the
observed values, w ith the lowest r2 value being 0.94.
N dife et al. (1998) developed a model to predict the dielectric constants and loss factors o f various
starches at temperatures between 30 and 95°C at a frequency o f 2450 MHz. To obtain their model they
used a least squares fit w ith two separate equations for e/ and e /' that were simple quadratic functions o f
temperature. The constants and coefficients o f the equations were different for every concentration o f each
starch investigated by the authors. The suitability o f the model to represent the measured data (as indicated
by the r2 values) also was different for each o f the different starch concentrations, varying between 0.66 and
0.99.
A review o f the literature reveals that the appearance o f m ultiple relaxation peaks is largely
dependent on the temperature and frequency at which the dielectric measurements are made. To
differentiate among the relaxations that each peak represents, they are named w ith Greek letters: a, P, y, 6,
and a. There is general agreement that the a-relaxation (also known as the prim ary relaxation) is due to the
movement o f the entire molecule, and, therefore, it can be observed only at or above the glass transition
temperature o f the substance. The P-relaxation, also known as a secondary relaxation, can be observed at
temperatures below the glass transition temperature, so the polymer is in a glassy state. This relaxation is
generally attributed to local chain motion, but, as Noel, et al. (2000) cautioned, “ The relationship between
molecular motion and the p-relaxation is somewhat controversial... a fu lly coherent view o f the prelaxation is yet to emerge.”
One confusing aspect o f much research about dielectric spectroscopy is the choice o f variables to
represent the data. In some cases, the authors choose to present their dielectric data as a series o f
isothermal curves as a function o f frequency (the approach followed in the present work). In this case the
relaxation peaks are identified as occurring at a certain frequency (fCnt) or, equivalently, as having a certain
relaxation time (t). Usually the critical frequency or relaxation time is then given as a function o f
temperature. In other cases, the data is presented as curves that are functions o f temperature at fixed
frequencies. In this case, the dielectric data is sometimes compared directly w ith thermal analysis data.
The confusion arises when some authors mention that there is an “ a-relaxation” or “ P-relaxation” at certain
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frequency, and then some other authors mention the same kind o f peak, but at a certain temperature. Since
dielectric relaxation is characteristic o f time-harmonic oscillation, it seems most reasonable to talk about a
critical frequency or relaxation time, and then study the effect o f temperature on this parameter. As
mentioned, this is the approach followed in the present study.
Einfeldt et al., (2001) performed a comprehensive dielectric relaxation spectroscopy study o f more
than a dozen different cellulosic materials and starches in the frequency range o f 10 mHz to 2 M Hz, and the
temperature range o f-135° to 180°C. Because these temperatures were below the glass transition range,
the authors did not observe an a-peak, but they did observe five other peaks. Acknowledging that
correlating these relaxations w ith their origins at the molecular level is “ a fundamental and, in literature,
controversially discussed problem” , the authors include a thorough discussion o f their interpretation o f their
results. Im portantly fo r this study is that the authors observed a relaxation process that they labeled y, and
attributed to the motion o f side groups, specifically hydroxyl groups and the m ethylol group in the glucose
monomers.
Noel, Parker, and Ring studied the dielectric behavior o f low molecular weight carbohydrates at
frequencies between 100 Hz and 100 kHz, (Noel, et al., 1996,2000). In general, they found that as the
amount o f water in the system increased, the temperature at which the a-peak was observed decreased,
whereas the magnitude o f the (i-peak increased. However, there were differences in the relaxation behavior
o f the various carbohydrate suspensions. The authors attributed the differences they observed to the
differences in molecular size, shape, and carbohydrate structure, including pendant groups distribution o f
dipoles.
Laaksonen and Roos (2001) studied the dielectric relaxations o f frozen wheat dough at
temperatures between -150° and 10°C and frequencies o f 0.1, 0.5, 1, 5, 10,20, 50, 100, and 1000 Hz.
Plotting their data as tan 6 vs. temperature, they observed three relaxations: the a-relaxation at the glass
transition temperature, and two secondary relaxations (P and y) that occurred at lower temperatures. The
authors suspected that these secondary relaxations “ were probably due to local mode rotations o f terminal
groups or other side chains o f gluten, starch, sugar, or other minor components.” The temperatures at
which a ll three relaxations occurred increased when the frequency was increased. The glass transition
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temperature was assumed to be the temperature o f the a-peak (which, when measured at 0.1 Hz, was noted
to be close to the values obtained by thermal and mechanical analyses), and at higher temperatures, this
coincided w ith the melting temperature o f ice. When sodium chloride was added to the mixtures, the aand (3-peaks occurred at lower temperatures, but the y-peak occurred at higher temperatures. Obviously,
conductivity played an important role in these changes.
Kumagai, et al. (2000) studied the dielectric properties o f gelatin w ith four different water contents
in the frequency range o f 100Hz - 1MHz and the temperature range o f-20° to 60°C. A t this temperature
range, the gelatin is in a glassy state, so the authors found only P-relaxation peaks in the curve o f e" vs.
frequency. These peaks shifted to higher frequencies at higher temperatures. The relaxation time
decreased w ith increased temperature and water content due to lowered viscosity that enabled the
molecules to reorient themselves more quickly.
Marsh and Wetton (1995) used dielectric relaxation spectroscopy to measure the dielectric
properties o f foods and polymers. The first system they discussed was that o f the polymer polyethylene
terepthalate at frequencies between 1 and 100 kHz, and in the temperature range o f -120°- 130°C. The
researchers observed an a-peak around 110°C (the glass transition temperature for the polymer), and a Ppeak at -20°C in the polymer’s glassy state. Theoretically, these two relaxation peaks are separated when
measurements are taken at low frequencies. However, the authors explain that the polymers are frequently
“ contaminated” w ith ions that contribute to the dc conductivity w ell into the glass transition region. (It is
important for this current study to note here that ionic “ impurities” are also frequently found in samples o f
commercial starch.) A t higher frequencies the dc conductivity is minimized, so it is easier to “ resolve” the
a-peak that occurs at the glass transition temperature. The area under this peak is proportional to the
number o f mobile dipoles in the system, but this information must be interpreted carefully for food because
o f compositional differences between crystalline and non-crystalline phases. The second system these
authors discussed was a mixture o f 40% amylopectin from pre-gelatinized waxy maize, 30% fructose, and
30% water. Measurements were taken at two different temperature ranges: 30 C to 70 C and -50 C to 0 C,
and at frequencies between 10‘3 - 106 Hz. In general, the loss peak shifted to higher frequencies as the
temperature went up. They observed a significant increase in the loss peak occurring between -30 C and -
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20 C, which they suspected was tied to the glass transition temperature o f -25 °C. The authors attributed the
relaxation to “ water molecules in association w ith hydroxyl groups on amylopectin.” The data suggested
that about 60% o f that water could relax when amylopectin is in the glassy state, and the remaining 40% is
released at the glass transition temperature. Between 30 °C and 40 °C, the loss peak o f the amylopectin
system displayed a deviation from normal to anomalous behavior that was consistent w ith a change in the
system’ s morphology. That is, the room temperature conditions had caused retrogradation, resulting in a
phase o f amylopectin crystals and a phase o f amorphous amylopectin, fructose, and water. The latter phase
was conductive due to ionic impurities. As the temperature increased above 40°C, the loss peak shifted
slightly to lower frequencies. The third system these authors tested was a commercial wafer composed o f
80% starch, 5% water, and 15% mixture o f sodium chloride, sodium bicarbonate, lecithin, vegetable o il,
and saccharin. They observed a p-relaxation a -40°C and 200kHz. A t temperatures greater than 0°C, this
system also displayed anomalous behavior, evidently due to the conductivity o f the sodium chloride. The
magnitude o f the anomalous peak increased w ith temperature, whereas its frequency location decreased
w ith temperature. When the magnitude and location o f this peak were plotted against temperature, the
authors observed a discontinuity at 25°C. They interpreted this as representative o f changes in “ structures”
that had formed in the wafer w hile it had been stored at that temperature before being used in the
experiments.
A group from the University o f Regensburg in Germany has performed several studies o f the
dielectric behavior o f electrolyte solutions and alcohols. In a particular study (Barthel et al., 1990a), the
researchers found that the best fit for their data was a double Debye model for water, and a triple Debye
model for the alcohols (methanol, ethanol, 1-propanol, and 2-propanol) in the frequency range o f 0.95 - 89
GHz at 25°C. They hypothesized that the fastest relaxation is due to the movement o f hydrogen bonding.
In a subsequent work (Barthel et al., 1990b), these researchers compared how three different models (ColeCole, and Debye w ith either 2 or 3 dispersion steps) fit the dielectric data their group had previously
measured fo r methanol. They found that the main dispersion seemed to be relatively stable; that is, its
values for dielectric perm ittivity and relaxation time did not change much, regardless o f how many
dispersion steps were included in the model. However, that was not the case for electrolyte solutions.
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Later (Buchner et al., 1999), this research group combined results from the literature w ith their own
laboratory results to get a better understanding o f the dielectric relaxation o f water in the temperature range
o f 0°C - 35°C, and the frequency range o f 0.2 - 410 GHz. They used a double Debye model that assumed
there were two relaxations:
e* = (e - e2) /( l + jcoTi) + (e2 - ew)/( l + jo)x2) + e*,
[3.94]
They found that the relaxation tim e for the first dispersion was too large to be attributed only to the rotation
o f water molecules. Therefore, the authors hypothesized that the time was related to the number o f
hydrogen bonds which had to be broken before the water molecules could be free to move.
Mashimo’s group (Mashimo et al., 1987) used time domain reflectometry to study the dielectric
behavior o f biological materials, including meats, cheese, m ilk, apples, tomatoes, and potatoes. They
observed that all o f the items exhibited two relaxation peaks: one around 100 M Hz (which they attributed
to “ bound water” ) and one around 20 GHz (which they attributed to “ free water” ). They modeled the
systems as the sum o f two Debye relaxations. In a later study (Mashimo et al., 1992), this group examined
the structure o f mixtures o f water w ith glucose, six polysaccharides (maltose, maltotriose, maltotetraose,
maltopentaose, maltohexaose, and maltoheptaose) and ascorbic acid over the wide frequency range o f
10kHz - 20 GHz at 25°C. They suggested that maltose might exhibit “ critical dielectric behavior” because
solutions o f sugars smaller than it showed only one peak, whereas solutions o f sugars larger than it showed
two peaks. Once again, they modeled the latter case as the sum o f two relaxations, and attributed the
higher-frequency peak to the relaxation o f water molecules. The lower-frequency relaxation was attributed
to the movement o f the polysaccharide molecules, and its relaxation time was dependent on the molecular
weight o f the molecule. Only one relaxation peak was observed for the glucose mixtures. The authors
suggested that perhaps the glucose molecule can fit into water’s lattice structure, form ing hydrogen bonds
that would inhibit it from rotating freely. However, once the lattice is broken up, “ the water and the
glucose molecules are released and orient cooperatively and simultaneously.”
In other research by the same group, M iura et al., (1994) used time domain reflectometry to study
dielectric relaxation o f ten globular proteins in aqueous solutions in the frequency range from 100 kHz to
10 GHz. They observed relaxation peaks at approximately 20 GHz and 10 MHz, so they attempted to
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model their data as the sum o f two relaxation processes. The high-frequency peak at 20 GHz obeyed the
Cole-Cole model, and they attributed this to the orientation o f free water molecules in the bulk o f the
solution, as did Mashimo et al. (1992). Since the low-frequency peak at 10 M Hz depended on the
molecular weight o f the protein, they hypothesized that it m ight be due to the rotation o f the complete
protein molecule and the movement o f counterions on the protein’ s surface. Around 100 M Hz, however,
the authors found that the data deviated from their model. This led them to suspect the existence another
relaxation process. They then adjusted their model to include a third term. They hypothesized that the
suspected relaxation process at this intermediate frequency was due to the “ orientation o f bound water
molecules on the protein surface, supplemented by fluctuation o f polar side groups on the surface.” They
further suggested that there could be even more than three relaxation processes occurring in this system.
Sun et al, (2004) found sim ilar results and made sim ilar conclusions in their study o f native ovalbumin
between 100 kHz and 20 GHz. The low-frequency relaxation they observed at 10 M Hz was considered to
be due to the rotation o f protein molecules; the middle-frequency relaxation they observed at 100 M Hz was
considered to be due to the orientation o f bound water; and the high-frequency relaxation they observed at
20 GHz was considered to be due to the orientation o f free water molecules.
Some representative relaxation parameters have been compiled in Table 3.5. The relaxation time
and the relaxation strength (when available) were tabulated along w ith the relaxation mechanism proposed
by the authors for the data. For most o f this data (w ith the exception o f pure glucose) there is one
relaxation process in the picosecond range that is attributed to orientation o f bulk or “ free” water
molecules. The relaxation processes reported in the nanosecond range are either attributed to bound water
or orientation o f relatively small molecules. Data from Lu (2005) is included in the table although the
focus o f that w ork was not a food material. This author measured the dielectric properties o f polymeric
membranes at different levels o f hydration, and fit a number o f relaxation peaks for the data. Most o f the
proposed relaxation processes were in the picosecond region, and were attributed to different states o f
m obility o f water molecules.
Haynes and Locke (1995) measured the dielectric properties o f cracker dough, gluten, and starch
at constant temperature and density, in the frequency range o f 0.2 to 20 GHz using an HP 85070B open-
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146
ended coaxial probe. They found that both e / and e /' fo r the cracker dough increased, but t decreased (i.e,
the frequency shifted to the right) as the amount o f water in the system increased. They observed only one
relaxation peak when they measured the dielectric properties o f cracker dough in the frequency range o f 2
to 10 GHz; this was consistent w ith Debye theory in that range. However, below 1 GHz they observed a
deviation from Debye’s model that was apparently caused by the conductivity o f salts in the cracker dough.
They saw another deviation from Debye’s model at frequencies above 10 GHz, especially at high moisture
contents. They noticed a second peak when they increased the moisture content in their samples to 64.7%,
much higher than the 20-40% moisture content that would be normally used in com m ercially processed
dough. DSC analysis confirmed that this second peak corresponded w ith an increase in the amount o f free
water in the dough. A graph o f dielectric loss as a function o f moisture content in the dough showed that at
moisture contents greater than 34.8%, the loss was higher at the relaxation frequency around 20 GHz
(corresponding to free water) than at the relaxation frequency around 6 GHz (which they attributed to
bound water). Therefore, the researchers concluded that the dough’s perm ittivity at lower moisture
contents was m ainly due to the relaxation o f bound water in the frequency range o f 6.0 GHz and 7.0 GHz..
Thus, their calorim etric measurements supported their observations o f distinct dielectric relaxations for
bound and free water. When the researchers performed the experiments on starch and gluten, they found
that the results from 0.2 GHz through 10 GHz were consistent w ith Debye theory since there were no salts
present in these substances to make a conductivity contribution at low frequencies. The researchers were
able to obtain the parameters in Debye’s model as a function o f composition, at constant density and
temperature by using equation 3.56.
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Table 3.5 Representative Samples o f Relaxation Parameters Reported in the Literature
System Studied
glucose
potato
human red cell ghosts
(0.192% w/w)
homogenized
m ilk
glucose in water
(75% w/w)
glucose in water
( 10% w/w)
maltotriose
(25% w/w)
maltoheptaose
(25% w/w)
ribonucrease A
hemoglobin
chicken
hydrated Nafion
membranes ( 100% r.h.)
hydrated Nafion
membranes ( 100% r.h.)
Proposed Mechanism
Reference
hydroxyl and m ethylol side group oscillation
bound water
free water
bound water
free water
bound water
free water
E infeldt et al., 2001
Relaxation
Tim e (t , sec.)
10* - 10*
1.36 x 10*
13.2 x 10' 12
1.85 x 10*
8.3 x 10' 12
2.02 x 10*
11.5x 10'12
Relaxation
Strength (A e)
29.8
67.5
1.04
72.3
6.26
70.3
25
617 x 10‘ 12
45.37
25
rotation o f water and glucose
Mashimo et al., 1992
10.7 x 10' 12
70.91
25
rotation o f water and glucose
Mashimo et al., 1992
0.794 x 10*
12.3 x 10' 12
6.17 x 10*
10.5 x 10' 12
20.0 x 10*
1.45 x 10*
9.12 x 10' 12
72.4 x 10*
2.69 x 10*
8.71 x 10‘12
0.288 x 10’6
0.646 x 10*
8.511 x 10' 12
94.19 x 1 0 12
21.46 x 10‘12
8.09 x 1012
77.11 x lO ' 12
18.09 x 10"12
5.69 x 10‘ 12
0.82
63.41
0.75
60.04
11.47
2.63
65.52
6.36
1.51
68.38
1040
12.6
49.6
10.06
2.23
6.52
14.45
1.89
6.12
Temp.
(°C )
-150 to -100
20
20
25
25
25
25
25
25
45
rotation o f maltotriose
rotation o f water
rotation o f heptaose
rotation o f water
rotation o f ribonucrease A
rotation o f bound water
rotation o f free water
rotation o f hemoglobin
rotation o f bound water
rotation o f free water
interfacial polarization
rotation o f bound water
rotation o f free water
water confined in hydrophobic environment
water in 2nd hydration layer o f polymer
rotation o f free water
water confined in hydrophobic environment
water in 2nd hydration layer o f polymer
rotation o f free water
Mashimo et al., 1987
Mashimo et al., 1987
Mashimo et al., 1987
Mashimo et al., 1992
Mashimo et al., 1992
M iura et al., 1994
M iura et al., 1994
M iura et al., 2003
Lu, 2005
Lu, 2005
4^
148
3.3 Objectives
Previous studies in the literature have reported how dielectric properties o f starch suspensions
change as functions o f temperature at constant frequency or, conversely, as functions o f frequency at
constant temperature. For those studies that do vary both, frequency and temperature, the focus o f study is
often systems w ith lim ited water or non-agitated systems measured at low frequencies and at temperature
ranges below their glass transition. A preferred method would take into account both the temperature- and
frequency-dependence o f the parameters o f one o f several theoretical models that can be used to predict the
dielectric properties. This would lead to a better understanding o f the nature o f the possible changes in
dielectric properties and a more effective means to predict them, rather than to sim ply describe changes in
their behavior. Such knowledge could then be extended to other sim ilar food systems to help w ith the
design o f food products and processes. W ith that in mind, the objectives o f this study were:
1.
To develop a technique to measure the dielectric properties o f well-m ixed liquids in-line
during heating and cooling;
2.
To measure the dielectric properties o f aqueous starch suspensions (both w ith and without
sodium chloride) during gelatinization as functions o f temperature and frequency;
3.
To fit the measured dielectric data to the Debye-Hasted model w ith an additional lowfrequency relaxation, and to study the temperature-dependence o f the fitted parameters.
4.
To study the effect o f added sodium chloride on the dielectric properties o f a waxy maize
starch suspension.
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149
3.4 Materials and Methods
In order to achieve the objectives o f this work, an experimental set-up was designed to heat, cool,
circulate, and measure the dielectric properties o f starch suspensions. The most important component o f
the set-up was the network analyzer that was used to make the dielectric measurements. A measurement
protocol was developed to allow calibration o f the instrument and measurement o f the suspensions at
continuously varying temperatures. Finally, a series o f computer programs was written to analyze the raw
data obtained from the experiments.
3.4.1 Materials
Three starches were chosen fo r the experiments, and were kindly supplied by two manufacturers:
the National Starch and Chemical Company o f Bridgewater, New Jersey, and Penford Food Ingredients
Company o f Englewood, Colorado. The names o f the starches, as w ell as some general inform ation about
them, are summarized in Table 3.6 below. Values listed as “ approximate” by the manufacturer are
represented w ith the symbol
before the number; other values were not available from the manufacturer.
Table 3.6 Data about Starches Used in Experiments
Type o f
Starch
National
Starch
&
Chem.
waxy
maize
AMIOCA™
common
MELOJEL® National Starch & Chem.
com starch
Penford
potato
PenCook™ 10
Starch Name
M anufacturer
Ash
M oisture
Content C ontent
~ 11%
-
Batch
GG5651
~ 11%
-
DG-8652
13-19%
-0.5%
control number 2895
Since the moisture contents listed by the manufacturers were approximate, the actual moisture
content o f each batch o f starch was determined using AACC Method 44-19 (1980). These moistures were
taken into account to prepare 3% (w /w ) starch suspensions for each experiment. Water for the suspensions
was purified by a multi-cartridge de-ionization system that provided water w ith a resistivity o f 15 to 18
M Q. This de-ionized water was boiled under vacuum to de-aerate it, and cooled in airtight containers. The
de-ionized and de-aerated water was used w ithin a few hours after preparation. Enough sodium chloride
was added to some o f the waxy maize starch suspensions to achieve a salt concentration o f 2% (w/w). The
starch and sodium chloride concentrations were chosen because they are in the range o f a variety o f foods
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150
like gravies, soups, sauces, and puddings. Another reason was that the cooked suspensions at 3% starch did
not generate excessive pressures as they were pumped through the apparatus described below.
Additionally, previous work conducted by researchers in this author’s laboratory found that beyond 2%
NaCl, further increase in sodium chloride concentration had little effect on the microwave heating profiles
o f liquids (Anantheswaran and Liu, 1994).
3.4.2. Apparatus
Figure 3.36 shows a schematic diagram o f the experimental setup, and Figures 3.37a and 3.37b are
photographs o f two different views o f that setup. The rectangular object on the bottom right o f Figure 3.36
represents the Tappan “ Space Saver” microwave oven (model 56-2277-10, Tappan Appliances, Columbus,
OH), the interior o f which is shown in the photograph in Figure 3.38. Two holes were drilled into the top
o f the le ft oven w all so that plastic tubing could be inserted. This tubing connected the heating/cooling
vessel inside the oven to the rest o f the apparatus outside the oven. However, as the experiments
progressed, it was found that the plastic tubing began to m elt where it crossed the oven w all, so that section
o f plastic was replaced w ith glass tubing. Unfortunately though, even after that problem was corrected, the
plastic tubing and connections inside the oven s till melted, so they were wrapped in moist paper towels so
that the water, rather than the plastic, would absorb the microwaves.
A beaker o f ballast water was placed inside the microwave oven in a marked position, and the
amount o f water was adjusted so that the oven would provide a heating rate o f 5°C/min to the circulating
fluid. The electric power supplied to the oven was regulated by a Sola power conditioner (EGS Electrical
Group, Sola/Hevi-Duty, Rosemont, IL 60018) to lim it the voltage fluctuations. A variable transformer
(Powerstat type 3PN126DP124202, Superior Electric Company, Bristol, CT 06010) was used to control the
voltage supplied to the oven at 115V.
Throughout the experiments, the voltage was monitored by a digital
voltmeter. The 115 V setting has been used in most o f the previous work in this author’ s laboratory to
prevent voltage fluctuations from surpassing 120 V, which could possibly damage the magnetron o f the
microwave oven. This danger was almost eliminated by using the power conditioner, but the 115 V was
kept to allow continuity w ith the previous research.
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151
wall outlet
power
conditioner
computer
B
network analyzer
voltage
regulator
data logger
thermocouple
wires
multimeter
peristaltic pump
I I I I El
power strip
microwave oven
dielectric measurement cell
coaxial probe
heating/cooling vessel
beaker of ballast water
Figure 3.36 Diagram of experimental set-up for the measurement of dielectric properties of liquids during
heating.
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152
Figure 3.37a View of experimental setup with network
analyzer on the left.
Figure 3.37b View of experimental setup with
microwave oven on the right.
The pump schematically shown in Figure 3.36
was a Masterflex peristaltic pump w ith an I/P pump
Company, Vernon H ills, IL ). It pumped the flu id from
the heating/cooling vessel into the bottom o f the test
measurement cell and out through its top. This
measuring cell (shown in Figure 3.39a) was customFigure 3.38 Interior of Tappan microwave
oven with heating/cooling vessel, tubing, and
beaker of ballast water.
Inc., Coming, N Y), and it was designed to provide
thermocouple
r
sealing
clamp
coaxial
probe
in
Figure 3.39a Close-up of dielectric measurement
cell.
Figure 3.39b Dielectric measurement cell with
insulating cover.
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153
adequate m ixing and to closely fit the m ounting nut o f the dielectric probe. Figure 3.39b shows the cell
w ith its insulating cover made from fiberglass and duct tape to minimize heat loss. Sim ilar insulating
covers were also used to prevent heat loss from most o f the tubing and connections.
The dielectric probe used to measure the complex perm ittivity was a Hewlett-Packard model HP
85070B (now Agilent Technologies, Inc., Palo A lto, CA 94306). The probe was secured to the
measurement cell w ith its mounting nut and a rubber gasket, and on the lower end it was connected to a
Hewlett-Packard network analyzer (model 8752C) by a rig id coaxial cable. Both the cable and the stand
holding the cell were securely attached to the desk on which they sat by duct tape to avoid any movement
that would invalidate the instrument’s calibration. The network analyzer was connected to the controlling
computer that ran the instrument through software provided w ith the dielectric coaxial probe measuring kit.
Two T-type thermocouples (Omega Engineering, Stamford, CT) were used to measure the
temperature o f the circulating flu id in two places: just outside the microwave oven and inside the
measurement cell, as shown in Figure 3.36. The thermocouples were introduced through rubber septa that
were tightened using compression fittings on brass “ tees” which were connected to the system’s plastic
tubing. The position o f the thermocouple in the measuring cell was adjusted so that it would be as close as
possible to the coaxial probe without interfering w ith the measurements. This was assured by performing
test measurements w ith and without the thermocouple, and adjusting its position o f until both measurments
were the same.
The thermocouples were connected to
a data logger (model 2 IX , Campbell
Scientific, Inc., Logan, U T 84321-1784) that
measured the temperatures and provided a
“ time stamp” for these measurements. The
data logger was programmed to measure the
temperatures every 10 seconds, and it was
connected to the controlling computer that
recorded the time-temperature data with a
Figure 3.40 Circulating bath.
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154
program written in Qbasic 4.5 (M icrosoft Corporation, Redmond, W A). This program is shown in
Appendix C. Cooling was achieved by submerging the heating/cooling vessel into a refrigerated
circulating bath containing 70% propylene glycol in water as shown in Figure 3.40 (model 1157, VW R
International, West Chester, PA 19380). The circulating bath was set at a temperature o f -30°C.
3.4.3 Calibration
Before every experiment, the network analyzer-coaxial probe combination was subjected to a
three-step calibration, as suggested by the manufacturer. For the first two steps o f the calibration, the
standard procedure described in the network analyzer’s manual was used. The first step was a calibration
against air, since air has approximately the dielectric properties o f a vacuum, e/ = 1 and s r" = 0. The
second step was a calibration against a m etallic short circuit block that was supplied in the coaxial probe
kit.
For the third step, a modification o f the manual’ s procedure was used. The objective o f this third
step was to obtain calibrations for the flow ing system as the temperature varied continuously. The flow
system was assembled and fille d w ith de-ionized, de-aerated water. This water was taken from a
previously weighed quantity, so that it could be used later to prepare the starch suspensions (see below).
The software was set to sweep between 0.3 and 3 GHz, which was the entire range available to the network
analyzer. Then the water circulation was started, and the temperature cycle was initiated. This temperature
cycle (which was the same as that used for the actual experiments) increased from 30 to 80°C during the
heating cycle. A fter 80°C, the heating cycle ended and the cooling cycle began by removing the
heating/cooling vessel from the oven and submerging it into the circulating bath set at -30°C. During the
cooling cycle the temperatures decreased back down to 30°C. Throughout the temperature cycle, when the
thermocouple indicated that a target temperature had been reached, the network analyzer was calibrated
w ith dielectric perm ittivity values for the 150 uniform ly spaced frequencies between 0.3 and 3 GHz. These
calibration files were saved in the computer in increments o f 5°C.
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155
3.4.4 Water reference measurements
A fter the system was calibrated, the water that had been used for the third calibration step
remained in the apparatus so that its complex perm ittivity could be measured as a reference. The
calibration data for 30°C o f the heating cycle was loaded into the computer, then the water’s complex
perm ittivity at 150 uniform ly spaced frequencies between 0.3 and 3 GHz were measured at that
temperature. A fter the measurements were performed at 30°C, the dielectric measurement files were saved,
and the calibration files corresponding to the next target temperature o f 35°C were loaded from the
computer into the network analyzer. Then a frequency sweep was started in the network analyzer once
again. This procedure was repeated throughout the entire temperature cycle: when each o f the target
temperatures was reached, a frequency sweep was started in the network analyzer; after each temperature,
the dielectric measurement files were saved, and the calibration files corresponding to the next target
temperature were loaded from the computer into the network analyzer. Thus, the water’s complex
perm ittivity at 150 uniform ly spaced frequencies between 0.3 and 3 GHz was systematically measured at
5°C intervals from 30°C to 80°C, and 75°C to 30°C.
3.4.5 Starch Suspension Measurements
Before the complex perm ittivities o f the starch suspensions could be measured at the same
frequencies and temperatures as the water measurements, the starch had to be carefully added to the waterfille d system. In order to avoid air bubbles, the plastic tubing inside the microwave oven was disconnected,
and the two resulting ends were connected to tubing extensions and inserted into an Erlenmeyer flask. This
flask contained the rest o f the weighed water that was used fo r calibration and reference water
measurement (see above). W ith the water flask thus connected to the rest o f the system, the pump was
slowly started, and any entrapped air was carefully purged from the apparatus. It was necessary to ensure
that a ll bubbles were removed because they greatly interfered w ith the measurement and were not visible
through the translucent starch suspension. A measured amount o f starch was added to the Erlenmeyer
flask, and the suspension was circulated through the system to achieve homogeneity before the microwave
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156
oven was started to heat the suspensions. A fte r a ll the starch was added and was completely mixed, the
tubing was re-connected in its original configuration, and the measurements were carried out as before.
Three experiments were performed fo r water, fo r suspensions o f waxy maize, common com
starch, and for a mixture o f waxy maize and sodium chloride. The only exception was potato starch. The
rapid increase in viscosity that potato starch underwent during gelatinization (that w ill be discussed later)
may have produced mechanical oscillations in the apparatus. This occasionally caused the connections to
burst apart, invalidating the experiment. In the end, four experiments were completed for potato starch, but
prelim inary data analysis discovered that only two o f those sets were usable. The data collected from each
experiment included network analyzer measurements o f the complex dielectric perm ittivities at 150
frequencies, 11 heating temperatures (except fo r waxy maize-NaCl which was only heated up to 75°C and
therefore had only 10 heating temperatures), and 10 cooling temperatures. The dielectric measurement
files contained a time stamp (provided by the computer’ s clock that was previously synchronized w ith the
data logger’s clock). Since human reaction tim e was involved in the initiation o f the frequency sweeps, the
temperatures at which the network analyzer measured the dielectric properties did not correspond exactly,
in general, to the target temperatures. Compensation was made fo r this small discrepancy during data
analysis, as w ill be explained below.
3.4.6 Determination of Model Parameters
The software that was used included M icrosoft® Windows XP Professional Upgrade (M icrosoft
corporation, Redmond, W A, 98052-6399), M icrosoft® Excel 97 SR-2 (M icrosoft Corporation) and
M A TLA B ® student version 6.0.042a, release 12 (The MathWorks, Inc., Natick, M A , 01760-2098). The
latter was used to fit the data. This software is especially useful to store and manipulate data in the form o f
multi-dimensional matrices. For this work, the dielectric data was stored in 3-dimensional matrices format
w ith rows o f frequency, columns o f measured temperatures, and separate “ pages” containing the real e/
data and the imaginary
data. In addition, it can solve m ultiple simultaneous nonlinear equations by
using the Optimization Toolbox, version 2.1, release 12. Since the real and imaginary components o f the
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157
complex perm ittivity were m utually dependent in the chosen dielectric model, this approach was necessary
for the estimation o f the parameters.
The original plan for analyzing the data was to model the waxy maize, common com, and potato
starch suspensions according to the Debye equation, and to model the waxy maize-NaCl suspensions w ith the
Debye-Hasted model to take into account the increased conductivity o f the salt. However, preliminary plotting
o f the data revealed that both the relative dielectric perm ittivity and dielectric loss for all starches exhibited an
abnormal curvature at low frequencies. Both phenomena could be explained w ith the presence o f a low
frequency peak o f the kind described in the literature for a number o f polymeric materials (Mashimo et al.
1987, Mashimo et al. 1992, M iura et al 1994, M iura et al 2003, Einfeldt et al. 2001, Lu, 2005). The curvature
in the dielectric loss data could also be explained w ith the presence o f conductivity in the suspensions, in the
form o f electrolytes. Unfortunately, the effects could only be positively differentiated having data at the
frequencies corresponding to the hypothetical low frequency peak. It was hoped, however, that even though the
frequency range would most probably be too narrow to give a final answer, some light might be shed on the
matter by fitting a model incorporating both effects to the data collected in this work. For that reason, the
Debye-Hasted model with two relaxation peaks was used.
* =
) +
[l+O O Ti)]
(Ss2- 2 S c - y ) _ _ J a _
[1+ (|ff>T2)]
S0lO
The goal o f subsequent data analysis was to find the m atrix o f the scaled dielectric parameters
(sim ply called the a m atrix) for the Debye-Hasted model w ith two peaks. This m atrix contained one row
per test temperature, and six columns w ith the follow ing parameters, shown schematically in Figure 3.41:
1. a, = £s, - e s2(the low-frequency relaxation strength), where esl is the staticdielectric constant
fo r the lower-frequency dipole rotation (dipole rotation peak #1 in Figure 3.41) and es2 is the
static dielectric constant for the higher-frequency dipole rotation (dipole rotation peak #2 in
Figure 3.41);
2. a2 = (lx lO 9)!], where t, is the relaxation time for thefirst (lower frequency) dipole rotation;
3.
a$= zs2 /1 0 4. a4 = e oo2, the value o f the dielectric perm ittivity at high frequencies; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 5. a5 = ( I x l012) i2, where x2 is the relaxation time for the second (higher frequency) dipole rotation; 6. a6 = ( lx l03)(equivalent salt concentration) for the starch suspensions w ithout added salt, referring to the equivalent salt concentration that corresponds to the conductivity peak, or a6= 10(equivalent salt concentration) fo r the starch suspensions w ith added salt. Dipole rotation peak #1 52 Dipole rotation peak #2 Conductivity peak CO b) 0 10 ' ,2 10 4 10 10 10 10 12 1014 frequency (Hz) Figure 3.41 Parameter definitions for the Debye-Hasted Model with Two Peaks. It was necessary to scale all o f the parameters before the iterative process began because o f the variation among them in terms o f orders o f magnitude. A m atrix o f in itia l estimates (called a0), had to be supplied to start the iterative process. The iteration, subject to certain convergence criteria and bounds (which w ill be explained in greater detail later), was carried out through M ATLAB®’ s optimization technique. Several computer programs were written in M ATLAB to accomplish the various steps o f the iteration, including a main program that recursively processed the data by repeatedly calling on different subfunctions. These programs and accompanying flowcharts are listed in their entirety in Appendix E, but Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 Figure 3.42 is a schematic overall view illustrating the flow o f the data analysis and how certain programs are “ nested” inside o f one another. correcttable th2o recursivehc th2o LSQDHTP "th2o tauw SDHTP DHTP salt density Recursiveresults Figure 3.42 Diagram showing computer programs and their subfunctions. Programs encircled by dotted lines are optional. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 3.4.6.a “ readnatable” The in itia l data processing steps were carried out by the program “ readnatable,” shown in flowchart form in Figure D .l. The program “ readnatable” (named such because it read the data from the network analyzer files and put in a table) opened the data logger’s temperature files and the network analyzer files at every temperature fo r a specific starch experiment. Line by line (each line corresponded to a time o f measurement), it extracted the data regarding the temperature in the measurement cell and the exact date and tim e at which it was measured. The temperature data was stored in the T21x vector, and the date-time inform ation was stored in the time21x vector. The program then opened the network analyzer files for both starch and water experiments, the names o f which were based on the target temperatures. Beginning w ith the heating cycle and ending w ith the cooling cycle, the program read each data line sequentially and extracted the data regarding the date and time o f the measurements and the measured values o f dielectric properties. This data was assigned to variables w ith names that included specific letters to indicate whether the measurements pertained to water (w) or starch (s), and whether they were obtained during the heating (h) or cooling (c) cycle. Thus, the measured perm ittivity and loss data was stored in matrices esmeash and esmeaSc for the starch files, and in matrices £wmeash and swmeasc fo r the water files. The time data was stored in vectors timenah and timenac. Because the network analyzer’ s sweep through at all 150 frequencies took time, the actual measurement time was considered to be about 5.5 seconds after the middle o f the sweep. A fter assigning the data from a given temperature to the matrices, the program closed the current starch and water files before moving on to the next temperature. When a ll o f the data was assembled, “ readnatable” compared the time o f the network analyzer files (timenah and timenac) w ith the time and temperature o f the data logger {time21x and T21x), interpolating to determine the actual temperatures (Th and Tc, respectively) at which the network analyzer had measured the dielectric properties. 3.4.6.b “interpolate” There was a concern (mentioned earlier in this section) that, due to human reaction time, the temperatures at which the network analyzer took measurements might not necessarily be those recorded by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the thermocouples. Although it turned out that they typically were w ithin 0.2°C o f the target temperatures (30°, 35°, 40°, 50°, 55°, 60°, 65°, 70°, 75° and 80°C), an effort was made to avoid the accumulation o f errors for repetitions o f the experiments. Thus, the program “ interpolate” (Figure D.2) was w ritten to interpolate the dielectric data extracted from the network analyer files by “ readnatable” to the target temperatures. The target heating and cooling temperatures were contained in the vectors Thi and Tci, respectively. The program then cycled through a ll 150 frequencies, interpolating w ith Th, Tc, esmeash, and Ssmeasc to find the dielectric data values that correspond to the target temperatures Thi and Tci. These interpolated measured dielectric perm ittivity and loss values o f starch for the heating and cooling cycles were stored in the matrices esmh, and esmci. Figure 3.43 illustrates graphically how the linear interpolation was done. Interpolation of temperature data at 3 different frequencies o 25 30 35 40 45 50 55 60 65 70 75 80 85 Temperature (°C) o fn+1 (measured) - • — fn+1 (interpolated) □ fn (measured) fn (interpolated) fn-1 (measured) fn-1 (interpolated) Figure 3.43 Interpolation of temperature data at 3 different frequencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 3.4.6.C “correcttable” In order to obtain reference measurements, the relative dielectric properties o f pure water were in itia lly measured. A comparison was then made between these measured values from the network analyzer and the theoretical values fo r water (obtained from the program “ th2o” described below .) The program “ Correcttable” (Figure D.3) defined correction factors for both the heating and the cooling cycles as: COrrh —Swt Cwmeash [3.96a] COrrc Cwmeasc [3.96b] £wt “ Correcttable” then applied these correction factors to the data at each temperature and frequency in the matrices ssmhi and esmci. By doing so, it created new matrices o f corrected, interpolated data fo r the heating and cooling cycles, £/,„ and ecci: Shci = eSmhi + corrh [3.97a] + corrc [3.97b] Scci = E sm ci Figure 3.44 illustrates the results o f the correction graphically. (It is worth mentioning that the option Correction Factor Determined from Water Data o 0.0 0.5 2.0 2.5 3.0 Frequency (GHz) Measured \«lues for w a t e r ---------Theoretical values for water Measured values for s t a r c h ---------Corrected values for starch Correction factor Figure 3.44 Correction factor determined by water data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 contained in the program “ Correcttable” for data that had not been interpolated was never used.) 3.4.6.d “th2o” This function (Figure D.4), which takes temperature and frequency as arguments, is called by the “ correcttable” program. It contains a lis t o f the theoretical values (as reported by Hasted, 1972b) o f the static dielectric constant, optic dielectric constant, relaxation time, and the alpha variable from the ColeCole model at 30°, 40°, 50°, 60°, and 75°C. The function interpolates this data to determine the values o f the parameters that correspond to the target temperatures. (This interpolation was necessary because o f a lack o f data in the literature about the temperature-dependence o f dielectric parameters o f water.) The interpolation/extrapolation was conducted by M A T LA B ’ s Piecewise Cubic Hermite Interpolating Polynomial (abbreviated PCHIP). The PCHIP function was chosen over the spline function, even though both use cubic polynomials for the interpolation. According to the explanation given in M A T L A B ’ s HELP file , spline is the more accurate function i f the data follow s a smooth curve. However, a graph o f the data given in Hasted (1972b) indicated that not a ll o f the data followed such a curve. Therefore, the PCHIP function seemed to be the better choice because o f its a bility to maintain the shape and the m onotonicity o f the data, as w ell as to avoid overshoots and to minimize oscillation when follow ing data that is not smooth. The program “ th2o” then used this interpolated data in the Cole-Cole model to calculate the theoretical relative complex perm ittivity for water at the input temperature and frequency, storing the values in the m atrix ewl. 3.4.6.e Bounds In order to prevent the iterative process from returning unrealistic answers, upper and lower bounds were imposed on the scaled parameters in the problem. D ifferent possibilities were written into the program throughout its tria l runs, and those not chosen to be used in the final runs were left in the program but were set aside as comments so as not to interfere w ith the flow o f the program. The upper bound for es, - es2 (the scaled difference esi - es2) was set to 50 in this study. The lower bound for a, was set equal to 1, based on the results o f Mashimo et al. (1987) and M iura et al. (1994, 2003) for certain long-chain biopolymers. The lowest value o f es, - es2 that they reported in this series o f papers was 1.04 for “ human Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 red cell ghost (25°C, 0.192 wt% )” (Mashimo et al., 1987). The values o f esi - es2 for a ll o f the other biological materials were much greater than 1. The lower bound for a2 (the scaled t,) was set to 0.53. This was equivalent to the unsealed value o f 0.53xl0"9 s which corresponded to the value fo r the lowest frequency investigated in these experiments, 0.3 GHz. Its upper bound was set to in fin ity, thereby not lim itin g the computer’s calculations. The unusual range 0.1818*n < n < 1.8182*n was chosen to be the lim its on both a3 and as (scaled scaled x2, respectively) so that there is one order o f magnitude between the lower bound (0.1818*/!) and the upper bound (1.8182*«), w ith the range centered on n since (0.1818 + 1.812)12 = 1. The number n could be chosen to be either the in itia l estimate or the value for water. The lower and upper lim its imposed upon a4 (the scaled eoo2) were 99.9% and 100.1%, respectively, o f the theoretical value o f the perm ittivity o f water at 1 x 10100 Hz. The lower lim it on a6s (the scaled equivalent salt concentration) was chosen to be the Matlab-reserved variable called realmin, the smallest positive floating point number, w ith a value is 2.225xl0 '16. The upper lim it on a6s was set to in fin ity. 3.4.6.f Convergence Criteria As mentioned earlier, in itia l estimates in the form o f the m atrix a0 had to be supplied to begin the iterative process. Each “ pass” through the recursive algorithm returned new values o f the parameters. Some form o f comparing the new values w ith the old was necessary to determine i f the iteration had converged, or i f more “ passes” were necessary. Thus, two convergence criteria and a maximum number o f iterations were imposed upon the system. The first convergence criterion was for the sum o f the squares o f the errors (SSE), i.e., the sum o f the squared difference between the measured, interpolated, and corrected data and the fitted values from the model. This criterion indicated i f the a m atrix calculated for an individual experiment was an improvement over the a0 m atrix used to start the iteration. The second convergence criterion was the mean convergence criterion. This indicated i f the means o f the a matrices fo r a ll the experiments converged. The benefit o f a low SSE criterion would allow each individual experiment to iterate more until the SSE reached a low value, but this low SSE value would come at the expense o f the results o f each experiment getting further away from the other experiments. The benefit o f a low mean convergence criterion is that it could permit the results for all the experiments o f a single starch Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to be more sim ilar. However, a decrease in either criterion would lead to more iterations and more time required for calculations, w ith the possibility that the number o f iterations would increase to the maximum level without ever achieving convergence. It was found that the final results for the iterative process were more influenced by the in itia l estimates o f the scaled quantities ai0 = (e^ - e^),, and a2o = t i 0 and the stringency o f the convergence criteria, than by the other parameters in the a0 m atrix. Thus, a procedure was undertaken to investigate this influence more thoroughly. Different orders o f magnitude o f the two convergence criteria, as w ell as different possible in itia l estimates o f (es] - es2) and xh were stored in separate “ combination” files fo r each starch. The combination files were run to calculate the intermediate a matrices, and the results were examined. I f the results exceeded or seemed “ pegged” at the pre-determined lower and/or upper lim its for the values o f (Eji - es2) and the starting combination that led to that result was immediately eliminated. Once those combinations were eliminated, the SSE o f the remaining results were examined, and the combinations that best fit the data were considered to be viable possibilities. Sometimes the results that yielded the lowest SSE values were the ones that very near or at the lower bound imposed on the algorithm. In some cases, the parameters at lower temperatures were almost constant and exactly at the lower bound, im plying that, given the opportunity, the algorithm probably would have lowered the parameters in search o f a better fit. In these cases, the set o f parameters was discarded as inappropriate. In this way, the range o f the (esl - es2) ’ s and t f s in these combinations were then successively narrowed down. A fter some prelim inary runs, a value o f 0.002 for both convergence criteria was chosen to be used for all starch experiments as a compromise between stringency and running time, as lower values o f the criteria did not seem to significantly improve the accuracy o f the results. 3.4.6.g “ recursivehc” “ Recursivehc” is the main program employed in the data analysis (see flowchart in Figure D.5). The letters “ he” refer to the fact that this program used both heating and cooling data. The purpose o f this program was to take the corrected and interpolated data from each starch experiment and to solve recursively for a solution to the a m atrix o f parameters for the Debye-Hasted model w ith two peaks. This Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 solution involved continuous iteration u n til the mean square error o f each experiment and the mean o f a ll the experiments met two specific convergence criteria, w ithout exceeding the pre-set lim it on the maximum number o f iterations. D uring this process, “ recursivehc” called several functions, which, in turn, called other sub-functions. In each iterative step, the previous a m atrix became the new in itia l estimate (a0) matrix. Starting w ith the in itia l a„ m atrix for the complete set o f experiments fo r a particular starch, “ Recursivehc” began an iterative outer loop that would run continuously until the mean o f those experiments (designated ameari) converged. (A lim it was imposed on the number o f iterations, so i f that lim it was reached, “ recursivehc” immediately stopped the loop and simply returned the values in the current a m atrix.) W ithin the outer loop were two other nested loops (referred to here as the middle and innermost loops) that sought convergence on the SSE, the m atrix which contained rows for each iteration and columns for each experiment. Using amean as a0 fo r the first experiment, the middle loop called upon the function “ LSQDHTP” (explained below) to calculate a new a m atrix and SSE. As long as that SSE was greater than the SSE convergence criterion, the algorithm entered the innermost loop. Once again that loop called upon “ LSQDHTP” to calculate a new a m atrix and SSE. It then calculated a quantity called SSEConv to compare the previous value o f the SSE w ith the current value to see how much it had changed during the iteration. That is, SSEConv was the coefficient o f variation o f the SSE values, defined as „„„„ standard deviation o f the previous SSE and the current SSE SSEConv = ---------------------------- j - r -------- . * OCI,— j - r ------------------------------------mean o f the previous SSE and the current SSE [3.981 1 1 “ Recursivehc” then checked to see i f this SSEConv was s till greater than SSEConvCrit. I f so, it resumed the innermost loop; i f not, it returned to the middle loop and repeated the procedure fo r the next experiment. When the SSE’s converged for a ll o f the experiments, the algorithm calculated a new amean. It then calculated a quantity called comMean to compare the previous value o f the amean w ith the current value to see how much changed had occured during the iteration. That is, comMean was the coefficient o f variation o f the amean values, defined as , . convMean= standard deviation o f previous amean and current amean -------------------------------- 7------ . mean o f previous amean and current amean Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F3.991 L J 167 “ Recursivehc” then checked to see i f this convMean was s till greater than MeanCovCrit and i f the maximum number o f iterations was s till w ithin the number permitted. I f so, the program returned to the outer loop to repeat the iterations; i f not, it exited the loop and called upon the function “ recursiveresults” (described below) to display the solution. This whole cycle was repeated fo r each combination o f in itia l estimates for at and a2. 3.4.6.h “ LSQ DHTP” The “ LSQDHTP” function (Figure D .6) uses the least-squares method to find an estimate o f the parameters for the Hasted-Debye model w ith two peaks, given the measured dielectric data as a function o f temperature and frequency. Thus, this is the program that actually calculates the intermediate (and final) a matrices. It can also return some statistical outputs. I f “ LSQDHTP” does not receive the firs t a0 m atrix from the “ recursivehc” program, it w ill assemble one. The in itia l estimates fo r esl- es2 and i i were based on the w ork o f Mashimo et al. (1987) and M iura et al. (2003). The in itial estimates for es2 and x2 were obtained from linear regression to fit a simple straight line through the data, according to equation 3.56 and the method used by Haynes and Locke (1995) that was mentioned in section 3.2.6. (The program also contained the option to set x2 equal to the theoretical relaxation time for for water by calling on the function “ tauw” shown in Figure D.7.) The in itia l estimate for sx2 was equal to the theoretical value fo r water at a frequency o f lx lO 100 Hz. The effective concentration o f sodium chloride was assumed to be constant throughout the experiments, and it was in itia lly set to an arbitrary value o f 5. “ LSQDHTP” sets the previously mentioned lower and upper bounds for the iteration o f the parameters. It then uses M A T LA B ’s optim ization toolbox and the pre-defined function LSQCURVEFIT to perform a least squares curve fit on the subfunction “ SDHTP.” “ LSQDHTP” returned the resulting a matrix and SSE to “ recursivehc” , and then displayed and plotted those results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 3.4.6.1 “tauw” This function (Figure D.7) is included as an option. It may be called by the “ LSQDHTP” program to determine the relaxation tim e o f water at the given input temperature to use as a5o. It uses pchip to interpolate the relaxation time data given by Hasted (1972b). 3.4.6.j “SDHTP” The name o f the function “ SDHTP” is an abbreviation for scaled Debye-Hasted model w ith two peaks. It is used as an argument its e lf in LSQCURVEFIT, and it takes as arguments frequency, temperature, and the a m atrix o f scaled parameters. It “ un-scales” these parameters, restoring them to their original orders o f magnitude so that they could be used by the sub-function “ DHTP” to calculate the complex perm ittivity. Then “ SDHTP” separated the real and imaginary parts o f the complex perm ittivity into the dielectric perm ittivity and loss, respectively. The SDHTP flowchart is found in Figure D.8. 3.4.6.k “DHTP” The name o f the function “ DHTP” is an abbreviation for the Debye-Hasted model w ith two peaks. This function is called by the “ SDHTP” function, and takes as inputs the a m atrix (w ith parameters in their original orders o f magnitude), frequency, and temperature. This model used the correction factor 2 SC described in section 3.2.10, where C is the molar concentration o f an electrolyte. For this study, it was assumed that a ll o f the starch suspensions would exhibit some conductivity. The value o f C used in this equation was that o f a simple NaCl solution that would exhibit the same conductivity as the starch suspension. As mentioned in section 3.2.10, the value o f 8 for a sodium chloride solution is -5.5. “ DHTP” then calculated the value o f the relative complex perm ittivity according to equation 3.95. The DHTP flowchart is found in Figure D.9. 3.4.6.1 “SaltDens” The “ DHTP” function (Figure D.9) uses NaCl concentration in m olarity. However, during the experiments, the NaCl was constant only in a (w/w) basis. In order to keep the equivalence w ith NaCl Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 solutions, the density o f NaCl in water solutions was needed for the conversion from (w /w ) to (w /v) concentration. The program “ SaltDens” (Figure D.10) was w ritten to determine the density o f NaCl solutions (in g/m l) at different concentrations and temperatures using regression equations based on concentration and temperature. It takes the temperature, concentration, and units o f concentration (i.e., either weight percent, m olarity, or m olality) as arguments. The regression equations were based on data obtained from the CRC Handbook (Lide, 1999-2000). The Handbook's inform ation on the specific volum e o f aqueous sodium chloride solutions as a function o f temperature and m olality, along w ith its inform ation on the density o f pure water at 10°C intervals between 0° and 100°C was used. (The data on pure water was necessary for interpolation because the converted concentration o f the solutions used in this study was less than 0.10 mol/kg o f NaCl, which was the lowest concentration for which specific volume inform ation was given in the Handbook.) The data was regressed using M icrosoft EXCEL’s Solver to find the parameters for an empirical equation that could be used to predict density o f sodium chloride solutions as a function o f temperature (T) and concentration (C). The resulting equation was o f the form: density = c0 + ct ,*T + c ^ T * 72) + cMC + cMi*(C eM1) + cM2*(C eM2) + cT3M3*(T eT3)*(C eM3) [3.100] where the constant, coefficients, and exponents change depending on whether the concentration was expressed in terms o f m olarity or m olality. The values used in each case are listed in the actual program “ SaltDens” given in Appendix D. The regression returned a value o f r2 = 0.999982391 for the case in which concentration is expressed in m olality, and r2 = 0.999936424 for the case in which concentration is expressed in molarity. 3.4.6.m “conductivity” The “ conductivity” function (Figure D .l 1) is called by the “ DHTP” function (described below). It takes the temperature in Celcius and the concentration in m olarity as arguments, and returns the corresponding value o f the conductivity (in S/m) o f the suspension. This function was necessary because, in order to calculate the complex perm ittivity, it had been assumed that all o f the ionic conductivity fo r the suspension could be expressed by the equivalent conductivity o f a simple solution o f just sodium chloride Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 and water. The effect o f the conductivity was introduced according to the Debye-Hasted equation (equation 3.80). The “ conductivity” function used data from three different sources. The first source was the International Critical Tables (Washburn, 1929), which listed the electrical conductivity o f aqueous NaCl solutions (for which 1 m ol = 1 equivalent) at concentrations o f 0.5, 1, 2, 5, 10, 20, 50, 70, 80, 100,200, and 500 m m ol/liter and at temperatures o f 18°, 25°, 50°, 100°, and 140°C. The data was given in terms o f A, which was defined as A = 106k/C k [3.101] = specific conductance in units o f (D cm )'1, and C = concentration in m illiform ula-w eights per lite r o f solution (m m ol/liter) Since the conductivity was the term o f interest for this study, equation 3.101 had to be rearranged as k = A C x 10"6 [3.102] and typed in an EXCEL spreadsheet to calculate the specific conductance. The second source o f conductivity data was Light (1984). The data o f interest from this paper was the resistivity o f water w ith traces o f sodium chloride impurities o f 1, 2, 10, 50, 100, and 1000 ppb at temperatures o f 25°, 50°, 75°, and 100°C. Before this inform ation could be used, the concentrations were converted to m olarity (by using the definition that 1 ppb was equal to 1 pg/kg), and the reciprocal o f all the resistivity values was calculated in EXCEL. The third source o f conductivity data was Bevilacqua, Light, and Maughan (2004). The data from this paper that was used was the conductivity o f pure water (in pS/cm) at temperatures between 20 and 100°C in increments o f 5. Because these three sources reported different concentration and temperature ranges, there were “ gaps” in the data. (For example, Washburn and Light give no conductivity data fo r temperatures in the range between 25° and 50°C, so their data could not be directly correlated w ith that o f Bevilacqua et al. in that range.) Therefore, the data had to be interpolated using PCHIP cubic interpolation to “ f ill in the gaps” and create a complete set o f conductivity data at temperatures between 20° and 100°C in increments o f 5°, and fo r concentrations that were in a roughly logarithmic sequence designed to stay as close as possible to the values actually given in the literature. That interpolated data was programmed into three matrices (one Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 for temperature, one for molar concentration, and one for conductivity in S/cm) in the function “ conductivity.” The function then interpolated this data using MatLab’s internal INTERP2 command (using pchip as before). 3.4.6.n “recursiveresults” This program (Figure D.12) plotted the results for the SSEConvCrit, MeanConvCrit, convMean, number o f iterations, and SSE. 3.4.7 Error Analysis and Goodness of Fit In an effort to measure the “ goodness o f fit” o f the model to the measured data, some error analysis was performed. As part o f that error analysis, a separate mean value was calculated for e /(f) and 8 r" (f) at every temperature. Three statistical quantities (the meanings o f which are illustrated in Figure 3.45) were then examined. The first, the “ error sum o f squares” (SSE), was calculated as the sum o f the squares o f the differences between every point and the regression line corresponding to each temperature for both e / and e," (this is the same SSE used in “ recursivehc” for the SSEconv). In Figure 3.45 that corresponds to the vertical solid lines. The second quantity, the “ total sum o f squares” (SSTO), was calculated as the sum o f the squares o f the differences between every point (either for e/ or e /') and its corresponding mean value. In Figure 3.45 that corresponds to the vertical dashed lines. The final SSE and SSTO that were reported are the sums o f all o f the above SSEs and SSTOs fo r a ll o f the s / and e /' regressions at every temperature. The third quantity, the “ regression sum o f squares” (SSR) is the sum o f the squares o f the differences between corresponding points on the fitted line and the mean. Because o f the partition o f sums o f squares, it is found that SSTO = SSR + SSE [3.103a] SSR = SSTO-SSE [3.103b] Therefore, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 172 In Figure 3.45, SSR would correspond to the length o f the dashed lines minus the length o f the solid lines. Then the R2 values, the coefficient o f m ultiple determination were calculated fo r each experiment o f each starch according to the follow ing equation: p2 _ exp SSRexn SSTOexp _ SSTOexp —SSEex„ SSTOexp SSE and SSTO fo r e’ at T, mean er' at T t SSE and SSTO fo r e’ at T- mean er' at T : SSE and SSTO fo r e” a t T, mean e” at T, SSE and SSTO fo re ” at T mean e” at T- Frequency • — Temperature 1 ■— Temperature 2 Figure 3.45 Illustration of the definitions of statistical quantities. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . L J 173 Some other quantities that were calculated for each experiment and are reported in section 3.5.2 were the number o f observations, the number o f parameters, and the mse. Each o f these is defined below: No. o f observations = (No. o f frequencies) * (No. o f dielectric properties)* (No. o f temps) [3.105] The number o f frequencies was 150, and the number o f dielectric properties was 2 (s / and s /') for a ll experiments. The number o f temperatures was 20 for the waxy maize-NaCl suspension, but 21 fo r the rest o f the starch suspensions. number o f parameters = 5 * (number o f temps) +1 [3.106] The coefficient o f “ number o f temps” in equation 3.106 comes from the fact that there were 5 parameters calculated for every temperature. The term “ + 1” on the right-hand side o f equation 3.106 refers to the one parameter that was calculated only once fo r each experiment, the conductivity. degrees o f freedom = number o f observations - number o f parameters mse= SSE j e(. ,----------degrees o f freedom [3.107] [3.108] L J 3.4.8 Data Analysis for waxy maize-NaCl suspensions Prelim inary analysis o f the data fo r the waxy maize-NaCl suspensions showed that this data was very different from the data for the other suspensions. To begin with, the relative loss factor for these suspensions was found to be over an order o f magnitude higher that the corresponding values for the other starch-water suspensions. This difference was not surprising since the conductivity o f the waxy maizeNaCl suspensions was considerably higher because o f the high electrolyte concentration, but it did indicate that the previous methods used to analyze the data might have to be adjusted for these suspensions. The prelim inary analysis also revealed that the fit fo r the suspensions w ith added NaCl was particularly poor. Upon further examination, it was discovered that the optimization algorithm tended to fit 8/ ' preferentially over Sr'. It was hypothesized that this occurred because o f the great difference in numerical values between Sr" and 5,' for this data. It was thought that corrections in the fitted 6/ ' would reduce the residual faster than corrections to the fitted 8/. Therefore, the algorithm produced an optimized NaCl concentration at the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 expense o f the other parameters, which were highly erratic and very different from the results fo r the previous suspensions. In order to compensate for the higher conductivity o f these waxy maize-NaCl suspensions and to avoid misleading results, a two-step approach to data analysis was followed. First, an in itia l “ low conductivity” parameter m atrix was determined by mathematically elim inating the effect o f most o f the NaCl from the data (see below) before performing the recursive optimization. Then, the NaCl was mathematically added back to the low conductivity parameter m atrix and this “ high conductivity” parameter m atrix was used as an in itia l estimate for a second recursive optimization. The result from this second optimization was the desired parameter m atrix w ith the fu ll effect o f conductivity. The low conductivity parameter m atrix was obtained by subtracting an em pirically found concentration o f NaCl from the data. This was achieved by reversing the effect predicted by the DHTP model. Recall that in the Debye-Hasted equations 3.78-3.80, the quantity 2 8 c was considered to be negative, so it was added to the static dielectric constant o f water to decrease it and to lower the total relative perm ittivity. Now, however, in order to reverse the effect, the correction term 2 8 c was subtracted from the total relative perm ittivity, thereby increasing e/. Accordingly, the values fo r s," were decreased by a/(cos0) (see equation 3.79). The relatively small effect o f the factor (1+ jcor) in the denominator o f the DebyeHasted model was ignored in this first step. The concentration o f NaCl was chosen to be as high as possible without making the &/' data reach zero at any frequency or temperature. The fit obtained w ith this procedure was much better than the fit obtained when the original data was directly used fo r the optimization, as demonstrated by a much lower SSE. The modified versions o f the programs (saltrecursivehc and saltLSQDHTP) are show in Appendix D. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 3.4.9 Broad Spectrum Dielectric Measurements A fter the original data set had been analyzed and modeled, there s till remained uncertainty about what type o f phenomenon was causing a slight upward curvature at the lower frequency range fo r both the relative dielectric perm ittivity, and the relative dielectric loss. A new set o f experiments that could better investigate the dielectric properties in the lower frequency range was proposed. These experiments were performed using a network analyzer (model HP 85 IOC, now from A gilent Technologies, Palo Alto, CA), kindly made available by the Center fo r Dielectric Studies at the Materials Research Laboratory o f the Penn State U niversity. This instrument had a broader frequency range (45 M Hz - 26.5 GHz) than the one used in the first part o f this study. It was used in conjunction w ith the same dielectric probe that was used in the first set o f experiments (HP 85070B, also from Agilent Technologies). The frequency range recommended by the manufacturer for this probe was 0.2 - 20 GHz. The lower lim it o f this range became a concern fo r the broad frequency experiments, but a prelim inary measurement o f water seemed to produce adequate results and the measurements were performed at the complete range for the starch suspensions. Waxy maize starch was chosen for these experiments, and once again, 3 % (w /w ) suspensions were prepared w ith deionized water. The experiments were only exploratory, and therefore the experimental setup was much simpler than the previous one described in section 3 .4 .2 . During the heating portion o f the experiment, the starch suspensions were sim ply contained in a 50-ml beaker, stirred w ith a stirring rod, and heated on a hot plate. The hot plate was accommodated on a “ lab jack” platform that allowed the hot plate to be lifted towards the probe. The temperature in the beaker was monitored by a thermocouple, and at the appropriate temperatures, the platform was lifted , submerging the dielectric probe in the hot starch suspension. A t this point, the network analyzer was instructed to take the measurements. Between the starch suspension measurements, the network analyzer was calibrated w ith deionized water at the appropriate temperature. A fter calibration, the dielectric properties o f water were taken to allow for data correction as detailed in section 3.4.6.C. A fter heating, the viscosity o f the suspensions increased visibly, and it became d iffic u lt to keep a uniform temperature in the 50-ml beakers. To solve this problem, a larger amount o f suspension was Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 prepared and heated in a 250-ml beaker. The suspensions were heated to 80°C and then cooled w ith ice to the target temperatures. Then a portion o f the suspension was transferred to an insulated 50-ml beaker, and the measurements were taken as before. D uring these measurements the heating rate was not controlled, but this was not thought to be important given the exploratory nature o f the experiments. The corrected dielectric data was fitted w ith the LSQDHTP program (discussed in section 3.4.6.h). It was noticed that the higher frequency data was very noisy and reached an overflow value in the equipment. Therefore the data was cropped at 20 GHz, instead o f the maximum 26.5 GHz achievable w ith the instrument. The upper and lower bounds fo r the fitted parameters were changed from those used for the narrow frequency fittin g as follows: a) the lower bound for the relaxation tim e at low frequency was set at 3.54 ns to correspond w ith 45MHz (the lowest frequency o f measurement); and b) the lower and upper bounds fo r equivalent sodium chloride concentration were set at 3.61x10-3 % and 5.57x10-3 %, respectively. The former value is 5% lower than the value corresponding to the lowest measured conductivity, 3.8xl0"3 % (as w ill be explained in the follow ing section), and the latter value is 5% higher than the value corresponding to the highest measured conductivity, 5.3x1 O'3 %. 3.4.10 Conductivity Measurements The broad frequency measurements were complemented w ith conductivity measurements intended to provide the range o f conductivities expected as the starch suspensions were heated. A Hanna Instruments model H I 8633 conductivity meter (Hanna Instruments Sri, Italy) was used to measure the conductivity o f the 3% waxy maize starch suspensions. This instrument had various ranges o f operation, and two o f them (0.0 to 199.9 pS/cm and 0 to 1999 pS/cm) were needed to measure the conductivity o f the starch suspension at a ll the temperatures o f interest. Once again, the starch suspension was prepared in a beaker, and a hot plate/magnetic stirrer was used to agitate and heat it from 25 to 80°C. A fter heating, the suspension was cooled w ith ice to 25°C. The conductivity probe and a thermocouple were immersed in the starch suspension and the conductivity was measured as the suspension reached the target temperatures. The equivalent sodium chloride concentration needed to achieve the measured conductivities at the target temperatures was calculated using the conductivity function detailed in section 3.4.6.m. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 3.4.11 Time-Temperature Superposition In order to obtain additional insight from the data from the narrow-frequency (0.3 - 3GHz) experiments, an attempt was made to use the time-temperature superposition (TTS) principle to (potentially) broaden the range o f the measurements. According to Olsen et al (2001), TTS can be applied i f a functions N and (p exist such that the response function e ((o ,T )*= N (T )c p [(o t(T )]. e(co)* vs. (e(co)* in this case) can be expressed as Graphically, this is equivalent to a translation o f the curves in a log-log plot o f frequency. In order to achieve an optimal translation o f the curves, a slightly m odified DHTP program (discussed in section 3.6.k) was used. In this case, a ll the rows o f parameters in the a m atrix were forced to be equal, and columns were added with m ultipliers fo r the frequency and for e ( co) * . The m ultipliers fo r the frequency, provided for horizontal translation in the log-log plot, and the m ultiplier for e ( co) * provided vertical translation. The function LSQCURVEFIT from MatLab was then used on this m odified function to optimize the one set o f parameters (esr£ S2, Ti, eS2, £*,, t 2, and NaCl concentration) and the set o f m ultipliers using a least squares error algorithm. 3.4.12 Statistical Procedures Statistical analysis o f e SI-£ s2, T i, e s2, and t 2were conducted w ith linear regression using M initab Release 11 for Windows (M initab Inc., State College, PA). The predictors used were temperature, temperature range (30°C<T<65°C, and T>65°C), cycle (heating or cooling), and kind o f starch (waxy maize, common com, potato, and waxy maize w ith added sodium chloride). Except fo r the temperature, the predictors were represented by indicator variables (Neter et al, 1990). The levels o f the predictors were as follows: Temperature (°C): = 30, 35,40,45, 50, 55, 60, 65, 70, 75, and 80 Temperature range: = 1 i f T>65°C, and 0 i f 30°C<T<65°C Cycle = 1 i f cooling cycle, 0 otherwise Common Starch = 1 i f starch is common com, 0 otherwise Potato Starch = 1 i f starch is potato, 0 otherwise Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 Waxy Maize w ith NaCl = 1 i f starch is waxy maize w ith added NaCl, 0 otherwise In this scheme, four kinds o f starch were represented by three indicator variables; the data fo r waxy maize corresponded to zero values o f the indicator variables fo r the other three starches. The coefficients for a ll predictors and their two-way and three-way interactions (except fo r interactions between starch indicator variables, which are always zero) were estimated using the “ regress” command in M initab. A dditionally, a 95% confidence prediction interval for the fitted values o f esi-Ss2, Ti, es2, and x2 were also obtained from the regression output. Using this regression approach, the dielectric parameters o f the different starch suspensions were compared w ith each other, and their behavior as a function o f temperature was studied. Changes in the dielectric parameters were studied in two different ways. First, differences in the values o f parameters at a specific temperature were compared using the 95% confidence prediction interval (i.e. the values were considered different at P<0.05 i f the intervals did not overlap). Secondly, the behavior as a function o f temperature was studied by comparing the slopes and constants (y-intercepts) o f the different curves at the different temperature intervals. The regression predictors that were used resulted in four different temperature intervals (30 to 60°C and 65°C to 80°C for the heating cycle, and the same intervals fo r the cooling cycle). Behavior was considered different between two starches or temperature intervals i f the slope, the constant or both were significantly different as indicated by the F-test in the regression analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 3.5 Results and Discussion 3.5.1. Recognizing the Differences Between the Debye Model and the Data Plotted against temperature and frequency in Figures 3.46a and b are the relative perm ittivity and loss curves, respectively for the waxy maize starch suspension (solid lines) during the heating cycle. Also given for the sake o f comparison are the theoretical curves for water (dotted lines) following a Debye model. (The plot fo r water was actually generated by using the Cole-Cole model, but the a values were so small, between 0 and 0.013, that the behavior essentially followed the Debye model, as explained in section 3.2.7). The corresponding plots for the other starch suspensions for both the heating and cooling cycles are given in Appendix H. These plots show averages for all repetitions o f the measured dielectric data (interpolated and corrected as discussed in section 3.4.6.b and c) for each starch. Based on the plots, it was hypothesized that the Debye model could not replicate the dielectric behavior o f the starch suspensions because two main differences are seen between the curves for water and starch. The first is that the relative dielectric perm ittivity tends to curve downward more noticeably fo r starch than for water as frequency increases, especially at low temperatures (Figure 3.46a). The second is that the relative dielectric loss o f starch suspensions tends to curve upwards at frequencies below 1 GHz, especially at higher temperatures (Figure 3.46b). When looking at the graph going from high to low frequency, it is noted that the relative dielectric loss appears to reach a minimum value before curving up at the lowest frequencies. Thus, the loss curve displays concavity at low frequencies. This behavior, which was exhibited by the loss curves o f all the starch suspensions, was compared w ith that o f water in Figure 3.29 during prelim inary analysis. Unlike the waxy maize curve in Figure 3.46b, the Debye model o f the relative dielectric loss curve for water asymptotically approaches zero at low frequencies and does not show any concavity. Thus, as already mentioned in section 3.4.6, the in itia l attempt to fit the data to a simple Debye model had to be abandoned. It was then suspected that the data might be showing the effect o f conductivity. Conductivity has been found to have a profound effect on the dielectric properties o f polymers in the presence o f water (Einfeldt, 2001; Lu, 2005). There are a few potential sources o f conductive behavior in the starch suspensions. First, pure water its e lf has some conductivity, but it is very low because o f the low concentration o f ionized species (the theoretical proton concentration is 10'7 m ol/L). However, excess Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 protons are highly mobile in water and can be transported through the formation o f the hydronium ion (H 30 +). The hydronium ion can then produce conductivity by various mechanisms, including classical hydrodynamic m otion or the transference o f a proton to a neighboring water molecule. These mechanisms are present in acid solutions and in hydrated polymer-electrolyte membranes or proteins (Komyshev et al., 2003). Conduction should also occur in starch suspensions through these mechanisms, since starch polymers can contribute some protons from the hydroxyl side groups (Oosten, 1982). Another possible source o f conductive species is the presence o f electrolytes in the native starch or le ft on the starch granules from their commercial processing. Starch analysis does normally render some ash content; Swinkels (1985) reports ash contents o f 0.1% for waxy maize and common com starches, and 0.4% for potato starch. However, not a ll o f this ash content is necessarily contributing to conductivity, since not a ll o f it is in the form o f ionic species. Conductivity is usually taken into account by introducing a term in the model, as done in this present work. The Debye model was m odified by adding the correction suggested by Hasted et al. (1948). The effect o f this correction fo r water is shown in Figure 3.47, that was obtained by theoretically calculating the dielectric properties o f water w ith a sodium chloride concentration o f 10"2 % (w /w ). The increased conductivity causes a concavity in the curve at lower frequencies: it displays high values o f the loss at extremely low frequencies, then it decreases to a minimum at approximately the lowest end o f the frequency range studied here, then finally it increases to form the relaxation peak. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 (a) e/ heating ------- • - —Wy. 803x10s 75- frequency (Hz) 7065 60 10 109 50 Frequency (Hz) Temperature (°C) 80 10° starch water (b) s," heating 50 Frequency (Hz) Temperature (°C) 80 10 water Figure 3.46 Comparison of the a) relative dielectric permittivity and b) relative dielectric loss of waxy maize starch suspension and water during the heating cycles. Inset details the relative dielectric permittivity at 30, 35 and 40°C. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 To make the choice o f models more d iffic u lt, it was noted that a sim ilar m inimum would appear if, instead o f enhanced conductivity, another relaxation process occurred at low frequencies, as shown in Figures 3.48 and 3.49. Figure 3.48 is a plot o f water’s relative dielectric perm ittivity and loss curves that have been modified to take into account an additional relaxation process at a critical frequency o f about 108 Hz. Figure 3.49 is analogous to 3.48, but the additional relaxation process occurs at about 106 Hz. These curves were modeled w ith two Debye equations. Once again, there is no asymptotic approach to zero at low frequencies; instead, the relative dielectric loss curve displays two maxima separated by a minimum. In addition, the “ static” dielectric perm ittivity is no longer flat, but instead shows an additional inflection point. It was difficu lt to discern whether or not the data’s deviation from Debye behavior was the effect o f conductivity (arising from electrolytes) and/or an additional relaxation peak at low frequencies. It was also wondered i f this could have simply been an artifact, but that is not very probable since the deviation appeared in a ll the data. As Adamec and Calderwood (1989) eloquently stated, When searching for a true property o f a material and for the correct interpretation o f an observed phenomenon, it is necessary to know whether the observed phenomenon is a consequence o f the process about which inform ation is being sought, or is produced by unwanted processes, e.g. irreversible changes in the composition or in the structure o f the specimen under investigation. Such unwanted processes may occur during the preparation o f the specimen, the application o f the electrodes during preconditioning, or even during the experiment itself. It was finally decided that the best way to attempt an interpretation o f the observed phenomena would be to take into account the possible effects o f both conductivity and a second relaxation peak. It w ill be seen later that the data measured in the present research can indeed be modeled by a combination o f the model represented by Figure 3.47 and the model represented by either Figure 3.48 or 3.49. Therefore, the DebyeHasted model w ith two relaxation peaks was applied to this data. It was clear, however, that no definitive answer could be given based on the frequency range studied. This results and discussion section w ill proceed to a discussion o f the combinations o f initial estimates used in the recursive program. It w ill then describe how closely the model fits the data. After comparing the behavior o f the perm ittivity and loss o f each starch suspension with that o f water, the focus w ill be on the actual Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 100 90 • ft- * 80 £r' ooo 70 freq. range studied — 915 and 2450 MHz . 60 50 40 30 20 £ r" 10 106 10B 10,Q 1012 10u frequency (Hz) Figure 3.47 Data for water theoretically modified to take into account the addition of electrolytes in the concentration of 10'2 % (w/w) sodium chloride 100 90 80 Sr' 70 60 50 40 30 20 £r" 10 106 108 1010 1012 frequency (Hz) Figure 3.48 Data for water theoretically modified to take into account an additional relaxation at lower frequencies with es2- esl = 7 and x2 =1.5 ns. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 184 sk>ooo ooo freq. range studied 915 and 2450 MHz 1(T frequency (Hz) Figure 3.49 Data for water theoretically modified to take into account an additional relaxation at lower frequencies with - eS] = 16 and x2 ~ 16 ns. parameters for the Debye-Hasted model w ith two peaks and the effect o f added sodium chloride. The last topics to be addressed w ill be the results o f the broad frequency experiments, the conductivity measurements, and the time-temperature superposition modeling. 3.5.2 Selection Process for Initial Estimates In order to arrive to the final sets o f parameters for the Debye-Hasted model w ith two peaks, a series o f initial estimates was screened. The “ best” sets from these were chosen for each starch by trying to balance the “ goodness o f fit” w ith the appropriateness o f the result. Figures 3.50a-c are graphical representations o f this balance between achieving low error (expressed as the SSE o f the regression) and preventing the algorithm from returning values consistently equal or very close to the imposed bounds. Such values were thought to be artificially “ pegged” at the bounds, and they were not considered valid solutions. These invalid solutions were denoted in the figures by blue asterisk markers, whereas the acceptable values were denoted with green Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 diamonds. Figures 3.50a-c show the SSE values for waxy maize, common com and potato starches, respectively, that were obtained when different combinations o f initial estimates o f E s l- e s2 and T i were used for data fitting. (There is no figure for the waxy maize-NaCl mixture because all initial estimate combinations resulted in the same set o f parameters.) The SSE represented by these figures is the average o f the SSE for three sets o f data for both waxy maize and common com starch, and o f two sets o f data for potato starch. The individual SSE values for the data sets with their corresponding initial estimate combinations were tabulated in Appendices E-G. The valid combination with the smallest SSE was chosen as the best combination, and was marked in the figures with a red star. Table 3.7 shows some information about the chosen initial estimate combinations (denoted in the following discussion by the subscript o after the variable name) for all starch suspensions and for the one combination for waxy maize-NaCl suspension. The chosen combinations for waxy and common com starches are similar, whereas the chosen combination for potato starch has a similar (esi - £52)0 but a much higher t i 0. It is important to note that the initial estimates do not, in general, relate to the final results in a simple manner. Rather, the initial estimates are the first point in a usually contorted path that leads to the final result. From the table it is also apparent that the data was well fit, in spite o f some variability that w ill be discussed later. The lowest SSE values were obtained for common starch, followed by those for waxy maize starch, potato starch, and finally the waxy maize-NaCl combination. The waxy maize-NaCl data was unique in that it was characterized by extremely high values o f e," caused by the high conductivity. Because o f this high 8/', the residuals (the difference between the measured data and the fitted values) were numerically higher than those for the other starch suspensions, but usually these residuals were still very small compared with the magnitude o f the measured data. Table 3.7 shows two additional indicators o f the “ goodness o f fit” for the model. The mean square error (MSE) values for all the data follows the same pattern as the SSE values, but they give a better idea o f the small magnitude o f the residuals for the fitted models, even in the case o f the waxy maize starch-NaCl suspensions. The last indicator in Table 3.7 is the coefficient o f multiple determination R2. A ll o f the R2 values for the different sets o f data are over 99%, except for one o f the data sets for potato starch that has an R2 o f 98.8%. These results indicate that the fitted model accounted for most o f the variation in the data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 (a) Waxy maize SSE I (Ssl **2 )0 ®0 0 50 0 Tlo (P S ) (b) Common com 150. 100N SSE 50 s v 0 “X *— °s .X 10 ttftl (£sl — £S2>o Tlo (PS) ♦ = acceptable values; * = n o n -a c c e p ta b le va lu e s; * = chosen c o m b in a tio n Figure 3.50 Graphical summary presentation of the selection of initial estimates for a) waxy maize; b) common com starch. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 187 c) Potato starch SSE 100 ♦ acceptable values; * = non-acceptable values; ♦ = chosen combination Figure 3.50 (cont.) Graphical summary presentation of the selection of initial estimates for c) potato starch. Table 3.7 Summary o f Details o f the Chosen Combinations Starch Waxy Common Corn Potato (Ssl“ Ss2)o 25 32 34 1 Tlo 8 1 45 1 SSEexpi 62.67 33.23 65.31 649.05 88.61 612.26 Waxy + NaCl SSEexp2 37.56 23.50 SSEexp3 mean SSE 31.98 23.42 44.07 26.72 76.96 516.27 # of observations for each exp’t 6300 6300 6300 6000 101 287.50 # of parameters 106 106 106 degrees of freedom 6194 6194 6194 5899 mseexpi 1.01E-02 5.37E-03 1.05E-02 1.10E-01 mseeip2 6.06E-03 3.79E-03 1.43E-02 1.04E-01 ®S®exp3 5.16E-03 3.78E-03 R expl 99.59% 99.59% 99.20% 99.995% R2exp2 99.51% 99.71% 98.88% 99.995% R exp3 99.58% 99.71% 4.87E-02 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99.998% 188 3.5.3 Comparison of the Dielectric Properties of Starch and Water The previous section referred to Figures 3.46a and b (and the graphs fo r the other starches in Appendix H ) to explain that the original plan to fit the data w ith the Debye model had to be abandoned because o f the non-Debye behavior o f the data. This section w ill return once again to those graphs to focus more specifically on comparing the starch suspensions’ dielectric properties w ith those o f water, and to speculate on possible causes fo r the observed deviations. In general, the dielectric behavior for all three starches is very similar, follow ing the trend o f water, w ith no obvious differences among them. That is similar to what RyySnen et al. (1996) reported when they studied native potato, wheat, com, and waxy com starches. This group found that water content was the most influential factor on the dielectric behavior. However, referring once again to the graphs in Figures 3.46 and Appendix H, it is clear that the data from waxy maize, common com, and potato starch do differ from water in a few aspects: 1) The e/ values for the starch suspensions are less than the values for water. 2) The e/ data at low frequencies exhibit a very slight upward curvature (concavity) that can not be fitted w ith a simple Debye model. This curvature is more clearly observed in the inset for Figure 3.46a that compares the data to curves generated w ith the Debye equations. 3) The e/ data at high frequencies has a more pronounced downward curve than that o f water. 4) The Er" data exhibits an upward trend at low frequencies that is more noticeable at higher temperatures. These characteristics are reproducible w ithin the starches and among the starches, occurring for all o f them, although w ith some variations. Some differences are more evident in the parameters o f the Hasted-Debye model w ith two peaks, rather than in the curves o f the dielectric properties themselves. These parameters w ill be the topic o f section 3.5.4. For now, differences among the starches and water that are visible in the perm ittivity and loss curves w ill be addressed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 3.5.3.a Relative Permittivity Curves There are two main trends for e/ fo r all the starch suspensions: a) a decrease w ith increasing temperature and frequency; and b) values that are lower than the values for water. Both trends can be observed in Figure 3.46a fo r waxy maize suspensions during heating. (The perm ittivity curves fo r both the heating and cooling cycles can be found in Figures H .l, H.3, and H.5 o f Appendix H fo r the waxy maize, common com, and potato starches, respectively.) A t about 75°C, the perm ittivity o f waxy maize approaches that o f water, but drops once again to values below that o f water during the cooling cycle. As mentioned before, the downward trend w ith frequency is more pronounced fo r the starch suspensions than fo r water. This stronger curvature causes the separation between starch suspensions and water to be more pronounced at higher frequencies, especially at lower temperatures during both heating and cooling. This same trend occurs for the other starches as well. The curves fo r the relative dielectric perm ittivity o f common com starch suspensions (Figure H.3) are slightly different from those for waxy maize in that they never approach the curves for water. The corresponding plot fo r potato starch suspensions (Figure H.5) is slightly “ noisy” at low temperatures during the heating cycle. It also shows an irregular “ dip” in the curve at about 65°C, which may be related to the changes that occur during gelatinization. Between 75 and 80°C in the heating cycle the perm ittivity for potato starch suspensions surpasses the value o f water. It decreases during the cooling cycle, being approximately equal to the value for water at 75°C, and lower than water by about 70°C. Since this particular change in dielectric perm ittivity occurs upon cooling, it is obviously not caused by gelatinization. The fact that the perm ittivity values o f waxy maize are below the values for water at low temperatures is an interesting result that deserves some discussion. Waxy maize contains only amylopectin (and almost certainly some impurities), and therefore it has no amylose that could leach out during heating. ( I f there were any amylose in solution, the decrease in perm ittivity could have been attributed to its presence.) Moreover, the crystalline regions w ithin the starch structure are not expected to exert a great influence in the dielectric properties o f the system. One possible explanation for this behavior is that the water is “ diluted” by the presence o f the starch granules. Based only on the waxy maize results presented here, this seems to be a viable possibility because dilution by the starch would im ply that the perm ittivity Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 would decrease w ith respect to that o f water. This dilution effect, however, does not explain the stronger curvature for the starch suspensions. A second possibility is that water and starch are interacting in some way. According to Umbach et al. (1992), there is very little interaction between dry starch granules and electromagnetic radiation. Therefore, any changes in curvature caused by the addition o f waxy starch must originate from an interaction between water and the amorphous phase (thought to be the m ajority o f the starch granule, according to Hizukuri, 1996) w ithin the non-gelatinized granule. A third possibility is that some electrolytes may have been introduced during the processing o f the starch, and these m ight leach out at low temperatures, affecting the perm ittivity. Examination o f the relative loss curves can provide further insight into these possibilities. 3.5.3.b Relative Loss Curves For all starches, the main trend is for e /' to increase w ith increasing frequency, and for the slope o f the e /' curves to decrease w ith increasing temperature. A smaller, but important, trend is the upward curvature o f the loss curves at low frequencies, especially at the highest temperatures. Figure 3.46b is the plot o f the relative dielectric loss during the heating cycle for both waxy maize starch and water. The fact that the observed loss for waxy maize is always higher than that fo r water, does not lend support to the hypothesis that the only effect o f starch granules is to “ dilute” the water because dilution would im ply that the loss should decrease. The curves for the waxy maize, common com, and potato starch suspensions at both cycles can be found in Appendix H: Figures H.2, H.4, and H.6, respectively. These, too, show that the values o f e /' for all the starches are above those o f water, but they are closer to those o f water at high frequencies. The plots also display the aforementioned obvious curvature at low frequencies. This curvature was the reason that the Debye-Hasted Model w ith two peaks was chosen. In the next section, the focus w ill be on how w ell that model actually fits the data. 3.5.4 The Fit of the Debye-Hasted Model with Two Peaks The result o f the recursive algorithm was to return the value o f the a matrix containing the six dielectric parameters to fit the Debye-Hasted model w ith two peaks, defined in Figure 3.41 o f section 3.4.6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 191 The fitted parameters o f the Debye-Hasted model w ith two peaks were used to construct plots o f the fitted dielectric properties against frequency and temperature. Examples o f such plots are shown in Figures 3.5 la and b for waxy maize starch during the heating cycle, where the fitted curves are represented by black lines. A ll sets o f the measured data (interpolated and corrected as explained before) are plotted as colored lines for comparison. (The plots for the other starches and cycles can be found in Appendix I.) The model fits the data well, and differences between replications and deviation from the measured data were observed in only a few instances. One o f those instances o f deviation is evident in the plot o f e/ for waxy maize w ith added salt shown in Figure 1.8. It is hypothesized that this deviation is caused by the inability o f the chosen model to predict the phenomenon o f electrode polarization that w ill be discussed later. An instance o f some variation between repetitions is shown by the potato starch curves (as evidenced especially by the “ double lines” in the e/ curves in Figure 1.5). This variation is especially prevalent at higher temperatures during the heating cycle and throughout the cooling cycle. The potato starch data also shows some noise in the e/ curves during the heating cycle at 65°C. As mentioned earlier, it is hypothesized that the experimental runs conducted w ith potato starch developed a much higher viscosity faster than the waxy maize and common com starch. This is supported by the report from K okini et al.(1992) that potato swells more rapidly than com, and by the findings o f Muzimbaranda and Tomasik (1994) that their microwaveheated starch gels had a much higher viscosity for potato (3000 cPs) than common com starch (1600 cPs). Further corroboration is given by Biliaderis et al. (1980), who recorded the “ in itia l pasting temperature” (the temperature at which their visco-amylograph first detected an increase in viscosity o f 10 Brabender units) for various starch suspensions. They reported that the in itia l pasting temperature was 51 °C for potato starch, but 72°C and 74°C for waxy maize and common com starch, respectively). McPherson and Jane (1999) reported that the onset gelatinization temperature o f normal potato starch was 60.8°C, a temperature that they claimed was lower due to the presence o f phosphate monoesters in potato. The noise and variation between repetitions seem to be worse at the higher temperatures, and seemed to improve as the temperature decreased during the cooling cycle. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 a) 80 8/ heating 75 70 65 80 10’ Frequency (Hz) Temperature ( C) 80 10° (b) e /' heating 65 50 10 40 10’ Frequency (Hz) Temperature ( C) Colored curves = measured data; 80 35 10° black curves = fitted data. Figure 3.51 Comparison of the fitted curve with the measured data for the a) relative dielectric permittivity and b) relative dielectric loss during the heating cycle for 3% waxy maize starch suspension Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 3.5.5 Parameters for the Debye-Hasted Model with Two Peaks There is still uncertainty about whether the relative dielectric loss data at low frequencies exhibits an increase because o f conductivity (as shown in Figure 3.47), because o f a relaxation peak (as shown in Figures 3.48 and 3.49), or because o f a combination o f the two effects. However, any o f these options must be associated with an obvious “ main” peak o f a greater magnitude at frequencies above 20 GHz. Comparison w ith the theoretical dielectric spectrum o f water given in Figure 3.29 indicates that this main peak is attributable to the relaxation o f water. The frequencies o f measurement examined in this study (0.3 to 3 GHz, as indicated by the dashed lines in Figures 3.47-3.49) are between the high-frequency relaxation peak and the process occurring at lower frequencies. This range o f frequencies caused most o f the data collected here to correspond to the behavior o f the lower frequency “ tail” o f the high frequency peak, and only the in itial manifestations o f the possible low-frequency “ process” are visible in the data. This means that changes in the high-frequency peak were the main influence on the behavior o f the measured dielectric properties for the starch suspensions without added NaCl. Changes in that high-frequency peak, as w ell as changes in the lower-frequency process, can best be studied by examining the fitted model parameters from the a matrix. In fact, the parameters can be thought o f as a “ distilled” version o f the data that can be more useful than the measured data when it comes to understanding and predicting dielectric behavior. Given the considerable influence o f frequency on the results, the parameters w ill be discussed according to whether they come into play at frequencies that are high (e^, t 2, and e*,) or low (Ssi - e^, Ti, and equivalent salt concentrations), relative to the data collected. Figure 3.52 is a “ fish bone” representation o f the complex relationship between these dielectric parameters and the relative dielectric properties. As Figure 3.52 indicates, most o f the parameters are influenced by several other variables, including temperature, concentration, and initial estimates used in the recursive algorithm. Four o f the parameters are plotted for each starch as functions o f temperature in Figures 3.53,3.54, and 3.55. The parameters for waxy maize, common com, potato, and waxy maize-NaCl suspensions are represented by red circles, green squares, blue diamonds, and magenta stars, respectively. Error bars in these plots represent a 95% confidence prediction interval. The fitted curves are represented by black lines. For Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 194 comparison, Figures 3.53a and b also show the parameters for pure water (light blue triangles). The focus o f this section is to compare the parameters resulting from the best combinations for the waxy maize, common com, and potato starch suspensions w ith those o f water. A discussion o f the waxy maize-NaCl suspension results w ill be reserved for section 3.5.6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. equivalent NaCl concentration starch X temperature^ temperature £ s2 equivalent NaCl concentration starch starch SSE / X temperature frequency cooling cycle I starch heating cycle cooling cycle p " starch fcr \ frequency temperature NT y e e of water heating cycle fitted values of o ' br starch added NaCl equivalent NaCl concentration £00 high frequency* electrolytes in starch starch electrolytes in starch SSE ' ■added NaCl temperature equivalent NaCl concentration SSE concentration conductivity y low frequency \ (Ssl £ s2) o ' pegged9 y \ \ starch £sl “ e s2 X SSE temperature Xl Tlo / pegge d? X e q lliv a le IIt N ac l (£si — es2)o SSE pegged9 X T pegged? temperature Figure 3.52 Diagram illustrating the variables affecting the influences on the relative dielectric properties of the starch suspensions. 196 (a) £s2 s2 68 50 40 Cooling Heating (b ) t 2 x ,(P S ) 30 o waxy maize; 40 50 Heating 70 80 T(°C) 70 common corn starch: 0 potato starch; *waxv maize + NaCl; 50 40 Cooling 30 — fitted curves Figure 3.53 Parameters of importance at high frequencies: a) &,2; b) x2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 197 (a) £sl- es2 s1 Heating Cooling 20 18 16 14 12 10 8 6 4 2 0 30 40 50 60 70 Heating o waxy maize; □ common corn starch: 80 T (°C) 0 70 60 potato starch; 50 40 Cooling 30 —fitted curves Figure 3.54 Parameters of importance at low frequencies: a) esl- es2; b) t, for starch suspensions without NaCl Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 198 100 90 80 70 to c ,• 50 40 30 0.5 20 10 Heating Cooling Figure 3.55 Parameters of importance at low frequencies: ss]- es2and T] for waxy maize starch suspensions with added NaCl 3.5.5.a H igh Frequency Parameters The parameters that most strongly influenced the behavior o f the data measured at the frequency range studied were and i 2 (Figures 3.53a and b). Since the data at this frequency range was well within the “ area o f influence” o f the high-frequency process, these two parameters were not sensitive to changes in the initial estimates. They were, however, sensitive to other variables, as indicated by Figure 3.52 and as described below. Figure 3.53a shows the results for for all the starch suspensions and for water. These plots contain fitted lines and error bars that correspond to a 95% confidence prediction interval. The most important trend shown is that the values o f decrease with increasing temperature during heating, and then increase with decreasing temperature during cooling. The value o f ss2directly affects the value o f s/, and therefore, the decrease in ss2 with temperature means that e/ also decreases with temperature as shown in Figure 3.51a (and in the other figures like it in Appendix I). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 199 Regression analysis showed that the slopes and constants o f the es2vs. T curves were not constant throughout the temperature range studied. Instead, in comparison w ith their in itia l values, the slopes for all starches increased (became less negative) and the constants decreased during heating in the temperature range 65oC<T<80°C. These changes were significant (P 0.0005) for a ll starches, but they were most pronounced for potato starch. Upon cooling, in the same temperature range (65°C<T<80°C), the slopes decreased (became more negative) and the constants increased (P<0.0005). W ith further cooling to T<65°C, the slopes o f the ss2 vs. T curves returned to values that were not significantly different from their in itia l values (P>0.056). Throughout this temperature range the constants also approached their initial values; the difference between initial and final constants was less than 1% o f the in itial values. These differences were significant for waxy maize and common com suspensions (P<0.007), but they were not significant for potato starch and waxy maize starch w ith added NaCl, (P>0.435). The es2values at 30°C for heating and cooling were not significantly different, since their 95% confidence prediction intervals did not overlap. This analysis suggests that the ss2 for the starch suspensions changed very little (or not at all) after the gelatinization process. A qualitative look at Figure 3.53a shows that the values o f es2 for the starch suspensions appear to be very close to each other at every temperature. A quantitative examination using the 95% confidence prediction interval indicates that the values are significantly different (P<0.005) in only two cases (at 30°C heating and at 80°C). The curves for the starch suspensions are also close to and almost parallel to the curve for water, but always significantly lower (P<0.05). This concurs with the results from Ryynfinen et al. (1996) that were discussed in section 3.2.14c. The lower es2values are probably due, at least in part, to a dilution effect. It was interesting to notice that the addition o f 3% starch to water caused an approximate 3% decrease in the value o f the high-frequency relaxation strength (as defined in Figure 3.41), and that this relationship remained true up until about 60°C. That is, the ratio ( E s2 — £oo)starch/ (^ s “ £°o)water was only slightly higher than 97% up to 60°C where it abruptly increased, coinciding w ith the gelatinization temperature range and the change in slope and constant discussed above. Because £*, does not vary much w ith temperature, the reduction in relaxation strength must originate from a reduction in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 static perm ittivity o f water when starch is added. A reduction o f the relaxation strength o f water caused by the addition o f a polymer agrees w ith the findings o f M iura et al. (2003), though in that case the authors correlated the change in relaxation strength to the amount o f available “ free water.” This finding implies that the change in the static perm ittivity o f starch at low temperatures is a linear function o f water content, even though (as was discussed in section 3.2.14.e) dielectric properties o f mixtures are not usually additive. Based on this discussion, factors influencing the parameter are temperature and concentrations o f starch and sodium chloride, as shown in Figure 3.52. The curves for x2 vs. temperature for all suspensions and for water are shown in Figure 3.53b. Once again, these plots contain fitted lines and error bars that correspond to a 95% confidence prediction interval. Like es2, x2 for all suspensions decreased with increasing temperature during heating, and increased with decreasing temperature during cooling. This indicates that the critical frequency for the main relaxation peak increases with temperature. Figure 3.53b shows that the general shape o f the curves follows the behavior o f water, but that the curves for the starch suspensions are a ll higher than the curve for water, w ith that o f waxy maize with added sodium chloride being the highest curve o f all. These qualitative observations were found to be statistically significant since the prediction interval for the starch suspensions do not overlap the relaxation times for water at any temperature for any starch, and since the relaxation times for waxy maize starch suspension with added NaCl are higher than those for the other suspensions at all temperatures (P<0.0005). Another feature visible in the plots in Figure 5.53b is a the curves for the starch suspensions without added NaCl overlapped each other, except during the heating cycle in the temperature range between 65 and 80°C. A t this temperature range, which coincided w ith the gelatinization temperature range, the x2 plots for the starch suspensions underwent a change in curvature. In order to investigate this change in curvature, regression analysis was performed. For this analysis, the x2 data was transformed into ln(x2) to make it linear. The results showed that in the gelatinization temperature range, the change in curvature did cause a significant change in constant and slope for the ln(x2) vs. T lines for all starches (P<0.0005). These changes were most pronounced for potato and waxy maize starch suspensions, and least pronounced for the waxy maize suspension with added NaCl. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 201 The regression analysis was also used to study the behavior o f the curves after reaching 80°C and starting the cooling cycle. Figure 5.53b shows that the curves began to revert to their in itia l shapes, but the regression analysis revealed that the slopes and constants were s till different from their in itia l values between 80 and 65°C (P<0.0005). Between 60 and 30°C the curves for the starch suspensions without added NaCl overlapped again, but the curves for the waxy maize suspensions with added NaCl s till remained significantly higher than the others. The slopes remained significantly higher than the in itial values (P<0.033), except for common com starch suspensions (P=0.511). The constants remained greater than their in itial values (P<0.044), except for waxy maize starch suspensions w ith added NaCl (P=0.103). In spite o f the differences in slope and constants, the prediction intervals for the initial and final values o f x2 for all the starch suspensions overlapped, indicating that there is no significant difference in the relaxation times after the gelatinization process. The increase in the relaxation time o f water observed in this study upon the addition o f starch was also observed by Miura et al. (2003) upon the addition o f agar, gelatin, and ovalbumin. In that work, an increase in relaxation time was observed as a function o f polymer concentration. The decreased relaxation time found in the present research explains why the e/ curves for the starch suspensions have a stronger downward curvature at high frequency than the curve for water. Since the longer relaxation times correspond to a lower critical frequency, the main relaxation peak for starch occurs at a lower frequency than the corresponding peak for water. Thus, the higher loss values and the stronger curvature at high frequency could both be explained by the behavior o f the relaxation time, even i f there is a dilution effect that causes the overall decrease in the e/ curves. Based on the observations in this study, factors influencing the parameter x2 are temperature and concentration o f starch and salt, as shown in Figure 3.52. The third parameter describing the process at high frequencies is e*,. However, the frequencies used for this study were much lower than the range in the spectrum where the relative perm ittivity decreases and become constant at the e*, value. The calculations were very sensitive to the values for e*,, so it was decided to set the bound for the values o f e*, for all o f the starches very close to the values for water. Therefore, no graph was plotted for £«, values. This subsection has been devoted to discussing the behavior o f the parameters that come into play at high frequencies: es2-em x2, and £*,. Understanding their behavior can help explain the observed behavior o f the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 202 relative dielectric loss. For example, as shown theoretically in Figures 3.47-49, as temperature increases, the critical frequency o f the high-frequency relaxation peak ( f 2= 1/2jit2) increases, and the height o f the peak decreases. In turn, the position o f the relaxation peak (determined by the critical frequency) and its height (determined by the relaxation strength, ss2-eM) influence the slope o f the dielectric relaxation curve on the high frequency side o f the range o f frequencies studied (the region between the vertical dotted lines in the plot). The fact that the critical frequency increased and the relaxation strength decreased w ith temperature (Figure 3.53) explains why the slope o f the high frequency portion o f the dielectric relaxation curve decreased. 3.5.5.b Low Frequency Parameters In contrast to the high-frequency parameters, the two parameters that describe the behavior o f the data at lower frequencies, the low-frequency relaxation strength (&,! - e^) and relaxation time (xt), were very sensitive to initial estimates. A good example o f this sensitivity is shown in Figure 3.54a and b, the plots o f the fitted difference esi - e s2and o f i i versus temperature for the waxy maize, common and potato starch suspensions. (Water does not have a low-frequency peak, so its curve is not plotted for comparison.) As w ill be discussed later, the difference between the curves for waxy maize suspensions and the curves fo r common com and potato starch suspensions can be traced to the initial estimates chosen in the parameter-fitting scheme. Figure 3.54a shows the plots o f esl-e s2 vs. temperature for waxy maize, common and potato starch suspensions. The error bars representing a 95% confidence prediction interval indicate that the curves for waxy maize suspensions are much higher than the curves for common and potato starch suspensions. The ssi-e s2 values for the waxy maize suspensions increased slightly but significantly (the slope is positive, P<0.0005) during heating, and then decreased, also slightly, during cooling. The initial and final values o f ss rss2for waxy maize were not significantly different as indicated by the 95% confidence prediction interval. The slopes and constants did not change significantly at any temperature (P>0.087). The curves for common com and potato starch suspensions show an almost constant value near 2 at heating temperatures from 30° to 60°C. The values for esi-e s2were not significantly different between the two starches in this temperature interval, and the slopes for both starches were not significantly different from zero (P>0.519), indicating that the values did not change substantially with temperature. As the temperature reached Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 203 the gelatinization range (at T>65°C), there was a sudden change in the curvature o f both the common com and potato starch plots that gave them a sigmoidal shape, sim ilar to that observed in the data for particle size analysis o f chapter 2. The slope in this temperature range was significant for both starches (P<0.0005), but the potato starch suspensions exhibited a greater change in curvature than the common com starch suspensions, perhaps due to the phosphate monoesters occurring naturally in the potato starch and their effect on granule swelling mechanisms. The slope for potato starch during cooling down to 65°C was significantly smaller than during heating (P0.0005), but the slope for common starch was not significantly different than during heating (P=0.275). On further cooling to temperatures between 60 and 30°C, the slope o f the curves greatly decreased, but was s till significantly different from zero (P<0.006). The initial and final values o f s^-e ^ were not significantly different, as indicated by the 95% confidence prediction interval. Figure 3.54b shows the fitted Ti values versus frequency. The plots are similar to the plots for esi - es2. Common com and potato starch suspensions showed a generally sigmoidal curve, w ith the tendency for i j to increase with increasing temperature during heating, and to decrease with decreasing temperature during cooling. This trend was not obvious for waxy maize suspensions, which, once again, had a curve significantly higher than the curves for the other starches. For common com and potato starch, the curves were flat up to 60°C (slopes were not significant, P>0.111) and then they showed a change in curvature at the gelatinization range (the change in slopes was significant, P<0.0005). During cooling, the curves for common and potato starch suspensions returned to values that were not significantly different from the in itial values. The curves for the waxy maize starch suspensions had significant (but erratic) changes in slope and constant (P<0.043) for the gelatinization region and during cooling. In any case, the increase in Ti w ith temperature means that the critical frequency for the low-frequency relaxation decreases w ith temperature. The curves o f 851-652 and t 2 for waxy maize, common com, and potato starch increased with temperature during heating, and then decreased during the cooling cycle. This indicates that the magnitude and critical frequency o f the low frequency peak, i f it exists, increases with increasing temperatures. For common com and potato starch suspensions there was a sudden change in curvature for both 851-852 and t 2 at the gelatinization temperature range. Marsh and Wetton (1995) observed a similar sudden increase in relaxation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 204 strength for a flour wafer and for a mixture o f amylopectin, fructose and water. They attributed this increase to the sudden increase o f the number o f dipoles after the glass transition. Some insight may be obtained by comparing the results from the present study w ith other measurements o f dielectric properties o f polymers at different temperatures. Lu (2005) measured the dielectric properties o f polymeric membranes in the presence o f water. The polymeric membranes contained both hydrophilic (polar) and hydrophobic (non-polar) regions at the molecular level. (This system is analogous to non-gelatinized starch granules with amorphous and crystalline regions.) The author reported two or three relaxation processes for the membranes at frequencies lower than the relaxation frequency for water. These processes were attributed to different populations o f water molecules w ith restricted m obility. The polymeric membranes studied by Lu did not exhibit a phase transition at the conditions reported; a ll the relaxation times and most o f the relaxation strengths decreased w ith temperature with no obvious change in curvature. I f the low frequency process suspected in this present study was caused by a mechanism sim ilar to the processes reported by Lu, the dielectric strength and the relaxation time would be expected to vary smoothly, and the relaxation time would be expected to decrease w ith temperature. As mentioned above, this was not the case for the starch suspensions. I f the process observed at low frequencies in this current study is indeed a low frequency relaxation peak, the mechanism involved could be similar to that reported by Marsh and Wetton (1995) for a system undergoing phase transition. It is hypothesized that when the crystalline order o f the starch granules is disrupted through gelatinization, at least two different processes with relaxation times higher than the relaxation o f water increase in importance. One o f these processes involves the outer chains o f the amylopectin molecules, which become able to interact w ith water. A population o f water with restricted m obility would appear (or increase its size i f already present). This population o f water molecules would have a relaxation time higher than pure water (as explained by Lu, 2005). The second type o f process is the oscillation o f side groups, including hydroxyl and methylol, and the movement o f sections o f the molecules w ith respect to the rest o f the molecule (Einfeldt et al., 2000). Both the interaction with water and the oscillation o f the side groups could occur to some extent before heating, but the population o f restricted water molecules and the number o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 205 oscillating side groups and molecule segments would suddenly increase during gelatinization, causing the dielectric strength ta i-e ^) to increase. The decrease o f both the relaxation strength and relaxation time upon cooling is not easy to interpret. Perhaps, as the starch polymers become less flexible at lower temperatures, the movement o f the side groups becomes more restricted, and the relaxation process becomes both faster and weaker. Another possibility is that the starch molecules start interacting w ith each other and with water, and this restricts the movement o f both water and side groups. It was mentioned above that the differences between the fitted low frequency parameters for waxy maize and for common com and potato starches could be traced to the initial estimates. This issue deserves some discussion. The fitting scheme, as may be remembered, was designed to balance the goodness o f fit (as measured by the SSE) against the suitability o f the results. This later criterion (the “ suitability” o f the results) was tackled with the aid o f some results in the literature. In particular, Mashimo’s group has published a series o f results about relaxation parameters (relaxation strength and relaxation time) for a great variety o f aqueous biopolymers in a wide range o f frequencies (Mashimo et al., 1987 and M iura et al., 1994,2003). A survey o f these researchers’ results provided the range o f relaxation strengths and relaxation times expected for the potential low frequency relaxation peak in the measured data. In their results, the lowest value for the relaxation strength (esl —852) reported was 1.04 (for human red cell ghosts). The upper values for the relaxation strength were much higher that this (around 153), but most o f their results were below 40, and even this value was never approached by the data in this present study. The values for the relaxation time (t0 reported by the same group ranged from 0.65 to 19.5 ns, but most o f the data was centered around 1 to 2 ns. Based upon these results, the lower bound for the relaxation strength in the present study was set at 1. The lower bound for the relaxation time was set at 0.53 ns to correspond with the lowest frequency studied (0.3 GHz). This bound was set under the assumption that there were no critical relaxation frequencies within the range o f the data because there were no visible maxima in the e/' curves. Thus, the lowest frequency at which a m a x im u m could possibly occur was the minimum frequency o f measurement (0.3 GHz). However, even that value would be unlikely, because maxima occur in areas where the slope o f the curve approaches zero, and the data at 0.3 GHz never appeared to approach this condition. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 206 During the development o f the fittin g algorithms for the present study, the curvature in the dielectric loss at low frequency was discovered. Upon further examination, it was also discovered that the relative dielectric perm ittivity curves had a slight upward curvature in the same frequency range. It was originally thought that if a low frequency process existed, its relaxation strength would have to be at least a few units in order to be observed w ith the instrumentation used. However, upon consulting the literature, it was discovered that small relaxation peaks (w ith relaxation strengths as low as 1-2 units) and relaxation times around 1 ns were possible, so the lower bound for the relaxation strength was then lowered to 1. This reduction in the lower bound o f the relaxation strength presented the danger o f having the algorithm superimpose a small relaxation peak over the main one in order to fit its shape better, instead o f fitting a real separate relaxation peak. Because o f this danger, the results obtained from the algorithm were checked to see if the values o f the relaxation strength were consistently equal or very close (“ pegged” ) to the lower bound. Since these pegged values occurred simultaneously for the relaxation strength and the relaxation time, it was thought that they indicated that the algorithm was progressing towards the superimposed peaks that were not suitable. Therefore, the sets o f initial estimates that generated such values were discarded. It is obvious by looking at Figures 3.53 and 3.54 that waxy maize’s fitted parameters behave differently than those o f common com and potato starch, both in value and in the shape o f their curves. W ith its relaxation strength around 16 to 19 units and its t! values around 16 ns, waxy maize is clearly far from both lower bounds. The plots for the other suspensions, in spite o f not being quite as low as the lower bound, do stay at low values close to 1 unit for the dielectric strength and 1 ns for Ti. In general, runs that returned parameter values close to the lower bounds also resulted in lower SSE values. These corresponded to the set o f parameters w ith a small peak at low frequencies, having Ti values in the same range as Mashimo’s group. In the case o f waxy maize, however, the parameter sets that resulted in Tt values around those o f Mashimo’ s group resulted in both - £52 and i i being “ pegged” at lower bounds, so these were therefore discarded. Other in itial estimate combinations for waxy maize starch resulted in t i values much greater than 1; there were no in-between values. The in itia l estimate combination that was fin a lly chosen resulted in Esi - sS2 and %i values that were much greater than 1 (at some temperatures they were an order o f magnitude higher than the values o f common com and potato starch). This higher value for the relaxation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 207 strength and the relaxation tim e corresponds to a taller relaxation peak occurring at a lower frequency. This explains why the waxy maize starch parameters are different, and the SSE higher than it should be, given the low noise level in the data for this starch. It seems unlikely that the values o f these parameters would be so different between the tw o groups o f starch, but nevertheless it was thought that the criteria used to fit the data were reasonable, so they were not changed. However, it is possible that the true esi - £52 and i i values for waxy maize starch are actually close to or below 1 (i.e., lower than those o f common com and potato). It is also possible (although it seems less probable) that common com and potato could have had a taller lower frequency peak. The size and position o f the peaks cannot be definitively determined from the results o f this study. N or can a scientific reason why m ight waxy maize act so differently be completely discounted. The one important difference between waxy maize starch and common com starch is the presence o f amylose in common com starch. It could be hypothesized that amylose may have an effect on the low frequency range o f the data o f common com starch suspensions, which is not present fo r the waxy maize suspension data. Since potato starch also has amylose, this same effect would also be observed in its dielectric properties at low frequencies. 3.5.5.C Using the Parameters to Predict D ielectric Behavior at Extended Frequencies In order to obtain a better idea o f the significance o f the fitting parameters obtained, they were used to predict the dielectric behavior o f the starch suspensions at extended frequencies. Figures 3.56 and 3.57 are plots o f the measured data and the fitted model for waxy maize and common starch suspensions, respectively. It is obvious from the plots for waxy maize starch suspensions that the low-frequency relaxation peak has a greater magnitude, and that it occurs at an even lower frequency than the peak for the common com starch. It is also interesting to note that, since conductivity is higher for the waxy maize starch (as w ill be discussed later), the loss peak for this starch is almost completely overpowered by the conductive loss at low frequency. It must be pointed out that the vertical axis for the loss curves is logarithmic because the features o f interest occur at different orders o f magnitude. For example, the conductive loss at low frequency for waxy maize is several orders o f magnitude larger than the low frequency peak for common com starch. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ■ (a) s/ 80 heating 75 Temperature ( C) Frequency (Hz) (b) e ," heating Temperature ( C) Frequency (Hz) Colored curves = measured data; black curves = fitted data Figure 3.56 Extended results: a) changes in the relative dielectric permittivity and b) changes in the relative dielectric loss during the heating cycle for 3% waxy maize starch suspension. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 208 209 80 (a) e/ cooling I 70 75 Temperature ( C) Frequency (Hz) (b) 8," cooling 80 I 75 70 65 60 null 55 50 45 40 / Frequency (Hz) Temperature ( C) 12 80 Colored curves = measured data; black curves = fitted data Figure 3.57 Extended results: a) changes in the relative dielectric permittivity and b) changes in the relative dielectric loss during the cooling cycle for 3% common corn starch suspension. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 30 210 3.5.6 Equivalent Sodium Chloride Concentration and the Effect of Electrolytes As shown in Figure 3.52, conductivity is affected by the equivalent NaCl concentration and the temperature. The equivalent NaCl concentration depends on the presence o f electrolytes (added or not) and in the presence o f starch. Obviously, the waxy maize-NaCl suspensions were deliberately made to have a higher salt content than the other starch suspensions. However, when compared with those o f water, the relative dielectric loss curves o f the other starch suspensions showed possible evidence o f increased conductivity. The addition o f starch apparently affects conductivity, and this interaction was modeled in this study the parameter equivalent salt concentration. 3.5.6.a Equivalent Sodium Chloride Concentration Parameter The equivalent salt concentration is the third parameter that affects dielectric properties especially at low frequencies. This parameter takes into account the contribution o f electrolytes to the complex perm ittivity by representing the amount o f salt that would be required to produce the conductivity value needed to fit the data. In other words, any effect that starch might have on the conductivity o f a suspension is modeled exclusively as a variation on electrolyte (i.e. NaCl) concentration. This concentration was assumed constant through the experiments, and therefore there is only one concentration per experiment. (This is also the reason that there is no graph o f the behavior o f this variable versus temperature). The model then converts this equivalent concentration o f NaCl to a conductivity that varies as a function o f temperature. This assumption was thought reasonable, since there are some naturally occurring salts in the starch, and then some more are introduced during extraction and isolation. The electrolytes can be assumed small enough to diffuse through ungelatinized starch granules. Obviously, in the case o f the waxy maize-NaCl suspensions a high concentration o f electrolytes was introduced purposely to examine their effect on the dielectric properties. Subsequent paragraphs w ill discuss this case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 211 Table 3.8 Equivalent Sodium Chloride Concentrations (% w /w ) for all Starch Suspensions Calculated from M atrix o f Parameters W axy M aizeW axy M aize Common C orn Potato N aCl 2.76E-03 5.87E-05 1.18E-05 1.40 2.53E-03 5.86E-05 4.60E-06 1.38 2.47E-03 5.86E-05 mean 2.59E-03 5.86E-05 8.20E-06 1.40 Standard dev. 1.26E-04 4.71E-08 3.60E-06 1.21E-02 C .V. % 4.88% 0.08% 43.90% 0.86% NaCl C oncentration (% w /w) 1.41 The equivalent concentrations o f NaCl for a ll the suspensions are shown in Table 3.8. These are not measured values, but rather values that were “ fitted” as part o f the a matrix o f parameters. Interestingly, the concentration for waxy maize, which had the highest values o f esl-es2 and ii, has also the highest equivalent concentration o f NaCl by several orders o f magnitude (not considering the waxy maize-NaCl suspension, o f course). This would imply that waxy maize’ s relative dielectric loss would be higher at low frequencies. In reality, probably the algorithm increased the salt concentration to fit the low frequency peak a little better. The lowest coefficient o f variation was for the common com starch, whereas the coefficient o f variation for the other starches increased in the following order: waxy maize-NaCl, waxy maize, and potato starch. The fact that the coefficient o f variation for potato starch was so high was probably due to irregularities caused by the increased viscosity. 3.5.6.b Effect o f Added Sodium C hloride The amount o f sodium chloride added to the waxy maize suspension to make the waxy maize-NaCl combination was 2%. However, as Table 3.8 indicates, the equivalent salt concentration for all three o f those suspensions averaged out to be about 1.4%, much lower than expected. This shows that there must be some interaction between the starch and the salt that changes expected dielectric behavior. It is hypothesized that starch may reduce the m obility o f the ions to a greater extent than water. Also there is the possibility that, as Oosten (1982) hypothesized, there is ionic interaction between the starch and the Na+ ions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 212 Comparison o f the individual parameters for the waxy maize-NaCl suspension w ith those o f the other starches reveals some interesting insights into the effect o f added salt. Figure 3.53a shows that £52 is virtually unaffected by the increased in salt concentration, since the curve for waxy maize-NaCl coincides w ith those o f the other starches. However, Figure 3.53b shows a higher t 2 value at every temperature than the other starches. Figure 3.55 shows that the values o f esl- es2for the waxy maize-NaCl suspensions were much higher that for the other suspensions. It is suspected from these values and from the abnormal downward concavity, that the data at the lower frequencies for this high conductivity sample are perturbed by electrode polarization. This would concur w ith the results o f Boughriet et al. (1999), whose data for e/ shows obvious abnormal curvature at NaCl levels o f 2% for frequencies below 0.8 GHz. The t i values for the waxy maize-NaCl suspensions (shown in Figure 3.55) are relatively constant at about 1-2 ns. They are in the same range as common com and potato starch, except that common com and potato starch show a maximum at about 80°C whereas waxy maize-NaCl curve shows a slight minimum at about 75°C at the beginning o f the cooling cycle. The data and the fitted curves for the waxy maize-NaCl suspensions are plotted for both the heating and cooling cycles for the relative dielectric perm ittivity (Figure 3.58) and for the relative dielectric loss (Figure 3.59). Figure 3.58 shows that the dielectric perm ittivity data at low frequencies is lower than that which is predicted by the model. This is probably caused by the inability o f the proposed model to describe the electrode polarization effect. Figure 3.59 shows that 8/ undergoes an upward trend at lower frequencies that is much more pronounced than for the other suspensions. The relative dielectric loss o f waxy maize-NaCl reaches extremely high values, indicated by the fact that the scale on the vertical axis goes to 325, whereas that o f Figure 3.51b and the other graphs o f e /' in Appendix I only go to 12. Conductivity is so high in the waxy maize-NaCl suspensions that its effect completely overwhelms any possible low-frequency dielectric relaxation peak. The effect o f conductivity is inversely proportional to frequency, and its value depends strongly on concentration and temperature. The data for waxy maize-NaCl was re-plotted using extended frequencies, and the result is shown in Figure 3.60 for the heating cycle. The pattern for e si - e s2 for these suspensions is visible in the low frequency region o f Figure 3.60a. In 3.60b the conductivity contribution to e," is so large, that the relaxation peaks were not visible in a linear scale, and the scale had to be changed to logarithmic. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 213 3.5.6.C The Im portance of Taking C onductivity in to Account when M odeling The waxy maize-NaCl suspensions are an extreme case o f the effect o f electrolytes on the dielectric properties. Comparing their results w ith those o f the regular starch suspensions helps to determine what is truly occurring at low frequencies and what changes in the observed dielectric behavior are caused by starch, by salts, or by a combination o f both. I f there is an additional relaxation process occurring at low frequencies, it would only increase the dielectric constant w ith respect to water. However, the data in this study exhibited a dielectric constant lower than that o f water. Electrolytes could decrease the dielectric constant, but at the same time they would increase the relative dielectric loss due to the increased conductivity. The data from the starches w ithout added NaCl in this study do not exhibit an increase in the relative dielectric loss quite as large as that shown at the low-frequency end in Figure 3.47. The curve in Figure 3.47 represents what would theoretically result from the addition o f the almost miniscule amount o f 10'2% (w /w ) sodium chloride to water. The increased conductivity o f the electrolyte suspension caused the relative dielectric loss to have a high “ ta il” at low frequencies, but the relative dielectric perm ittivity remained virtually unchanged from that o f water shown in Figure 3.29. Since the relative dielectric perm ittivity in this study has an observable decrease, it does not seem like ly that electrolytes could explain the whole dielectric behavior o f the waxy maize, common com, and potato starch suspensions. Thus, i f the decrease in relative dielectric perm ittivity cannot come either from a low-frequency relaxation process or from electrolytes, then it must be caused by a dilution effect, as was hypothesized before. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 214 (a) s/ heating Frequency (Hz) Temperature ( C) 80 10° (b) 6/ cooling 80 ■ 75 70 65 50 10’ Frequency (Hz) Temperature (°C) Colored curves = measured data; 80 10° black curves = fitted data Figure 3.58 Comparison of the fitted curve with the measured data for the relative dielectric permittivity during the a) heating and b) cooling cycles for waxy maize with NaCl. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 215 a) e/’ heating 80 75 70 65 325 300 275 250 225 200 g" 175 150 125 100 Frequency (Hz) Temperature (°C) (b) e/' cooling 75 325 300 275 250 225 200 g" 175 150 125 100 45 40 Frequency (Hz) 35 Temperature (°C) Colored curves = measured data; black curves = fitted data Figure 3.59 Comparison of the fitted curve with the measured data for the relative dielectric loss during the a) heating and b) cooling cycles for waxy maize with NaCl. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 216 (a) e/heating 75 140 120 100 70 Temperature (°C) Frequency (Hz) (b) heating 80 75 70 65 60 55 8 50 I 45 40 Temperature (°C) Frequency (Hz) 35 30 Figure 3.60 Extended frequency results for waxy maize with NaCl during the heating cycle: a) relative dielectric permittivity and b) relative dielectric loss. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 217 3.5.7 Broad Spectrum Dielectric Measurements The results from the broad spectrum measurements fo r heating and cooling are shown in Figure 3.61. In these plots, the measured dielectric data (corrected w ith the water measurements) is represented by the colored curves, and the fitted data is represented by the black curves. For the measured data, the different colors correspond to the different temperatures, as indicated by the color bar. These plots show some o f the same trends observed in the narrow frequency measurements. The relative dielectric perm ittivity decreases w ith increasing temperature and frequency. The relative dielectric loss increases w ith increasing frequency, and the slope o f the e," curves decreases w ith increasing temperature. For the broad frequency data, the upwards curvature at lower frequency is evident fo r the e /' curves, but it is not discernible in the s / curves. The frequency spectrum was extended an extra decade at the high frequency end o f the spectrum, and therefore more o f the main relaxation process (corresponding to water dipole rotation) is visible. A t the low frequency end, the spectrum is also one decade wider, but it must be kept in mind that frequencies below 0.2 GHz are outside the optim al frequency range o f the dielectric probe and the measurements are not as reliable as the previous narrow frequency measurements. Figure 3.62 shows the parameters fitted w ith the DHTP model fo r the broad frequency data, compared w ith the narrow frequency parameters. It is obvious in both cases that the broad frequency data is “ noisier” than the narrow frequency measurements. As before, the parameters at low frequency were sensitive to in itia l estimates, and therefore were not unequivocally determined. Since it is most probable that there exists another relaxation process at some low frequency (as for many other bio-polymers) it can be concluded that the critical frequency for this process is s till much lower than 45 MHz. I f there were a relaxation process w ithin the frequency range o f these measurements, it would have been obvious in the plots even i f the measurements were not completely accurate. The fitted parameters for the high frequency process shown in Figure 3.62b show reasonable agreement w ith the parameters fitted to the narrow frequency data, and are also close to the parameters for water. These parameters were not sensitive to in itia l estimates. The im plication is that ss2 and t 2 (which account fo r most o f the variation in the data) were appropriately fitted from the narrow frequency data, even i f the low frequency parameters (accounting for only a small portion o f the data) seem to s till be out Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 218 (a) heatmg ■ 1 0 Frequency (Hz) (b) cooling A Frequency (Hz) Colored curves = measured black curves = fitted data Figure 3.61 Comparison of the fitted curves with the measured relative dielectric permittivity and loss data from the broad frequency experiments for waxy maize starch suspension during the a) heating and b) cooling cycles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 219 a) 22 20 18 16 W- 14 U o 1? es1- ss2broad frequency t 1broad frequency es1- ej2 original data x1original data 10 CO CO 8 6 4 2 0 30 40 50 60 70 Heating 80 70 60 Temperature (°C) 50 40 30 Cooling (b) es2broad frequency t 2broad frequency ss2original data t 2original data es water x water 30 Heating 40 50 60 70 80 70 Temperature (°C) 60 50 40 30 Cooling Figure 3.62 Comparison of parameters for both the original data and the broad frequency data: a) esl - es2and t, curves; b) £,2 and r2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 220 o f the reach o f the instrumentation. Table 3.9 is sim ilar to Table 3.7 w ith the Table 3.9 Summary o f Details o f the Broad Spectrum Experiments Starch W axy M aize SSTO 7.8953 x 10" SSE 3.4883 x 103 # o f observations 12600 # o f parameters 106 results o f the statistics fo r the broad spectrum experiments. Also included in this table is the fitted equivalent NaCl concentration for the data. degrees o f freedom 12 494 mse 0.2792 R2 99.56% % NaCl in a m atrix 3.61 x 10'3 Notably, the SSE is much larger than the SSE for the narrow frequency experiments, but the MSE is s till very low because the broad frequency range provided many more observations. The R2 fo r the fitted values is very close to one, meaning that the variation in the data is satisfactorily accounted fo r by the parameters. Table 3.9 also shows that the fitted equivalent NaCl concentration was at the lower bound set for this parameter. This implies that the algorithm was attempting to fit more o f the curvature w ith the low frequency process, at the expense o f the NaCl concentration. From the conductivity measurements, it is known that this concentration cannot be far below 3.8 x 10"3 % (a maximum o f 5% under this value, or 3.61xl0'3 %, was set as the lower bound). Therefore, the fitted values for the low frequency process are suspected to be spurious at least for this set o f data and parameters. 3.5.8 Conductivity Measurements The conductivity measurements were performed in order to provide an idea o f the range o f conductivities expected during heating o f starch suspensions, and to complement the broad frequency dielectric measurements. The results o f these conductivity measurements for a suspension o f waxy maize starch are shown in Figure 3.63. In this plot, contour lines o f equivalent NaCl concentration are also shown. These contours represent the concentration o f NaCl in water that would produce the conductivities on the vertical axis at the temperatures indicated on the horizontal axis. Conductivity increased with temperature during heating between 8.96 mS/m at 25°C to 23.0 mS/m at 95°C. Upon cooling, the conductivity increased briefly to 2.33 mS/m at 75°C, and then decreased to 11.5 pS/cm at 25°C, never returning to its original low value o f 8.96 mS/m. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 221 x10J 0.024 I 0.022 11 10 0.02 E 0.018 > 0.016 T3 0.014 0.012 0.01 0.008 50 60 70 Temperature (°C) 100 Figure 3.63 Conductivity of a waxy maize starch suspension during heating and cooling. Colored contour lines and color bar indicate equivalent NaCl concentration in %(w/w). Processing the raw conductivity data w ith the conductivity function developed earlier indicated that the equivalent sodium chloride concentration increased slightly during heating (from 3 .8xl0"3 to 4 .2 xl0 "3 % ) during heating, then it increased rapidly, reaching up to 6.0 xlO '3 % at 60°C. Finally, the equivalent sodium chloride concentration decreased, and became roughly constant (average -5.3 x 10'3 %) during the last stages o f cooling. This behavior is consistent w ith some in itia l leaching o f electrolytes from the starch granules, followed by a further release upon gelatinization. The roughly constant conductivity during cooling from 95 to 60°C (and the apparent increase in equivalent sodium chloride concentration) was attributed to slow circulation o f the hot, gelatinized suspension through the conductivity sensor. It is hypothesized that, while the measured temperature in the bulk o f the suspensions was decreasing quickly (at about 8.2 °C/min in itia lly), the effective temperature at the sensing area o f the instrument was decreasing more slowly. As the cooling rate decreased during the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 222 last stages o f cooling (to about 1.2 °C/m in) the temperature equilibrated, and the conductivity decreased in response. Based on these results (and dismissing the points during the first stages o f cooling), the lower and upper bounds for equivalent salt concentration in the LSQDHTP program were adjusted fo r the broad frequency dielectric measurements. The bounds were set to 95% o f 3.8xl0'3 % and 105% o f 5.3xlO'3 % to account fo r possible inaccuracies in the measurement. It is interesting to note that the fitted parameters for the narrow frequency data predicted an equivalent concentration o f 2.59x10‘3 % fo r the waxy maize starch suspensions. This value is w ithin the same order o f magnitude as the concentration obtained from the conductivity measurements, but it is lower, indicating that the algorithm probably decreased the fitted equivalent concentration in order to preferentially fit a low frequency relaxation process. Since the low frequency relaxation process has two different parameters (esi - es2 and tO affecting it, the algorithm can fit the data closer manipulating these two parameters rather than manipulating the equivalent concentration. 3.5.9 Time-Temperature Superposition Figures 3.64 and 3.65 show the results o f the time-temperature superposition (TTS) fittin g for the relative dielectric perm ittivity and the relative dielectric loss, respectively o f waxy maize starch suspensions using the narrow frequency data. Figure 3.66 shows the m ultipliers fitted for the frequency and for the dielectric data. The fittin g shows very good agreement w ith the data at the higher frequencies, especially fo r the relative dielectric loss curves, but fails to correctly follow the curvature o f the data at lower frequencies. It should be pointed out that it is possible that the main relaxation process could follow the TTS principle even i f the process at lower frequency does not. Cases have been reported in the literature in which the TTS principle either does not apply or applies for only one relaxation process but not another (Papadopoulos, et al, 2004; Olsen et al. 2001). In general, the inability o f this method to fit the low frequency data for the relative dielectric perm ittivity may indicate that the observed curvature at lower frequencies may be indeed caused by a lower relaxation process, even i f at present the exact position and magnitude o f the process cannot be identified. Final resolution o f this question w ill require much wider frequency range measurements, probably using time domain measurements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 223 a) heating 61.5 s ' 60.5 50 45 40 59.5 135 Frequency (Hz) (b) cooling 61.5 b' 60.5 59.5 Frequency (Hz) Colored curves = measured data; black curves = fitted data Figure 3.64 Time-temperature superposition curves of the relative dielectric permittivity data for the waxy maize starch suspension narrow spectrum experiments during the a) heating and b) cooling cycles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 224 (a) 80 ■ heating 7S ■70 65 e " r 150 145 140 35 Frequency (Hz) (b) 80 9 cooling 75 8 70 7 65 6 60 E 55 50 4 145 3 140 2 35 130 10m Frequency (Hz) Colored curves = measured data; black curves = fitted data Figure 3.65 Time-temperature superposition curves of the relative dielectric loss data for the waxy maize starch suspension narrow spectrum experiments during the a) heating and b) cooling cycles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 225 2.5 frequency multiplier e f * multiplier 1.5 Heating T(°C) Cooling Figure 3.66 Multipliers for the frequency and relative complex permittivity for the time-temperature superposition curves for the waxy maize starch suspension broad spectrum experiments. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 226 3.6 Conclusions In this study, a method was developed to measure the dielectric properties o f liquids in-line during heating and cooling. This method was applied to dilute starch suspensions (both w ith and w ithout sodium chloride), and their dielectric properties were measured as functions o f both temperature (between 30° and 80°C) and frequency (between 0.3 and 3 GHz). The measured dielectric data was then successfully fitted w ith the Debye-Hasted model w ith two peaks. The temperature-dependent parameters o f this model turned out to be very useful for both explaining and predicting the dielectric properties o f the starch suspensions as functions o f both frequency and temperature. The results indicated that the presence o f starch lowered the dielectric perm ittivity o f water, and shifted the relaxation tim e o f the water dipole rotation to lower frequencies. It was also found, as suspected, that added sodium chloride did affect the dielectric properties o f starch suspensions. Interestingly, however, the effect o f this salt was less than would have been expected fo r pure water, indicating that starch inhibits conductive loss in this system. During data analysis it was discovered that some kind o f phenomenon that caused an “ upswing” in the dielectric loss curve was occurring at low frequencies. Much effort was put into trying to determine i f that phenomenon was a second relaxation peak, a conductivity-induced increase in the loss caused by residual salts le ft on the starch granule, or a combination o f both. The literature indicates that a great variety o f biopolymers shows a peak (or more) at lower frequencies in addition to the main water relaxation peak. It is also known that very small quantities o f electrolytes can greatly affect dielectric properties, especially the dielectric loss. In an attempt to find an answer, an extra set o f dielectric measurements w ith broader frequency range (between 45 M Hz and 20 GHz) was taken, and the conductivity o f the suspension was measured during heating. Unfortunately, the issue could s till not be definitively settled. The low frequency relaxation process, that most probably does exist and that would explain the curvature, was not obvious even w ith the w ider frequency range. In addition, the conductivity measurements did indeed point to the presence o f electrolytes, but these electrolytes by themselves could not explain the curvature in the dielectric perm ittivity. It is le ft as a suggestion fo r future research in Chapter 5 that an even lower frequency range be examined. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 227 In summary, the conclusions o f this study are as follows: 1. Starch lowers the dielectric perm ittivity (e /) o f water and increases its dipole rotation relaxation tim e ( t 2) . For starch suspensions w ithout added salt, the Debye-Hasted model w ith two peaks can be used to adequately fit the resulting dielectric properties at a ll frequencies, but especially at frequencies greater than 1 GHz. 2. There is an upward curvature at the low-frequency region o f both e / and e /' vs. frequency curves that could be explained w ith a relaxation peak at low frequencies and/or the presence o f electrolytes in starches (as supplied by their manufacturers). This curvature is visible starting at about 1GHz, but no peak-related maximum was observed at frequencies as low as 45 MHz. 3. The conductivity o f dilute waxy maize starch suspensions during heating and cooling indicates the presence o f electrolytes. 4. Time-temperature superposition can be used to fit the dielectric data at high frequencies, but fails to follow the curvature at low frequency. 5. The addition o f 2% sodium chloride to the waxy maize starch suspension depresses e/ and greatly increases e /' values, especially at lower frequencies. Starch moderates both o f these effects as compared w ith pure water. There is anomalous upward curvature for e / at low frequencies, consistent w ith the phenomenon o f electrode polarization 6. The behavior o f the parameters o f the Debye-Hasted model w ith two peaks as functions o f temperature can be used to infer the behavior o f dielectric properties as functions o f both frequency and temperature. Therefore, they should provide valuable insight about the behavior o f sim ilar food systems. 7. It was hypothesized that changes in the dielectric behavior o f water when starch is added could be could be caused by a combination o f the follow ing: a) dilution o f water caused by the presence o f the granules; b) electrolytes leaching out o f the granules; and c) water interacting first w ith the amorphous portions o f the starch granules at low temperatures, and then w ith all components o f starch after higher temperatures are achieved. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 CHAPTER 4 FINITE DIFFERENCE TIME DOMAIN MODEL OF THE ELECTROMAGNETIC FIELD DISTRIBUTION IN A DOMESTIC MICROWAVE OVEN LOADED WITH AN AQUEOUS STARCH SUSPENSION AT DIFFERENT TEMPERATURES 4.1 Introduction In 1986 Mudgett noted that “ One o f the major obstacles to successful development o f new industrial microwave food processes has been the lack o f predictive models relating the electrical properties o f foods to transient time-temperature profiles that determine m icrobial safety and product quality.” chapter 3 o f this current study focused on the measurement and prediction o f the dielectric properties o f starch suspensions w ith concentrations typical o f foods such as soups, sauces, and gravies. Because these foods are frequently heated in a microwave oven, it is o f interest to develop a model that could predict the time-temperature distribution in a starch system during microwave heating, thereby addressing the concern expressed by Mudgett. In order to model microwave heating o f any material, it is necessary to determine the amount o f heat generated at every point w ithin the volume o f the material. This inform ation can be obtained from M axw ell’ s equations, which most be numerically solve for all but the simplest geometries. Application o f the numerical solutions requires knowledge o f the geometry o f the system, the dielectric properties o f the material, and the nature o f the microwave source. This chapter describes the use o f a numerical method (Finite Difference Time Domain, FDTD) to model the electromagnetic field distribution and power absorption o f water and starch suspensions, given the dielectric properties measured in chapter 3, and the geometry o f a domestic microwave oven. This chapter had three main goals. The first was to create a detailed model o f the geometry o f a household microwave oven containing a bowl fille d w ith loads o f either water or starch suspensions. The second goal was to use the FDTD method to compute the three-dimensional electromagnetic field distribution inside the oven and bowl at different temperatures, using the data collected in chapter 3 as the dielectric properties o f the loads in the bowl. The third goal was to compare the effect o f the two different Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 229 loads and the various temperatures on the calculated specific absorption rate, (SAR), which is a quantity related to dissipated power and heat generation. To aid in the understanding o f this work, a b rie f explanation o f the power transmitted by a wave w ill be given in section 4.1.1, followed in section 4.1.2 by a classification o f models o f food processes. Then the literature review w ill begin in section 4.2.1 w ith a survey o f some attempts to model microwave heating. The Yee Cell and its application to the FDTD method w ill be introduced in section 4.2.2. The objectives o f this chapter w ill be listed in section 4.3, the materials and methods w ill be described in section 4.4, the results w ill be discussed in section 4.5, and the conclusions w ill be summarized in section 4.6. 4.1.1 Power Transmitted by a Wave When an electromagnetic wave encounters a material, some photons o f the incident wave’ s radiation is absorbed by atoms, ions, and/or molecules in the material, exciting them to a higher energy state. When this occurs, those frequencies o f radiation that were absorbed are removed from the original wave o f radiation, and the wave’s power is attenuated w ith increasing penetration depth (IFT, 1989). The ratio o f the wave’ s final power (Pz) to its in itia l power at the surface o f the material (P0) is known as the transmittance Tmo f the material (Skoog and Leary, 1992). That is, Tm= Pz/P0 [4.1] Some discussion o f power absorption is necessary in any treatment o f microwave heating. Microwave ovens themselves are labeled w ith their maximum power outputs, and many claim to have adjustable power levels. The truth is that the magnetron o f any given microwave always operates at the same power level. Any adjustments to the power level are achieved only through timers that turn the magnetron on and o ff periodically (Kudra, et al., 1991). The amount o f microwave power that a given food can absorb affects how it is cooked. For example, Barringer et al. (1995) found that power absorption was a very influential factor in the increased heating rate o f emulsions. They explained that the dispersion o f one phase in another created numerous interfaces where some o f the electromagnetic radiation would be internally reflected before exiting the sample. As the concentration o f the dispersed phase increased, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 230 number o f interfaces also increased, leading to more internal reflections. These increased reflections improved the power absorption, which, in turn, led to increased heating rates. O f fundamental importance, then, is how the absorption o f microwave power by a material can be modeled. Van Remmen, et al. (1996) mentions two general approaches. The first is the use o f electromagnetic fie ld equations o f M axwell, and the second is the use o f Lambert’ s third law. Lambert’ s third law is based on the fact that the intensity o f electromagnetic radiation decreases exponentially w ith distance as it travels through an absorbing medium, as shown in Figure 4.1 (Francis, 2001; Nicholson, 2001; Summons, 2002). That is, Iz = Ioe_kz [4.2a] where I0 is the in itia l intensity o f the radiation at the surface o f the medium, z is the distance it travels through the medium, k is a linear absorption coefficient, z Iz is the intensity o f the transmitted radiation after traveling a distance z in the medium. A variation o f this law, known as the Beer-Lambert law (Atkins, 1990), describes the reduction in intensity o f radiation as it travels through an absorbing solution o f concentration [c] Figure 4.1 Change in intensity of radiation passing through a material of thickness z and thickness (z): Iz = Ioe'K[c)z [4.2b] Johann Lambert first proposed his third law for light (Summons, 2002), and the Beer-Lambert law is applied in visible light spectrophotometry and spectroscopy (Atkins, 1990; Nicholson, 2001; Skoog and Leary, 1992). However, both laws are applicable for a study o f microwave radiation, in which case they describe changes in power, rather than changes in intensity (Van Remmen, 1996). The terms “ power” and “ intensity” are often used interchangeably, but Skoog and Leary (1992) explain that there is a subtle difference between them: “ The power P o f radiation is the energy o f the beam reaching a given area per second whereas the intensity I is the power per unit solid angle. These Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 231 quantities are related to the square o f the amplitude.” In its power form , Lambert’s third law is often w ritten as Pz = Poe'112 [4.2c] where P0 is the power at the surface o f the material. There is an obvious sim ilarity between equation 3.83b (which indicates that the intensity o f an electric field entering an infinite medium decreases exponentially) and equation 4.2c. Lambert’ s law is used frequently in the literature (some o f which w ill be reviewed in the next section) to model heating food by microwave radiation. Unfortunately, there are lim itations to Lambert’s Law. The first is that it only considers the power transmission, not the power dissipation. A second lim itation is that Lambert’ s Law assumes the sample is in fin ite ly thick, which is usually not the case for food materials (van Remmen, et al., 1996). A third lim itation, o f particular interest to this study, is that Lambert’s Law ignores standing wave patterns that often occur in a microwave oven. There are also lim itations to the accuracy o f the Beer-Lambert law w ith certain types o f solutes and solutions o f high concentrations (Skoog and Leary, 1992). For a more accurate description o f the power in a system, M axw ell’s equations must be used directly. Quick dimensional analysis shows that units o f power (watts per square meter) can be obtained by m ultiplying two quantities from M axw ell’s equations: E (w ith units o f volts per meter) and H, (w ith units o f amperes per meter): V m . _A_ = m J Cm . C ms = J m2 s = W m2 *■ J I f this m ultiplication is actually the cross product o f E and H , the resulting vector is known as the Poynting vector, and is given the symbol S in the notation o f Shen and Kong (1987): S(x,y,z,t) = E (x,y,z,t) x H (x,y,z,t) [4.4] The Poynting vector is named for the English physicist John Henry Poynting (1842-1914) who studied the transference o f electrical energy by electric and magnetic fields (Hayt and Buck, 2001; Ohanian, 1985; Sttsskind, C., 1968b. The Poynting vector represents the flow o f electromagnetic power per unit area that is carried by a uniform plane wave. Because it is found by the cross product o f the electric and magnetic fields, the Poynting vector is perpendicular to both. Since the energy flows in the direction o f propagation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 232 o f the plane electromagnetic waves, “ The direction o f the vector P indicates the direction o f the instantaneous power flo w at the point, and many o f us think o f the Poynting vector as a ‘pointing’ vector. This homonym, w hile accidental, is correct.” (Hayt and Buck, 2001). Edminster (1993) explains that part o f the usefulness o f the Poynting vector comes from the fact that it can be used to determine the directions o f fields or waves (be they incident, transmitted, or reflected) without specifying coordinates (1993), which is useful for studying waves. I f E is in the complex frequency domain rather than in the real time-space domain, it is represented by equation 3.83b. The corresponding equation for H is H = y ( E o /q V 1*2 [4.5] where q* is the complex intrinsic impedance o f the medium, which is defined in Appendix A. The magnitude o f the power carried by these time-harmonic electromagnetic waves at any given instant in time w ill vary because the waves’ amplitudes cycle between a minimum value o f zero and a maximum value. Knowing the average power delivered by the wave over several cycles would be more useful, so the time-average o f the Poynting vector is calculated to find the average flo w o f electromagnetic power per unit area (Shen & Kong, 1987): <S> = z (1/2)(E02/1 q* | )e'2kiz cos <p [4.6] where cp is the phase angle. I f cp is considered to be a constant that could be combined w ith the coefficients o f the exponential term into a generic constant related to the in itia l power (symbolized by P0), and i f the factors 2 and ki are combined into one generic constant k, equation 4.6 then takes on the form <S> = z (constant) e_kz [4.7] Equation 4.7 shows remarkable resemblance to Lambert’s law for the exponential decrease in the power o f an electric field entering a medium that was given in equation 4.2.c. Inherent in the derivation o f this form o f the time-average Poynting vector are the follow ing assumptions: 1) Only one electromagnetic wave is considered; 2) The wave has no source or charges; 3) The wave is a plane wave, varying only in the z-direction; 4) The wave travels undisturbed through space; there are no standing wave patterns; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 233 5) The medium is lossy so the wave simply decays when it impinges the medium; there are no reflections or interactions. A ll o f these assumptions are also inherent in the application o f Lambert’ s Law. 4.1.2 Classification of Models of Food Processes Puri and Anantheswaran (1993) classified methods to study food processes as either experimental or analytical. The experimental approach usually involves studying the processing steps in a laboratory or p ilo t plant setting. Often the result o f such work is the development o f empirical equations which may be useful in one particular setting, but that cannot be applied elsewhere. However, such experimental work should be undertaken to validate the analytical approach. The analytical approach uses physical principles to derive and solve equations that model the processing steps. Puri and Anantheswaran (1993) recognized three different types o f analytical approaches. The first type, known as exact or closed-form solutions, involves making assumptions and sim plifications to convert a complex problem into a more idealized one. This method is,useful for determining “trends and order o f magnitude o f variables.” The second type, approximate closed-form solutions, “ are attempts at solving real-world problems,” but they may be lim ited by the geometry o f the food, boundary conditions, and the steady- or unsteady-state processing conditions o f the system. The third type, numerical solutions, involves the computer simulation o f the process being studied. Two numerical techniques that are frequently used in the microwave literature are the finiteelement method (FEM ) and the finite-difference method. A special adaptation o f the finite difference method, which is called the finite difference time domain method (FDTD), is often used to solve electromagnetic problems, including the problem presented in section 4.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 234 4.2 Review of the Literature 4.2.1 Survey of Some Attempts to Model Microwave Heating One o f the earliest attempts at modeling microwave heating o f foods was conducted by Ohlsson and Bengtsson (1971). Their goal was to study dielectric heating, while also taking into account heat conduction. They designed a one-dimensional computer program to simulate the microwave heating o f an infinite slab o f meat. The computer program enabled them to calculate the temperature w ithin the meat at a certain depth or time by using the finite difference method to solve the Fourier heat transfer equation. In order to sim plify the problem, they assumed that the electromagnetic field w ithin the microwave oven was a plane wave, perpendiculary incident to the meat surface. They used Lambert’s law to describe the exponential decrease o f power w ithin the meat, and they estimated the power densities based on data from actual experiments. Ohlsson and Bengtsson validated their computer simulation w ith actual microwave experiments on beef, salted ham, and a simulated meat. A fter finding that the computer predictions for the temperature corresponded favorably w ith the measured values, they commented, “ It is surprising that such a simple model resulted in so good agreement between experimental and simulated heating, in spite o f the various approximations and lim itations inherent in the model.” Among the lim itations that they readily acknowledged were the assumptions o f the perpendicular plane wave, the infinite size o f the meat, and the insignificance o f edges, comers, and evaporative losses. In addition, they sim plified the problem by using linear approximations o f the derivatives in the computer program. Wei et al. (1985a) also used a one-dimensional model to represent a w ater-filled sandstone. Since they were studying a drying application, they incorporated a thermal energy equation, liquid and gas phase continuity equations, and a gas phase diffusion equation into their model. They applied Lambert’ s law for power absorption and used values o f the dielectric constants and loss tangents that had been measured as functions o f saturation level, but not as functions o f temperature. The nonlinear differential equations that they derived were solved w ith a finite difference formulation. The model’ s predictions for temperatures and evaporation rates over time were in good agreement w ith the measurements o f the authors had made during actual drying experiments in a microwave oven. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 235 Komolprasert and O foli (1989) used dimensional analysis and two different microwave oven models to develop two correlations to predict the heating time as a function o f many variables. The variables included temperature, power, density, dielectric loss, specific heat, in itia l and local temperatures, thermal conductivity, frequency o f the radiation, duration o f exposure, radius and height o f the container, and depth from the surface. One o f the correlations was more successful at predicting microwave heating times than the other. The authors acknowledged that differences between the two types o f ovens (including cavity size, power output, and presence or absence o f a mode stirrer) may have resulted in different electromagnetic field patterns, leading to different temperature profiles. In addition, they admitted that “ the assumption o f uniform electric field strength may also be invalid.” They suggested that future research should take into account d iffic u lt modeling issues such as the multi-component nature o f foods, the effect o f moisture content, the determination o f dielectric properties, and the changes that can affect microwave absorption, such as gelatinization and protein denaturation. Zhou et al. (1995) undertook a study to predict the temperature and moisture distribution in foods during microwave heating. They derived a system o f heat and mass transfer equations that took into account heat losses due to convection and evaporation, as w ell as moisture loss due to evaporation. Before using Lambert’ s law to represent the power absorption in the food, they obtained the value o f the necessary parameters by performing a regression analysis o f the data based on power and product weight. They estimated that the incident power was equal to the total absorbed power divided by the surface area, and assumed the dielectric constant and relative dielectric loss values were constant. The researchers simulated the process w ith a three-dimensional finite element method program. They then validated their program by both analytical and experimental approaches. The results from their analytical approach agreed w ith the results o f their FEM program. The experimental approach yielded results w ith sim ilar trends, but there were some differences. The researchers attributed these differences to several possible inaccuracies, including inaccuracies in the placement o f temperature probes, in the measurements o f absorbed power and surface evaporation rate, in the data they had used for the physical properties o f materials, in the amount o f moisture lost during the preparation o f the samples, and in the assumption o f uniform power and moisture distributions. One notable observation that they made was that the FEM predicted a higher temperature at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 236 the center o f a potato cylinder than what was actually measured. The authors explained that this was due to the assumption o f Lambert’s law that the electromagnetic radiation is perpendicularly incident on the surface o f the food. This means that a ll the energy would pass through the center o f the cylinder, causing higher temperatures. However, in actuality, “ the surface o f the food sample is not perfectly smooth which w ill cause the microwaves to deviate from the center. This deviation w ill lead to a reduction in power at the center.” Van Remmen et al. (1996) used fin ite difference approximation to solve the equations they derived in their efforts to qualitatively model the penetration o f microwave energy into foods and to predict the temperature distributions in three different geometries. They assumed that the foods were homogeneous, w ith temperature-independent material properties. They also assumed that the electric field w ithin the microwave oven was uniform , im pinging the food perpendicularly from a ll directions. They based their models on the Fourier heat transfer equations, assuming negligible mass transport and internal convection, although they did account fo r surface convection in the boundary conditions. Incident waves were assumed to be partly transmitted and partially reflected at air-food interfaces. The transmitted waves decayed exponentially according to Lambert’s Law, and power distribution was modeled as three superimposed passages o f electromagnetic radiation w ith internal reflections, an approach which they claimed had “ resulted in a good approximation o f the electric field distribution resulting from application o f M axw ell’s equations” in previous research done by the group. The power dissipation in the various layers o f the food was the basis for calculating internal heat generation terms in the model. When they sought to validate their model experimentally, they found that the model did not give an exact description o f the temperature distribution, but it did show that penetration depth was dependent on composition. It accurately predicted both the increased penetration depth in foods without salt and the resulting center heating fo r spheres and cylinders as described in section 3.2.12. It also predicted the reverse o f those heating patterns for foods w ith salt. Table 4.1 is a copy o f the table in van Remmen et al. (1996) that predicts how a given food’s temperature profile would change based on changes in various model parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 237 Table 4.1 Effect o f Changes o f Parameters on Temperature Profile A fter Microwave Heating (from van Remmen et al., 1996) M odel param eter electric field strength E dielectric constant 8' E ffect o f Param eter Increase increase o f total level o f temperature profile; more pronounced temperature differences less penetration causing surface/peripheral heating loss factor e" high absorption in outer product layer causing surface/peripheral heating thickness/diameter (D/R) lower penetration depth causing surface/peripheral heating heat transfer coefficient h thermal conductivity k temperature decrease in outer product layer flattening o f temperature profile E ffect o f Param eter Decrease decrease o f total level o f temperature profile; less pronounced temperature differences. higher penetration depth causing flattening o f profile (slab) or centre heating (sphere, cylinder). lower absorption in outer layers; higher penetration depth causing flattening o f profile (slab) or centre heating (sphere, cylinder) higher penetration depth causing flattening o f profile (slab) or centre heating (sphere, cylinder) temperature increase in outer product layer more pronounced temperature differences Oktay and Akman (2001) examined the microwave heating o f rubber rods to validate their numerical model. Instead o f directly modeling the electromagnetic distribution, they assumed a heat transfer equation for an empty single mode microwave cavity system. They calculated the electric field intensity based on the mode o f propagation o f the radiation and on the magnetron’ s power, and assumed that the rubber sample would be too small to perturb that electrical field. The authors accounted for external convective heat transfer in their boundary condition, and they considered the temperaturedependence o f the rubber’ s dielectric loss. However, they assumed that the thermal properties o f rubber were temperature-independent. They used an FDTD method to solve their two-dimensional model that would predict the temperature at the surface o f the rubber over time. The model’s predictions were in good agreement w ith the experimental results at absorbed power levels o f 17 and 10 W, but not at 8W. The authors attribute the model’ s failure at 8 W to the instability o f the microwave power source at low powers, and to the model’s assumption that the thermal properties o f rubber were temperature-independent. It is evident even from the b rie f preceding literature review, that Lambert’s law has frequently been used by many researchers in attempts to model microwave heating. However, as mentioned earlier, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 238 this law is really only applicable for substances w ith infinite thickness (van Remmen et al., 1996) which obviously is not the case fo r most foods heated in a microwave oven. M axw ell’s equations would be the better basis for a model, but they are often d iffic u lt to solve. The next section describes a practical method fo r solving M axwell’s equations. 4.2.2. The Yee Cell In 1966 Yee published a paper offering a numerical solution to M axw ell’s equations when the boundary condition is a perfect electrical conductor. The solution was based on a three-dimensional cell model (w ith dimensions Ax, Ay, and Az) that Yee used to represent the location o f the three Cartesian components o f the electric and magnetic fields in space. In the Yee cell shown in Figure 4.2, the point (i, j, k) is located at the far bottom right comer, and other points in the cell are found by translating a distance o f either Ax, Ay, and Az (or one-half o f those distances) along each o f the three axes, respectively. In Yee notation, Ex(i+ '/2, j, k) represents the x-component o f the electric field found at the point x = (i+!A)Ax, y = jA y, and z = kAz (which is located on the bottom face o f the cell towards the back), whereas the component E*(i+1A, j, k+1) represents the x-component o f the electric field on the opposite (front) side o f the Yee cell. Ex(i+'A, j+ l, k) EZ(H 1, j+ l. k+'A ) 0) i, +»/*. k t W i-ij+ 'A .k + y ) ,p.a+y,.iH-y,.k+i, Ey(i -1, j- A , k+1) E,(i, j, ki 'A) S ----j- ! •> / Ez(i+1. j. k + 'A ) ► x (i) z (k) Figure 4.2 Yee cell showing locations of the indexed field components used in FDTD. Adapted from Yee, 1966. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 239 The components o f the magnetic fie ld are displaced from those o f the electric field by h a lf a cell, as can be seen by Hy(i+ '/2, j, k+!4) located on the bottom face o f the cell between the two previously mentioned x-components o f the electric field. As a result, in a grid o f numerous Yee cells, each electric field component is surrounded by the four magnetic fie ld components required to calculate the curl in equation 3.10; and each magnetic field component is surrounded by the four magnetic field components required to calculate the curl in equation 3.11. This arrangement permits algebraic substitutions to replace the curl equations in M axw ell’s formulas w ith good accuracy. These algebraic substitutions form the basis o f the fin ite difference time domain method. Mathematically speaking, it consists o f replacing the differential operator d in M axw ell’ s partial differential equations w ith the operator A, which represents a finite difference. For a generic time-dependent quantity A that has two different values at ti and t2, the finite difference time domain uses the approximation dA „ at AA At _ A (t2) - A ( t 2) t2- t i [4.8] The approximation in equation 4.8 can be used in M axw ell’s curl equations after they are w ritten in the format o f partial differential equations. Equations 4.9 through 4.11 below is the set o f partial differential scalar equations for equation 3.5: aEv dz [4.9] [4.10] N ay ii 5BX at aBv at _ SE* dz aEz 5x aBz _ 5E* aEv ay ax at Yee (1966) also gives analogous equations fo r equation 3.6. Applying Yee’ s finite difference approximation in equation 4.8 to equation 4.9 yields Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [4.11] 240 Bxn+1/2(i, j+1/2, k+172) - Bx"~1/2(i, j+1/2, k+1/2) = At Ey11(i, j+1/2, k+1) - E j (i, j+1/2, k) Az _ Ezn (i, j+ l, k + 1 /2 )-E zn (i, j, k+1/2) Ay where the superscript “ n” represents the number o f time lapses considered in the study, such that the tim e o f interest is given by the expression t = nAt [4.13] In words, the numerator o f the le ft side o f equation 4.12 represents the difference between the x-component o f the magnetic flu x density (at one location in space) at h a lf a time increment before the tim e o f interest (n - Vi) and at h a lf a time increment after the tim e o f interest (n + Vi). This difference is then divided by the time increment. The numerator o f the first term on the right side o f equation 4.12 represents the difference between the y-components o f the electric fie ld at two different locations in space at the time o f interest. This difference is divided by the distance along the z-axis that separates the two points. Sim ilarly, the second term on the right side o f equation 4.12 represents the difference between the z-components o f the electric field at two different locations in space at the time o f interest. Once again, their difference is divided by the distance along the y-axis that separates the two points. When modeling an electromagnetic problem w ith the finite difference method, a series o f equations sim ilar to equation 4.12 must be solved by a computer. Such equations represent “ nearestneighbor interactions” as the electric and magnetic field propagate in “ discrete time steps” over a grid o f Yee cells (Kunz and Luebbers, 1993). This is the finite difference time domain (FDTD) technique, the subject o f the 1993 book published by Kunz and Luebbers. The authors claimed, “ O f the many approaches to electromagnetic computation, including method o f moments, finite difference time domain, finite element, geometric theory o f diffraction, and physical optics, the finite difference time domain (FDTD) technique is applicable to the widest range o f problems.” Its advantages include its ability to model a wide range o f field responses and locations, to represent systems in three dimensions, and to take into account the various conductivities and frequency-dependent variables that objects under study might possess. In addition, the method can be conveniently used on many computers, even PCs. M ur (1981) points out that the FDTD method can be easily applied to various objects with various shapes and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 241 characteristics (i.e, magnetic, dielectric, conducting, homogeneous, or nonhomogeneous). Chen et al. (2000), cite references that support their claim that “ the accuracy, efficiency, and overall u tility o f FDTD has been demonstrated in recent years fo r a variety o f complex high-speed interconnect design problems.” The first step in applying FDTD to solve electromagnetic problems is determining the size o f the cells and the time increment At. This is done by considering a plane wave propagating through a grid o f Yee cells. In any given time increment At, the “ wave must not pass through more than one cell, because during one time step FDTD can propagate the wave only from one cell to its nearest neighbors” (Kunz and Luebbers, 1993). Therefore, the cell lengths should be small, but not too small or they w ill increase the computational requirements. A “ rule o f thumb” that is frequently used is that the cell length should be onetenth (or less) o f the wavelength o f the highest-frequency electromagnetic radiation involved in the problem (Yee, 1966; Kunz and Luebbers, 1993). In some situations, greater accuracy may require onetwentieth o f the wavelength (Kunz and Luebbers, 1993). The wavelength can be calculated from the dielectric properties by the follow ing equation, derived in Appendix A: ^ co(p0E ')1/2 {1 + [ct/(( oe')]2} 1/4{ cos (1/2 tan'1 [ct/( co6')])} [4.14] The cell’s linear dimensions, the time increment, and the speed o f the electromagnetic radiation w ithin a medium (v) are related through an inequality known as the Courant stability condition. In Yee’ s (1966) notation, that condition is given as [(Ax)2 + (Ay)2 + (Az)2] 1/2> vAt = [l/(s p ) ]1/2 At [4.15] W ithin a medium, the speed depends on the dielectric perm ittivity and permeability,which are assumed to be constant in equation 4.14. In the case o f a vacuum, the speed is the speed o f light: c= id ? 14,61 The calculations can be sim plified by specifying that each cell in the grid is a perfect cube, meaning that Ax = Ay = Az. Equation 4.15 then reduces to a more manageable relation for the tim e increment’ s upper lim it (Chen et al., 2000; Kunz and Luebbers, 1993; Mur, 1981): Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 242 A. < - ^ 3~ [4.17] It is possible to use a time increment that is smaller than the one represented by this upper lim it; however, Kunz and Luebbers (1993) point out that this usually does not improve the accuracy o f the solution. On the other hand, they also warn that a time increment that exceeds this upper lim it leads to an unstable system. A fter determining cell size and tim e increment, the next step in applying XFDTD is to determine the size o f the problem space required to hold the object o f interest and the cells that separate it from the boundaries (Kunz and Luebbers, 1993). For practical computational purposes, it is assumed that these boundaries (the outer edges o f the grid o f Yee cells) absorb the radiation (M ur, 1981), even though the fields actually propagate without restriction in free space. A thorough discussion o f boundary conditions is beyond the scope o f this work, but M ur (1981) and Kunz and Luebbers (1993) offer more inform ation to interested readers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 243 4.3 Objectives The goal o f this w ork was to use a numerical solution to M axw ell’s equations to model the electric field distribution in a microwave oven w ith a load. The dielectric properties o f the load corresponded to water and 3% waxy maize starch suspension at temperatures from ambient to beyond the gelatinization temperature o f the starch. This kind o f modeling, coupled w ith heat and mass transfer models, could be used to model microwave heating, elim inating the need for “ guess work” in products and process designs. Therefore, the objectives for this chapter were: 1. To model the geometry o f a household microwave oven and its load. 2. To use the finite difference tim e domain method o f electromagnetic modeling to compute the three-dimensional electric fie ld distribution inside the oven, using loads o f water and starch suspensions at different temperatures. 3. To compare the effect o f the two different loads and the various temperatures on the calculated specific absorption rate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 244 4.4 Materials and Methods 4.4.1 Hardware and Software The modeling was performed using the software XFDTD Bio-pro, that was kindly supplied by its developer, Remcom Incorporated (State College, P A ). Prelim inary modeling was performed w ith version 5.3 o f the software; the final model was completed w ith version 6.O.6.9. The software is an implementation o f the finite difference time domain (FDTD) method used to calculate and visualize electromagnetic fields w ithin and around objects. The calculations were executed w ith the CalcFDTD version 6.0 that was included w ith the XFDTD 6.0.6.9 software. The software was run on a Dell™ Dimension™ 4600 Series PC (D ell Computer Corporation, Austin, T X ) w ith an Intel® Pentium® 4 processor operating at 2.66 GHz (Intel Corporation, Santa Clara, CA). The computer was equipped w ith a 40-GB, 7200-RPM hard drive and 1024 M B o f DDR SDRAM operating at 333 MHz. Its operating system was M icrosoft® Windows® XP Home Edition version 5.1.2600, service pack 1.0 (2002) (M icrosoft Corporation, Redmond, W A). Calculations and data visualization were performed using M ATLAB® student version 6.5.0.1924, release 13 (The MathWorks, Inc., Natick, M A). The microwave oven that was modeled was the Sharp Carousel Model #R-3W96 household microwave oven (Sharp Electronics Corp., Mahwah, NJ) shown in Figure 4.3. Its nominal capacity and power were 0.9 cubic feet and 800 W, respectively. Figure 4.3 Sharp Carousel microwave oven. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 245 4.4.2 Determining Cell Size The XFDTD software allows the user to define objects o f interest in the electromagnetic problem by their geometric shapes and dielectric properties. XFDTD is capable o f generating basic threedimensional geometric shapes such as solid or hollow cylinders, spheres, parallelepipeds, etc. It also gives the user the option to design other geometries by manually specifying the materials corresponding to the xy-, and z-components o f the Yee cells. A ll o f these geometries are bu ilt o f small three-dimensional cells, and one o f the most important steps in using XFDTD (and, indeed, any FDTD implementation) successfully is determining what size the cells should be. As mentioned in section 4.2.2, a general rule o f thumb is that the length o f a side o f the cells should be less than one-tenth o f the shortest wavelength o f radiation that occurs in any material in the model (Yee, 1966; Kunz and Luebbers, 1993). However, XFDTD offers fle x ib ility in cell size. A user has the option o f creating a main grid w ith one cell size, and then creating sub-grids w ith cell sizes that are either one-third or one-fifth that o f the main grid cell size to model components in which the radiation wavelength is smaller. The wavelength in a material can be calculated by substituting the material’s dielectric perm ittivity and loss into equation 4.18 below, which, as Appendix B shows, is equivalent to equation 4.14: 2ji * = © ( Z^ ^ { [ l +(^7> ] A+ 1}/2 g [4'18] Equation 4.18 was used to determine the wavelengths for the materials o f interest in this particular study, which included the water and starch suspensions that served as the two different loads, the glass bowl that contained the load, the glass turntable upon which the bowl sat, the plastic rotating ring upon which the turntable sat, and the ceramic portion o f the magnetron. Because the exact composition o f some o f these materials was unknown, some assumptions had to be made. It was assumed that the ceramic portion o f the magnetron was porcelain, and that the glass plate was Pyrex, just like the glass bowl. Since there are many different types o f plastics, it was d ifficu lt to identify the plastic that made up the ring, but it was fin a lly assumed to be polystyrene. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 246 The m etallic portions o f the oven and the air inside the oven cavity were modeled as perfect conductors and as free space, respectively, which were standard materials in the software w ith pre­ programmed values o f dielectric properties. The XFDTD software required the user to supply the relative dielectric perm ittivity and dielectric conductivity for non-standard materials. The value o f the dielectric conductivity was calculated from a re-arrangement o f equation 3.72 a = er"Eoto, [4.19] assuming that the dielectric loss included both polar and conductivity losses. The dielectric conductivity value proved to be so low for Pyrex, porcelain, and polystyrene, that it was effectively assumed to be zero. The values o f the dielectric properties for both the water and the waxy maize starch suspensions at different temperatures came from the data and computer code o f chapter 3. The program “ th2o,” (based on Hasted, 1972, as described in chapter 3) gave values e/ and for water at the six temperatures o f interest. The dielectric properties for the starches were obtained from the Debye-Hasted model w ith two peaks developed in this study, using the best estimate o f the m atrix o f parameters as described in chapter 3. These e / and e /' values were then used in equation 4.19 to calculate the corresponding dielectric conductivities. Once all o f the dielectric property data was listed, equation 4.18 was used to compute the wavelength o f the electromagnetic radiation in each material so that the appropriate cell size could be estimated. Table 4.2 contains the wavelengths and the dielectric property values that were used in the model. Unless otherwise noted, these tabulated properties were measured at 2.45 GHz. In order to minimize the “ staircasing” errors that would inevitably result from trying to represent smoothly curved surfaces by rectangular cells, XFDTD allows the use o f “ fuzzy” materials between two materials w ith different dielectric properties. These “ fuzzy” materials have properties that are intermediate between those o f the two materials surrounding them. In this study, one “ fuzzy” layer was used between the water/starch suspension and the Pyrex bowl, and another “ fuzzy” layer was used between the water/starch suspension and the air. A fter studying Table 4.2 and considering the combination o f several sizes, it was decided that the main grid cells for the model would be 6.1-mm cubes. O ne-fifth o f this size (1.22 mm) was used as the subgrid size for the bowl and its contents, and one-third o f this size (2.03 mm) was used as the subgrid for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 247 Table 4.2 D ielectric Properties o f Materials at 2.45 GHz (except as noted) and Calculated Wavelengths Values Used in XFD TD M odel *Source M odel Component relative p e rm ittiv ity E'r A ir (assumed to be free space in the model) Polystyrene (at 3 GHz) Porcelain (at 60 Hz, 1 M Hz) f Pyrex 1 C alculated wave­ length (m m ) relative loss e "r 0** dielectric conductivity S/m 0 0 ** 0** 0** 76.62 49.95 61.17 14.20 14.52 14.83 122.32 3% waxy maize susp. (30 °C) 3% waxy maize susp. (40 °C) 3% waxy maize susp. (50 °C) 2.55 6 4 73.9573 70.8124 67.9674 8.3111 6.4263 5.2242 0 0 0 1.1328 0.8759 0.7121 3 3% waxy maize susp. (60 °C) 64.8727 4.1021 0.5591 15.18 3 2 4 1 3 3 3 3% waxy maize susp. (70 °C) 62.5573 3.4745 0.4736 15.46 3 3% waxy maize susp. (75 °C) 61.6882 3.2892 0.4483 15.57 3 Water (30 °C) 75.7226 8.1218 1.1070 14.04 3 Water (40 °C) 72.5433 6.2311 0.8493 14.35 3 Water (50 °C) 69.5167 4.9890 0.6800 14.66 3 Water (60 °C) 66.2958 3.8401 0.5234 15.02 3 Water (70 °C) 63.3387 3.0917 0.4212 15.37 3 61.9605 2.8313 0.3859 15.54 Water (75 °C) * Sources: 1 = Hayt & Buck, 2001; 2 = Inan & Inan, 1998; 3 = c lapter 3 model data, 4 = Callister, 1991. ** The sources report relative loss values for these materials, but the;y were very small. Calculations o f the wavelengths done by both taking into account the!se losses and by excluding them did not differ out to at least the second decimal place. Therefoire, in the model these losses were assumed to be negligible. t This relative perm ittivity value was reported at frequencies w ell b slow the model frequency, but since it does not change in the reported frequency range, it was assurned to remain constant. Any error in the relative perm ittivity value was not expected to have a gre at effect on the model because o f the small size o f the porcelain piece in the magnetron. the magnetron. This was only the first o f several attempts, and it proved not to be the best approach because the CalcFDTD output indicated that instabilities had developed in the course o f the calculations. According to the XFDTD User Manual (Remcom, Inc., 1994-2001), these instabilities were most like ly “ due to the interpolation o f the fields between the main and subgrids.” Several attempts were made to correct the problem by increasing the size o f the load and “padding” both the main grid and the subgrids w ith empty cells, but all proved unsuccessful. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 248 It was then decided that the entire system should be modeled w ith a main grid cell size o f 1.22 mm to avoid the instability associated w ith Table 4.3 Settings for Calculations for the M odel cell size dimensions o f problem space (cells) tim e step equivalence (seconds) number o f tim e steps exit tim e (seconds) (1.22mm)a 303 x 349 x 179 2.349517 x 10'1" 25,000 5.873791 x 10'8 the subgrids. A ll component parts and surface features o f the microwave oven were carefully re-measured, and those o f a size greater than or equal to 1.22 mm were incorporated into the geometry o f the model created in XFDTD. The total problem space o f this new model was 303 by 349 by 179 cells, and the small size o f these cells greatly increased the amount o f processing time required. However, even w ith the burden o f increased processing time, this approach was much more successful. Table 4.3 lists the settings fo r this model w ith the 1.22-mm cells, and the details o f this model are the only ones that w ill be discussed in the next section. 4.4.3 Modeling the Oven The oven, at first appearing to be a simple rectangular metal box, actually had several parts that needed especially detailed modeling, such as the back w all, the floor, the rotating ring, the rotational coupler, the glass turntable, the magnetron, and the glass bowl. Every feature w ithin the oven was carefully measured (w ith calipers when necessary) to accurately construct the geometry file o f the problem in the XFDTD software. The information in the geometry file was later extracted by a program written in MatLab to render three-dimensional figures o f the various parts o f the model shown below. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 249 Figure 4.4 View of microwave oven cavity showing recessed back, location of waveguide (covered by a gray panel on the right), oven floor, and rotational coupler (center of floor). A photograph o f the back w all o f the microwave cavity is shown in Figure 4.4 Most o f the central portion o f this w all is recessed and connected to the rest o f the oven by inclined walls and a border. The comers o f the recessed panel and the corresponding inclined walls are rounded. In order to model these rounded comers, hollow, truncated cones were generated w ith XFDTD. Since only one-fourth o f each cone was needed, the remaining part was erased manually. Once the four comers were formed, they were joined by four inclined planes. The recessed section o f the oven was closed in the back by the outer boundary (which was set to be a perfect conductor). A section o f the resulting modeled component is shown in Figure 4.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 250 Figure 4.5 Model of the back, floor, rotational coupler, and rotating ring inside the microwave oven cavity. A close-up view o f the bottom o f the oven cavity is shown in Figure 4.6 The center o f the floor has a series o f steps designed to accommodate the turntable and the rotating ring. In the center o f the floor there is a truncated cone that holds the rotational coupler. The edges formed between the bottom and the side walls o f the oven are rounded (see Figure 4.4). The steps in the floor were modeled using truncated, hollow cones, and the spaces between steps were modeled using washer-shaped sections. The rounded edges were formed w ith two hollow cylinders. As before, only one-fourth o f each cylinder was used, and the rest was erased manually. The result is visible in Figure 4.5. Figure 4.6 also shows the rotational coupler (which transmits movement from a motor to the turntable), the rotating ring, and its attached wheels. The rotational coupler is basically shaped like a small cylinder w ith three rectangular projections. These three projections were modeled in XFDTD as three rotated parallelepipeds attached to the central cylinder, as shown in Figure 4.5. The rotating ring has a vertical cross section that is T-shaped. This was modeled using hollow cylinders, and the wheels were modeled by cylinders rotated to the appropriate angle. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 251 Figure 4.6 View of the floor, rotational coupler, and rotating ring of the microwave oven. Figure 4.7 is a photograph o f the underside o f the glass turntable. The turntable is essentially a disk, but it has several features that had to be included in the model. The rim o f the main disk is cupped to form a shallow inclined wall. This w all itse lf has a small ledge. The underside o f the turntable has a ridge designed to contain the wheels o f the rotating ring. The inner face o f this ridge forms a right angle w ith the turntable, and the outer portion slopes towards the rim . Also on the underside are three hemispherical “ bumps” on the periphery, and three raised curved pieces towards the center that closely fit over the rotational coupler. The XFDTD model is shown in Figure 4.8. The main portion o f the turntable was modeled w ith a disk-shaped cylinder. The rim and the ridge on the underside were modeled using Figure 4.7 Glass turntable (shown upside down). Figure 4.8 Model of glass turntable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 252 truncated hollow cones. The ledge fo r the rim was modeled w ith a washer-shaped hollow cylinder. The hemispherical bumps were modeled using spheres, and the top parts were manually erased. The rotational coupler was superimposed on a glass cylinder on the underside o f the turntable in order to recreate the curved pieces shown in Figure 4.7. This was achieved through a feature o f XFDTD that allows a newly created object to replace the material that had previously been in that same location. To take the photograph shown in Opening where the cone of, the magnetron is inserted in the waveguide. Figure 4.9, the outside cover o f the oven and waveguide * the magnetron were removed to show the waveguide and the opening in it where the magnetron is inserted. The magnetron its e lf is shown in Figure 4.10. The nozzle-like structure in the lower part is what protrudes into the waveguide opening shown in Figure 4.9. The nozzle consists o f a ceramic cylinder (the purple part in Figure 4.10) w ith a cylindrical-conical cap. Inside the ceramic Figure 4.9 View of inside of microwave oven and waveguide without magnetron. cylinder there is an antenna that fits into an end piece that is also roughly conical. This end piece fits inside the cap. Some o f these internal details are visible in Figure 4.11, which shows a cross section o f another magnetron without the cap. The antenna extends from its end piece, through the ceramic cylinder, and through an opening into the microwave generation cavity. The functioning aspects o f a magnetron are beyond the scope o f this work, but, in general, microwave oven magnetrons force “ clouds” o f electrons to rotate w ithin a strong magnetic field at the characteristic frequency o f the oven (2450 MHz for a household oven). An important part o f the device is a number o f m etallic vanes (eight in this case) arranged like the spokes in a wheel. These vanes change Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 253 polarity as the electron cloud rotates, and the antenna is connected to one o f them. In the center o f the cavity there is an electrode that is heated to produce the electrons. microwave generation cavity antenna ceramic cylinder ceramic cylinder cap Figure 4.10 View of magnetron from Sharp Carousel oven. Figure 4.11 Cross-sectional view of magnetron taken from McDunn and Quinn (2003) of the Stanford Linear Accelerator Center Virtual Visual Center. antenna’s end piece Although the outer dimensions o f the magnetron were carefully measured, the internal parts (such as the end piece fo r the antenna and its distance from the cap, as w ell as the cap’ s position w ith respect to the ceramic cylinder) could not be measured without destroying the unit. They had to be estimated from the external features o f the magnetron and from diagrams, drawings and pictures from other sources (Goldwasser, 1994-2004; Hochwald, 1998; McDunn and Quinn, 2003; Meyers, et al., 1993). Some parts o f the magnetron are regular shapes, and were modeled as such (see Figure 4.12). Other parts, like the external cap and the antenna and its end piece were manually b u ilt in the XFDTD graphical interface. Since magnetron modeling was not the objective o f this work, the microwave generating cavity was sim plified by reducing the number o f vanes from eight to four. This sim plification preserved the symmetry o f the magnetron, but greatly sim plified the modeling work. The pair o f vanes in the x-direction were connected to each other by a rod w ith a square cross-section measuring one cell. This rod was positioned at the bottom o f the vanes. The pair o f vanes in the y- direction were connected both to the center electrode and to each other at the top by a square rod. (Some o f these details are visible in Figure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 254 4.12.) Connecting these opposing pairs o f vanes ensured that their polarities varied simultaneously. x-directed vanes center electrode hole forantenna antenna ■rod connecting x-directed vanes ceramic cylinder antenna’s end piece waveguide Figure 4.12 Cross-section of model of magnetron in waveguide. The source o f power driving this model was a discrete voltage source located between the connecting rod for the x-directed vanes and the central electrode. This voltage source had a 5 0 -fi resistance in series. The waveform fo r this voltage source was a ramped sine wave w ith a frequency o f 2.45 x 109 Hz, an input amplitude o f 1 V/m , and a phase angle o f 0. The number o f periods (cycles) was 144. A 3-quart Pyrex® glass bowl (W orld Kitchen, Inc., Elmira, N Y ) was chosen to represent a typical heating vessel. Figure 4.13 shows this bowl, fille d w ith water, inside the oven cavity. The base o f the bowl was modeled w ith a cylinder. The curved profile o f the w all o f the bowl was modeled by a few hollow, truncated cones. The dimensions o f the cones were chosen to closely follow the curvature o f the real bowl. The ledge at the top o f the bowl was modeled w ith a washer-shaped hollow cylinder. Figure 4.14 shows the model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 255 4.4.4 Running the model Once the geometry file was completed in the graphical interface portion o f XFDTD, the actual FDTD calculations were performed by another XFDTD program called CalcFDTD. CalcFDTD was run in batch mode, i.e., from the DOS prompt in Windows®. As Figure 4.13 Water-filled bowl inside oven cavity many programs and services as possible were removed from the computer’s memory in order to reduce to a minimum the computing, recording, and memory load. CalcFDTD generated as part o f its output files the instantaneous value o f the electromagnetic field at selected sample points. Ten sample points were chosen so Figure 4.14 Model of water-filled bowl inside oven cavity as to be at “ extreme” locations throughout the oven and to be at least one-quarter o f a wavelength from the m etallic oven surfaces. The yellow dots in Figure 4.15 illustrate the location o f these points: one in each comer o f the oven, one in the middle close to the roof, and one in the middle o f the bowl o f water, which was expected to be the last point to reach steady state. (The coordinates o f the exact locations o f these points and o f the walls o f the microwave oven in terms o f cell number are listed in Appendix F). The instantaneous value o f the electric fie ld at these points was used to determine i f steady state had been achieved in the system, and it alerted the researchers to the instabilities in the first model. The final 1.22-mm model was run w ith the dielectric data for both water and the waxy maize starch suspension at temperatures o f 30°C, 40°C, 50°C, 60°C, 70°C, and 75°C. A ll results are presented in the next section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 256 4.4.5 Representing the Data In order to better compare these runs as w ell as to present the results for further runs, the volum etric data representation abilities o f M A TLAB were used. One such capability is the calculation o f the so-called “ isosurfaces.” Isosurfaces are the three-dimensional equivalents to two-dimensional contour plots. Given a set o f volumetric data, the program calculates a surface defined by a ll the points in the volumetric data that are equal to a certain given value (the Figure 4.15 Location of sample points isovalue). The surface obtained in this fashion is often d iffic u lt to interpret, and therefore the M atlab manual recommends the use o f so-called “ isocaps.” By closing the open spaces produced by the isosurface, the isocaps provide a context to interpret it. They enable the user to visualize a cross-section or an interior view o f the isosurface, and they can be color-scaled to indicate the relative magnitude o f the quantity o f interest at each point. In this study, isosurfaces and isocaps were used to visualize the electric fields at the end o f each experiment, as w ill be discussed in the next section. Incidentally, isosurfaces were also used to visualize the model oven and its components from the geometry file, as shown in Figures 4.5,4.8,4.12,4.14, and 4.15. Included in the output o f XFDTD was the specific absorption rate (SAR) for materials that have a non-zero, but finite, conductivity. In this study, materials in that category included only water and fuzzy layers. The calculation takes into account the density, conductivity, and root mean square amplitude o f the three directional components o f the electric field in a cell (XFDTD manual, Remcom Inc., 1994-2001). The SAR was calculated at every point in the water and the “ fuzzy” layers. The units o f the SAR, watts per kilogram , indicate that this quantity measures how much heat is generated per unit mass and unit time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 257 One other quantity derived from the SAR was the one-gram-average SAR. This quantity is found by first determining how many cells are needed to make up 1 g o f the material (in this case 550.7 cells, or about a cube w ith a side o f 8 cells). The one-gram average at each cell is then determined as the average SAR o f a one-gram cube centered on that cell. The resulting one-gram SAR average is “ coarser” (i.e., has less detail) than that o f the original SAR calculation performed for single cells over the entire material. The 1-g SAR is useful to estimate the real heating profile w ithin an object. Theoretically, the SAR results could be used in another program to calculate the temperature distribution w ithin the liquid, but this was beyond the scope o f this study. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 258 4.5 Results and Discussion 4.5.1 Confirmation of Steady State Using Electric Field Amplitudes The amplitude o f the electric fie ld at each o f the 10 sample points shown in Figure 4.15 was calculated from the XFDTD output by taking the magnitude o f the electric fie ld vector at two different times, | E (t[) | and | E (t2) | , separated by a quarter o f a wavelength. The electric field amplitude was then ( | E (ti) | 2 + | E(t2) 12)(1/2). Figures 4.16 and 4.17 contain the results o f these electric field amplitude calculations at 30°C for the runs w ith water and waxy maize starch suspension, respectively. (Sim ilar graphs fo r the other temperatures can be found in Appendix K ). Each figure contains three panes w ith graphs, each displaying 10 colored lines. The 10 colored lines represent the 10 sample points shown in (a) water at 30°C 10° E 0 > a) "O 10 20 30 40 50 60 — . — — ..... E 10° .a io' 20 o 30 40 50 60 30 40 50 60 0 111 10u ■ 10 11 N ' 10 20 t(ns) Figure 4.16 Electric field amplitudes at 10 sample points versus real oven time at 30°C for water Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 2 3 4 5 6 7 8 — 9 ----- 10 259 X in -2 10' E > a> ■o 0 10 20 30 40 50 60 0 10 20 30 40 50 60 3 a E < 2 > LU 0) o a> m 10° -2 10' t(ns) Figure 4.17 Electric field amplitudes at 10 sample points versus real oven time at 30°C for 3% waxy maize starch suspension. Figure 4.15. The three panes show the data fo r the x-, y-, and z-components, respectively, o f the electric fie ld amplitude at each o f those points versus the elapsed tim e. Most o f the figures show that from the beginning o f the experiments, up to about 10-15 ns, the electric field amplitudes increased first, then fluctuated. The in itia l increase was in response to the ramping o f the excitation source. The fact that some o f the curves do not begin exactly at 0 ns means that there was a time delay from the start o f the experiment when the radiation left the source until it reached the points corresponding to those curves. The fluctuations were evidence that the electric fields changed as the energy interacted w ith the walls o f the oven, the bowl, the liquid, and other components in the oven. A ll the curves seemed to reach a plateau by about 30 ns, indicating that steady state was achieved. The amplitude o f the electric field was lower for point number 10 (which corresponded to the center o f the bowl) than it was for the other nine points. There are a number o f possible explanations fo r this. One is that point 10 may be located at a symmetry point Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 260 where the electric field amplitudes are lower due to destructive interference. Another possible explanation is that the significantly higher dielectric properties o f water and starch suspension result in a lower amplitude. A third possibility is that since both the water and starch suspensions are known to have a nonzero dielectric loss factor, it is expected that the energy o f the wave w ill decrease upon penetrating these media. Thus, the electric field amplitude at the center o f the bowl would be lower. 4.5.2 Steady-State Electric Field Magnitudes in the Entire Oven One o f the main outputs from CalcFDTD was the magnitude o f the steady-state electric fie ld in the entire oven. The results are shown in Figures 4.18a and b for the runs w ith water at temperatures o f 30° and 75°C, and in Figures 4.19a and b fo r the starch suspensions at the same temperatures. (The results for the rest o f the temperatures are shown in Appendix K). The isovalue fo r these isosurface plots was set at 2.51 V/m. The isosurface plots were “ cut” into slices that were then separated slightly to reveal crosssectional views o f their interiors. This facilitated the observation o f the electric field distribution in the internal parts o f the problem space. Where the boundary plane o f the slices intersected the isosurface, isocaps were added to show the field distribution inside globular-shaped regions o f higher electric field intensity. The electric field intensities were represented on the logarithmic bar scale by colors ranging from blue to red. The red color indicates areas o f highest electric field intensity (i.e., local maxima), and it is immediately evident that such areas occur near the magnetron. The blue color represents the isovalue, and the isosurfaces enclose volumes w ith electric field intensity values greater than or equal to the isovalue. Portions w ith no color or w ith only a grayish color represent areas in which the intensity o f the electric field is less than that o f the iso value. The metal, liquid, and glass components o f the oven are such areas, and they were shown in these figures to provide context to the electromagnetic field. The change in dielectric properties from free space to Pyrex, and most importantly, to the liquid, seem to have modified the shape o f the regions o f high intensity that would naturally be globule-shaped, but now appear to be stretched or deformed by the shape o f the bowl. Because o f this, the field distributions in the areas around and below the bowl were not as regular (i.e, not as nicely “ globule-shaped” ) as the areas above and nonadjacent to the bowl. Even though the location o f the local maxima remained the same for both liquids at all temperatures, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 261 their intensities did not. The intensities increased w ith increasing temperature, but w ith no noticeable difference between the results fo r water and those for the starch suspension. 4.5.3 Electric Field Magnitudes in the Bowl Isosurfaces were made from the electric field magnitude w ithin the liquids. The isovalue fo r these figures was 0.1 V/m, which was not the same as that used in Figures 4.18 and 4.19, so a different (linear) color scale was used. The results are shown in Figures 4.20 and 4.21 for the water and starch suspensions, respectively, at 30°C and 75°C. (Sim ilar graphs fo r the other temperatures are shown in Appendix K). Figures 4.20a and 4.21a show the angle at which the bowl is viewed, w ith respect to the magnetron in the microwave oven. To facilitate visualization o f the electric fie ld in the water and starch suspensions, the space o f the bowl was cut by two vertical perpendicular planes in the x- and y-directions, and by two horizontal planes along the z-direction, producing 12 separate wedge-shaped “ slices.” From these slices it is evident that the electric fie ld inside the liquids forms a pattern o f concentric rings, or toroids. The circular cross sections o f some o f the toroids are visible in the figures w ith a red core surrounded by a yellow ring, or w ith a yellow core surrounded by a greenish blue ring. These color patterns reveal that the local maxima o f the electric field is located at the center o f each toroid. The separation between these centers was roughly estimated to correspond to h a lf o f the wavelength o f microwaves in water or starch suspension at the different temperatures. There is a semblance o f radial symmetry except for a "hot spot" in the section o f the bowl closest to the magnetron. Also, there is, in general, a decrease in electric field intensity from the outside surface to the inside center o f the bowl. This decrease in intensity, along w ith the concentric pattern o f the electric field, may explain how some researchers have been successful in modeling microwave heating using models based on Lambert’ s law and the assumption o f microwave energy penetrating uniform ly from the whole surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 262 Figure 4.18 Electric field magnitudes in the entire oven at the end o f the calculation with water as the load at a) 30°C and b) 75°C. Scale given in V/m. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 263 (a) 30 C 4 b) 75 C 4 Figure 4.19 Electric field magnitudes in the entire oven at the end of the calculation with 3% waxy maize starch suspension as the load at a) 30°C and b) 75°C. Scale given in V/m. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 264 a) (b) 30°C - 1.4 - 0.8 0.6 1 0.4 0.2 (C) 75°C 0.6 1 0.4 0.2 Figure 4.20 Electric field magnitudes in the bowl at the end of the calculation with water as the load: a) bowl position in oven, b) 30°C, c) 75°C. Scale given in V/m. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 265 (a) (b) 30°C 10.2 (c)75°C ■ 0.8 0.6 1 0.4 0.2 Figure 4.21 Electric field magnitudes in the bowl at the end of the calculation with 3% waxy maize starch suspension as the load: a) bowl position in oven, b) 30°C, c) 75°C. Scale given in V/m. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 266 The effect o f increasing temperature on the electric field inside the bowl was to increase the depth o f penetration o f the radiation. This effect is evident in Figures 4.20 and 4.21, that show that the area o f higher electric fie ld intensity penetrate deeper fo r the higher temperatures. The effect o f starch in the electric field is not as pronounced, but it is s till significant. Figures 4.20 and 4.21 show that the microwaves penetrated deeper into the water than into the starch suspension, as evidenced by relatively more red areas in the “ slices” fo r water at 75°C than for the corresponding “ slices” o f the starch suspension. The effects o f temperature and material can be understood in the light o f the dielectric properties reported in chapter 3 and listed back in Table 4.2. For both materials the dielectric loss decreases w ith temperature at the frequency o f interest (2.45 GHz). Therefore, at higher temperatures there is not as much power absorption in the liquids, and the energy can penetrate deeper. Moreover, the dielectric loss o f the starch was higher than that o f water at a ll temperatures, causing the starch suspension to have more power absorption near the surface o f the bowl than the water at the same temperature. 4.5.4 SAR Values in the Bowl Figures 4.22 and 4.23 show isosurfaces o f the SAR values fo r the water and starch runs, respectively, at temperatures o f 30° and 75°C. (Sim ilar figures for the other temperatures can be found in Appendix K .) The isovalue fo r these plots was 5.6 x 10‘6 W/kg. Since only water or starch suspension dissipates energy in this model, only areas occupied by the liquid and the “ fuzzy” areas adjacent to the liquid have a SAR. The pattern for the SAR follow s the same pattern as the electric field, w ith the exception that the decrease in intensity is faster, since absorbed power is proportional to the square o f the electric field. The SAR graphs seem to indicate that the “ hot spot” would be more pronounced than would be suggested by the electric field graphs. An interesting result o f the model is that the change o f the SAR distribution as a function o f temperature is less pronounced than the change for the electric field (compare Figures 4.22 and 4.23 w ith Figures 4.20 and 4.21). The reason for this difference is that the SAR is a function o f both the intensity o f the electric field and the dielectric conductivity o f the load. The penetration o f the electric field is greater at higher temperatures, but this effect is moderated in the SAR distribution by a decrease in dielectric Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 267 conductivity. Therefore the SAR distributions at 30 and 75°C are more sim ilar that the corresponding distributions fo r the electric fie ld magnitude. The SAR data is m ainly used to predict temperature increase in objects exposed to microwaves. Since electromagnetic phenomena and heat transfer are occurring simultaneously, it is very doubtful that the patterns observed in Figures 4.20 - 4.21 (electric field magnitudes in the bow l) and 4.22 - 4.23 (SARs in the bowl) would be reproduced in the temperature distribution measured in real life . In practice, such fine detail would be blurred very quickly by heat transfer. In order to report SAR values referring to the same mass, it is customary to take an average over a standard mass (a one-gram mass in this case). Figures 4.24 and 4.25 show the one-gram average SAR results for the water and starch, respectively, at 30° and 75°C. (Sim ilar figures fo r the other temperatures can be found in Appendix K .) These figures offer a “ blurred” version o f the non-averaged SAR. Instead o f the differentiated concentric rings, Figures 4.24 and 4.25 show a few "hot spots" that represent a relatively high SAR value w ith respect to the isovalue. For both SAR plots, the areas o f higher absorption penetrate deeper into the bowl as temperature increases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 268 (b) 30 C (c) 75 C Figure 4.22 SARs in the bowl at the end of the calculation with water as the load: a) bowl position in oven, b) 30°C, c) 75°C. Scale given in W/kg. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 269 (a) (b) 30°C (c) 75°C Figure 4.23 SARs in the bowl at the end o f the calculation with 3% waxy maize starch suspension as the load: a) bowl position in oven, b) 30°C, c) 75°C. Scale given in W/kg. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 270 Figure 4.24 1-g SARs in the bowl at the end o f the calculation with water as the load: a) bowl position in oven, b) 30°C, c) 75°C. Scale given in W/kg. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 271 (b) 30°C 9 8 7 6 5 4 3 2 1 (c) 75°C 9 6 7 6 5 4 3 2 1 Figure 4.25 1-g SARs in the bowl at the end o f the calculation with 3% waxy maize starch suspension as the load: a) bowl position in oven, b) 30°C, c) 75°C. Scale given in W/kg. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 272 4.6 Conclusions The purpose o f the work in this chapter was to use a numerical solution to M axw ell’s equations to model the electric fie ld distribution in a microwave oven w ith a load o f either water or starch suspension. Based on the results, the follow ing conclusions can be made: 1. The FDTD method is capable o f modeling electric fields inside a microwave oven w ith a load o f water and/or a very dilute starch suspension. 2. The plots o f electric field amplitudes vs. tim e show that steady state was achieved by the models at about 30 ns. 3. The steady-state electric field distribution in the oven for all temperatures and both liquids takes the shape o f globular regions. The area outside the bowl is very sim ilar for all conditions under study, w ith only m inor variations in intensities among temperatures, but almost no difference between water and starch suspensions at the same temperatures. 4. The steady-state electric field inside the liquids forms a pattern o f concentric toroids w ith a semblance o f radial symmetry except for a “ hot spot” in the section o f the bowl closest to the magnetron. The SAR distribution follows a sim ilar pattern. 5. The intensity o f the electric field and the SAR decrease from the outside surface to the inside center o f the bowl, but the SAR decreases faster than the electric field. This causes the “ hot spot” to be more pronounced fo r the SAR than for the electric field. 6. The penetration depth o f the microwaves increased as temperature increased for both water and starch. This effect is more pronounce for the electric field magnitude than for the SAR. 7. The microwaves penetrated more deeply into the water than into the starch suspension. 8. The model developed is useful to predict microwave heating patterns for the kind o f load used, and it is able to show areas o f possible overheating and heterogeneity o f heating. 9. More research is needed to couple the results from the electromagnetic modeling to heat transfer models in order to determine the actual temperature distribution for this kind o f system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 273 CHAPTERS CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH This study had three main parts: 1) dynamic measurement o f starch granule swelling during microwave heating (chapter 2); 2) in-line measurement o f the dielectric properties o f aqueous starch suspensions during microwave heating (chapter 3); and 3) using the finite difference time domain (FDTD) method to model the electromagnetic field distribution in a domestic microwave oven loaded w ith an aqueous starch suspension at different temperatures (chapter 4). A ll three parts have contributed to a better understanding o f the behavior o f starch during microwave heating. The dielectric measurement technique developed in the second part, and the model developed in the third part have potential practical use for developing better microwave-ready food products. The methodology and results o f this work point to a few suggestions for improvement and to other further studies that would be o f interest; these w ill be listed after the conclusions that were formed after studying a ll the results. 5.1 Conclusions The measurements o f the swelling behavior o f starch in dilute aqueous suspension during microwave heating were compared w ith previous results for conventional heating. For the experimental conditions used, no significant differences were found in maximum diameter and temperature o f maximum swelling rate. The plot o f D[v,0.5] against temperature showed the same features for both studies. Thus, no “ microwave effect” or “ athermal effect” was observed in this study. The Laser Scattering Particle Size Analysis developed in this work can be effectively used to study the swelling o f starch granules in dilute aqueous suspensions, in-line dining microwave heating. The well-m ixed system that was used overcame some o f the lim itations faced by previous researchers by allowing control o f the heating rate and a uniform distribution o f temperature and starch granules. In view o f the sim ilarities observed between microwave heating and conventional heating, it seemed reasonable that microwave heating o f dilute starch suspensions could be modeled using available numerical methods, i f heat generation data could be obtained. Heat generation for microwaves is a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 274 function o f the dielectric properties. The method developed in the first part o f this w ork was extended to the measurement o f dielectric properties o f dilute starch suspensions, in-line, during microwave heating. The dielectric properties o f the suspensions follow ed the dielectric properties o f water, but starch lowered the dielectric perm ittivity (e /) and increased its dipole rotation relaxation tim e (x2). An upward curvature observed for e/ and e /'a t low frequencies could be explained w ith a relaxation peak at low frequencies and/or the leaching o f electrolytes from the starches. The decrease in s/, the decrease o f x2, and the upward curvature were adequately fit using the Debye-Hasted model w ith two peaks. It was hypothesized that the observed changes in dielectric behavior o f water w ith added starch could be caused by a combination o f the follow ing: a) dilution o f water caused by the presence o f the granules; b) electrolytes leaching out o f the granules; and c) water interacting w ith the amorphous regions o f the starch granules at low temperatures and then w ith a ll components o f starch after higher temperatures were achieved. As expected, the addition o f 2% sodium chloride to waxy maize starch suspensions depressed e/ and greatly increased e," values, especially at lower frequencies. Starch, however, moderated both o f these effects as compared w ith pure water w ith the same concentration o f sodium chloride. Anomalous upward curvature for e/ at low frequencies was observed for the suspensions w ith added sodium chloride. This curvature was consistent w ith the phenomenon o f electrode polarization. This work highlighted the use o f the parameters o f the Debye equation and its variants (more specifically, what has been termed here the “ Debye-Hasted model w ith two peaks” ) as a useful tool to assess dielectric behavior as a function o f frequency and temperature. Complex perm ittivity data for a specific temperature and a range o f frequencies can be summarized by a small number o f parameters (such as esi, ii, es2, eo2, x2, and equivalent sodium chloride concentration). The study o f the behavior o f these parameters as functions o f temperature gives more inform ation about the nature o f the interaction between microwaves and food components than does isolated complex perm ittivity data w ith lim ited general applicability. The data presented in the format used in this work can be complemented w ith additional data from future studies to develop a better understanding o f phenomena occurring during microwave heating. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 275 The dielectric parameters can always be used to extract the specific complex perm ittivity values fo r a particular frequency and temperature, i f the need arises. Once dielectric data for a specific system is available at the temperatures and frequency o f interest, the calculation o f heat generation can be undertaken. During microwave heating this heat generation can be accurately described only through the use o f M axw ell’ s equations to find the electromagnetic energy dissipated. In this study a numerical method (FDTD) was used to model the electric fie ld and the SAR (directly related to the heat generation) inside a domestic microwave oven w ith a load o f either water or starch suspension in a bow l. This model was meant to be a first step in the development o f a microwave heating model that w ould include heat transfer and eventually mass transfer. An important feature in the modeling results is that the predicted electric fie ld distribution outside the load was almost identical for both materials and for all temperatures. Inside the loads, a pattern o f concentric toroids w ith a “ hot spot” in the area closest to the magnetron was predicted fo r both systems at all temperatures. The intensity o f the electric field and the SAR were predicted to decrease from the outside surface to the center o f the bowl, but the SAR decreased faster than the electric field. This caused the “ hot spot” to be more pronounced for the SAR than for the electric field. Inside the loads, the differences in dielectric properties had an effect in the predicted field distributions. In general, as expected, the depth o f penetration o f the microwaves increased w ith decreasing dielectric loss. Since the load w ith water had a lower dielectric loss than the starch suspension, the penetration fo r water was predicted to be deeper that the penetration for the starch suspension. The dielectric loss decreased w ith temperature for both water and the starch suspension, and the penetration depth for both loads was predicted to increase w ith increasing temperature. W ithout attempting to be a complete predictive model, the model developed in this work is useful to predict microwave heating patterns for the kind o f load used, and it is able to show areas o f possible overheating and heterogeneity o f heating. Upon availability o f more sophisticated models, the process o f measuring dielectric properties, modeling geometry, and applying a numerical method (or a combination o f numerical methods) should allow a more complete understanding o f microwave heating, and should provide a tool for the development o f microwave-ready products. An extension o f the method for the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 276 geometries o f industrial ovens could help w ith the design o f microwave heating equipment fo r industry applications. 5.2 Suggestions for Future Research 5.2.1 Suggestions for Dynamic Measurement of Starch Particle Size Behavior During Microwave Heating This work had a number o f lim itations that warrant additional investigation. The particle size measurements were performed for a very low concentration o f starch at a specific heating rate. Moreover, the particle size data provided information about only one aspect o f starch gelatinization. As mentioned in chapter 2, there were some differences between the experimental apparatus used in this experiment and the apparatus used by Ziegler et al. (1993). The differences were assumed unimportant in this work, but their elim ination in any further work would add to the certainty o f the results. The follow ing suggestions address the experimental lim itations, and also point to potential areas o f research that would increase the present understanding o f microwave heating o f starch-containing foods. 1) Future experiments comparing conventional and microwave heating could be performed in a dual oven capable o f operating w ith microwaves or conventional heating. This would allow the use o f exactly the same apparatus for both heating methods. 2) It was hypothesized that the presence o f curvature in the time-temperature treatment was caused by heat loss in the tubing and the measuring cell. Future experiments could insulate the tubing and measuring cell to minimize this problem. 3) Regardless o f the heating method, there is s till uncertainty about the swelling behavior o f specific sizes o f starch granules. Future experiments could separate different sizes o f starch granules (e.g. by their difference in settling time) and then measure the swelling behavior o f the individual fractions. 4) The analysis o f the particle size data is at present lim ited to the analysis o f the D[v,0.5] and the qualitative description o f the diameters o f the volume distribution. A parametric function could be derived to describe the entire size distribution (maybe the superposition o f two or more lognormal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 277 distributions). The swelling behavior could then be described through the behavior o f the parameters as a function o f temperature. 5) The present work was lim ited to different com starches (common, waxy and cross-linked waxy) and one heating rate. Future experiments w ith different botanical sources and types o f m odifications, and different heating rates would make the results obtained here more general. 6) The results in this work were lim ited to the particle size distribution during microwave heating. A better understanding o f starch gelatinization during microwave heating would require additional measurements o f different aspects o f this process. For example, a method for viscosity measurement in ­ line during microwave heating could be developed. Some authors (like Marsh and Wetton, 1995) have developed methods for estimating the crystallinity o f starch-containing systems using broad frequency dielectric relaxation spectroscopy. Such methodology could conceivably be combined w ith a system like the one developed in the second part o f this work to obtain inform ation during microwave heating. 5.2.2 Suggestions for In-line Measurement of the Dielectric Properties of Aqueous Starch Suspensions During Microwave Heating The second part o f this work reported results for the dielectric properties o f starch suspensions, at one starch concentration, one heating rate, and a lim ited range o f frequencies. Additional experiments studied the effect o f sodium chloride and expanded the frequency range. In the experiments w ith added sodium chloride, anomalous curvature, most probably caused by electrode polarization, interfered w ith the estimation o f the low frequency parameters. The expanded frequency range was not sufficient for a complete interpretation o f the data at low frequencies. The follow ing suggestions address these lim itations, and they fa ll into three categories: test materials and sample preparation, changes in the experimental method, and model. 5.2.2.a Test materials and sample preparation: 1) It was hypothesized that electrolytes associated w ith the starch granules had an effect on the conductivity o f the starch suspensions, and may have caused some o f the upward curvature in the low Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 278 frequency dielectric data. Future experiments could “ decouple” the effect o f the electrolytes from the dielectric properties o f the starches by washing the granules in double-distilled deionized water (as in M ille r et al., 1991) removing as much o f the electrolytes as possible. 2) The concentration o f starch in this w ork was lim ited to dilute suspensions (3% w /w ). Future experiments could measure the dielectric properties o f different concentrations o f starch to better elucidate the effect o f “ water dilution” by the starch. A higher concentration o f starch would also contribute to a greater understanding o f the interaction o f microwaves and starch at a w ider range o f possible food applications. 3) This work was lim ited to com starches (common and waxy) and potato starch. Further understanding o f the interaction between foods and microwaves could be obtained by measuring the dielectric properties o f other botanical sources o f starch, and other food materials o f interests such as proteins and fats. 5.2.2.b Experimental method: 1) The coaxial probe reflection method used in this work has a lim ited range o f frequencies o f application. Other measuring fixture designs (such as transmission cells) and different types o f signal processing (such as time domain methods) could be used to explore a wider frequency range. Future research could, for example, focus on collecting data in the lower-frequency region to eliminate the uncertainty about whether there is a low-frequency peak or a conductivity effect. 2) The triggering o f the frequency sweep in the network analyzer was done manually, and therefore the temperatures o f data collection were slightly different. This problem was addressed w ith data post­ processing, but better repeatability would be possible w ith new software for the instrument that could automatically control the loading o f calibration files, start o f the frequency sweep, and data saving. 3) The heating rate was controlled in this work using ballast water. Future experiments could develop a more direct control over the temperature using “ o n /o ff’ automatic control o f the magnetron. This control could be implemented via software,-!- as in Ziegler et al., 1993. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 279 5.2.2.C M odel 1) It was hypothesized that electrolytes leached out o f the starch granules, affecting the conductivity o f the starch suspensions. Future work could include the effect o f this leaching by developing a model o f the effective electrolyte concentration as a function o f tim e and temperature. 2) Develop a model to account for electrode polarization, allow ing interpretation o f the low frequency data for systems containing added sodium chloride. This would help to further investigate the apparent reduction o f NaCl conductivity caused by the addition o f starch. 3) Instead o f determining the a m atrix o f parameters for the Debye-Hasted model w ith two peaks, empirical functions o f temperature could be developed to describe the behavior o f each individual parameter in the model. This would reduce the number o f curves o f interest from 21 to 6. 4) Use models other than the Debye-Hasted model w ith two peaks to estimate the dielectric properties; investigate differences among the models. In some cases this can be done by sim ply setting some terms in the Debye-Hasted model w ith two peaks equal to zero. The Debye-Hasted model w ith two peaks and four other possible models are shown schematically in Figure 5.1, and their names and their respective parameters are listed below: a. Hasted-Debye model and an extra Debye relaxation (also known as the Debye-Hasted model w ith two peaks) w ith parameters o f Eji, Tj, es2, x2, and £«,, and a b. Simple Debye model w ith parameters o f e*, c. Cole-Cole model w ith parameters o f e s, e *,, and x x, and a. d. Two Debye relaxations w ith parameters o f e si , e. Hasted-Debye model w ith parameters o f e s, KA A. frequency (a) (b) X i, e s2, t 2, e^,, t , and £* and a A (C) (d) tel Figure 5.1 Sketches of the general trends in graphs of e' and e" versus frequency for different dielectric models: a) Hasted-Debye with an extra Debye relaxation; b) simple Debye model; c) Cole-Cole model; d) two Debye relaxations; e) Hasted-Debye model Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 280 5.2.3 Suggestions for the FDTD Model of the Electromagnetic Field Distribution The model developed in this w ork was lim ited in its application because it only provided the SAR. There were also a few m inor lim itations in the geometric modeling o f the oven. The follow ing suggestions address these lim itations, and fa ll into two categories: im proving how the model is constructed and run in XFDTD, and applying the results to obtain inform ation about microwave heating. 5.2.3.a Improving how the model is constructed and run in XFDTD 1) The geometry o f the magnetron was estimated w ith pictures and diagrams. For future modeling, a magnetron could be cut open so that the parts could be measured and modeled better in XFDTD, paying particular attention to how the cap fits, how long the antenna is, where the source could be located, etc. 2) Future modeling could be improved i f the actual material used in the glass plate, the ceramic part o f the magnetron, and the plastic ring and wheels could be determined so that more accurate values o f their dielectric properties could be used as inputs to the model. 3) The use o f a faster computer or parallel computers (Bellanca et al., 2001) would decrease the computing time, allowing more variables to be modeled. 4) In practice, domestic microwave ovens use the “ turntable” to rotate the load and avoid “ hot spots.” Further modeling attempts would have to take into account this movement in order to accurately represent the functioning o f a real microwave oven. 5.2.3.b Applying the results to obtain information about microwave heating 1) In order to model the temperature distribution inside the load, the SAR information provided by XFDTD would have to be used in future heat transfer models as a “ forcing function.” 2) Once the temperature distribution is modeled, measurements o f the temperature distribution during microwave heating should be performed to validate the model. For the sim plified model developed here, these measurements would have to be conducted for short heating times to avoid confounding the effect o f the electromagnetic field w ith the effects o f conduction and convection. For Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 281 this stage o f model development, the effect o f rotation (from the turntable) would have to be eliminated by disabling this function in the oven. 3) A next stage in model development would have to take into account the non-uniform temperature distribution during microwave heating. Such a model could run recursively between XFD TD and a heat transfer modeling program. The entire “ problem” would be broken into small incremental steps, beginning at the in itia l temperature and using the SAR inform ation to calculate a first temperature distribution. From this firs t temperature distribution, a dielectric property distribution could be calculated. 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APPENDIX A:
DERIVATION OF THE WAVE EQUATIONS
AND
PENETRATION DEPTH
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This appendix contains the details o f the derivations o f the electric fie ld and magnetic fie ld wave
equations, as w ell as the expression for penetration depth given in section 3.2.12. It w ill be shown that the
terms associated w ith the dielectric perm ittivity result in a sinusoidal wave w ith constant amplitude, and
that the terms associated w ith the loss factor result in an exponential decay o f the amplitude o f the wave.
This section contains a lot o f mathematics, especially the mathematics o f vectors, complex numbers, and
differential equations. The equations that are derived w ill be applicable for dielectric materials that are
isotropic and homogeneous.
A .l Derivation of the Electric Field Wave Equation
I f constitutive relationship 3.23 is substituted into equation 3.34, a new version o f Faraday’ s Law
is obtained:
VxE=-jfflpW
[A .l]
Sim ilarly, i f constitutive relationship 3.38 is substituted into equation 3.37, a new version o f Gauss’ electric
law is obtained:
V • s*E = pv
[A.2]
The pv term in equation A.2 and the J0term in equation 3.35 represent the sources o f the electromagnetic
radiation, like the tip o f the antenna in Figure 3.1. However, assuming that the electromagnetic fields o f
interest are in an area farther away from the source (such as the area in the white box in Figure 3.1), J0 in
equation 3.35 and pv in equation A.2 can be set equal to zero. In addition, since magnetic properties are
negligible for this current study, it can be assumed that the permeability (p) in equation A. 1 is equal to the
permeability o f free space, p,,. The four M axwell equations (3.34 - 3.37) can then be sim plified to:
V x E = - jcop0W
[A.3]
V x H = ]m e *E
[A.4]
V *S = 0
[A.5]
V • E= 0
[A .6]
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300
Equations A.3-A.6 can be used to derive a general, second-order differential “ wave equation” that
can then be used to model the behavior o f various media in the presence o f an electromagnetic field. This
wave equation can then be used to model the behavior o f various media in the presence o f the field.
Shen and Kong (1987) began the derivation by taking the curl o f equation A.3:
V x ( V x E ) = V x ( - jroPoW)
V x (V x E)= - joop0 (V x H)
[A .7]
Substituting equation A.4 into equation A.7 yields
V x (V x £ ) = - jcop0 (j<ds*£)
V x (V x E) = co2p0 e *£
[A .8]
The left-hand side o f equation A .8 is o f the same form as the left-hand side o f the follow ing general vector
identity:
V x (V x Z ) = V (V • Z ) -V 2 Z
[A .9]
This means that the right side o f equation A .8 can be equated to the right side o f equation A .9 i f the generic
variable Z is replaced by E:
V ( V * E ) - V 2 E=a>2p0e*E
[A . 10]
According to equation A .6, the dot product o f the del vector and E is 0, so equation A. 10 reduces to
-V 2 E = ©2p0 e*E
V 2E + co2p0 e* E = 0
Equation A. 11 (3.82 in
text) is the
[A . 11]
second order differential equation for the vectorE,and a
sim ilar equation w ill be derived later in this appendix for H. Referring to this equation, Hayt and Buck
(2001) said, “ It is fa irly formidable when expanded, even in rectangular coordinates, because three scalar
phasor equations result, and each has four terms.” Therefore, they, like Shen and Kong (1987), sim plify
the system by examining an E field that is a function only o f z and is parallel to the x-axis (i.e., a traveling
plane wave). A schematic, two-dimensional representation o f such a wave is given in Figure 3.32 in
section 3.2.12. The wave equation for this sim plified system is given by
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301
<?Ex
.
5x2
+
#EX
,
+ ~
S2 Ex
^
L~
,
,
+ “
E* “ °
[ A ' 12]
Since this x-component o f the electric field does not vary w ith either x or y, equation A. 12 can be reduced
to
+ co2|i0s* Ex = 0
[A . 13]
Equation A. 13 is a second-order homogeneous linear differential equation that is linear in Ex (an
unknown function) and its derivatives (Kreyszig, 1993) Such a differential equation can have numerous
solutions. In fact, a fundamental theorem for homogeneous equations (Kreyszig, 1993) states, “ For a
homogeneous linear differential equation (2), any linear combination o f two solutions on an open interval I
is again a solution o f (2) on I. In particular, for such an equation, sums and constants m ultiples o f solutions
are again solutions.” The unknown expression for Ex must be a function o f z w ith a second derivative that
satisfies equation A. 13. The general solution is o f the form:
Exi = eqz
[A . 14]
where q is a constant that must be determined from equation A.13. The first derivative o f equation A.14 is:
3EX
8z
_
qz
qe,z
[A . 15]
= _ 2 qz
[A . 16]
and the second derivative is:
&EX
dz2
q eq
Substituting equations A.14 and A. 16 into equation A.13 yields,
q2eqz + (o2|i0E*eqz = 0
q2+a>2pos* = 0
[A. 17]
The quadratic formula can be used to solve for q in equation A. 17:
q = ± Vi (-4co2p0e*)1/2 = ± co(-p0s* ) 1/2
q = ±j<D(p0e* ) 1/2
[A. 18]
There are two choices for the sign o f q, but the negative one w ill be used here because it indicates that the
wave dampens exponentially as it travels into a medium. The coefficient o f the j term on the right-hand
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302
side o f equation A. 18 is defined as a new constant, k *, known as the complex propagation constant (given
in the text as equation 3.84):
k* =
© ( h 0e * ) 1/j
[A . 19]
Since k* is a complex quantity, it can be broken down into its real and imaginary parts, kR and ki,
respectively, (given in the text as equation 3.85):
k * = kR- j k 1
[A.20]
q = ~ jk *
[A.21]
Thus, q in equation A. 18 represents
Substituting equation A.21 into equation A.14 yields one solution to equation A.13:
Ex = e"jk*z
[A.22]
However, according to the fundamental theorem (Kreyszig, 1993), another solution to the differential
equation A.13 can be formed by simply m ultiplying equation A.22 by a constant. The constant used here is
E0, the in itia l amplitude o f the wave at z = 0 and t = 0 (Hayt and Buck, 2001).
Ex = E0 e-jk*z
[A.23]
Equation A.23 is the solution in the frequency domain for the sim plified electric field wave equation A.13.
In vector notation this electric field could be represented as
E = xEx = xE0e‘jk*z
[A.24]
A.2 Derivation of Penetration Depth
An expression for the penetration depth can be derived from equation A.24, but first the complex
propagation constant k* must be rewritten as the sum o f its real and imaginary parts. This yields the
follow ing equation for the electric field:
£ = xE0e-j(kRjk.)z
£ = xE0e'k,z e'jkRz
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[A.25]
303
Then the complex method must be “ reversed” by firs t m ultiplying the frequency domain expression in
equation A.25 by the quantity e*®' (Shen and Kong, 1987; Hayt and Buck, 2001) and then taking the real
part (as explained in section 3.2.2). That is,
Ex (z,t) = R e { E 0e-jkRz-k,z ei“ ‘ }
Ex (z,t) = Re{E0e'kiz ei(cot- kRz >}
[A.26]
By Euler’ s identity (equation 3.27), equation A.26 is transformed into
Ex (z,t) = Re{E0 e"kz [cos (cot - kRz) + j sin (cot - kRz)]
[A.27]
The real part o f equation A.27 contains the cosine term. Therefore, equation A.27 sim plifies to
Ex (z,t) = E0 e'kz cos (cot - kRz)
[A.28]
Equation A.28 (3.86 in text) has a negative sign in front o f ^ (which has a positive value in the
case considered here). This indicates that the wave w ill decay exponentially as it penetrates deeper (w ith
increasing z) into a material. For this reason, kr is known as the attenuation factor (Hayt and Buck, 2001).
The original amplitude o f the wave at the surface o f the material (at z = 0) is E0. When the wave penetrates
a distance z into the material such that the amplitude o f the wave has decayed to 1/e (approximately 36.8%)
o f its original E0 value, the wave has reached what is known as its penetration depth, dp, defined as (Shen &
Kong):
dp = 1 /^
[A.29]
Equation A.29 (3.87 in text) can be used to calculate the penetration depth once k[ is known, but
before ki can be found, the definition o f s* in equation 3.50 must be substituted into the definition o f k * in
equation A. 19:
k* = <o[p0(e' - js " ) ]1/2
[A.30]
Substituting equation 3.71 for e" in equation A.30 results in
a 1/2
k* = co[p0( s '- j—)]
co
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[A.31]
304
The derivation continues by squaring both sides o f the equal sign in equation A .3 1 to manipulate
the square root, factoring the expression, and then taking the square root to solve for k * in a format that has
recognizable real and imaginary parts:
k *2- « . w
[1-j^r]
1/2
k* = o>(n0s')1/2 [
[A.32]
Equation A.32 (3.90 in text)contains a complex quantity that is being raised to the 'A power. It can be re­
w ritten in terms o f sines and cosines, using De M oivre’ s theorem (Moore,1975) fo rcomplex numbers. De
M oivre’ s Theorem states that for any complex number z (expressed in polar notation) being raised to the nth
power,
z“ = r"(cos n0 + j sin n0)
[A.33]
According to this theorem, therefore, the square root o f a complex number can be found from:
zm = r 1/2(cos 0/2+j sin 0/2)
[A.34]
To effectively use equation A.34 to solve equation A.32, the complex term in brackets in equation A.32
[1 - j CT/(coe')] must be converted into polar notation. This can be done by finding r 1/2 and 9 as follows:
r = { l 2+[a/(coe')]2} 1/2
r 1/2 = {1 + [cy/(ooE')]2} 1/4
[A.35]
and
tan 0 = -a/((nef)
0 = tan-1 [-a/(coe')]
[A.36]
Equations A.35 and A.36 can be used to re-write the complex quantity in the brackets in equation A.32 that
is raised to the 'A power as
1/2
(1-j
^ -7 )
= (1 + [a/(coe')]2} 1/4{cos (1/2 tan~' [-ct/(cos')]) + j sin (1/2 tan-1 [-a/(a>e')])}
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[A.37]
305
By applying the trigonom etric identity (Spiegel, 1968):
tan' 1 (-A ) = -tan"1(A )
[A.38]
equation A.37 becomes
1/2
(1 - j- ^ - )
v
to e ''
=
{1 + [cr/(coe')]2} 1/4{cos (-1/2 tan-1 [ c t / ( c o s ') ] ) + j sin (-1/2 tan-1
[ c j/( c o e ') ] ) }
[A .39]
The follow ing trigonom etric identities (Spiegel, 1968)
cos (-A ) = cos A
[A.40]
sin (-A ) = -sin A
[A .4 1]
enable equation A.39 to be rewritten as
1/2
( 1 - j —— )
v
= {1 +
[c t /(< d e ')] 2} 1/4{ c o s
(l/2tan_1 [ c t / ( ( o e ') ] ) - j sin (1/2 tan-1 [a/(o)s')])}
[A.42]
COE '
The quantity k* can be found by substituting equation A.42 into equation A.32:
k* = co(p 0e ' ) 1/2{ 1 +
[ c t/( c o e ') ] 2 } 1/4{ c o s
(1/2 tan-1 [a/(cos')]) - j sin (1/2 tan-1 [a/(cos')])}
[A.43]
The real part o f equation A.43 is
kR= co(p 0e ')1/2{ 1 + [a/(cos')]2} 1/4{cos (1/2 tan-1 [ c t /( c o e ') ] ) }
[A.44]
To find ki, the imaginary part o f equation A.43 is selected:
k! = co(n 0e ' ) 1/2{ 1 +
[ c j/( c o s ') ] 2 }
1/4{sin (1/2 tan" 1 [a/(coe')])}
[A.45]
Equation A.45 (3.92 in text) is the general equation for k[. By taking its reciprocal (according to equation
A.29), the penetration depth can be calculated from:
co(p 0e ' ) 1/2 { 1 + [ c r /( o o s ') ] 2 } 1/4{sin ( l/ 2tan_1 [ c t / ( c o e ') ] ) }
which was given as equation 3.93 in the text.
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[A.46]
306
A.3 Simplification for a Slightly Conducting Medium
Equation A.46 (3.93 in text) can be sim plified fo r the special case o f a material that is only slightly
conducting can be sim plified. In this case, a is so small that the quantity
ct/
( cos' )
«
1 (Shen and Kong,
1987). Therefore, equation A.35 can be approximated as:
r 1/2 = {1 + [a/(coe')]2} 1/4 ® 1
[A.47]
The follow ing small angle approximations (Dahlquist and BjOrck, 1974; Spiegel, 1992)w ill be
needed to continue the derivation:
tan 0 = 0
[A.48]
cos 0 « 1
[A .49]
sin 0 * 6
[A.50]
Using equation A.48, 0 in equation A.36 can be approximated as
0 = tan"1 (-a/(<b e ' ) ) » -a/(cos')
[A .5 1]
Applying the approximations o f equations A.47 and A.51, equation A.37 can be approximated as
[1 - j (a/cos')]172« {cos l/2[-a/(coe')] + j sin l/2[-a/(coe')]}
[A.52]
Using the trigonometric identities in equations A.40 and A.41, equation A.52 can be reduced to
[1 - j ( ct/ cos')]1/2 ~ {cos [a/(2(os')] - j sin [c/(2cos')]}
[A.53]
Applying the small angle approximations A.49 and A.50 to equation A.53 yields
[1 - j (cr/oe')]1/2« 1 - j o/(2coe')
[A .54]
Substituting equation A.54 into equation A.32 produces the new approximation for the complex
propagation constant for a slightly conducting material
k* « co(p 0e')1/2{ 1 -j o/(2cae')}
[A.55]
kR = co(p0e')1/2
[A.56]
The real part o f equation A.55 is
and the imaginary part o f equation A.55 is
k x = G)(p0e ') 1/2{a/(2(D s')}
kI = (a/2)(p0/e')1/2
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[A.57]
307
By taking the reciprocal o f equation A.57 according to equation A.29, the penetration depth fo r this special
case o f a slightly conducting medium is found to be
dp= (2/c)(s7p0)1/2
Note that in this special case o f equation A.58 that was derived using the assumption that cr/(cos') «
[A .58]
1, the
penetration depth o f the electromagnetic wave is independent o f the wave’ s angular frequency. However,
this was not true for the general case, described by equation A.46.
A.4 Derivation o f the Magnetic Field Wave Equation
A sim ilar procedure can be followed to derive the magnetic field wave equation. I f constitutive
relation 3.23 is substituted into equation A.5, the result is
V • pH = 0
V• H=0
[A .59]
Taking the curl o f equation A.4
V x ( V x H ) = V x (j(08*£)
V x (V x H) = jcoe*(V x £ )
[A.60]
and substituting equation A.3 into equation A.60 yields
V x (V x H)= jcDs*(-jcop0 H)
V x (V x H) = o)2p0 £* H
[A.61]
The left-hand side o f equation A.61 is equivalent to the le ft hand side o f the vector identity given in
equation A.9, so the right-hand side o f equation A.61 can be equated to the right side o f equation A.9 i f the
generic variable Z is replaced by H:
V (V • H) -V2 H = £02pos* H
However, according to equation A.59, the dot product o f the del vector and H is 0, so equation A.62
reduces down to
-V2 H = Q2p0 e *H
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[A.62]
308
0 = V 2 H + co2p0 8* H
[A.63]
Equation A.63 is the second order differential equation fo r the vector H. Since the del vector involves
derivatives in a ll three Cartesian coordinates, Shen and Kong (1987) sim plify the system by examining an
H field that is a function only o f z and is parallel to the y-axis, as shown in Figure A. 1. Since only the
partial derivative o f H w ith respect to z is applicable in this case, the wave equation for this sim plified
system can be written as
<fHv
+ co p 0E *
0Z2
Hy = 0
[A.64]
The follow ing is a solution to this differential equation, w ith a form analogous to equation A.14:
Hy = eqz
[A.65]
where q is a constant that must be determined from
H
^
equation A.64. The first and second derivatives o f
equation A.65 are:
z
Figure A.l Magnetic field vector as a function
of z, parallel to the y-axis (adapted from Hayt
and Buck, 2001).
OZ
& H ,X
=qeqz
-
0Z2
=
[A .66]
qe
[A.67]
n 2p qz
Substituting equations A.65 and A .66 into equation A.64 yields the same quadratic expression given in
equation A. 17. The steps outlined in equations A.18-A.22 produce a solution to differential equation A.64
that is analogous to equation A.22:
Hy = e'jk‘ z
[A .68]
Once again, another solution can be found by simply m ultiplying equation A.56 by a constant. The desired
constant is H0, the in itia l amplitude o f the magnetic field wave. Thus, the final solution for the sim plified
wave equation for Hy (equation A.64) is
Hy = H0e'jk*z
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[A.69]
309
In vector notation the fie ld could be represented as
H = y Hy = yH 0e'jk*z
[A .70]
Substituting equation A.70 into equation A.3, and assuming that £ = f(x ) and H = f(z) so that the del
operator reduces to partial derivative w ith respect to z yields
J ^ = -jc o p 0H0 e-jk*z
[A .71]
Taking the partial derivative o f equation A.23 w ith respect to z, and substituting it on the left-hand side o f
equation A.71 yields
- jk * E0e-jk*z = - j(op0H0e-jk*z
[A.72]
Now equation A.72 can be rearranged to find an expression for the constant H0:
k* E0 = topoHo
Ho =
k *E°
(op0
[A .73]
Substituting the definition o f k* given in equation A. 19 into equation A.73 yields
H0 = (e*/p0) 1/2E0
[A.74]
A special term known as q*, the complex intrinsic impedance o f the medium can be used to
sim plify equation A.74. The units o f q* are ohms, and it is defined as:
q* =
( P o / e * ) 1/2
[A.75]
Substituting equation A.75 into A.74 yields
H „ = E(/q*
[A.76]
I f equation A.76 is in turn substituted into equation A.70, the result is the follow ing solution for the wave
equation A.64 for the sim plified system in which H is parallel to the y-axis and is a function only o f z
H = yHy = y (E „/q*)e-jk*z
[A.77]
Just as the electric field vector equation was re-written in equation A.25 to take into account both
the real and the imaginary parts o f the complex perm ittivity, so, too, can the magnetic field vector be
w ritten. However, the additional complex quantity o f q * must also be considered. In the phasor notation
o f Shen and Kong (1987), q* can be represented as
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310
x]*= | n* I ei<p
[A .78]
where cp is the phase angle. Substituting equations A.20 and A.78 into equation A.77 yields
H = y (Eq/ | q* | ) e"kiz e'jV e"j9
[A .79]
A.5 Calculation of the Wavelength in a Medium
Returning to the real part o f the complex propagation constant, it is evident from its position in
equation A.28, that kR is the phase constant o f the wave. The velocity o f the wave is then given by (Hayt
and Buck, 2001):
v = co/kR
[A . 80]
Substituting equations 3.1 and 3.26 into equation A.80, yields
X = (2n)/kR
[A.81]
Replacing k Rin equation A.81 w ith its definition in A.44 produces
, _ ______________________ 2n______________________
co(p0e ')1/2 {1 + [ ct/(( de')]2} 1/4{ cos (1/2 tan"1 [cr/(coe')])}
Equation A.82 (4.19 in the text) states that i f the real value o f the complex propagation constant can be
determined, then the length o f the electromagnetic wave can be calculated. Two different expressions for
kR are found in the literature. Appendix B reconciles them, proving that they both are equal.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX B:
RECONCILING DIFFERENT VERSIONS OF
k! AND kR
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
312
As mentioned in section 3.2.12, a survey o f the literature w ill reveal that authors frequently use
other variables and expressions in place o f the ones used here. The purpose o f this appendix is to reconcile
some o f the expressions for kR, ki, and penetration depth.
Hayt and Buck (2001) use the terms a and P in their version o f the wave equation that is
equivalent to equations A.23 and A.25
Ex = Exoe-jkz = Exoe “ e-jPz
[B .l]
jk = a + jp
[B.2]
k = p -ja
[B.3]
where
so that
Equation B.3 is equivalent to equation 3.85 i f P = kRand a = ^
The follow ing steps prove that this is
indeed true.
Rearranging equation A.30 yields
Fi i
1/2
k * = co(p0e') 1/2 [ 1 - j ^ - ]
[B.4]
The complex term in the brackets can be re-written using the same procedure outlined in equations A.33
through A.42 to yield
(
p,i
i - j - j j r ) = { l + [e " /e f} 1/4{cos (1/2 ta n ~ V 7 e ']) - j sin (1/2 tan' 1 [£ "/e '])}
[B.5]
so, the quantity k* can be found by substituting equation B.5 into equation B.4:
k* = (D(p 0e')1/2{ 1 + [e'Vs']2} 1/4{cos (1/2 tan’ 1[e"/e']) - j sin (1/2 tan' 1 [e "/s '])}
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[B .6]
313
imaginary
Based on equation 3.51, the quantity {tan-1 [e '7 s ']} in equation
B .6 represents an angle, 8, that has a tangent equal to e"/e'. Such
an angle can be represented in the complex plane as shown in
real
Figure B .l. However, equation B .6 actually requires the sine and
-e"/e'
cosine o f 8/2, so the follow ing half-angle identities (Spiegel,
r = [ l + (-s"/s')2] 1/2
1992) must be used:
Figure B .l. Representation
of the angle 8 .
sm
[B .7]
1/2
[B .8]
cos Y = +
An expression for cos 8 can be found from Figure B .l:
cos 8 = 1 /
V
1+
[B.9]
[ e " /s ' ] 2
By substituting equation B.9 into equations B.7 and B.8, and then substituting those resulting equations into
equation B.6, a new expression for k * can be derived:
C
k* = £0(p os ')1/2{ l + [e'7e']2} 1/4 I I
I”
1 4-
1
— J L — p—
V 1 + [e'Ve'l2
j
k* = m (p0e72)1/2{ l + [e"/e']2} 1/4
1 + V 1+
[e " /e '] 2
1/2
T
^
I
- jl
J
_
1“
,
1
1 —
V l + rs'Ve'l2
1/2
j
1-V
1+
I
i
J|
1/2
[e " /e '] 2
[B .10]
The real part o f equation B.10 is
r ■+ .
2> 1/4
kR= o)(p 0e72)1/2{ 1 + [ e" / e']2}
1/41 1 +
L
1 T°
V i + [ e " /e '] 2 J
1/2
kR = co(p0s72)1/21" {1 + [e"/e']2} ,/2( 1 + -------1----- Y \
L
V y 1 + [e'7e'] / J
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
314
kR = co(p 0e'/2) 1/2 [ {1 + [8"/s ']2} 1/2+ 1] 1/2
[B . 11]
The right side o f equation B .l 1 is equal to the definition o f (3 used by Hayt and Buck (2001). Therefore,
kR= p.
The imaginary part o f equation B.10 is
k, = co(|i0s72)1/2{ l + [e '7 e f}1/4 1" 1 - -------1----- . 1 1/2
L Vl + [e'7e'] J
k, = ff>(p0e72)1/2 f {1 + [e'7e']2} 1/2^ 1 - -------1----- - \ V
L
V
V1h- [ e " / e f] J j
1/2
k, = G)(p 0e72)1/2 [ {1 + [e'7e']2} 1/2- 1]
[B . 12]
The right side o f equation B. 12 is equal to the definition o f a used by Hayt and Buck (2001). Therefore,
k^a.
Another expression that is represented differently in the literature (Hayt and Buck, 2001; Mudgett,
1982, 1986; Umbach, et al., 1992; von Hippel, 1954) is the expression fo r penetration depth. One common
expression, used by Mudgett, (1982, 1986), Umbach, et al. (1992), and von Hippel (1954) is given by:
dp = l/(a )
[B.13]
where
a = (2n/X0)
[ ( 1 / 2 ) k '( 1
+ tan2 5) 1/2 - 1] 1/2
[B. 14]
and k ' is their symbol for the dielectric constant. That is,
K' = e7e0
[B .l 5]
Daniel (1967) explains that k is the notation usually used by electrical engineers, who refer to it as the
relative perm ittivity, whereas 8 is the notation usually used by physicists and chemists, who call it the
dielectric constant.) In this appendix it w ill be shown that the version o f penetration depth in equation B.13
is in fact equal to the one given here using the notation o f Shen and Kong (1987) in equations A.29 and
A.46 (equations 3.87 and 3.93 in the text).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
315
Substituting the trigonometric identity
tan2 8 + 1 = f — W ' l
v cos 8 /
[B.16]
in equation B.14 yields:
a = (2 n l\0) [ ( 1/2) k ' { ( ^ ) 1/2 - i } ] ' /2
1/2
a = (2ji/A,0) [ ( 1/ 2 )k ' { ^ 5
- l } ]
Shifting the 2 from the denominator o f the k ' term to the term in braces yields
• 1- c o s 8
1/2n
]
P -.7 I
The first term in parentheses in equation B. 17 w ill be addressed first. In a vacuum, the wavelength is X0
and the wave travels w ith the velocity o f light. W ith these variables, equation 3.1 can be re-written as
c = X.0f, so
K = c /f
[B .l 8]
f= co/(27t)
[B.19]
X0 = 27tc/co
[B.20]
From equation 3.26,
so
However, substituting the alternative definition o f c given in equation 4.16 into equation B.20 yields:
271
^•o
/
, 1/2
or,
271 _ . ,
,1/2
,
A,0
Substituting equation B.21 in equation B.17 yields,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[B.21]
316
[B .22]
but applying the definition o f k ' in equation B. 15 results in
n = n , , ,,
ot
co (s0^i0)
\I/2 / I j i COSS
\ ( s /s0)
a = CO(p0e')l/2
'I 1/2-.
2cos 5
J
}
1 - cos 5 y 1/2
2cos 8
|
[B.23]
Equation B.14 has been sim plified to equation B.23. Now it w ill be shown that the expression fo r kt given
in equation 3.91 can also be reduced to equation B.23.
Equation 3.91 gave the follow ing expression for the loss tangent:
tan 8 —
[B.24]
(08
Finding the angle 8 requires taking the inverse o f equation B.24
8 = tan 1( -^ 7 )
[B.25]
(08
so equation A.45 (3.92 in text) can be written as
k, = (o(p0e')1/2 [ l + ( ^ 7) 2] 1/4s in ( l/2 8)
[B.26]
Using the trigonometric identity given in equation B.7, equation B.26 can be rewritten as
r
/
ct \ 2 i i / 4 / l
- c o s 8 \ 1/2
k,-®(Wf)“ L i+ f e ) J (— T “ )
[B.27]
Substituting the left-hand side o f equation B.24 for the quantity o/((oe') produces
k, = (m£’),/2 [l +(BnJ8) ] “ ( i ^ ) ' “
[B.28]
Substituting equation B.16 in equation B.28 yields
1 1 1 / 4 / 1 - COS 8 \
k, = (o(n0s ')1/2 [ ^ r g ] 1/4(
2
/
1/2
Applying the rules o f exponents to the term in brackets, equation B.29 can be rewritten as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[B.29]
317
[B.30]
Rearrangement o f the terms raised to the h a lf power yields
,\H 2
)
(
l - C O S 5 -J
I
2cos8~ }
1/2
[B .3 1]
Note that the right-hand side o f equation B .31 is the same as the right-hand side o f equation B.23,
therefore, the left-hand sides must be equal as w ell. That means
a = kI
so the two equations fo r penetration depth, equations B.13 and A.29 (3.87 in text) are also equal.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[B.32]
318
APPENDIX C:
COMPUTER PROGRAM TO
COLLECT TIME-TEMPERATURE DATA FROM THE
DATA LOGGER
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
319
DECLARE SUB COLLECT (TE1!, TE2!, TE3!, TE4!, pr!, t, t$) DECLARE SUB COLLECT (TE1, TE2, TE3, TE4, pr, t, t$)
q$= CHR$(34): ' ASCII code for "
CLS
SCREEN 8: KEY OFF
REM
REM
REM
REM
REM
REM
REM
REM
FNMX AND FNMY ARE FUNCTIONS TO CALCULATE THE SLOPE OF THE
LINEAR RELATION BETWEEN XP AND XT AND YP AND YT
FNXP AND FNYP CALCULATES XP AND YP, THE COORDINATES TO PLOT
FROM XT AND YT, THE COORDINATES IN TIME AND TEMPERATURE
REM
DEF FNMX (XTH, XTL) = (602)
DEF FNXP (MX, XT) = MX * XT
DEF FNMY (YTH, YTL) = ( - 1 7 4 ) /
DEF f n y p (my, YT, YTL)= my
DEF FNT = VAL(MID$(TIME$, 1,
60 + VAL(MID$(TIMES, 7, 2 ) ) 170 IND = 1 REM CLS REM /(XTH - XTL) + 30 (YTH - YTL) * (YT - YTL) + 174 2 ) ) * 3600 + VAL(MID$(TIMES, 4,
2))
*
REM ACQUIRE DATA FOR THE PLOT AND THE STORAGE FILE
REM
REM
REM ====== These are the default values
pr = 2:
REM number of probes
XTH = 1200
REM max. time for the graph
YTL = 0:
REM min. temperature for the graph
YTH = 100:
REM max. temperature for the graph
SUBDIRS = "c:\hpna_cal\salt\": REM default directory
SF$= "test .dat": REM name of the storage file REM save graph? SG$ = " n " :
REM name of the picture file
GF$= "test, pic' REM everything correct? EC$ = " n " :
SA$= " y " : REM collect start again? RF$ = " n " :
REM replace storage file?
RG$= " n " : REM replace picture file? R E M ------370 LOCATE 2, 1: PRINT "HOW MANY PROBES ARE YOU USING (1-4) pr LOCATE 2, 37: INPUT pr$
IF pr$= "" THEN 410 pr = VAL(pr$)
410 IF pr < 1 OR pr > 4 THEN BEEP: GOTO 370
REM
430 LOCATE 4, 1: PRINT "ENTER THE MAXIMUM TIME FOR THE GRAPH (seconds)
"; XTH
LOCATE 4, 47: INPUT XTH$IF XTHS = "" THEN 470 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. XTH = VAL(XTH$)
470 IF XTH <= 0 THEN BEEP: GOTO 430
REM IX = TIME BETWEEN GRID LINES
IX = XTH / 10
R E M ----------------------------------------------------------510 LOCATE 6, 1: PRINT "ENTER THE MINIMUM TEMPERATURE FOR THE GRAPH "
YTL
LOCATE 6, 44: INPUT YTL$IF YTL$ = "" THEN 550
YTL = VAL(YTL$) 550 IF YTL < -999 THEN BEEP: GOTO 510 R E M ---------------------------------------------------------570 LOCATE 8, 1: PRINT "ENTER THE MAXIMUM TEMPERATURE FOR THE GRAPH " YTH LOCATE 8, 44: INPUT YTH$
IF YTH$= "" THEN 620 YTH = VAL(YTH$)
REM IY = DEGREES BETWEEN GRID LINES
620 IF YTH < YTL OR YTH > 999 THEN BEEP: GOTO 570
IY = (YTH - YTL) / 10
R E M ---------------------------------------------------------650 LOCATE 10, 1: PRINT "TYPE THE NAME OF THE STORAGE FILE "; SF$LOCATE 10, 34: INPUT sfl$
IF LEN(sf1$) > 12 THEN BEEP: GOTO 650 IF sf1$ = "" THEN 690
SF$= sf1$
690 ON ERROR GOTO 7 90
SF$= SUBDIR$ + SF$OPEN SF$ FOR INPUT AS #1
ON ERROR GOTO 0
CLOSE #1
IF SF$= SUBDIR$ + "test.dat" THEN RF$= " y " : SG$ = " n "
720 LOCATE 11, 1: PRINT "Replace "; SF$; " "; RF$
LOCATE 11, 9 + LEN(SF$): INPUT RF1$
IF RF1$= "" THEN 760 RF$ = RF1$760 IF RF$ = " n " OR RF$= "N" THEN 650 IF RF$ <> " y " AND RF$<> "Y" THEN BEEP: GOTO 720 GOTO 810 790 RESUME 810 R E M ---------------------------------------------------------810 ON ERROR GOTO 0 LOCATE 12, 1: PRINT "DO YOU WANT TO SAVE THE GRAPH IN A FILE (Y/N) "; SG$
LOCATE 12, 46: INPUT SG1$IF SG1$ = "" THEN 850
SG$= SG1$
850 IF SG$= "N" OR SG$ = " n " THEN EC = 14: GOTO 1050: REM LINE FOR
NEXT QUESTION
IF SG$<> "Y" AND SG$ <> " y " THEN BEEP: GOTO 810
R E M ---------------------------------------------------------GF$= LEFT$(SF$, LEN(SF$) - 3) + "pic"
880 LOCATE 14, 1: PRINT "TYPE THE NAME OF THE PICTURE FILE "; GF$LOCATE 14, 34: INPUT gfl$
IF LEN(gf1$) > 12 THEN 880 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 321 EC = 16 IF gf1$ = "" THEN 930
GF$= gf1$
930 ON ERROR GOTO 1030
OPEN GF$FOR INPUT AS #1 ON ERROR GOTO 0 CLOSE #1 960 LOCATE 15, 1: PRINT "Replace GF$; "
RG$LOCATE 15, 9 + LEN(GF$): INPUT RG1$IF RG1$ = "" THEN 1000
RG$= RG1$
1000 IF RG$= "n" OR RG$ = "N" THEN 880
IF RG$<> "y" AND RG$ <> "Y" THEN BEEP: GOTO
960
GOTO 1050
1030 RESUME 1050
R E M --------------------------------------------------------1050 ON ERROR GOTO 0
LOCATE EC, 1: PRINT "Is everything correct (Y/N)
EC$LOCATE EC, 28: INPUT EC1$
IF EC1$= "" THEN 1090 EC$ = EC1$1090 IF EC$ = "N" OR EC$= "n" THEN 370 IF EC$ <> "Y" AND EC$<> "y" THEN BEEP: GOTO 1050 CLS : PRINT "When you see the graphic screen, hit any key to start" PRINT "To end data collection, again hit any key" PRINT "To restart or quit after data collection, hit the letter 'e '" 1108 IF INKEY$ = "" THEN 1108
REM
REM
REM
REM
PREPARE THE AXES FOR THE GRAPH
CLS
MX = FNMX(XTH, XTL)
my = FNMY(YTH, YTL)
REM
Draw the grid at time intervals of IX (set in line 460)
FOR L = XTL TO XTH STEP IX
LINE (FNXP(MX, L), fnyp(my, YTL, YTL))-(FNXP(MX, L), fnyp(my, YTH,
YTL) )
NEXT
REM Draw the grid at temperature intervals of IY
FOR L = YTL TO YTH STEP IY
LINE (FNXP(MX, XTL), fnyp(my, L, YTL))-(FNXP(MX, XTH), fnyp(my, L,
YTL) )
NEXT
LINE (FNXP(MX, 0), fnyp(my, 30, YTL))-(FNXP(MX, 600), fnyp(my, 80,
YTL) )
REM Draw a box
for the
legend
LINE
(32, 4)-(90,
33), 0, BF
LINE (32, 4)- (90,
33), 16, B
LINE (35, 8)- (50,
8), 1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
322
IF pr > 1 THEN LINE (35,
IF pr > 2 THEN LINE (35,
IF pr > 3 THEN LINE (35,
15)-(50, 15),
22)-(50, 22),
29)-(50, 29),
2
3
5
REM
Print the numbers for the temperature axis
LOCATE 1, 1: PRINT MID$(STR$(YTH), 2, 3)
LOCATE
22, 1: PRINT MID$(STR$(YTL), 2, 3)
LOCATE
11, 1: PRINT MID$(STR$((YTL + YTH)/ 2), 2, 3)
REM Print the numbers for the time axis
LOCATE
23, 4: PRINT MID$(STR$(XTL), 2, 4)
LOCATE
23, 42: PRINT MID$(STR$((XTL + XTH)/ 2), 2, 4)
LOCATE
23, 76: PRINT MID$(STR$(XTH), 2, 4);
REM Print the name of the file
LOCATE
21, 30: PRINT SF$1440 IF INKEY$ = "" THEN 1440
REM
REM
REM
OPEN COMMUNICATIONS PORT #2
OPEN "COM2:9600, N, 8,1" FOR RANDOM AS #1: LINE INPUT #1, a$REM REM REM OPEN OUTPUT FILE OPEN SF$ FOR OUTPUT AS #2
REM
REM
REM
START DATA COLLECTION
1 tO = TIMER: REM TO = FNT
14 60 REM
REM
REM COLLECT WAITS FOR DATA TO COME FROM THE 2IX UNIT AND RETURNS THE
REM TEMPERATURES AND THE TIME (t)
CALL COLLECT(TE1, TE2, TE3, TE4, pr, t, t$) REM REM print the data to IF pr = 1 THEN PRINT IF pr = 2 THEN PRINT IF pr = 3 THEN PRINT IF pr = 4 THEN PRINT file #2, t$,
#2, t$, #2, t$,
#2, t$, TE1 TE1, TE2 TE1, TE2, TE3 TE1, TE2, TE3, TE4 REM IF pr = 1 THEN PRINT #2, USING REM IF pr = 2 THEN PRINT #2, USING REM IF pr = 3 THEN PRINT #2, USING TE1; TE2; TE3 ###.#"; t$; TE1
###.#, ###.#"; t$; TE1; TE2 ###.#, ###.#, ###.#"; t$;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
323
REM
IF pr = 4 THEN PRINT #2, USING
t$; TE1; TE2; TE3; TE4 REM ###.#, ###.#, ###.#, ###.#"; Print the data to screen LOCATE 6, 1: PRINT USING "####.#"; t;: PRINT t$
LOCATE 7, 1: PRINT USING "###.#
TE1
IF pr > 1 THEN LOCATE 8,1: PRINT USING "###.#
TE2
IF pr > 2 THEN LOCATE 9,1: PRINT USING "###.#
TE3
IF pr > 3 THEN LOCATE 10, 1: PRINT USING "###.#
TE4
REM Plot the data
YP1 = fnyp{my, TE1, YTL)
YP2 = fnyp(my, TE2, YTL)
YP3 = fnyp(my, TE3, YTL)
YP4 = fnyp(my, TE4, YTL)
XP = FNXP(MX, t)
IF Z <> 1 THEN 1740
LINE (XP, YP1)-(XA, YA1), 1
IF pr > 1 THEN LINE (XP, YP2)-(XA, YA2), 2
IF pr > 2 THEN LINE (XP, YP3)-(XA, YA3), 3
IF pr > 3 THEN LINE (XP, YP4)-(XA, YA4), 5
1740 Z = 1
REM Save the values ofthe coordinatesto use forthe line in next
time
YA1 = YP1
YA2 = YP2
YA3 = YP3
YA4 = YP4
XA = XP
IF INKEY$<> "" THEN 1820 GOTO 14 60 REM Close the files 1820 CLOSE #1: CLOSE #2 1830 a$ = INKEY$: IF a$ <> "E" AND a$<> "e" THEN 1830 IF SG$ <> "Y" AND SG$<> "y" THEN 1870 DEF SEG = &HB800 BSAVE GF$, 0, &H4000
187 0 CLS : SCREEN 8
1880 LOCATE 12, 1: PRINT "Do you want to start again
LOCATE 12, 27: INPUT SA1$IF SA1$ = "" THEN 1920
SA$= SA1$
1920 IF SA$= "n" OR SA$ = "N" THEN END
IF SA$<> "Y" AND SA$ <> "y" THEN BEEP: GOTO 1880
CLS
to = 0
CLEAR
GOTO 170
END
STOP
SA$SUB COLLECT (TE1, TE2, TE3, TE4, pr, t, t$) STATIC
DIM t {4)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
324
q$= CHR$(34): ' ASCII code for "
LINE INPUT #1, a$nowh$ = MID$( a$ , 15, 2)
nowh = VAL(nowh$) nowm$ = MID$(a$, 17, 2)
nowm = VAL(nowm$) nows$ = MID$(a$, 26, 4) + "0"
nows = VAL(nows$) now = nowh * 3600 + nowm * 60 + nows REM IF nows > = 1 0 THEN REM t$ = q$+ MID$(STR$(nowh), 2, 2)+ ":" + MID$(STR$(nowm), 2, 2) + + MID$(STR$(nows), 2, 4) + "0" + q$
REM
ELSEIF nows < 10 AND nows >= 1 THEN
REM
t$= q$ + MID$(STR$(nowh), 2, 2) +
+ MID$(STR$(nowm), 2,
2) + ":0" + MID$(STR$(nows), 2, 4) + "0" + q$REM ELSEIF nows < 1 THEN REM t$ = q$+ MID$(STR$(nowh), 2, 2) + + MID$(STR$(nowm), 2, 2) + ":00" + MID$(STR$(nows), 2, 4) + "0" + q$
REM
END IF
t$= q$ + nowh$+ + nowm$ +
+ nows$+ q$
FOR p = 1 TO pr
t(p) = VAL(MID$(a$, 25 + 10 * p, 5))
NEXT
IF tflag = 0 THEN
tO = now
tflag = 1
END IF
t = now - tO
TE1 = t (1)
IF pr >1 THEN
IF pr >2 THEN
IF pr >3 THEN
TE2 = t (2)
TE3 = t (3)
TE4 = t (4)
SOUND 1600, 2
END SUB
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
325
APPENDIX D:
FLOWCHARTS AND CODE FOR
COMPUTER PROGRAMS
USED IN PROCESSING DATA FROM CHAPTER 3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
326
This first section opens and names the data logger
files, extracts the time-temperature data, and stores
it in matrices.
Begin
I
Open the data logger file containing measured
time/temperature data for specific starch.
I
n21x is the record counter fo r the data
logger; initialize it to 0. The follow ing
loop extracts the tim e- temperature
data from the data logger file and
assembles time-temperature vectors.
n21x = 0
r
W hile not
end o f data
true
n21x = n21x + 1
Close the data
logger file
I
Extract time and temperature data from the «27xth record
and store it
This loop extracts the dielectric data for the starches at each
temperature from the network analyzer files. Then it
assembles the data in m atrix form.
FOR horc
from
1 to 2
For each experiment there are 11
heating and 10 cooling temperatures.
HorC =1 indicates heating cycle;
HorC = 2 indicates cooling cycle.
fileH orC is a
the file name
HleHorC
fileHorC
Figure D .l Flowchart for program to read data files
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
327
This loop assembles the names o f the
network analyzer files. For each
temperature there is a starch and a water file.
FOR temp 1
to number
o f temps
exit
loop
temp is the temperature number
(between 1 and \ \ ) , filetem p is
the temperature itself._________
filetem p = (30 + (temp - 1)*5)]
The program goes
back and assembles
the file name fo r the
next temperature.
Assemble the name o f the starch and water
files including fileHorC, and filetemp
Open the starch and water files
nNA = 0
Close the starch and
water files
exit
nNA is the number o f the
record o f the network analyzer
file.
W hile
not end
o fN A
data
loop
/
This loop goes through the records
(lines) o f the network analyzer
files, extracting pertinent data.
r
nNA = nNA + 1
I
Extract the nNAtt line from the
starch and water files
Figure D.l (cont.) Flowchart for program to read data files
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
328
The time
data in the
network
analyzer
files
(timenah for
heating and
timenac for
cooling) is
interpolated
to determine
the corres­
ponding
temperatures
(Th for
heating and
Tefor
cooling) in
the data
logger files.
To account fo r
instrument sweep time,
5.5 seconds is subtracted.
r is the record
number fo r the
dielectric data.
5<nNA<14
or nNA < 4
=5
nNA-14
HorC
Interpolate
to find Th
and Tc at
timenah
and
timenac
using
tim e21x
and T21x.
Line 4 has the tim e o f
measurement and line 5 has the
date. D ielectric data begins at
line 15 and continues fo r 150
lines, corresponding to the 150
frequencies.
Extract the
e' and e"
for starch
measured
during
heating;
store them
in m atrix
Extract the
e' and e"
fo r starch
measured
during
cooling;
store them
in matrix
£smeash
^smeasc
I
Extract the
month= mo
Extract the
hour = h
Extract the
day = d
Extract the
minutes = m i
Extract the
year = y
Extract the
seconds and
subtract 5.5
sec. for
sweep time
=s
Calculate date
number for
each exp’t.
I
Extract the
sands'
fo r water
measured
during
heating;
store them
in m atrix
Extract the
sands'
for water
measured
during
cooling.;
store them
in matrix
^wmeash
^wmeasc
Calculate
timenah
Calculate
timenac
Figure D .l (cont) Flowchart for program to read data files
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329
%
% This program reads the data from the 21X data logger
% (temperature files) and the Network Analyzer(measured dielectric
data)
%
and builds matrices with this data to be used for further
% processing.
% Revised 4/9/05
%
The first section reads the temperature file.
%
%
%
%
%
%
Form a file "prefix" that contains the part of the file names
that is common to all the files for a particular experiment.
There is more than one prefix defined in the next lines, but
only the one that is not commented (with % in the first
column) will be used.
fileprefix='C :\THESIS DATA & DOC\Salt\Salt Net. An. Data\3';% for salt
data
%fileprefix='C:\THESIS DATA & DOC\PENCOOK\PenCook Net. An. Data\14';
%fileprefix='C:\THESIS DATA & DOC\Amioca\Amioca Net.An. Data\6';
%
Now form the name of the temperature file by adding "temp.dat"
% to the prefix. ;
filename=[fileprefix 'stemp.dat'] % for salt data
%filename=[fileprefix 'temp.dat']
%
Tfid is the Temperature "file id".
%
fopen with 'rt' option opens filename in "read" and "text" mode.
Tfid=fopen(filename,'rt');
%
Initialize the 21x temperature record (line) number.
n21x=0;
%
The while command executes a loop as long as a mathematical
%
equality is true. When "While 1" is used, the mathematical
%
condition is set to true (1), so the loop cycles until it
%
is interrupted by a break command,
while 1
%
Update the 21x temperature record number.
n21x=n21x+l;
%
The fgetl function returns the next line of the file as a
%
string (text line)
tline = fgetl(Tfid);
%
If the next line is not a character, break the loop,
if -ischar(tline), break, end
%
%
%
Each record has a time stamp and temperatures measured in 2
different places. The temperature of interest is the first
one in the record.
%
%
%
%
%
The command strread extracts separate pieces of data from a
string. The format string '%q%n%n' looks for one item
between double quotation marks (%q) which correstponds to the
time stamp (time21xString) and two numbers (%n%n) which
correspond to the two temperatures (T21x and T2).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
330
%
%
The temperature of interest is the first one (T21X) and it is
stored in a vector. The other temperature is not used.
%
%
%
%
The datenum function converts date and time data into date
number. Date numbers are serial days where, for example,
1 corresponds to l-Jan-0000, and 2.5 corresponds
to 2-Jan-0000 at 12 noon.
time21x(n21x,1)=datenum(time21xString);
end;% end of 21x temp loop
fclose(Tfid);
%
%
%
** *** *** *** *** *** *** *** *** *** ************** *********************
The next section reads the dielectric data from the
analyzer files.
% Start a loop to process
for HorC=l:2; %HorC = 1 for
network
heating and cooling data
heating or 2 for cooling.
% temp is the temperature number (not the temperature itself)
for temp=l:11*(HorC==l)+10*(HorC==2)
% For each experiment there are 11 heating and 10 cooling
% temperatures. When heating, HorC = 1, the first term in
% parenthesis is 1 (true), the second term is 0 (false), and so
% the first 11 temps are chosen.
When cooling, HorC =2, the
% first term in parentheses is 0 (false), the second term is 1
%
(true), and so the 10 temperatures are chosen,
if HorC==l
% suffix 'h1 (heating) for the file name
fileHorC='h';
elseif HorC==2
% suffix 'c' (cooling) for the file name
fileHorC='d';
end
%
%
%
The following lines assemble the names of the network analyzer
files by creating a string with the temperatures from 30 to 80.
For each temperature there is a starch file and a water file.
%
First form the temperature suffix for the file names.
filetemp=[num2str(30+(temp-1)*5)];
%
Suffix "s" on filenames means "starch."
filenames=[fileprefix 's' fileHorC filetemp '.prn'];
%
Suffix "w" on filenames means "water."
filenamew=[fileprefix fileHorC filetemp '.prn'];
%
Display the starch filename
disp(filenames)
fs=fopen(filenames,'rt');% file id number for starch
fw=fopen(filenamew,'rt');% file id number for water
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331
%
Initialize the Network Analyzer (NA) record number
nNA=0;
%
%
Start a loop that goes through the
network analyzer files
records (lines) of the
while 1
nNA=nNA+l;
tlines = fgetl(fs); % next record for starch
tlinew = fgetl(fw); % next
record for water
if -ischar(tlines)|-ischar(tlinew), break, end
%
%
%
The first few lines in the network analyzer files contain date
and time information, as well as some other non-dielectric
data.
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
The following lines extract different parts of the tlines
string to obtain the time.
Line 4 is the time of measurement in the network analyzer
files.
if nNA==4
The hour is in the 7th and 8th characters.
h=str2num(tlines(7:8));
The str2num function converts a string variable to a numeric
variable.
h=str2num(tlines(7:8))+2;% melojel 6 has a 2 h error
The minutes are in the 10th and 11th characters.
mi=str2num(tlines(10:11));
The seconds are in the 13th to the penultimate character.
(the last one is a quotation mark).
A delay time (5.5 seconds) is subtracted to account for
instrument sweep time.
s=str2num(tlines(13:size(tlines,2)-1))-5.5;
Line 5 is the date of measurement.
elseif nNA==5
The month is in the 7th and 8th characters.
mo=str2num(tlines(7:8));
The day is in the 10th and 11th characters.
d=str2num(tlines(10:11));
The year is in the 13th to the penultimate characters.
y=str2num(tlines(13:size(tlines,2)—1))+1900;
The following lines calculate the date numbers for heating
and cooling separately. The formula used avoids data errors
for experiments that have two different dates (i.e. with
times past midnight).
if HorC==l % Times for heating
timenah(1,temp)= datenum(y,mo,d,h,mi,s)-datenum(y,mo,d);
elseif HorC==2 % Times for cooling
timenac(1,temp)= datenum(y,mo,d,h,mi,s)-datenum(y,mo,d);
end
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
332
%
Dielectric data begins in line 15
elseif nNA>14
%
%
%
%
%
Create a record number (r) for the dielectric data. The 15th
line of the network analyzer file is the 1st record for the
dielectric data. There is a total of 150 records,
corresponding to measurements of the relative complex
permittivity at 150 frequencies from 0.3 GHz to 3 GHz.
r=nNA-14;
%
%
%
%
%
%
%
%
%
The following lines extract the frequency, dielectric and
sensitivity data from the network analyzer files. The
dielectric data is then put in a matrix that has one row
for each frequency, one column for each temperature, and
two "pages" (the third dimension) used for the real and
the imaginary parts of the dielectric permittivity. A
"dummy" variable is included at the end of the variable
list to account for a "tab" character at the end of each
record.
if HorC==l % Heating
[f(r,l) esmeash(r,temp,1) esmeash(r,temp,2)...
[f (r,1) ewmeash(r,temp,1) ewmeash(r,temp,2)...
elseif HorC==2 % Cooling
[f (r,1) esmeasc(r,temp,1) esmeasc(r,temp,2)...
[f(r,l) ewmeasc(r,temp,1) ewmeasc(r,temp,2)...
end % end for HorC conditions
end % end of nNA (==4, ==5, and >14) conditions
end % end of "while 1" loop through the records of network analyzer
%
files
%
Close the starch and water files,
fclose(fs);
fclose(fw);
%
The program then goes back and assembles the file name for the
%
next temperature.
end % For "temp" temperature loop.
end % for HorC = 1 or 2
^*****************************************************************
%
%
%
This section uses the time data in the network analyzer files
to look for the corresponding temperature in the 21X
temperature files. This is accomplished by interpolation.
Th=interpl(time21x,T21x,timenah,'cubic', 1extrap1);
Tc=interpl(time21x,T21x,timenac,'cubic','extrap1);
%
Clear unnecessary variables.
%clear time* temp* d fi* fs fw n* r* m* y h* s tline* Tfid T2 dummy
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333
%
Create a variable containing the processing steps performed
%
on the data,
if exist(1ProcSteps1)==1
ProcSteps=strcat(ProcSteps,', Extracted from file1)
else
ProcSteps='Extracted from file'
end
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334
D.2 “interpolate”
Begin
Interpolate
This program interpolates the
target temperatures.
Thi= vector w ith target
heating temperatures.
’ci = vector w ith target
cooling temperatures.
m is the counter for the 150 frequencies.
exit
fo rm
from
1 to 150
loop
Calculate esmhi, the matrix o f
perm ittivity values at Thi, by
interpolating w ith Th and £smcash
Update and display
processing steps.
esmh, and esmci are the
interpolated measured
perm ittivity values o f
starch for the heating and
cooling cycles,
respectively.
Calculate £smc„ the matrix o f
perm ittivity values at Tci, by
interpolating w ith Tc and £smeas
Figure D.2 Flowchart for “interpolate” program
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
335
Interpolate .m
%
%
%
%
%
This program interpolates the dielectric data extracted from the
Network Analyzer files by "readnatable" to the target temperatures
(30,35,...80 C).
The interpolation proceeds across temperature columns, for
every frequency row.
%
Thi=(30:5:80); % Thi is the vector with target heating temperatures.
Tci=(30:5:75); % Tci is the vector with target cooling temperatures.
%
%
%
%
%
%
%
%
In the following loop, m is the counter for the 150 frequencies.
The function interpl uses the following syntax:
yi = interpl(x, Y, xi, linear, 'extrap'),
meaning that it returns the value yi corresponding to the value xi
by linearly interpolating with the value x and its corresponding Y
value. The function extrapolates for values that are slightly over
or under the target temperatures, such as 80.1 or 29.9.
%
for m=l:150
esmhi(m,:,1)=interpl(Th, esmeash(m,:,1),Thi,'linear1, 1extrap1);
esmhi(m,:,2)=interpl(Th,esmeash(m,:,2),Thi,'linear', 'extrap');
esmci(m,:,1)=interpl(Tc,esmeasc(m,:,1),Tci,1linear','extrap1);
esmci(m,:,2)=interpl(Tc,esmeasc(m,:,2),Tci,'linear','extrap1);
end
if exist('ProcSteps')==1
ProcSteps=strcat(ProcSteps,', Interpolated1)
else
ProcSteps='Interpolated'
end
clear m
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
336
D.3 “correcttable”
The program calls function
“ th2o” to get the theoretical
values o f the relative
complex perm ittivity o f
water at the target
temperatures and
frequencies. The values
are stored in the m atrix ewl.
This program corrects the starch
dielectric data using a correction
factor based on water
perm ittivity.
Begin
correcttable
£w t
Calculate correction factors for
heating and cooling. swmeash and
Swmeasc are the matrices assembled in
measured perm ittivity values o f
water during heating and cooling,
respectively.
= th2o (T ,f)
1r
and
are the
vectors
containing
the corrected
values o f the
perm ittivity
for the
heating and
cooling
cycles,
respectively.
C O IT h
Swt
C O IT c
Swt
Ewmeash
1r
Swmeasc
I f the data was interpolated, the
variable esmhi exists.
Esmhi
exists
Ehc
Esmeash t
C O ITh
Esmhi + corrh
Sec
Esmeasc
C O IT
Esm ci
+ corr
Ehd and ecci are the
vectors containing
the corrected
interpolated values
o f the perm ittivity
fo r the heating and
cooling cycles,
respectively.
Update and display data
processing steps.
Plot the data
(
end
)
Figure D.3 Flowchart for program to correct the data
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
337
%
%
%
%
Correctable .m
Corrects starch dielectric data using
correction factor= (theoretical -measured) values for
permittivity.
%
%
%
This program may be used directly on the raw data (after
"readnatable") or on the interpolated data (after
"interpolate") .
%
%
%
%
%
The following lines calculate the theoretical relative complex
permittivity for water using the user-defined function
"th2o". The function takes temperature and frequency as inputs
and uses the Cole-Cole model with parameters taken from
the literature.
water
ewt
= real(th2o(ones(150,1)*(30:5:80),f‘ones(1,11)));
ewt(:,:,2) = -imag(th2o(ones(150, 1)*(30:5:80),f*ones(1,11)));
% Calculate the correction factors
corrh = ewt
- ewmeash;%Correction factor for heating
corrc = ewt(:,1:10,:)- ewmeasc;%Correction factor for cooling
%
%
%
if
Corrected complex permittivity for starch
The function "exist" returns true (1) ifa variable exists
and false otherwise.
exist('esmhi')==1 % If data was interpolated
ehci = esmhi + corrh;%heating
ecci = esmci + corrc;%cooling
else % If data was not interpolated
ehc = esmeash+corrh; % heating
ecc = esmeasc+corrc; % cooling
end
if exist('ProcSteps')==1
ProcSteps=strcat(ProcSteps,', Corrected')
else
ProcSteps='Corrected'
end
%
If plotGraph is 1, then graph the data
plotGraph=l;
if plotGraph==l
if exist('esmhi')==1
subplot(2,2,1);surf(Thi,f',ehci(:,:,1))
title(strcat('\epsilon\prime - Heating data
ProcSteps))
xlabel('Temperature')
ylabel('frequency')
zlabel('\epsilon\prime')
set(gca,'YScale', 'log')
view(45,30);
axis([30 80 3e8 3e9 60 80])
subplot (2,2,2); surf(Tci,f',ecci(:,:,1))
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
title(strcat('\epsilon\prime - Cooling data - ',.
ProcSteps))
xlabel('Temperature')
ylabel('frequency')
zlabel('\epsilon\prime')
set(gca,'YScale', 'log')
view(45,30);
axis([30 80 3e8 3e9 60 80])
subplot (2,2,3); surf(Thi,f',ehci(:,:,2))
title(strcat('\epsilon\prime\prime - Heating data
ProcSteps))
xlabel('Temperature')
ylabel('frequency')
zlabel('\epsilon\prime\prime')
set(gca,'YScale','log')
view(45,30);
axis([30 80 3e8 3e9 0 12])
subplot (2,2,4); surf(Tci,f',ecci(:, :, 2))
title(strcat('\epsilon\prime\prime - Cooling data
ProcSteps))
xlabel('Temperature')
ylabel('frequency')
zlabel('\epsilon\prime\prime')
set(gca,'YScale','log')
view(45,30);
axis([30 80 3e8 3e9 0 12])
else
subplot(2,2,1);surf(Th,f', ehc(:, :, 1))
title(strcat('\epsilon\prime - Heating data - ',.
ProcSteps))
xlabel('Temperature')
ylabel('frequency')
zlabel('\epsilon\prime')
set(gca,'YScale','log')
view(45,30);
axis([30 80 3e8 3e9 60 80])
subplot (2,2,2); surf(Tc,f',ecc(:,:, 1))
title(strcat('\epsilon\prime - Cooling data - ',.
ProcSteps))
xlabel('Temperature')
ylabel('frequency')
zlabel('\epsilon\prime')
set(gca,'YScale','log')
view(45,30);
axis([30 80 3e8 3e9 60 80])
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
subplot (2,2,3); surf(Th,f',ehc(:,:,2))
title(strcat('\epsilon\prime\prime - Heating data
ProcSteps))
xlabel('Temperature')
ylabel('frequency')
zlabel('\epsilon\prime\prime')
set(gca,'YScale', 'log')
view(45,30);
axis([30 80 3e8 3e9 0 12])
subplot (2,2,4); surf(Tc,f',ecc(:,:,2))
title(strcat('\epsilon\prime\prime - Cooling data
ProcSteps))
xlabel('Temperature')
ylabel('frequency')
zlabel('\epsilon\prime\prime')
set(gca, 'YScale', 'log')
view(45,30);
axis([30 80 3e8 3e9 0 12] )
end
end
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
340
D.4
“th 2 o ”
This function takes temperature and
frequency as arguments. It calculates the
theoretical relative complex perm ittivity
values fo r water using the Cole-Cole model
and data given in Hasted, 1972b.
Function
c = th2o(T,f)
Tj = [30,40, 50, 60, 75]
e* = [76.8, 73.2, 70.0, 66.6,62.1]
Coo, = [4.20,4.16,4.13,4.21,4.49]
r, values have magnitude lx lO ' 11 seconds
Ti = [0.72,0.58, 0.48,0.39, 0.32]
a; = [0.012, 0.009, 0.013, 0.011, 0]
30 < T < 80
The data is interpolated to
determine the values o f the
parameters that correspond to the
target temperatures.
T > 80 or
T < 30
Display
warning
message
Interpolate to find ^ at T using Ti and es.
Interpolate to find £«, at T using Tj and e^i
Interpolate to find t at T using T; and tauj
Interpolate to find a at T using Tj and ctj.
to = 2n f
c = E00 + (es- £ 00)/[l+(ic»T)(1 a)]
Calculate the relative complex
perm ittivity according to the Cole-Cole
model.
Figure D.4 Flowchart for “th2o” function
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
341
th2o
function c=th2o(T, f)
%
th2o
%
Calculates theoretical relative complex permittivity for
% water using the Cole-Cole model
% th2o(T, f) (T in C and f in Hz)
%
%
%
%
%
%
%
%
%
%
T
--- > temperature (C)
es --- > static dielectricconstant
einf -- > "optic" dielectric constant (= e 1 at infinite
freq.)
tau --- > relaxation time
al --- > alpha in Cole-Cole model
The "i" after variable names refers to values used for
interpolation.
PlotGraph is a flag. If its value is 1, results will be
graphed; if its value is 0, results will not be graphed.
%
%
%
%
%
The following data was taken from Hasted(1972)in table Va,
p. 277 of chapter 7 of the book
Water:
A Comprehensive Treatise, Volume 1:The Physics and
Physical Chemistry of Water, edited by Felix Franks and
%
Initialize variables
Ti = [30 40 50 60 75];
esi =[76.8 73.2 70.0 66.6 62.1];
einfi =[4.20 4.16 4.13 4.21 4.49];
taui = [0.72 0.58 0.48 0.39 0.32]; % tau, *le-ll
ali =[0.012 0.009 0.013 0.011 0];
plotGraph=0;
% This part displays a warning if T<30 or T>80
if sum((T<30)+(T>80))>0;
textl= ['Warning! ! ', num2str (sum(TOO) ), ...
' temps lower than 30 C, and ',...
num2str(sum(T>80)),' temps, higher than 80 C'];
if plotGraph==l;plot(T);end
warndlg(textl)
end
% The interpolation is performed using Matlab's Piecewise
% Cubic Hermite Interpolating Polynomial (pchip).
es =interpl(Ti,esi,T,'pchip', 'extrap');
einf=interpl(Ti,einfi,T,'pchip','extrap');
tau=interpl(Ti,taui,T,'pchip','extrap');
tau = (tau * le-11);
al=interpl(Ti,ali,T,'pchip','extrap');al=al.*(al>=0) ;
w=2*pi*f;
c=einf+(es-einf)./(1+(i*w.*tau).A (1-al));
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
342
D.5 “recursivehc”
StarchName takes a value
between 1 and 5:
1 = Amioca; 2 = M elojel
3 = Pencook
4 = Amioca w ith 2% NaCl
5 =arbitrary data for testing
Begm
Recursivehc
Input
StarchName
assume that the data
for Amioca
exp’ts
#3, 5, & 6
from disk
Assign
exp’t #.,
freq., temp.,
dielectric
perm, and
loss data to
m atrix e
1=2
fo r M elojel
exp’ts
#7,9, & 10
from disk
StarchName
=3
fo r Salt
exp’ts
#1,2, & 3
from disk
Pencook
exp’ts
#11,13, 14, & 15
from disk
Assign
exp’t #.,
freq., temp.,
dielectric
perm . and
loss data to
m atrix e
The variable SSEConvCrit
is the residual convergence
criterion. The SSEConv
variable is set to a large
value to ensure that at least
1 iteration is completed.
MeanConvCrit is the mean
convergence criterion
Assign
exp’t #.,
freq., temp.,
dielectric
perm, and
loss data to
m atrix e
Assign
exp’t #.,
freq., temp.,
dielectric
perm, and
loss data to
m atrix e
Initialize matrices a and a0 to 0
Initialize SSEConvCrit
Initialize
SSEConv =100 * SSEConvCrit
~
%
=5
r__
Input arbitrary
numbers fo r
parameters
otherwise
'D isplay
error
message;
Calculate
dielectric
properties using
SDHTP
Assign
exp’t #., freq.,
temp., dielectric
perm, and loss
data to m atrix e
This algorithm w ill iterate
until the sum o f the square
errors o f each experiment
and the mean o f a ll the
experiments have met
specified convergence
criteria and/or the
maximum number o f
iterations has been
exceeded.
Initialize MeanConvCrit
maxI ter is the maximum
number o f data iterations
Initialize maxlter = 300
Set the iteration counter for
each experiment to 1
Initialize SSEIter = 1
Figure D.5 Flowchart for “recursivehc”
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
343
Set the iteration counter fo r
the mean to 1.
Initialize meanlter = 1
sparcity pattern
ExN is the counter for the
experiment number. There are
4 experiments for Pencook and
3 for Amioca & M elojel.
loop calculates the
1st iteration for
each experiment
because there is no
a0 input
For ExN = 1
to # o f expt’s
Each experiment
has an a matrix.
The mean o f all
a matrices fo r a
given starch is
amecm.
Display “ Now
starting iteration
w ith means.
The LSQDHTP function
calculates a, a0, and SSE
fo r each experiment.
Display
st iteration fo r data”
and the SSE value
amean = mean o f a s o f a ll expt s
Set the convergence
mean fo r the current
mean iteration to a
large value to force at
least 1 iteration.
Display
SSEIter
I
convMeanmeanIter —
MeanConvCrit *100
The follow ing loop calculates a for the 3
expt’ s, forms amean, and checks i f amean has
converged. I f not, amean is used as a0 for a
new iteration for each o f the 3 experiments.
W hile
convMeanmeanIter
> MeanConvCrit
and the
max(SSEIter)
< maxlter
The m atrix
SSE has
columns for
each exp’t
and rows for
each
iteration.
prevMean = amean
t
For ExN = 1
to # o f expt’s
max(SSE
Iter) >
maxlter?
recursiveresults
graphs the
solutions
meanlter
Display
Maximum
number o f
iterations
reached.
Ending. . .
Check i f
maximum
no. o f
iterations
has been
exceeded.
prevSSE = SSE
meanlter +1
SSEIter = SSEIter +1
Calculate
new
amean
amean
mean o f a s o f a ll expt s
convMeanmeanIter=
std dev(prevMean & amean)
mean (prevMean & amean)
Display value o f
convMeanmeanIt,r
LSQDHTP
calculates the
matrices a and SSE
fo r each
experiment using
amean as a0
mseConv =
100*SSEConvCrit
Figure D.5 (cont.) Flowchart for “recursivehc”
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
344
The
program
returns to
the next
experiment.
e x it.
Display
ExN,
SSEIter, and
SSE for that
iteration o f
that
experiment.
Using the a m atrix
generated above as a0, the
another a m atrix. This a
m atrix then becomes a0 if
the loop needs to repeat
again because SSEConv is
s till greater than the SSE
convergence criteria.
W hile
SSEConv >
SSEConvCrit
prevSSE =SSE
SSEIter = mselter + 1
LSQDHTP
calculates the
a m atrix and SSE
std dev(prevSSE and SSE)
SSEConv = -------- , * OOI-,— .
mean (prevSSE and SSE)
The variable
SSEConv is the
coefficient o f
variability
between the
previous SSE and
the current SSE.
It is found by
dividing the
standard
deviation o f those
quantities by their
mean.
Figure D.5 (cont.) Flowchart for “recursivehc”
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
345
%
Recursivehc .m
%
% Revised 4/23/2005
% Revised 4/9/2005
% Revised 3/28/2005, 3/26/2005, 1/9/05, 9/30/04, 8/8/04, 10/17/03
%
%
%
%
%
%
%
%
%
This program takes the interpolated experimental data and
solves for a solution to the "a" matrix of parameters by
iterating until the sum of square errors of each experiment
and the mean of all the experiments have met specified
convergence criteria and/or until the maximum number of
permitted iterations has been exceeded. The criterion that
is checked for convergence is the coefficient of variation,
the standard deviation divided by the mean.
%
%
%
%
This program calls upon various subfunctions in order
to model the data according to the Debye-Hasted model with
two peaks. In each iterative step, the previous "a" matrix
becomes the new initial estimate ("ao") matrix.
%
%
%
%
%
The following lines load the interpolated data from all the
experiments performed for each starch (there were 4 for
Pencook and 3 for the others) and stores the data in matrix
"e". Matrix "e" has a "book" of data for each experiment,
and each book contains 2 "pages": one for e 1 and one for e".
% StarchName
= 1 is
for Amioca
% StarchName
= 2 is
Melojel
% StarchName = 3 is
Pencook
% StarchName = 4 is
Amioca with2%
% StarchName = 5 is
generated data
StarchName=3;
switch StarchName
case 1
e=ehci;
e (:,12:21,:)=ecci;
e (:,1:11,:,2)=ehci;
e (:,12:21,:,2)=ecci;
e (:,1:11,:,3)=ehci;
e (:,12:21,:,3)=ecci;
case 2
e=ehci;
e (:,12:21,:)=ecci;
e (:,1:11,:,2)=ehci;
e (:,12:21,:,2)=ecci;
e (:,1:11,:,3)=ehci;
e (:,12:21,:,3)=ecci;
NaCl
fordebugging
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
346
case 3
e=ehci;
e (:,12:21,:)=ecci;
e (:,1:11,:,2)=ehci;
e (:,12:21,:,2)=ecci;
%e(:,1:11,:,3)=ehci;
%e(:,12:21,:,3)=ecci;
%e(:,1:11,:,4)=ehci;
%e(:,12:21,:,4)=ecci;
case 4
e=ehci;
e (:,12:21,:)=ecci;
e (:,1:11,:,2)=ehci;
e (:,12:21,:,2)=ecci;
e (:,1:11,:,3)=ehci;
e (:,12:21,:,3)=ecci;
case 5
Thi=(30:5:80);
ag=ones(11,6);
a g (:,1)=(11:-1:1)';
%ag(:,1) = (1:1:11) ';
%ag(:,2)— (11:—1:1)'./l;
a g (:,2)=(1:1:11)'./10;
a g (:,3)=real(th2o(Thi',lei))/10;
a g (:,4)=real(th2o(Thi',lelOO));
a g (:,5)=10*tauw(Thi');
a g (:,6)=10
f=logspace(loglO(3e7),loglO(3el0),150)';
e (:,:,:,1)=SDHTP(ag,f,Thi');
rand('state',0)
e (:,:,:,2)= e (:,:,:,1)+(rand(150,11,2)-0.5)./100;
e (:,:,:,3)= e (:,:,:,1)+(rand(150,11,2)-0.5)./100;
otherwise
error('Unknown starch chosen')
end; % StarchName
T=[Thi Tci];
nTemp=size(T,2);
% doMainLoop is a flag that is manually set to 1 to proceed
% with the main loop, and any other number to skip it.
% When the main loop is skipped, the data is loaded with no
% further calculations.
doMainLoop=l;
fileRoot='E :\Pencook Recursive\PencookComb3' % 3\26\2005
for Ccount=l:size(combination,1);
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
347
disp(['
'
1
'
desl= 1 num2str(combination(Ccount,1))...
taul= 1 num2str(combination(Ccount, 2)).. .
MeanConvCrit= ' num2str(combination(Ccount,3))...
sseConvCrit= ' num2str(combination(Ccount,4))] )
MeanConvCrit=combination(Ccount, 3); % Mean convergence criterion
sseConvCrit=combination(Ccount,4); % Residual convergence
criterion
if doMainLoop==l
% Initialize some variables
a=zeros(nTemp,6,size(e,4));
% Result parameter matrix
a (2:nTemp,6)=NaN;
aO=zeros(nTemp,6,size(e,4));
% Initial estimate matrix
aO(2:nTemp,6)=NaN;
sseConv=100*sseConvCrit; % Initial sse convergence variable
sse=0;
convMean(1)=MeanConvCrit*100;
maxlter=300;
% Maximum number of data iterations
sseIter=ones(1,size(e,4)); % Number of iterations for each exp.
meanlter=l;
% Number of iterations with mean
% Jacobian sparcity pattern (variable
jp)
nmaxlter=0;
% Number of combinations reaching
maxlter.
%
%
%
Calculate the initial estimates for each experiment using
the LSQDHTP function without specifying the aO matrix.
ExN is a counter for Experiment Number
for ExN=l:size(e,4)
%
Use the following two commented lines to obtain
%
parameters that can be used for troubleshooting. Note
that
%
the corresponding lines in LSQDHTP also have to be
changed.
%
[a(:,:,ExN) aO(:,:,ExN) sse(sselter(ExN),ExN)
residual ...
%
exitflag output] =
LSQDHTP(f,T,e(:,:,:,ExN));
%
init is the initial estimates for esl-es2 and taul that
%
were specified in the combination file
init=[combination(Ccount, 1) combination(Ccount,2)];
[a(:,:,ExN) aO(:,:,ExN) sse (sselter(ExN),ExN)] = ...
LSQDHTP(f,T,e(:, :,:,ExN), [],jp,init);
disp(['lst iteration for data ' num2str(ExN) ...
'. sse = ' num2str(sse(sselter(ExN),ExN)) '.'])
end;% ExN=l:size(e,4)
a (2:nTemp,6,:)=NaN;
disp('Now starting iteration with means.')
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
348
amean=mean(a,3);
while
(convMean(meanlter)>MeanConvCrit)& (max(sselter(:))<maxlter)
prevMean=amean;
for ExN=l:size(e,4)
prevSSE=sse(sselter(ExN),ExN);
sselter(ExN)=sselter(ExN)+1;
%[a(:,:,ExN) aO(:,:,ExN) sse(sselter(ExN),ExN) residual exitflag
%output] = L S Q D H T P ( f , T , e E x N ) ,amean,jp);
[a(:,:,ExN) aO(:,:,ExN) sse (sselter(ExN),ExN)] = ...
LSQDHTP(f,T,e(:,:,:,ExN),amean,jp);
% The following internal loop is repeated numerous times
for
% each experiment of each starch. It calculates the a
matrix
% and sse.
If the sse is greater than the sse
convergence
% criterion, the loop repeats again.
The a matrix from
the
% previous loop then becomes the ao matrix for the next
% set of calculations,
a (2:nTemp,6,:)=NaN;
sseConv=100*sseConvCrit; % re-set sseConv
while sseConv>=sseConvCrit
prevSSE=sse(sselter(ExN),ExN);
sselter(ExN)=sselter(ExN)+1;
% [a(:,:,ExN) aO(:,:,ExN) sse (sselter(ExN),ExN) residual
% exitflag output]=LSQDHTP(f,T,e(:,:,:,ExN),...
% a (:,:,ExN),jp);
[a(:,:,ExN) aO(:,:,ExN) sse(sselter(ExN),ExN)]= ...
LSQDHTP(f,T,e (:,:,:,ExN),a(:,:,ExN),jp);
sseConv=std([prevSSE sse(sselter(ExN),ExN)])/...
mean([prevSSE sse(sselter(ExN),ExN)]);
end; % sseConv>=sseConvCrit
disp(['Experiment ' num2str(ExN)...
'. ' num2str(sselter(ExN)) 1 iterations.
'sse = ' num2str(sse(sselter(ExN),ExN))])
end; % for ExN=l:size(e,4)
meanlter=meanlter+l;
amean=mean(a,3);
% Calculate the st. dev. / mean
convMeanM=amean;
convMeanM(:,:,2)=prevMean;
convMeanM=std(convMeanM,0,3)./mean(convMeanM+eps,3);
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
'...
349
convMean(meanlter)= mean(convMeanM(1:5*nTemp+l));
% convMean(meanlter)= (convMean(meanlter1)+mean(convMeanM(1:5*nTemp+l)))/2;
disp([1Convmean = ' num2str(convMean(meanlter))])
end; % convMean>MeanConvCrit
disp(['Iterations: ' num2str(sselter)])
if max(sselter(:))>maxlter
disp('Maximum number of iterations reached. Ending...1)
nmaxlter=nmaxlter+l;
end
close gcf
recursiveresults
end % doMainLoop==l
disp(['Final ConvMeans = ' num2str(convMean)])
disp('Final sse = ');
disp(num2str(sse))
%filename=['e :\Amioca RecursiveV fileRoot num2str(Ccount) '.mat'];
filename=[fileRoot num2str(Ccount) '.mat'];
disp(filename)
combin=combination(Ccount,:);
disp(['Combination was ' num2str(combin)])
save(filename,'e ','a ','combin','sse',...
'convMean','sseConv','nmaxlter')
close all
end
%plotrecursive
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
350
D.6 “LSQDHTP”
Function
LSQDHTP(f, T, e, an, jsp, in it)
This function uses the least
squares method to find an
estimate o f scaled parameters for
Hasted-Debye model w ith two
peaks. The parameters are scaled
to the same order o f magnitude.
ntemp is the
number o f
temperatures.
Create m atrix
ntemp
The first three inputs are mandatory; the last three
(described below) are optional.
-a0 is the m atrix o f in itia l estimates o f parameters.
It contains one row per test temperature and 6
columns (indicated by subscripts below) w ith the
®lo ~ £ sl—es2 5
a2 0 = H ;
d? o = e s2
a 4o =
£od2
a5o = *2
a6o = Cequiv = equivalent salt concentration
-init is the m atrix containing in itia l estimates for
only aJoand a2o
-jsp is the Jacobian used in the regression to
minimize errors.
L*o3
yes
&o6
specified?
Initialize aB
a0= 0
Set ctj0 and a2o
equal to values
chosen in the
range o f
Mashimo’s
(1987) data.
(*o l
(*o2
specified?
aIo = init,
a2o = in it2
a2o= 1.6
In itia l estimates for
a3o and a5o are
found through
linear regression.
Form matrix
toe" = 2jifs "
Loop through ntemp
temperatures to return
values for es2 and x2.
Figure D.6 Flowchart for “LSQDHTP” function
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E xit
fo rn
from 1 to
ntemp
In itia l estimate for
s^ 2 set equal to that
o f water at very
high frequency.
Loop
Compute the least
squares estimate o f
E' = 852 -
tco e" .
I
J§
X
a4o ^co2
th20(T, 1
eS2 = a3o = intercept
T2o= -aso = -slope
I
I
display step is
optional and w ill
not occur unless
“ un-commented”
in the program.
..............
T
II
£
Set equivalent salt
concentration to
arbitrary, empirical
value.
T"
I
_!
1. ____ .(optional).___
The follow ing optional step w ill not
occur unless it is “ un-commented” in
the program. It sets the in itia l value
o f t2 to the theoretical value for
water, instead o f calculating it from
the linear regression.
Display
a„
I
Set lower and upper boundaries for
the iterative process.
Set lower bounds (lb) for
all parameters.
Set upper bounds (ub) for
all parameters.
I
a and mse =
LSQCURVEFIT
(SDHTP, a0, f, e , lb,
ub, options, T).
Compute the least squares
estimate o f the parameters a,
through a6 fo r the data, using
the function “ SDHTP.” This
function “ un-scales” the
parameters for use in the
Debye-Hasted-Two-Peak
model.
Display and
plot the results.
Figure D.6 (cont.) Flowchart for LSQDHTP function
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
352
%
LSQDHTP.m
%
%
%
Revised 4/23/05
Revised 1/9/05, 1/12/05, 1/16/05,
7/12/04
%
%
%
%
%
%
%
%
This function uses the least squares method to find an
estimate of the parameters for the Hasted-Debye model with
two peaks, given the measured dielectric data as a function of
temperature and frequency.
The result is a parameter matrix "a" with a column for each
parameter and a row for each temperature.
Thisfunction
can also return some statistical outputs.
%
%
%
%
%
%
The matrix "a" contains one row per test temperature, and
6 columns with different parameters as listed below.
These parameters, as used by LSQDHTP, are scaled so that
all their values are approximately the same order
of magnitude.
%
%
a (:,1)/I
%
%
%
%
%
%
a (:,2)*le-9
a (:,3)*10
a (:, 4)
a (:,5)*le-12
a (1,6)*le-3
a (:,6)*le-l
=
=
=
=
=
=
_
esl - es2
taul
es2
einf2
tau2
equivalent NaCl concentration % (w/w)
ir
ii
it
m
(for salt experiments)
%
%
%
%
%
In order to use these parameters, they have to be
"un-scaled" by an auxiliary function "SDHTP" that
returns them to their real order of magnitude.
%
%
%
%
%
%
The vertical vector "f" contains the test frequencies (Hz).
Thehorizontal vector "T" contains the test temperatures (C).
The matrix "e" contains the dielectric data organized in
one column for every temperature, one row for each
frequency, one page for e', and a second page for e".
%
%
%
%
%
%
%
The least squares function used to estimate the parameters
(LSQCURVEFIT) requires an initial estimate of the parameters
(the "aO" matrix). The "aO" matrix can be specified
when LSQDHTP is called, or otherwise the
The size of "aO" corresponds to the size of "a."
%
%
%
%
%
%
%
%
%
The optional argument "jsp" is the "Jacobian sparcity matrix."
The algorithm used by MatLab for LSQCURVEFIT uses a
very large matrix that contains all posible derivatives
of variables with respect to each other. The Jacobian sparcity
matrix specifies which of these derivatives are non-zero.
In this case, only derivatives with respect to frequency and
temperature are needed. By specifying a Jacobian sparcity
matrix, the execution time for this function is greatly reduced.
%
%
Usage is:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
353
%
[a,aO,sse] = LSQDHTP(f,T,e,aO,jsp,init)
function [a,aO,sse] = LSQDHTP(f,T,e,aO,jsp,init); % aO, jsp, and init
%
are optional. The matrix "init" contains the first two columns
% of the aO matrix (aOl = esl-es2 and ao2 = taul).
% The following commented line is used to obtain
% additional parameters that can be used for troubleshooting
%function [a,aO,sse,residual,exitflag,output,jacobian]=...
%
in particular, the jacobian output is used to obtain the
%
the jacobian sparcity matrix (jsp) by running LSQDHTP once with
% the data with the appropriate inputs. Then, jsp can be found
% using
jsp=(jacobian~=0) to find the non-zero elements in
% the jacobian matrix. Usually jsp is calculated once and saved
% in a file for later use.
^
• k ' k ' k ' i e ’k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' j r ' k ' k ' k ' k ' k ' k ’k ’k ' k ' k ' k ' k ' k ' k ' j r ' k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' j e ' k i e - j e ' k ' k - k - k ' k ' k j r i e ' J c
%
Assemble the matrix aO with the initial parameter estimates.
T=T';% Transpose the T vector to make it vertical.
ntemp=size(T, 1);% ntemp is the number of temperatures.
%disp(['nargin=' num2str(nargin)])
% Nargin is a MatLab reserved variable that is set to the
% number of input arguments used when a function is called.
% In this case, if nargin=3, then the aO matrix was not
% specified, and it has to be calculated. If 5 or more inputs
% were supplied but the aO matrix was left empty, then an esti% mate still must be calculated for it.
%disp(['nargin= ' num2str(nargin)])
if (nargin==3)|((nargin>=5)&isempty(aO))
%disp('Generating aO matrix')
aO=zeros(ntemp,6); % Initialize aO.
%
%
The first and second columns of "aO" (esl-es2 and taul) were
chosen in the range of Mashimo's (1987) results,
if nargin~=6
a0(:,l)=30; % From Mashimo, 1987
a0(:,2)=1.6; % Around 100 MHz from Mashimo, 1987
% 1/(2*pi*100e6) = 1.6e-9
elseif nargin==6 % If all 6 inputs are supplied, the values
a O (:,l)=init(1); % of the first two columns of aO are
aO (:,2)=init(2); % taken from init.
end
%
%
%
%
The inital estimates for es2 and tau2 are obtained by
using linear regression to fit a simple straight line
through the data. The interpolation formula is:
e 1 = es2 - tau2 * w * e"
%
%
%
For some data (particularly for experiments with NaCl)
the measurements at low frequency are very "noisy."
The variable "drop" can be used to skip a number of data points.
drop=l;
%
%
drop=l skips no data
To assemble some of the matrices in this function, a vector
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
%
is multiplied by a matrix of ones as follows:
%
%
%
fl
f2
%
. . .
%
fn
X[ 1 1 1 . . . ]
=
f1 fl fl f1 ...
f2 f2 f2 f2 ...
.................................
fn fn fn fn ...
%
%
%
%
In the following line, this approach is used to form the
matrix with the product w*e" (w=2*pi*f, and e" is in the
second page of the e matrix).
well=2*pi*(f*ones(1,ntemp)).*e(:,:,2);
%
%
%
%
%
%
%
%
%
%
%
The LSQCURVEFIT function solves least square problems, given
a function.
In this case, the inline function y = b(l)-b(2)
is used, where b(l) andb(2) are the intercept and the slope
respectively. For this case, b(l) corresponds to es2 (a3),
b(2) corresponds totau2 (a5), the
"x" data is the product w*
and the "y" data is e'.
The following lines loop through the temperatures and
return the estimates for the 3rd and 5th columns of "aO"
(es2 and tau2). The parameters are "un-scaled" to make the
function work. The lower and upper bounds are set by the
elements [0 0] and [], respectively.
%
First specify some options for LSQCURVEFIT
options=optimset('Disp1,'off');
%
Loop through "ntemp" temperatures;
for n=l:ntemp
aO(n,[3 5])=...
LSQCURVEFIT(inline('(b(1).*10)-b(2).*le-12.*x',...
'b','x'),[1 .1],well(drop:150,n)
e(drop:150,n,1),[0 0],[],options);
end
%
%
The initial estimate for the fourth column of "aO"
(einf2) is set to the theoretical value for water.
aO(:,4)=real(th2o(T,lel00));
%a0(:,4)=real(th2o(T,lelOO))*0.9;% for sens, analysis
%
%
%
Uncomment this line to set the fifth column (tau2) to the
theoretical value for water, rather than calculating it
from linear regression.
%a0(:,5)=10*tauw(T)
%
%
%
%
%
The first row of the sixth column of aO contains the
equivalent NaCl concentration. The other rows of that
column are set to "NaN" (Not a Number) so that no
parameter is calculated. The initial estimate for this
parameter is an arbitrary, empirically chosen value.
aO(:,6)=NaN;
aO(1,6)=5;
% a O (1,6)=0.5; % For salt experiments
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
355
end;
%
end for "if (nargin==3)|((nargin>=5)&isempty(aO))"
%display(aO)
^
************************************************************
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
The following lines specify the lower and upper bounds
(lb and ub matrices) for the parameter estimates.
The size of lb and ub correspond to the size of "a."
When iterative methods are used, the final result is usually
strongly dependent on the bounds imposed in the problem.
In the following lines there may be more than one kind of
bound for the same parameter. The bound that is in use for
a particular parameter is the only one that should NOT be
commented. All the other bounds were left in the program
as examples, but were commented to disable them.
The unusual range
0.1818 * n < n < 1.8182 * n
was calculated so that there is one order of magnitude
between lb (0.1818 * n) and ub (1.8182 * n) and the
range is centered in n, since (0.1818+1.8182)/2 = 1.
The number "n" can be chosen to be the initial estimate,
or the value of the parameter for water, for example.
% Initalize the lower bound matrix (lb).
lb=zeros(ntemp,6);
% Lower bound for esl-es2
lb (:, 1) =1;
% Lower bound for taul
lb(:,2)=0.53; % tau corresponding to 0.3 GHz,
% the lowest frequency in the exps1.
%
lb(:,2)=0; % For salt experiments
% Lower bound for es2
lb(:,3)=a0(;,3).*0.1818;
% Lower bound for einf 2
lb (:, 4) = (real(th2o(T,lelOO))/l) .*0.999;
% this lb was set very close to water
%lb(:,4)=a0(:,4).*0.1818;
%lb(:,4)=(real(th2o(T,lelOO))/l).*0.9*0.999; % for sensitivity analysis
% Lower bound for tau2
l b (:,5)=a0(:,5).*0.1818;
%lb(:,5)=10*tauw(T)*0.1818;
%
%
%
%
% tauw(T) is tau for water
Lower bound for equivalent salt concentration.
The number
realmin is the smallest positive floating pointnumber; its
value
is 2.225 x 10A (--16) . Realmin was chosen because it is a very
small number, but not zero.
%lb(1,6)=realmin;%4/9/2005
l b (1,6)=realmin;%4/10/2005
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
356
%lb(1,6)=2000;%4/9/2005 for Salt runs
%lb {1, 6) =2 .9;
lb(2:ntemp,6)=NaN;
%
Initialize the upper bound matrix (ub).
ub=ones(ntemp,6);
%
Upper bound for esl-es2
% #3 1/11/05 ub(:,l)=a0(:,1)*10;
%ub(:,1)=100;
% #1
1/11/05 u b (:,1)=realmin+eps;
% eps is the smallest "delta" value one can use = 2.2204 x 10A (-16).
%ub(:,1)=10;
% #4
1/15/05 u b (:,1)=inf;
% #2
1/11/05 u b (:,1)=50;
u b (:,1)=50;
%ub(:,1)=100; % For salt experiments
%
Upper bound for taul
%
#2 1/11/05
u b (;,2)=10;
%
#3 1/11/05
u b (:,2)=a0(:,2)*10;
%
#1 1/16/05
ub(:,2)=inf;
%
#1 1/11/05
ub(:,2)=realmin;
%
#1 4/10/05
u b (:,2)=a0(:,2)*10;
%
#1 4/23/05
u b (:,2)=20;
u b (:,2)=a0(:,2)*10;
%ub(:,2)=20; % For salt experiments
%
Upper bound for es2
u b (:,3)=a0(:,3).*1.8182;
%ub(:,3)=real(th2o(T,1));
%
Upper bound for einf2
ub(:,4)=(real(th2o(T,lelOO))/l).*1.001;
%
this ub was set very close to water
%ub(:,4)=a0(:,4).*1.8182;
%ub(:,4)=(real(th2o(T,lelOO))/l).*0.9*1.001; % for sensitivity analysis
%
Upper bound for tau2
u b (:,5)=a0(:,5).*1.8182;
%ub(:,5)=10*tauw(T)*1.8182; % For salt experiments
%
Upper bound for equivalent salt concentration
u b (2:ntemp,6)=NaN;
% #1 1/15/05 ub(1,6)=inf;
% #1 1/11/05 ub(1,6)=realmin+eps;
%ub(1,6)=le-l;
% #1 1/16/05 u b (1,6)=1;
%ub(1,6)=inf;4/9/2005
% #1 4/23/05 ub(1,6)=3000;
u b (1,6)=inf;
%ub(l,6)=30; % For salt experiments
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
%disp(lb)
%disp(ub)
^
%
%
%
•ie-k'k'k'k'k-ic'kjt'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'Jc'k'k'k'k'k'k'k-k-k'k-k'k'Je
This section calls LSQCURVEFIT with the function
SDHTP (listed later in this appendix) and
the matrices aO, f, e, lb, ub, and T.
%
First specify some options for LSQCURVEFIT.
%options=optimset('MaxFunEvals', 8000, 'Disp','iter');
% #1 1/15/05 if nargin==5; % If the Jacobian sparcity matrix was
specified
% #2 1/15/05 if nargin>=5; % If the Jacobian sparcity matrix was
specified
if nargin>=5; % If the Jacobian sparcity matrix was specified
options=optimset('MaxFunEvals',8000,'Disp1,'off','tolfun',.01,.
'DerivativeCheck','off', 'Diagnostics','off', 'Jacobian','off',..
'JacobPattern',jsp);
%disp('using jacobian')
else
options=optimset('MaxFunEvals',8000,'Disp','off','tolfun',.01,.
'DerivativeCheck', 'off', 'Diagnostics','off', 'Jacobian','off');
end
%options
% Now call the actual function.
% The next commented line calls LSQCURVEFIT with extra output
%
parameters that can be used for troubleshooting.
% [a sse residual exitflag output lambda jacobian]=...
[a sse] = LSQCURVEFIT(0SDHTP, aO,f,e,lb,ub,options,T);
^
'k'k'k'k'k-k'k'k'ie'k'k'k'k'ic'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'ie'k'k'k'kic'k'k'k'kie'k'k'k'k'k'k'k'k'k'k'if'k'k'k'k'k'k'k
%
This section displays and plots the results.
%display(a)
%disp(['sse = ' num2str(sse)]);
hold on
plot(T,a)
legend('\epsilon_{si}','\tau_l',...
'\epsilon_{s2}','\epsilon_\infty_2','\tau_2','NaCl Cone.',-1)
xlabel('Temperature')
ylabel('\epsilon_s, \epsilon_\infty, \tau, or NaCl Cone.')
%axis([30 80 0 20])
%axis([30 80])
drawnow;
Tcolor=jet(ntemp);
graph=0;
if graph==l; % if graph = 1, the program plots the graph,
figure
efit=SDHTP(a,f,T);
hold on
for temp=l:ntemp
plot(f,efit(:,temp,1),'-','color',Tcolor(temp, :))
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
358
end
legend('30AoC', '35 ', '40 ', '45 ', '50', '55', '60', '65', '70', ...
'75', '80' ,-1)
for temp=l:ntemp
plot(f,e (:,temp,1),':1,1color',Tcolor(temp,:))
end
xlabel('frequency')
ylabel('\epsilon\prime')
set(gca,'XScale', 'log')
axis([3e8 3e9 60 76])
figure
hold on
for temp=l:ntemp
plot(f,efit(:,temp,2),'-','color',Tcolor(temp,:))
end
legend('30AoC', '35', '40', '45 ', '50', '55', '60', '65', '70', ...
'75','801,-1)
for temp=l:ntemp
plot(f,e (:, temp,2
'
color',...
Tcolor(temp,:))
end
xlabel('frequency')
ylabel('\epsilon\prime\prime')
set(gca,'XScale','log')
axis([3e8 3e9 0 12])
figure;plot(efit(:,:,1),residual(:,:,1))
xlabel('\epsilon\prime_{fitted}')
ylabel('Residual')
figure;plot(efit(:,:,2),residual(:,:,2))
xlabel('\epsilon\prime\prime_{fitted}')
ylabel('Residual')
end
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
359
D.7 “tauw”
Function
x = tauw(T)
Create vectors w ith values from literature:
T in Celcius and t in lx lO 'u s
This function
determines the value
o f t fo r water at a
given temperature.
Ti = [30,40, 50, 60, 75]
Tj = [0.72, 0.58, 0.48, 0.39, 0.32]
Interpolate to find x at T using t j and Tj
end
Figure D.7 Flowchart for “tauw” function
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
360
%
Tauw.m
function t=tauw(T)
% This function determines the value of tau for water at a
% given temperature. It interpolates the tau data given
% by Hasted(1972b)in(table Va,p. 277 of)chapter 7 of the book
% Water:
A Comprehensive Treatise, Volume 1: The Physics and
% Physical Chemistry of Water, edited by Felix Franks and
%
%
%
The interpolation is performed using Matlab's Piecewise
Cubic Hermite Interpolating Polynomial (pchip).
%
%
%
Magnitude and units of taui and t:
Usage:
tauw(T), where T is in C
tau, *le-ll seconds
Ti = [30 40 50 60 75];
taui = [0.72 0.58 0.48 0.39 0.32];
t=interpl(Ti,taui,T,1pchip','extrap');
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
361
D.8 “SDHTP”
Function
SDHTPfa.f.Tf
This function restores the parameters
in the a m atrix to their original orders
o f magnitude, then inputs them to the
“ DHTP” function to calculate the
complex perm ittivity.
e* = DHTP(unscaled parameters, f, T)
The m atrix e* contains the
complex perm ittivity data
organized in one column for
every temperature, and one row
for each frequency.
e' = real(e*)
The program separates the
complex perm ittivity data
into one page for s ' and a
second page fo re ".
e" = -imag(E*)
Figure D.8 Flowchart for the “SDHTP” function
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
362
%
%
%
%
%
%
%
%
%
%
SDHTP.m
Revised on 4/23/05
This function is to be used as an argument in LSQCURVEFIT.
It "un-scales" the parameters in the "a" matrix,
returning them to their real order of magnitude.
The un-scaled parameters in the "a" matrix,
the frequency, and the temperature are then used as inputs
to the DHTP function which calculates the complex
permittivity using the Debye-Hasted model
with two relaxation peaks.
Revised 1/9/05
%
% The matrix "a" contains one row per test temperature, and
% 6 columns with different parameters as follows:
% a (:,1)/I
= esl-es2
% a (:,2)*le-9
= taul
% a (:,3)*10
= es2
% a (:,4)
= einf2
% a (:,5)*le-12 = tau2
% a(:,6)*le-3
= equivalent salt concentration % (w/w)
%
a (:,6)*le-l =
"
"
"
"
for salt
experiments
a
~6
%
%
%
%
%
%
%
%
%
%
%
%
%
The vector "f" contains the test frequencies (Hz)
The vector "T" contains the test temperature (C)
The parameters in the "a" matrix are restored to
the original values and used by the DHTP function
to calculate the complex permittivity values.
The DHTP function is used to calculate "et", an intermediate
variable that contains the dielectric data in complex
number format.
The output "e" contains the dielectric data organized in
one column for every temperature, one row for each
frequency, one page for e', and a second page for e".
Usage is:
e=SDHTP(a,f,T)
function e=SDHTP(a,f,T)
%#1 4/11/05 et=DHTP([a(:,1)/I a(:,2).*le-9 a(:,3).*10 ...
%#1 4/11/05
a (:,4) a (:,5).*le-12 a (:,6)*le-3],f,T);
%#1 4/23/05 et=DHTP([a(:,l)/l a(:,2).*le-9 a(:,3).*10 ...
%#1 4/23/05
a (:,4) a (:,5).*le-12 a (:,6)*le-l],f,T);
et=DHTP([a (:,1)/I a(:,2).*le-9 a(:,3).*10 ...
a (:,4) a (:,5).*le-12 a (:,6)*le-3],f,T);
e (:, :, l)=real(et);
e (:, :, 2)=-imag(et);
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
363
D.9 “DHTP’
This function takes the a m atrix o f
parameters in their original order
o f magnitude, the frequency and
the temperature to calculate the
relative complex perm ittivity
using the Debye-Hasted model
w ith two relaxation peaks.
Function
s* = D H TP (a,f,T),
M olar is the value o f the molar
concentration. It is calculated from the
equivalent salt concentration in weight
percent (Cequiv) by means o f the the
molecular weight o f NaCl and the density.
The density at the given temperature and
equivalent salt concentration is determined
by the function “ SaltDens.”
Calculate
molar = 10*Cequiv* [SaltDens(T,C,equiv.,)]/58.4427
The matrix for each parameter
has one row for each
frequency and one column for
each temperature.
A pply the correction factor
from the Debye-Hasted model.
£*2 = 852 - 11 *m olar
Arrange the data for each
parameter in its own matrix
£sl _ £s2
S sl
£ s2
Xi
Es2
£cc2
t2
The conductivity function
determines the conductivity at the
given temperature and molar
concentration.
cond = conductivity (T, molar)
a = 2nf
Define the perm ittivity o f free
space.
Eo = 8.854187817x10"
Calculate the relative complex
perm ittivity
e* = ( f e i - es2)/(l + itox,)) + emf2 + ((es2-
+ io)T2)) - (i*cond)/(Soto)
( e"d )
Figure D.9 Flowchart for “DHTP” function
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
364
DHTP.m
Calculates the relative complex permittivity using the
Debye-Hasted model with two relaxation peaks.
NOTE: This function takes parameters in their original
order of magnitude. The output matrix has the complex
permittivity in complex number format, with one column
per temperature and one row per frequency.
The matrix "a" contains one row per test temperature, and
6 columns with different parameters as follows:
a (:,1)
a(:,2)
a (:,3)
a (:,4)
a(:,5)
a (1,6)
=
=
=
=
=
=
esl - es2
taul (seconds)
es2
einf2
tau2 (seconds)
equivalent Salt concentration % (w/w)
Usage is:
e=DHTP(a,f,T), f in Hz, and T in Celcius
function e=DHTP(a,f,T)
%
%
%
%
%
%
The Debye-Hasted model has a correction factor for es
equal to 2 * delta * molar concentration (Hasted et al.1948).
Delta (the average hydration number) is -5.5 for NaCl, so the
factor is 2 * (-5.5) * molar concentration = -11 * molar
concentration. The molar concentration is calculated
from % (w/w) using:
%
%
molar = 10*%(w/w)*density/58.44277
%
%
%
%
The density in this conversion is a function of temperature
and concentration. The function saltdens (also listed in
this appendix) performs this calculation.
molar=(10*a(1,6)*saltdens(T,a(l,6),3)/58.44277);
a (:,3)= a (:,3)-ll*molar;
%
%
%
%
In order to assemble the e matrix with the right dimensions,
each of the parameter columns is transposed and multiplied
by a vertical vector of ones with the same length as the
frequency vector. An example of this operation follows:
%
a (1,1)
%
a (2,1)
temps,1)]
Transpose
>
[a(1,1) a (2,1) ... a(# of
%
%
a (# of temps,1)
1
...
X [a(1,1) a(2,l) ... a(# of temps,!)]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
365
%
1
%
%
%
%
%
%
a{l,l) a(2,l)
a (1,1) a(2,l)
. . .
. . .
a (1,1) a(2,l)
... a(# of temps,1)
... a(# of temps,1)
...
... a(# of temps,1)
%
%
%
%
%
%
This process is used for each parameter column, in such a
way that each of them corresponds to the format with
one column per temperature and one row per frequency. The
calculation assembles six of such matrices (one for
each parameter).
s=ones(size(f));
eslmes2=s*a(:,1)';
taul=s*a(:,2)';
es2=s*a (:,3)';
einf2=s*a(:,4)';
tau2=s*a(:,5)';
%
%
%
% form the vector of ones
The conductivity term is a function of concentration and
temperature. The function conductivity, also listed in
this appendix, performs this calculation.
%cond = s*conductivity(T',a (1,6));
cond = s*conductivity(T1,molar')';
%cond = s*conductivityl(T',a (1,6));
%
Calculate the angular frequency and define the permittivity
% of free space.
w=2*pi*f*ones(size(a(:,1)'));
e0=8.854187817e-12; % Hayt and Buck, 2001, p. 536
% Calculate the complex permittivity
e = ((eslmes2)./(l+i.*w.*taul))+...
einf2+((es2-einf2)./(1+i.*w.*tau2))-(i*cond)./(e0*w);
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
366
D.10 “SaltDens’
This function determines the density o f
NaCl solutions (g/m l) using regression
equations based on concentration and
temperature. It takes the temperature,
concentration, and units o f concentration as
arguments.
Function
d = SaltDens(T,C,u),
T > 100
or T <10
Display
error
message
u stands for the units in which
concentration is expressed:
1 = m olarity,
2 = fo r m olality,
3 = weight percent
I f u = 3, concentration is first converted
to m olality.
C >2 or
I
i rr
\
Display
error
message
“>
u=3
else
ir
d = NaN
Display
error
message.
tr
C=
1000C/((100C)*58.44277)
u=2
ir
Define parameters
used in regression
equation:
c0, cTh cT2, cM , cM i ,
cM 2, cT3M 3, eT2, eT3,
eM ], eM2,3 M 3
u= 1
tr
Define parameters
used in regression
equation:
c0, cTi, cT2, cM , cM ,,
cM 2, cT3M 3, eT2, eT3,
eM i, eM2, 3M 3
u=2
d - c 0 + cT iT +
cT2(TA eT2) + cMC +
cM ^C^eMO +
cM 2(CA eM2) +
cT3M 3(T a
eT3)(CAeM3)
o
( end)
d —c0 + cT ]T +
eT2) + cMC
+ c M ^ C ^ M j) +
cM 2(C a eM2) +
cT3M 3(T A eT3)(CA
eM3)
cT2(T a
r
i r ________
Figure D.10 Flowchart for “SaltDens” function
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
r
SaltDens.m
function d=SaltDens(T,C,u)
% Calculates the density of NaCl solutions (g/ml)
% at different temperatures and concentrations
% Usage is
% SaltDens(T,C,u)
% where T is temperature in Celcius: 10 <T <100;
% C is concentration, and u indicates conc. units:
% u=l for molar, 2 for molal, 3 for % (w/w)
% Molarity is mol NaCl/liter of solution.
% Molality is mol NaCl/kg of water,
if T<10 | T>100
error('Error, T is outside of limits. Use 10<=T<=100.
end
%
if C<0 | C>2
%
error('Error, C is outside of limits. Use 0<C<=2')
%
end
if u==3
% If the concentration is in %, convert it into molality and
% use the molality equation, making u=2
C=1000*C./((100-C).*58.44277);
u=2;
end
if u == 1
% Density as a function of T and molarity
% rA2=0.999936424 (7/7/02)
cO = 1.0009578467441;
cTl = 1.81316651755606E-04;
cT2 = -9.59059833539708E-05;
cM = 3.47432044944228E-02;
cMl = -1.57107133413689E-04;
cM2 = 37.2486102768913;
CT3M3 = -37.2405548667706;
eT2 = 1.39654751419053;
eT3 = 2.14724572664865E-05;
eMl = 2.33422063027813;
eM2 = 0.629837566750513;
eM3 = 0.629809428001669;
d = cO + cTl .* T + cT2 .* (T ,A eT2) + cM .* C.
+ cMl .* (C .A eMl) + cM2 .* (C .A eM2) +...
CT3M3 .* (T .A eT3) .* (C .A eM3);
elseif u == 2
% Density as a function of T and molality
% rA2=0.999982391 (7/7/02)
cO = 0.998884347879093;
cTl = 0.00082169701378;
cT2 = -4.38102149096671E-04;
cM = 0.03635911198767;
cMl = 12.5873256587741;
cM2 = -1. 4489042681151E-03;
CT3M3 = -12.5779981617408;
eT2 = 1.2223644625702;
eT3 = 1.24470744280831E-04;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
368
eMl
eM2
eM3
d =
= 0.73955881538566;
= 1.85437330468906;
= 0.73952679659455;
cO + cTl .* T + cT2 .* (T .A eT2) + cM .* C ...
+ cMl .* (C .A eMl) + cM2 .* (C .A eM2) +...
CT3M3 * (T .A eT3) .* (C .A eM3);
else
d = NaN;
error('Error. Third parameter is 1 (molar), 2(molal),or 3(%w/w)')
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%c0 = 1.00454361181696
%ct = -2.3038 9150328251E-04
%cm = 3.98809232058226E-02
%ct2 = -2.24841233790767E-06
%cm2 = -5.45574432790848E-04
%ct2m2 = -7.41438681020142E-09
%c=cO+ct.*T+cm.*C+ct2.*(T.A2)+cm2.*(C.A2)...
%
+ct2m2.*(T.A 2) .*(C .A2);
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
369
D .ll “conductivity”
The function takes the
temperature in Celcius and
the concentration in
m olarity as arguments.
Function
c =conductivity(T,C)
Data on conductivity o f
water and sodium chloride
at certain concentrations
and temperatures was taken
from Bevilacqua, Light, and
Maughan, 2004; Light.
(1984); and Washburn,
(1929). These values were
interpolated to f ill in the
missing values needed to
create a complete set o f
conductivity data that was
then arranged in 3matrices.
Initialize the molar concentration m atrix
Cm= [concentration values listed in and interpolated from the literature]
Initialize the temperature matrix
Tm= [20,25, 30,35,40,45, 50, 55, 60,65, 70, 75, 80, 85, 90, 95, 100]
Initialize the conductivity matrix
km= [conductivity values listed in and interpolated from the literature corresponding to Cmand Tm]
The data was interpolated
using M atlab’ s PCHIP function
to calculate the conductivity
corresponding to the input
temperature and
concentration.
Interpolate the data in
matrices Cm, Tm, and km
to find the conductivity at
T and C
end
Figure D .ll Flowchart for “conductivity” function
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
370
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
conductivity
This function is used to determine conductivity values to be used
in calculations of complex permittivity. It uses data from 3
different sources that had been interpolated to provide a
complete set of conductivity values at certain temperatures and
molar concentrations. The information on the zero concentration
conductivities was taken from Bevilacqua,Light, and Maughan.
2004. Ultrapure Water. 21(2):17-22. The information on the
conductivities in the ppb range was taken from Light. 1984.
Analytical Chemistry. 56(7):1138-1142. The information on the
conductivities at sodium chloride concentrations of 0.5, 1, 2,
5, 10, 20, 50, 70, 100, 200, 500, 700, and 1000 mmol/liter was
from the Critical Tables [Washburn, E. W., (Ed.) 1929.
Electrical conductivity of aqueous ionic solutions. In
International Critical Tables of Numerical Data, Physics,
Chemistry and Technology, Vol. 6, 1st edition, pp. 229-233.
National Research Council, USA. McGraw-Hill Book Company,
Inc. New York.]
%
% Because the literature was taken from three sources, some of
% which used certain concentrations and temperatures, whereas
% others did not, the data was not in the format that could be
% interpolated for all temps and concentrations by the computer.
% Therefore, the data was interpolated to "fill in the gaps."
% Then it was put in 3 separate matrices: one for molar
concentration,
% one for temperature, and one for conductivity. Once in these
complete
% matrices,the data could be interpolated by MatLab's "interp2"
function,
% using cubic interpolation (pchip).
%
%
%
%
%
Cm = molar concentration; concentration matrix
Tm = temperature matrix
km = conductivity in units of S/cm
All matrices have 18 columns and 17 rows
%
%
%
Usage:
c=conductivity(T,C)
%
%
With c in S/m (converted from S/cm), T in Celcius, and C in molarity
CM
1
d)
CM
—
CM
1
CD
\ 1
00
1
Q)
O
7. Oe-2
8.4e-7
7.0e-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
l . l e - 6
1.0e-l
1.7e-5
2.0e-l
1.7e-5
2.0e-l
1.7e-5
2.0e-l
1.7e-5
2.0e-l
1.7e-5
2.0e-l
CJ1
1.7e-6
1.0e-l
1.7e-6
1.0e-l
1.7e-6
1.0e-l
1.7e-6
1.0e-l
le-3
5e-l;.. .
5.0e-4 le-3
5e-l;..
5.0e-4 le-3
5e-l;..
5.0e-4 le-3
5e-l;..
5.0e-4 le-3
5e-l;..
0
<D
1
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
<D
1
Cm=[...
0 1.70e-8 3.3e-8
5..Oe-3 le-2 2e-2
0 1.70e-8 3.3e-8
5.,Oe-3 le-2 2e-2
0 1.7 Oe-8 3.3e-8
5..Oe-3 le-2 2e-2
0 1.70e-8 3.3e-8
5.,0e-3 le-2 2e-2
0 1.70e-8 3.3e-8
5.
CO
function c=conductivity(T, C)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2e-3
2e-3
2e-3
2e-3
2e-3
371
7. Oe-2
8.4e-7
7. Oe-2
8 .4e-7
7.Oe-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
8.4e-7
7.Oe-2
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-6
1. 0e-l
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
1. 7e-5
2. Oe-1
5. Oe-4
5e-1 .
5. Oe-4
5e-1 J ,
5. Oe-4
5e-1 ,
5. Oe-4
5e-1 ,
5. Oe-4
5e-1
5. Oe-4
5e-1
5. Oe-4
5e-1
5. Oe-4
5e-1
5. Oe-4
5e-1
5. Oe-4
5e-1 r •
5. Oe-4
5e-1 r •
5. Oe-4
5e -1] ;
20
20
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20
25
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90
90
90
90
CD
1
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.0e-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
1.7e-7
5.Oe-2
00
0 1.70e-8 3.3e-8
5.Oe-3 le-2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le-2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le- 2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le- 2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le- 2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le- 2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le- 2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le- 2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le- 2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le- 2 2e-2
0 1.70e -8 3.3e-8
5.Oe-3 le- 2 2e-2
0 1.70e-8 3.3e-8
5.Oe-3 le- 2 2e-2
Tm=[...
20 20 20 20 20
20; ...
25 25 25 25 25
25; ...
30 30 30 30 30
30; ...
35 35 35 35 35
35; ...
40 40 40 40 40
40;...
45 45 45 45 45
45; ...
50 50 50 50 50
50;...
55 55 55 55 55
55; ...
60 60 60 60 60
60; ...
65 65 65 65 65
65; ...
70 70 70 70 70
70;...
75 75 75 75 75
75;. ..
80 80 80 80 80
80; ...
85 85 85 85 85
85; ...
90 90 90 90 90
90;. ..
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
le- 3
2e-3
f
}
f
f
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
372
95 95
95; ...
100
100
100
100
95
95
100
100
95
95
100 100
100 ];
95
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95
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95
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95
100
95
100
km=[ . . .
4.193e-8 4.3807373663948952e-8 4.557 55204 47666829e-8
0715406831698582e-8 1 ,3476052623013980e-7 2. 2981162960945468e-7
9223390251845340e-6 5 .6000295552202428e-5 1. 1125756964894940e-4
2052519076397922e-4 5 .4191991048471125e-4 1. 0645172969825664e-3
0803862542669413e-3 4 .9947858956719375e-3 6. 8707911598051025e-3
6002798566083587e-3 1 ,8294998342680162e-2 4. 2144129603084682e-2;
501e-8 5.6811008338047899e-8 5.9020595859404151e-8
6260480588396522e-8 1.6121943784590484e-7 2.7024492220931399e-7
2097560893990845e-6 6.2500000000000001e-5 1.2412000000000000e-4
2.4605999999999999e-4 6.0439999999999995e-4 1.18 62999999999999e-3
2.317000000000000Oe-3 5.5539999999999999e-3 7.6306999999999998e-3
1.0666000000000000e-2 2.0310000000000002e-2 4.6655000000000002e-2;
7.098e-8 7.3301661015647880e-8 7.5493985106173184e-8
9.4265914248033148e-8 1.8607421168071078e-7 3.0392735021072736e-7
2.4024024313409584e-6 6.9376299968813794e-5 1.37 66656196498528e-4
2.7294668430964989e-4 6.7027453687275242e-4 1.3140005202828995e-3
2.5625627396901759e-3 6.12877 64816125691e-3 8.405159439870068Oe-3
1.1759399496034183e-2 2.2379050854513530e-2 5.1261650338960636e-2;
9. 019e-8 9.2735039404086397e-8 9.5147946590964738e-8
1 . 1580846439521619e-7 2. 1684895329701438e-7
3.4654271578420325e-7
2 . 653884804612927le-6 7. 6331632856456975e-5
1.5162934723451211e-4
3. 0065893930230145e-4 7. 3823100902922767e-4 1.4452234693495694e-3
2 . 8131186567945639e-3
6. 7112770219294729e-3 9.1965943196102053e-3
1. 2866159880245431e-2 2. 4472037129510370e-2
5.5893197253655248e-2;
1.1845004150566645e-7
1. 130e-7 1.1581104783852334e-7
1. 4104651374816238e-7
.5155710258275912e-7 3. 9341209260219351e-7
2 . 918150058 610032 9e-6
,3373706451920105e-5 1. 6593685152154630e-4
3. 2904185214012807e-4
.0786021274932649e-4 1. 5795661582747897e-3
3. 0689932040538648e-3
,3049893214400929e-3 1. 0009269208300010e-2
1. 3993620516439587e-2
,6611497977250445e-2 6. 0650918998869548e-2;...
1 . 398e-7
1.4283082998410837e-7 1.4570037636864862e-7
1 . 7027098769100538e-7
2.9043693590227273e-7 4.4468734327708871e-7
3. 1903157017607905e-6 9.0510228544193742e-5 1.8051757053905368e-4
3. 5794050998530307e-4 8.7875294431295032e-4 1.7166258981333399e-3
3. 3305118342087767e-3 7.9134010806338074e-3 1.0840089759480274e-2
1 . 5149120768422494e-2
2.8819972549993682e-2 6.5636093829389230e-2;...
1 , 709e-7
1.7035797610546626e-7 1.7344786192079046e-7
2 . 0100408383869057e-7
3.3569840101628415e-7 5.0704314602352549e-7
3, 5659024054992245e-6 9.7748906922268472e-5 1.9530000000000000e-4
3. 8719999999999998e-4 9.504 9999999999996e-4 1.8560000000000000e-3
3. 5980000000000001e-3 8.5400000000000007e-3 1.1697000000000001e-2
1 , 634 0000000000000e-2
3.1119999999999998e-2 7.0949999999999999e-2;...
2.066e-7 2.1018153817444939e-7 2.1354152750469738e-7
4231221068339277e-7 3. 8303549774803178e-7 5.6371376420178215e-7
7868850385228534e-6 1. 0509744937513485e-4 2.1023391443546007e-4
1672557747434764e-4 1. 0228708313720710e-3 1.9971800131154700e-3
8703729484635858e-3 9. 1819696036886025e-3 1.2574219835657778e-2
7560870292541171e-2 3. 3499886720906848e-2 7. 6567535319723753e-2; .
474e-7 2.5122634244501652e-7 2.5482549581816800e-7
2.8564 45207 9450213e-7 4.3639811372845008e-7 6.2998231040706783e-7
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
373
4 .0884234471060474e-6 1.1256356369178345e-4 2.2532190923134433e ■4
4 .4653549032983334e-4 1.0959313384645074e-3 2.1399780947997547e •3
4 .146440214856037Oe-3 9.8344571515079152e-3 1.3466596481039452e -2
3.5934112349087360e-2 8.2398653344748635e - 2 ;
1 . 8802003425250242e-2
2 . 935e-7 2.9759310006207076e-7 3.0143290101570273e-7
3. 3431281430182338e-7 4 .9515404765212950e-7 7. 0170800878026645e- 7
4 .3947853300589456e-6 1 ,2015495766120484e-4 2. 4057171138287199e- 4
4 .7664431467566798e-4 1 ,1697169902678327e-3 2. 2844724682465935e- 3
4 .4263686842487297e-3 1 ,049789594743487Oe-2 1, 4374736146193745e- 2
3 ,8424475900905017e-2 8. 8447215950609767e- 2 ;
2 . 0066059111813890e-2
3. 453e-7 3.4964127448105999e-7 3.5372358983439004e-7
3. 8868039409163886e-7 5. 5968799340199843e-7 7. 7931389714531010e- 7
4 .7090522220851989e-6 1.2787933907238959e-4 2. 5599104788526239e- 4
5. 0706662662106257e-4 1.2442632557725707e-3 2. 4307413566497238e- 3
4 .7103252417130415e-3 1.1172719295446405e-2 1. 5299245041169383e- 2
4.097277 6392723280e-2 9. 4717085012842220e- 2 ;
2 . 1348373853406523e-2
4 .035e-7 4.0810954356336340e-7 4.1243102235031510e-7
4. 4943467064327810e-7 6.3068486038607728e-7
.6280904568988600e- 7
.7158764573373471e- 4
5. 0088355190246890e-6 1.3574441571432823e-4
,5788629832028853e- 3
5. 3781700227522786e-4 1.3196056039692454e-3
4 ,9984767723203483e-3 1.1859360499519448e-2
,6240729376015089e- 2
4.358081284 0905626e-2 1.0121212240698109e-l;
2 2655268969970940e-2
4 ,668e-7 4.7170163187483321e-7 4.7626466815121565e-7
5, 1533574841140003e-7 7.0641541212930818e-7 9.5168699540515012e- ■7
5, 3161979999725633e-6 1.4375789537601134e-4 2.8736923192350828e- ■4
5 -6891001774737513e-4 1.3957795038483805e-3 2.7289155710998160e- ■3
5 2909901611420274e-3 1.2558252863630937e-2 1.7199795360779587e- •2
2 3985742568937683e-2 4.6250384261815519e-2 1.0793619000856147e- • l ; .
5 369e-7 5.4217446654077113e-7 5.4699098303897593e-7
5 8823278933046670e-7 7.8993618670289221e-7 1.0488630918130151e- •6
5 6598012547077376e-6 1.5192748584642955e-4 3.0334353344980232e- ■4
1.4728204244 004 997e-3 2.8809773435342543e- ■3
6 0036024914671485e-4
5 5880322932494557e-3 1.3269829691757802e-2 1.8177049205511605e- ■2
2 5340623560865354e-2 4.8983289671816435e-2
1.1489314969311845e- ■ 1 ;.
6.134e-7 6.1888869211044248e-7 6.2396329937577435e-7
.674150209867 6661e-7 8.7992637171740248e-7 1.1527269939910353e- ■6
.0106399471166134e-6 1.6026089491457334e-4 3.1951827730783612e- •4
.3218227258245835e-4 1.5507638346161268e-3 3.0351265236999384e- ■3
.8897700537140097e-3 1.3994524287876980e-2 1.9173097120259861e- ■2
.6720740856312562e-2 5.1781328087271844e-2 1.2208686333618715e- ■ l ; •
.961e-7 7.0205082926391742e-7 7.0738919000142350e-7
.5309924391627127e-7 9.7665353214582466e-7 1. 2636251338430014e- •6
.3730209242339089e-6 1.6876583036943332e-4 3. 3590119049282897e- ■4
.6439066416381655e-4 1.6296452034857858e-3 3. 1914413347906077e- ■3
6 .1963703276070644e-3
1.4732769955965400e-2 2. 0188545315073088e- ■2
2 .8126923365837910e-2
5.4646298524545209e-2 1. 2952119281330263e- ■ l ; •
7.848e-7 7.6924569181371016e-7 7.7526063241562809e-7
8.2871656703751664e-7 1.0814567349508852e-6 1.4053310371930212e- 6
7.1613713272854785e-6 1.7745000000000000e-4 3.5250000000000006e- 4
6.97 00000000000014e-4 1.7095000000000001e-3 3.34 99999999999997e- 3
6.507 9999999999999e-3 1.54 85000000000002e-2 2.1224000000000000e- 2
2.9560000000000003e-2 5. 7580000000000006e-2 1.3719999999999999e- 1 ] ;
.
% The interp2 command expects its arguments to be two-dimensional, or
if not, it expects them
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
374
% to be a row vector and a column vector that it will use to build the
two-dimensional matrix.
% In this case, only a vector was needed, so the command had to be
"fooled" by using two columns of
% repeated values as the arguments ([C' C'] and [T" T'])«
% multiply by 100 to convert from S/cm to S/m
c=(100*interp2(Cm,Tm,km,[C1 C'],[T' T 1],'cubic'));
c—c (:,1);
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
375
D.12 “recursiveresults”
Begin
recursiveresults
Display the value o f SSEConvCrit
Display the value o f MeanConvCrit
Display the value o f convMean
Display the value o f SSEIter
Display the final SSE value for
each o f the 3 experiments
Plot a graph o f
amean vs. Temperature
Graph the results in the a matrix
for each o f the 3 experiments
T itle the graph w ith the values o f
SSEConvCrit and MeanConvCrit
Plotmse
T itle the graph w ith the values o f
mseConvCrit and MeanConvCrit
Plot convMean
T itle the graph w ith the values o f
SSEConvCrit and MeanConvCrit
Figure D.12 Flowchart for “recursiveresults” program
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
%
%
%
Recursiveresults.m
This program plots the results for the sseConvCrit,
MeanConvCrit, convMean, number of iterations, and sse.
disp(['sseConvCrit= 1 num2str(sseConvCrit)])
disp(['MeanConvCrit= ' num2str(MeanConvCrit)])
disp('convMean= ')
disp(convMean)
disp('Number of iterations')
disp(sselter)
disp('minimum sse s')
%disp([sse(sselter(1),1) sse(sselter(2),2) sse(sselter(3),3)
disp([sse(sselter(1),1) sse(sselter(2),2)])
figure;
subplot(3,3,1);plot(T,amean,'k')
hold on
subplot(3,3,1);plot(T,a(:,:,1))
subplot(3,3,1);plot(T,a(:,:,2),'-+')
%subplot(3,3,1);plot(T,a(:,:,3),'-o')
title(['sseConvCrit= ', num2str(sseConvCrit), ' ,
'MeanConvCrit= ' num2str(MeanConvCrit)])
subplot(3,3,5);;plot(sse)
title(['sse for sseConvCrit= ', num2str(sseConvCrit), ' , '
'MeanConvCrit= ' num2str(MeanConvCrit)])
subplot(3,3,9);;plot(convMean)
title(['convMean for sseConvCrit= ', num2str(sseConvCrit),
'MeanConvCrit= ' num2str(MeanConvCrit)])
figure
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
D.13 “saltrecursivehc”
%
saltrecursivehc.m
% Revised 4/9/05 (Salt runs), 3/28/05, 3/26/05, 1/9/05, 9/30/04,
8/8/04, 10/17/03
% This program takes the interpolated experimental data and
% solves for a solution to the "a" matrix of parameters by
% iterating until the mean square error of each experiment
% and the mean of all the experiments have met specified
% convergence criteria and/or until the maximum number of
% permitted iterations has been exceeded. The criterion that
% is checked for convergence is the coefficient of variation,
% the standard deviation divided by the mean.
%
%
%
%
This program calls upon various subfunctions in order
to model the data according to the Debye-Hasted model with
two peaks. In each iterative step, the previous "a" matrix
becomes the new initial estimate ("ao") matrix.
%
%
%
%
%
The following lines load the interpolated data from all the
experiments performed for each starch (there were 4 for
Pencook and 3 for the others) and stores the data in matrix
"e". Matrix "e" has a "book" of data for each experiment,
and each book contains 2 "pages": one for e' and one for e".
% StarchName = 1 is for Amioca
% StarchName
= 2 is Melojel
% StarchName
= 3 is Pencook
% StarchName
= 4 is Amioca with 2% NaCl
% StarchName
= 5 is Amioca with 2% NaCl minus conductivity effects
1.3% NaCl
% StarchName = 6 is generated data for debugging
StarchName=4;
switch StarchName
case 1
e=ehci;
e (:,12:21,:)=ecci;
e (:,1:11,:,2)=ehci;
e (:,12:21,:,2)=ecci;
e (:,1:11,:,3)=ehci;
e (:, 12:21,:,3)=ecci;
case 2
e=ehci;
e (:,12 :21,:)=ecci;
e (:,1:11,:,2)=ehci;
e (:,12:21,:,2)=ecci;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
378
e (:,1:11,:,3)=ehci;
e (:,12:21,:,3)=ecci;
case 3
e=ehci;
e (:,12:21,:)=ecci;
e (:,1:11,:,2)=ehci;
e (:,12:21,:,2)=ecci;
%e(:,1:11,:,3)=ehci;
%e(:,12:21,:,3)=ecci;
%e (:,1:11,:,4)=ehci;
%e(:,12:21,:,4)=ecci;
case 4
e=ehci;
%e(:,12:21,:)=ecci;
e (:,11:20,:)=ecci;
e (:,1:11,:,2)=ehci;
%e(:,12:21,:,2)=ecci;
e (:,11:20,:,2)=ecci;
e (:, 1:11, :,3)=ehci;
%e(:, 12:21, :, 3)=ecci;
e (:, 11:2 0,:,3)=ecci;
%e(:, 11, :) = (e(:, 10,:)+ e (:,21,:))/2;
%e(:,11,:,:)= e (:,10,:,:) + (e(:,10,:,:)—e(:,9,:, :));
case 5
load('\\91q0901\e in nogo\Salt RecursiveXe minus 1_35
NaCl.mat','enocond')
e=enocond;clear enocond
case 6
Thi=[(30:5:80) (30:5:75)];
ag=ones(21,6);
a g (:,1)=[(40:4:80) (39:4:75)]';
%ag(:,1) = (1:1:11) ';
%ag(:,2)=(11:-1:1)'./l;
ag(:,2)=[(1.5:0.05:2) (1.5:0.05:1.95)]';
ag(:,3)=(real(th2o(Thi',lei))/10)-2;
ag(:,4) = (real(th2o(Thi',lelOO)))*0. 95;
ag{:,5)=10*tauw(Thi');
a g (1,6)=14;
f=logspace(loglO(3e8),loglO(3e9),150)';
e (:,:,:,1)=SDHTP(ag,f,Thi');
rand('state', 0)
e (:, :, :, 2)= e (:, :,:,1) + (rand(150,21,2)-0.5)./10;
e (:,:,:,3)= e (:,:,:,1)+(rand(150,21,2)-0.5)./10;
otherwise
error('Unknown starch chosen')
end; % StarchName
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
379
%T=[Thi Tci];
T=[Thi(1:10) Tci];
nTemp=size(T,2);
% doMainLoop is a flag that is manually set to 1 to proceed
% with the main loop, and any other number to skip it.
% When the main loop is skipped, the data is loaded with no
% further calculations.
doMainLoop=l;
%fileRoot='MelojelComb2';
%fileRoot='amiocatestcomb';
%fileRoot='E:\Pencook Recursive\PencookComb3' % 3\26\2005
fileRoot='\\91q0901\e in nogo\Salt Recursive\saltComb5'
for Ccount=l:size(combination, 1);
disp([' desl= ' num2str(combination(Ccount,1))...
' taul= ' num2str(combination(Ccount,2))...
' MeanConvCrit= ' num2str(combination(Ccount,3))...
' sseConvCrit= ' num2str(combination(Ccount,4))] )
MeanConvCrit=combination(Ccount,3); % Mean convergence criterion
sseConvCrit=combination(Ccount, 4); % Residual convergence
criterion
if doMainLoop==l
% Initialize some variables
a=zeros(nTemp,6,size(e,4));
% Result parameter matrix
a (2:nTemp,6,:)=NaN;
a0=zeros(nTemp,6,size(e,4)) ;
% Initial estimate matrix
a O (2:nTemp,6,:)=NaN;
%
sseConvCrit=le-3;
% Residual convergence criterion
sseConv=100*sseConvCrit; % Initial sse convergence variable
%
MeanConvCrit=le-3;
% Mean convergence criterion
sse=0;
convMean(1)=MeanConvCrit*100;
maxlter=300;
% Maximum number of data iterations
sseIter=ones(1, size(e, 4));
% Number of iterations for
each exp.
meanlter=l;
% Number of iterations with mean
% Jacobian sparcity
pattern (heating)
nmaxlter=0;
% Number of combinations reaching
maxlter.
% Calculate the initial estimates for each experiment using
% the LSQDHTP function without specifying the aO matrix.
% ExN is a counter for Experiment Number
for CondLoop=l:2
if CondLoop==l
e0=8.854187817e-12; % Hayt and Buck, 2001, p. 536
w=2*pi*f;
econd=e;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
380
molar=(10*1.35*saltdens(T,1.35,3)/58.4427 7);
cond = conductivity(T,molar);
molar=ones(150,1)*molar;
cond=ones(150,1)*cond';
condloss=cond./((w*ones(1,nTemp))*e0);
e (:, ,1,1)= e (:, ,1,1)+ll*molar;
e (:, /1/2)=e (:, ,1,2)+ll*molar;
e (:, /1» 3)= e (:, ,1,3)+ll*molar;
e (:, ,2,1)= e (:, ,2,1)-condloss;
e (:, ,2,2)=e (:, ,2,2)-condloss;
e {:, ,2,3)=e (:, ,2,3)-condloss;
elseif CondLoop==2
enocond=e;
e=econd;
a(l,6, :)=a (1, 6, :)+13 .5;
end %CondLoop==l
if CondLoop==l
for ExN=l:size(e,4)
%
Use the following two commented lines to obtain
troubleshooting.
%
parameters that can be used for
Note that
%
the corresponding lines in LSQDHTP also have to
be changed.
%
[a(:,:,ExN) aO(:,:,ExN)
sse(sselter(ExN),ExN) residual ...
%
exitflag output] =
LSQDHTP(f,T,e(:,:,:,ExN));
init=[combination(Ccount,1) combination(Ccount,2)];
[a(:,:,ExN) aO(:,:,ExN) sse(sselter(ExN),ExN)] =
saltLSQDHTP(f,T,e(:,:,:,ExN),[],jp,init);
disp(['lst iteration for data ' num2str(ExN) ...
'. sse = ' num2str(sse(sselter(ExN),ExN))
' . '] )
end;% ExN=l:size(e,4)
elseif CondLoop==2
sseConv=100*sseConvCrit;
convMean(meanlter)=1;
end %CondLoop==l
a (2:nTemp,6,:)=NaN;
%a(1,6,:)=8;
%a=a*l.1;
disp('Now starting iteration with means.')
amean=mean(a, 3);
while
(convMean(meanlter)>MeanConvCrit)&(max(sselter(:))<maxlter)
prevMean=amean;
for ExN=l:size(e, 4)
prevSSE=sse(sselter(ExN),ExN);
sselter(ExN)=sselter(ExN)+1;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
%
[a(:,:,ExN) aO(:,:,ExN)
sse(sselter(ExN),ExN) residual exitflag ...
%
output] =
saltLSQDHTP(f,T,e(:, :, :, ExN),amean,jp);
[a(:,:,ExN) aO(:,:,ExN) sse(sselter(ExN),ExN)] =
saltLSQDHTP(f,T,e(:,:,:,ExN),amean,jp);
%
The following internal loop is repeated
numerous times for
each experiment of each starch. It calculates
the a matrix
and sse.
If the sse is greater than the sse
convergence
criterion, the loop repeats again.
The a
matrix from the
previous loop then becomes the ao matrix for
the next
%
set of calculations,
a (2:nTemp,6,:)=NaN;
sseConv=100*sseConvCrit; % re-set sseConv
while sseConv>=sseConvCrit
prevSSE=sse(sselter(ExN),ExN);
sselter(ExN)=sselter(ExN)+1;
%
[a(:,:,ExN) aO(:,:,ExN)
sse(sselter(ExN),ExN) residual ...
%
exitflag
output]=saltLSQDHTP(f,T,e(:,:,:,ExN),...
%
a (:,:,ExN),jp);
[a(:,:,ExN) aO(:,:,ExN) sse(sselter(ExN),ExN)]
saltLSQDHTP(f,T,e(:,:,:,ExN),a(:,:,ExN),jp);
sseConv=std([prevSSE
sse (sselter(ExN),ExN)])/...
mean([prevSSE sse(sselter(ExN),ExN)]);
end; % sseConv>=sseConvCrit
I
disp(['Experiment ' num2str(ExN)...
'. 1 num2str(sselter(ExN)) ' iterations.
'sse = 1 num2str(sse(sselter(ExN),ExN))])
end; % for ExN=l;size(e,4)
meanlter=meanlter+l;
amean=mean(a,3);
% Calculate the st. dev. / mean
convMeanM=amean;
convMeanM(:,:,2)=prevMean;
convMeanM=std(convMeanM,0,3)./mean(convMeanM+eps,3);
convMean(meanlter)= mean(convMeanM(1:5*nTemp+l));
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
%
convMean(meanlter)= (convMean(meanlter1)+mean(convMeanM(1:5*nTemp+l)))/2;
disp(['Convmean = ' num2str(convMean(meanlter))])
end; % convMean>MeanConvCrit
disp([1Iterations: ' num2str(sselter)])
if max(sselter(:))>maxlter
disp('Maximum number of iterations reached. Ending..
nmaxlter=nmaxlter+l;
end % max(sselter(:))>maxlter
close gcf
recursiveresults
end % CondLoop
end % doMainLoop==l
disp(['Final ConvMeans = ' num2str(convMean)])
disp('Final sse = ');
disp(num2str(sse))
%filename=['e :\Amioca Recursive\' fileRoot num2str(Ccount) '.mat
filename=[fileRoot num2str(Ccount) '.mat'];
disp(filename)
combin=combination(Ccount,:);
disp(['Combination was ' num2str(combin)])
save(filename,'e ','a ','combin','sse',...
'convMean','sseConv','nmaxlter')
end
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
383
D.14 “saltLSQDHTP”
%
SaltLSQDHTP.m
%
%
Revised on
1/16/05, 1/12/05, 1/9/05, 7/12/04
%
This function uses the least squares method to find an
estimate of the parameters for the Hasted-Debye model with
two peaks, given the measured dielectric data as a function of
temperature and frequency.
The result is a parameter matrix "a" with a column for each
parameter and a row for each temperature.
Thisfunction
can also return some statistical outputs.
The matrix "a" contains one row per test temperature, and
6 columns with different parameters as listed below.
These parameters, as used by LSQDHTP, are scaled so that
all their values are approximately the same order
of magnitude.
,
a ( 1)/1
a ( ,2)*le 9
a ( , 3)*10
a ( ,4)
a ( ,5)*le-12
a (1,6)*le-3
a (:,6)*le-l
=
=
=
=
esl - es2
taul
es2
einf2
tau2
equivalent NaCl concentration % (w/w)
IV
II
II
II
(for salt experiments)
In order to use these parameters, they have to be
"un-scaled" by an auxiliary function "SDHTP" that
returns them to their real order of magnitude.
The vertical vector "f" contains the test frequencies (Hz).
The horizontal vector "T" contains the test temperatures (C).
The matrix "e" contains the dielectric data organized in
one column for every temperature, one row for each
frequency, one page for e', and a second page for e".
The least squares function used to estimate the parameters
(LSQCURVEFIT) requires an initial estimate of the parameters
(the "aO" matrix). The "aO" matrix can be specified
when LSQDHTP is called, or otherwise the
The size of "aO" corresponds to the size of "a."
c^o
o\°
cAP
cAP
o\°
cA°
oY3 cAP
cAP
cA°
o\P
c^°
cAP
CAP
o\P
c^o
c^P
o\°
o\o
o\°
%
%
%
%
%
%
%
The optional argument "jsp" is the "Jacobian sparcity matrix."
The algorithm used by MatLab for LSQCURVEFIT uses a
very large matrix that contains all posible derivatives
of variables with respect to each other. The Jacobian sparcity
matrix specifies which of these derivatives are non-zero.
In this case, only derivatives with respect to frequency and
temperature are needed. By specifying a Jacobian sparcity
matrix, the execution time for this function is greatly reduced.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
384
%
%
%
Usage is:
[a,aO,mse] = LSQDHTP(f,T,e,aO,jsp,init)
function [a,aO,mse] = saltLSQDHTP(f,T,e,aO,jsp,init); % aO, jsp, and
init
%
are optional. The matrix "init" contains the first two columns
% of the aO matrix (aOl = esl-es2 and ao2 = taul).
% The following commented line is used to obtain
%
additional parameters that can be used for troubleshooting
%function [a,aO,mse,residual,exitflag,output,jacobian]=...
%
Assemble the matrix aO with the initial parameter estimates.
T=T';% Transpose the T vector to make it vertical.
ntemp=size(T,1);% ntemp is the number of temperatures.
%disp(['nargin=' num2str(nargin)])
%
Nargin is a MatLab reserved variable that is set to the
% number of input arguments used when a function is called.
% In this case, if nargin=3, then the aO matrix was not
% specified, and it has to be calculated. If 5 or more inputs
% were supplied but the aO matrix was left empty, then an esti%
mate still must be calculated for it.
%disp(['nargin= ' num2str(nargin)])
if (nargin==3)|((nargin>=5)&isempty(aO))
%disp('Generating aO matrix')
aO=zeros(ntemp,6); % Initialize aO.
%
%
%
No method for estimating the first and second columns
of "aO" (esl-es2 and taul) was found, so an arbitrary,
empirically established, initial estimate is used,
if nargin~=6
a0(:,l)=30; % From Mashimoto, 1987
a0(:,2)=1.6; % Around 100 MHz from Mashimoto, 1987
% 1/(2*pi*100e6) = 1. 6e-9
elseif nargin==6 % If all 6 inputs are supplied, the values
a O (:,1)=init(1); % of the first two columns of aO are
a O (:,2)=init(2); % taken from init.
end
%
%
%
%
The inital estimates for es2 and tau2 are obtained by
using linear regression to fit a simple straight line
through the data. The interpolation formula is:
e' = es2 - tau2 * w * e"
%
%
%
For some data (particularly for experiments with NaCl)
the measurements at low frequency are very "noisy."
The variable "drop" can be used to skip a number of data points.
drop=l;
%
%
%
drop=l skips no data
To assemble some of the matrices in this function, a vector
is multiplied by a matrix of ones as follows:
%
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
385
fl
f2
%
. . .
dP
%
%
X
[111...]
dP
dP
dP
dP
fn
fi
fi fi
fi ...
= f2
f2 f2
f2 ...
.................................
fn
fn fn
fn ...
In the following line, this approach is used to form the
matrix with the product w*e" (w=2*pi*f, and e" is in the
second page of the e matrix).
oY3 d\°
o\P
cAP
well=2*pi*(f*ones(1,ntemp)).*e (:, :,2);
%
%
%
%
%
%
%
The LSQCURVEFIT function solves least square problems, given
a function. In this case, the inline function y = b(l)-b(2)x,
is used, where b(l) and b(2) are the intercept and the slope
respectively. For this case, b(l) corresponds to es2 (a3),
b(2) corresponds to tau2 (a5), the "x" data is the product w*e",
and the "y" data is e '.
The following lines loop through the temperatures and
return the estimates for the 3rd and 5th columns of "aO"
(es2 and tau2). The parameters are "un-scaled"
to make the
function work. The lower and upper bounds areset bythe
elements [0 0] and [], respectively.
%
First specify some options for LSQCURVEFIT
options=optimset('Disp','off');
%
Loop through "ntemp" temperatures;
for n=l:ntemp
aO(n, [3 5])=.. .
LSQCURVEFIT(inline('(b(1).*10)-b(2).*le-12.*x',...
'b','x'),[1 .1],well(drop:150,n)
e(drop:150,n,1),[0 0],[],options);
end
%
%
The initial estimate for the fourth column of "aO"
(einf2) is set to the theoretical value for water.
aO (:,4)=real(th2o(T,lelOO));
%a0(:,4)=real(th2o(T,lelOO))*0.9;% for sens, analysis
%
%
%
Uncomment this line to set the fifth column (tau2) to the
theoretical value for water, rather than calculating it
from linear regression.
%a0 (:,5)=10*tauw(T)
%
%
%
%
%
The first row of the sixth column of aO contains the
equivalent NaCl concentration. The other rows of that
column are set to "NaN" (Not a Number) so that no
parameter is calculated. The initial estimate for this
parameter is an arbitrary, empirically chosen value.
aO (:,6)=NaN;
aO (1,6)=0.5;
end;
%
end for "if (nargin==3)|((nargin>=5)&isempty(aO))"
%display(aO)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
386
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
The following lines specify the lower and upper bounds
(lb and ub matrices) for the parameter estimates.
The size of lb and ub correspond to the size of "a."
When iterative methods are used, the final result is usually
strongly dependent on the bounds imposed in the problem.
In the following lines there may be more than one kind of
bound for the same parameter. The bound that is in use for
a particular parameter is the only one that should NOT be
commented. All the other bounds were left in the program
as examples, but were commented to disable them.
The unusual range
0.1818 * n < n < 1.8182 * n
was calculated so that there is one order of magnitude
between lb (0.1818 * n) and ub (1.8182 * n) and the
range is centered in n, since (0.1818+1.8182)/2 = 1.
The number "n" can be chosen to be the initial estimate,
or the value of the parameter for water, for example.
% Initalize the lower bound matrix (lb).
lb=zeros(ntemp,6);
% The lower bounds for esl-es2 and taul were chosen to be 0, so
% lb(:,1) and lb(:,2) need not be set again,
lb (:, 1)=1;
% #1 4/11/05 l b (:,2)=0.53; % tau corresponding to 0.3 GHz, the lowest
frequency in the exps'.
lb (:, 2)=0;
% Lower bound for es2
lb (:,3)=a0(:,3).*0.1818;
% Lower bound for einf 2
l b (:,4)=(real(th2o(T,lelOO))/l).*0.999; % this lb was set very close to
water
%lb(:,4)=a0(:,4).*0.1818;
%lb(:,4)=(real(th2o(T,lelOO))/l).*0.9*0.999; % for sensitivity analysis
% Lower bound for tau2
% #1 4/10/05 lb(:,5)=a0(:,5) .*0.1818;
lb (:,5)=10*tauw(T)*0.1818;
%
%
%
%
Lower bound for equivalent salt concentration. The number
realmin is the smallest positive floating point number; its value
is 2.225 x 10A (-16). Realmin was chosen because it is a very
small number, but not zero.
%lb(1,6)=realmin;%4/9/2005
lb (1,6)=realmin;%4/10/2005
%lb(1,6)=2000;%4/9/2005 for Salt runs
%lb(1,6)=2.9;
lb (2:ntemp,6)=NaN;
% Initialize the upper bound matrix (ub).
ub=ones(ntemp,6);
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
387
%
Upper bound for esl-es2
% #3 1/11/05 ub(:,l)=a0(:, 1)*10;
%ub(:,1)=100 ;
% #1 1/11/05 ub (:,1)=realmin+eps;
% eps is the smallest "delta" value one can use = 2.2204 x 10*(-16).
%ub(:,1)=10;
% #4 1/15/05 u b (:,1)=inf;
% #2 1/11/05 u b (:,1)=50;
% #1 4/10/05 u b (:,1)=50;
% #1 4/16/05 u b (:,1)=inf;
% #1 4/18/05 u b (:,1)=50;
ub (:,1)=100;
%
Upper bound for taul
%
#2 1/11/05
u b (:,2)=10;
%
#3 1/11/05
u b (:,2)=a0(:,2)*10;
%
#1 1/16/05
u b (:,2)=inf;
%
#1 1/11/05
u b (:,2)=realmin;
%
#1 4/10/05
u b (:,2)=a0(:,2)* 10;
% #2 4/10/05ub(:,2)=20;
u b (:,2)=2 0;
%
Upper bound for es2
u b (:,3)=a0(:,3).*1.8182;
%ub(:,3)=real(th2o(T,1) );
%
Upper bound for einf2
u b (:,4)=(real(th2o( T , lelOO))/l).*1.001;% this ub was set very close to
water
%ub(:,4)=a0(:,4).*1.8182;
%ub(:,4)=(real(th2o( T , lelOO))/l).*0.9*1.001; % for sensitivity analysis
%
Upper bound for tau2
% #1 4/10/05 ub(;,5)=a0(:,5).*1.8182;
u b (:,5)=10*tauw(T)*1.8182;
%
Upper bound for equivalent salt concentration
u b (2:ntemp,6)=NaN;
% #1 1/15/05 u b (1,6)=inf;
% #1 1/11/05 ub(l,6)=realmin+eps;
%ub(1,6)=le-l;
% #1 1/16/05 u b (1,6)=1;
%ub(1,8)=inf;4/9/2005
% #1 4/16/05 u b (1,6)=2;
u b (1,6)=30;
%disp(lb)
%disp(ub)
^
'k'k'k'k'k'k'k-k-k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'ie ie 'k'k'k'k'k'it'k'k'k'k'k'k-k'jt'kie 'k'jc'jc'jc
%
%
This section calls LSQCURVEFIT with the function
SDHTP (listed later in this appendix) and
the matrices aO, f, e, lb, ub, and T.
%
First specify some options for LSQCURVEFIT.
%
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
%options=optimset('MaxFunEvals', 8000, 'Disp','iter');
% #1 1/15/05 if nargin==5; % If the Jacobian sparcity matrix was
specified
% #2 1/15/05 if nargin>=5; % If the Jacobian sparcity matrix was
specified
if nargin>=5; % If the Jacobian sparcity matrix was specified
options=optimset(1MaxFunEvals',8000, 'Disp','off', 'tolfun',.01,.
'DerivativeCheck','off','Diagnostics','off','Jacobian','off1,..
'JacobPattern',jsp) ;
%disp('using jacobian')
else
options=optimset('MaxFunEvals',8000,'Disp','off','tolfun',.01,.
'DerivativeCheck','off','Diagnostics','off','Jacobian','off');
end
%options
%
Now call the actual function.
%
The next commented line calls LSQCURVEFIT with extra output
%
parameters that can be used for troubleshooting.
%[a mse residual exitflag output lambda jacobian]=...
[a mse] = LSQCURVEFIT(0SDHTP,aO,f,e,lb,ub,options,T);
^
' i t i e' k' k' l e' k' k' k' j c' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' k' i e' k' k' k' k' k' k
%
This section displays and plots the results.
%display(a)
%disp(['mse = ' num2str(mse)]);
hold on
plot(T,a)
legend('\epsilon_{si}','\tau_l',...
'\epsilon_{s2}','\epsilon_\infty_2','\tau_2','NaCl Cone.',-1)
xlabel('Temperature')
ylabel('\epsilon_s, \epsilon_\infty, \tau, or NaCl Cone.')
%axis([30 80 0 20])
%axis([30 80])
drawnow;
Tcolor=jet(ntemp);
graph=0;
if graph==l; % if graph = 1, the program plots the graph,
figure
efit=SDHTP(a,f,T);
hold on
for temp=l:ntemp
plot(f,efit(:,temp,1
)
color',Tcolor(temp,:))
end
legend('30AoC','35','4 0', '45 ', '50 ', '55', '60','65', '70 ',. ..
'75','80',-1)
for temp=l:ntemp
plot(f,e (:,temp,1),':','color',Tcolor(temp,:))
end
xlabel('frequency')
ylabel('\epsilon\prime')
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
set(gca, 'XScale','log')
axis([3e8 3e9 60 76])
figure
hold on
for temp=l:ntemp
plot(f,efit(:,temp,2),'-',1color',Tcolor(temp,:))
end
legend('30AoC','35', '40 ', '45', '50', '55 ', '60 ', '65', '70',
'75','80',-1)
for temp=l:ntemp
plot(f,e (:,temp,2
)
'
color1, ...
Tcolor(temp,:))
end
xlabel('frequency')
ylabel('\epsilon\prime\prime1)
set(gca,'XScale','log')
axis([3e8 3e9 0 12])
figure;plot(efit(:,:,1),residual(:,:,1))
xlabel('\epsilon\prime_{fitted}')
ylabel('Residual')
figure;plot(efit(:,:,2),residual(:,:,2))
xlabel('\epsilon\prime\prime_{fitted}')
ylabel('Residual')
end
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
390
APPENDIX E:
RECURSIVE RESULTS FOR WAXY MAIZE STARCH
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
391
Table E.1 Recursive Results for Waxy Maize Starch
Init. Estimate
Final Result OK?
SSE
File set uomo. NO.
1
2
3
11
®s1"£s2
*1
6.1"£.2
1
1
0
0
37.42 20.85 19.57
1
1
1
1
75.89 42.03 30.64
1
2
1
10
1
20
0
1
76.58 42.18 33.84
1
3
1
1
30
0
78.06 42.96 34.05
1
4
1
1
1
40
0
90.10 49.88 39.18
5
1
0
1
1
50
87.49 48.43 37.90
6
1
1
0
48.14 21.26 15.39
7
10
1
1
1
10
10
91.95 52.40 37.32
8
1
20
1
71.64 42.02 32.40
1
9
10
1
1
92.64 54.06 38.75
1
10
30
10
1
1
40
87.64 49.65 36.51
1
11
10
1
1
1
12
10
50
99.26 58.16 40.34
1
48.64 21.35 15.44
1
13
20
1
1
1
14
20
10
125.40 68.85 55.54
20
1
1
89.79 52.14 37.54
1
20
15
1
1
84.75 48.34 34.62
1
20
30
16
1
1
40
72.53
40.37 32.19
1
17
20
1
1
1
20
50
70.76 41.38 32.85
18
1
49.30 21.56 15.48
1
19
30
1
1
71.04 40.25 32.28
1
10
20
30
1
1
1
21
30
20
75.16 42.09 32.79
1
1
1
22
30
30
86.73 49.47 36.12
1
1
40
1
75.88 43.30 33.72
23
30
1
1
1
24
30
50
73.61 41.18 33.13
1
48.37 21.31 15.42
1
40
25
1
1
1
26
40
10
70.83 40.30 32.09
1
1
1
27
40
20
70.80 40.45 32.46
1
1
1
40
30
93.27 53.39 38.54
28
1
1
40
40
1
29
82.53 49.11 35.78
1
1
40
50
1
83.52 47.75 35.01
30
1
1
50
49.70 21.70 15.52
31
1
1
1
32
50
10
66.70 38.18 31.49
1
50
20
1
1
33
92.09 52.55 38.08
1
1
34
50
30
1
80.13 50.09 36.23
1
50
40
1
1
35
77.18 43.84 33.17
1
50
50
1
1
36
87.93 49.40 36.71
2
1
1
25
8
1
62.67 37.56 31.98
2
2
25
11
1
1
67.61 37.78 31.32
2
14
1
1
3
25
77.17 43.28 33.38
2
4
17
1
25
1
80.13 44.43 33.18
2
5
25
20
1
1
74.08 42.25 32.39
2
25
23
1
6
1
79.76 45.77 34.24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mean
25.95
49.52
50.87
51.69
59.72
57.94
28.26
60.56
48.68
61.82
57.93
65.92
28.48
83.26
59.82
55.90
48.36
48.33
28.78
47.86
50.01
57.44
50.96
49.31
28.37
47.74
47.90
61.73
55.81
55.43
28.97
45.46
60.91
55.48
51.39
58.01
44.07
45.57
51.27
52.58
49.58
53.26
392
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
7
8
g
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Table E.1 Recursive Results for Waxy Maize Starch
1
1
30
8
74.63 42.27
11
1
1
76.46 42.04
30
1
14
1
69.80 39.27
30
17
1
1
72.95 40.99
30
1
20
1
75.16 42.09
30
1
1
76.71 46.09
23
30
1
1
74.42 42.37
8
35
1
11
1
83.32 47.86
35
14
1
1
76.81 41.75
35
1
17
1
100.06 57.84
35
1
1
35
20
84.99 48.50
1
23
1
79.68 45.44
35
1
1
66.42 38.17
40
8
1
11
1
70.10 39.47
40
14
1
1
78.09 44.60
40
1
1
17
74.28 42.56
40
1
1
20
70.80 40.45
40
1
1
40
23
79.78 45.73
1
1
73.42 40.36
8
45
1
11
1
66.36 39.05
45
14
1
1
77.45 42.80
45
17
1
1
68.86 38.87
45
20
1
1
68.50 39.06
45
1
1
23
72.13 42.77
45
1
1
50
8
72.70 41.10
1
1
11
73.60 41.10
50
14
1
1
50
77.15 43.88
17
1
1
78.47 43.05
50
1
1
50
20
92.09 52.55
1
1
50
23
67.89 38.71
7
1
1
25
69.36 40.74
1
1
62.67 37.56
25
8
9
1
1
25
77.08 42.88
1
1
72.84 42.44
25
10
11
1
1
25
67.61 37.78
12
1
1
25
69.54 39.16
7
1
1
26
71.28 40.97
1
1
26
8
69.14 39.82
26
9
1
1
64.11 38.44
10
1
1
26
66.02 38.87
26
11
1
1
71.66 39.69
26
12
1
1
77.50 43.45
27
7
1
1
71.45 40.63
27
8
1
1
69.11 39.06
27
9
1
1
65.00 38.37
27
1
10
1
68.99 39.12
32.18
32.75
31.62
32.49
32.79
33.95
32.09
33.82
32.90
41.41
35.21
33.86
31.34
32.08
33.73
32.65
32.46
33.71
32.15
32.15
33.25
32.35
32.11
32.61
31.87
32.38
34.37
33.67
38.08
32.05
49.69
50.41
46.90
48.81
50.01
52.25
49.63
55.00
50.49
66.44
56.23
52.99
45.31
47.22
52.14
49.83
47.90
53.07
48.64
45.85
51.17
46.70
46.56
49.17
48.56
49.03
51.80
51.73
60.91
46.22
31.45
31.98
33.84
32.53
31.32
31.56
31.36
31.23
32.32
31.85
31.76
31.87
31.24
31.18
32.12
32.15
47.18
44.07
51.26
49.27
45.57
46.75
47.87
46.73
44.95
45.58
47.70
50.94
47.77
46.45
45.16
46.75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
393
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
Table E.1 Recursive Results for Waxy Waize Starch
1
1
27
11
74.13 43.27
1
1
27
12
80.63 44.93
7
1
1
28
72.09 41.56
1
1
77.11 43.88
28
8
1
1
77.92 43.02
28
9
1
10
1
72.24 40.80
28
1
1
28
11
85.95 49.95
12
1
1
73.36
28
41.43
1
7
1
65.70 38.88
29
1
1
75.42 42.51
29
8
9
1
1
65.24 39.04
29
10
1
1
65.35 38.48
29
1
29
11
1
69.90 40.22
12
1
1
74.82 41.72
29
7
1
1
65.25
30
38.33
1
8
1
74.63 42.27
30
1
1
30
9
68.91 39.31
1
1
30
10
71.04 40.25
11
1
1
30
76.46 42.04
1
1
12
72.36 41.40
30
7
1
1
68.92 38.93
31
1
1
66.52 38.67
31
8
9
1
1
31
68.19 40.57
1
10
1
70.28 40.23
31
11
1
1
75.22 42.37
31
1
12
1
76.19 42.56
31
7
1
1
70.77 38.66
32
1
1
32
8
72.61 42.27
1
1
32
9
79.21 43.87
1
1
32
10
78.51 43.65
32
11
1
1
70.97 40.07
32
12
1
1
75.76 42.43
1
1
33
7
71.09 39.88
33
8
1
1
75.30 42.10
1
1
33
9
71.92 40.27
33
1
1
10
67.61 38.77
33
11
1
1
76.91 41.10
12
1
33
1
74.30 42.06
34
7
1
1
67.86 39.17
34
1
8
1
74.74 41.56
34
1
1
9
63.62 38.24
34
10
1
1
68.83 39.51
34
11
1
1
78.21 42.81
34
12
1
1
76.88 42.58
35
7
1
1
64.29 38.49
35
8
1
1
74.42 42.37
32.31 49.91
33.17 52.91
31.32 48.32
32.48 51.16
32.79 51.24
32.05 48.36
35.25 57.05
32.31 49.03
31.20 45.26
32.25 50.06
33.08 45.79
32.58 45.47
31.90 47.34
33.11 49.88
31.97 45.18
32.18 49.69
31.98 46.74
32.28 47.86
32.75 50.41
32.16 48.64
31.75 46.53
32.00 45.73
31.48 46.74
32.20 47.57
32.80 50.13
32.81 50.52
31.45 46.96
32.00 48.96
33.07 52.05
33.11 51.75
31.99 47.68
32.26 50.15
31.45 47.48
32.60 50.00
32.37 48.19
32.21 46.19
33.21 50.41
32.09 49.48
30.99 46.01
31.95 49.42
32.50 44.79
31.97 46.77
32.82 51.28
32.78 50.75
32.68 45.16
32.09 49.63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
394
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
Table E.1 Recursive Results for Waxy IVaize Starch
35
9
1
1
64.14 38.57
10
1
1
35
68.20 38.97
11
1
1
83.32 47.86
35
12
1
1
35
77.58 42.65
7
1
1
36
94.81 54.70
36
8
1
1
78.50 43.62
1
36
9
1
74.39 41.10
1
1
36
10
67.80 39.15
36
11
1
1
77.46 43.97
36
12
1
1
73.72 41.50
37
7
1
1
66.47 38.43
37
8
1
1
72.94 40.88
37
1
1
9
69.66 41.18
37
10
1
1
75.75 42.51
37
11
1
1
84.92 47.87
37
12
1
1
77.15 43.64
38
7
1
1
72.86 40.80
38
8
1
1
65.57 38.28
1
1
38
9
74.48 41.89
38
10
1
1
64.06 38.53
11
1
1
38
65.99 39.09
12
1
1
38
77.50 44.39
7
39
1
1
74.78 41.66
39
8
1
1
70.83 40.75
39
9
1
1
64.33 38.14
39
10
1
1
64.89 38.59
11
39
1
1
76.62 43.46
39
12
1
1
73.80 45.74
40
7
1
1
74.71 41.93
40
8
1
1
66.42 38.17
40
9
1
1
91.99 53.11
40
10
1
1
70.83 40.30
40
11
1
1
70.10 39.47
40
12
1
1
78.45 44.86
41
7
1
1
71.26 39.72
41
1
1
8
73.13 41.60
41
9
1
1
76.60 42.34
41
10
1
1
69.27 39.38
41
11
1
1
78.58 43.55
41
12
1
1
79.71 44.60
42
7
1
1
71.20 39.88
42
8
1
1
74.09 41.65
42
9
1
1
76.53 42.13
42
10
1
1
77.31 43.52
42
11
1
1
78.53 44.09
42
12
1
1
79.74 45.65
32.78
31.82
33.82
33.23
38.78
32.78
32.53
32.15
32.85
33.00
31.39
32.26
32.49
33.00
35.02
33.92
31.46
31.84
32.31
32.23
31.96
34.20
31.36
31.69
32.17
32.24
33.83
32.92
31.88
31.34
38.71
32.09
32.08
34.21
31.73
31.54
32.47
30.46
33.46
34.04
31.47
32.36
32.35
32.42
33.10
34.35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45.16
46.33
55.00
51.15
62.76
51.63
49.34
46.36
51.43
49.40
45.43
48.69
47.78
50.42
55.94
51.57
48.38
45.23
49.56
44.94
45.68
52.03
49.27
47.76
44.88
45.24
51.30
50.82
49.51
45.31
61.27
47.74
47.22
52.51
47.57
48.76
50.47
46.37
51.86
52.78
47.52
49.37
50.34
51.08
51.91
53.25
395
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
Table E.1 Recursive Results for Waxy Maize Starch
7
1
43
1
68.15 39.23
43
1
1
8
76.03 42.05
1
43
9
1
68.83 38.45
1
1
43
10
74.65 42.10
1
43
11
1
72.23 41.22
43
12
1
1
81.05 43.81
44
7
1
1
75.06 42.82
44
8
1
1
73.62 41.35
44
1
9
1
69.70 39.33
44
1
10
1
65.04 38.69
44
1
11
1
65.61 38.74
44
12
1
1
81.22 44.97
45
7
1
1
76.86 42.81
45
1
1
8
73.42 40.36
45
9
1
1
68.91 37.68
45
10
1
1
69.52 39.00
1
1
45
11
66.36 39.05
45
12
1
1
76.51 42.88
46
7
1
1
89.03 52.82
1
46
8
1
70.91 39.80
46
1
1
9
66.56 37.46
46
1
1
10
71.66 39.53
46
11
1
1
65.65 38.74
1
46
12
1
88.43 48.68
47
7
1
1
69.55 40.99
47
1
8
1
66.49 38.17
47
1
9
1
73.86 40.92
47
1
10
1
75.00 41.70
47
11
1
1
76.03 42.94
47
12
1
1
84.73 46.42
48
7
1
1
73.78 40.89
1
48
8
1
73.70 40.57
48
1
1
9
74.88 41.46
1
48
10
1
74.55 41.20
1
48
11
1
81.97 46.46
48
12
1
1
70.50 40.88
7
1
49
1
65.47 38.74
49
8
1
1
67.39 38.52
49
9
1
1
70.48 38.80
49
10
1
1
69.61 39.36
1
49
11
1
71.22 39.43
49
12
1
1
72.45 41.63
50
7
1
1
76.90 42.95
50
8
1
1
72.70 41.10
50
1
9
1
66.38 38.24
50
1
10
1
66.70 38.18
31.36
32.20
31.66
32.98
32.72
34.12
32.10
32.26
31.57
32.42
32.25
34.33
32.30
32.15
31.62
31.62
32.15
32.93
36.86
31.51
31.37
31.69
32.67
35.65
31.91
31.92
32.07
32.19
33.24
34.69
32.36
31.60
32.16
32.27
34.32
32.41
32.21
31.89
32.02
31.62
31.74
32.95
32.19
31.87
31.75
31.49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46.24
50.09
46.31
49.91
48.72
52.99
49.99
49.08
46.87
45.38
45.53
53.51
50.66
48.64
46.07
46.72
45.85
50.77
59.57
47.41
45.13
47.63
45.69
57.59
47.48
45.53
48.95
49.63
50.74
55.28
49.01
48.62
49.50
49.34
54.25
47.93
45.47
45.93
47.10
46.86
47.46
49.01
50.68
48.56
45.46
45.46
396
3
3
155
156
Table E.1 Recursive Results for Waxy Waize Starch
11
1
1
50
73.60 41.10
50
12
1
1
66.02 38.47
32.38
32.15
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49.03
45.55
397
APPENDIX F:
RECURSIVE RESULTS FOR COMMON CORN STARCH
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
398
Table F.1 Recursive Results for Common Corn Starch
Final Result
Init. Estimate
SSE
File set Comb. No.
OK?
1
2
3
Es1"Es2
6s1"Es2
*1
*1
1
1
1
1
0
0
27.55
18.05
18.48
1
2
1
1
1
10
44.19
36.56
34.32
1
1
1
3
20
1
45.58
37.03
34.53
1
4
1
1
1
30
50.16
39.59
36.69
1
1
40
1
1
5
46.23
38.05
35.32
1
6
1
50
1
1
47.62
37.90
35.56
1
7
1
0
0
28.54
10
17.02
17.55
1
1
8
10
10
1
51.89
41.05
38.22
1
10
20
1
1
45.62
9
36.71
35.48
1
10
10
30
0
0
28.60
17.02
17.57
1
11
10
40
0
0
28.53
17.02
17.56
1
12
10
50
0
0
28.53
17.02
17.57
1
1
13
20
1
1
36.44
29.46
28.25
14
1
1
1
20
10
44.30
37.66
35.42
1
1
1
15
20
20
83.47
71.25
65.34
1
1
16
20
30
1
50.15
41.31
39.33
1
17
20
40
0
0
28.60
17.02
17.57
1
18
20
50
0
0
28.56
17.03
17.58
1
1
1
1
19
30
40.81
32.62
31.21
1
1
20
30
10
1
47.88
37.99
35.94
1
21
30
20
1
1
57.32
44.09
39.99
1
22
30
30
0
0
28.61
17.02
17.57
1
1
1
23
30
40
80.54
73.03
66.33
1
24
30
50
0
0
28.54
17.03
17.56
1
25
40
1
0
0
30.30
19.94
19.99
1
1
26
40
10
1
66.65
58.57
53.31
1
27
1
1
40
20
73.91
62.19
57.82
1
40
30
1
1
28
82.15
75.61
67.85
1
29
40
40
1
1
79.18
67.48
62.07
1
30
40
50
0
0
28.51
17.07
17.56
1
31
50
1
1
1
137.90 137.31
143.72
1
1
32
50
10
1
49.98
40.20
37.60
1
33
50
20
1
1
82.41
74.90
67.95
1
34
50
30
1
1
83.17
70.01
64.30
1
35
50
1
1
40
78.88
65.90
60.62
1
36
50
50
0
0
31.54
20.96
20.71
2
1
15
1
0
0
28.61
17.02
17.57
2
2
15
2
1
1
44.94
34.28
31.14
2
3
15
3
0
0
28.60
17.02
17.57
2
4
15
4
1
1
45.51
35.85
34.13
2
5
15
5
1
1
43.29
36.40
34.40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mean
21.36
38.35
39.05
42.15
39.87
40.36
21.04
43.72
39.27
21.06
21.04
21.04
31.39
39.13
73.35
43.60
21.06
21.06
34.88
40.60
47.13
21.07
73.30
21.04
23.41
59.51
64.64
75.20
69.58
21.05
139.64
42.59
75.09
72.50
68.46
24.40
21.07
36.79
21.06
38.50
38.03
399
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
1
2
3
4
5
6
7
8
9
10
Table F.1 Recursive Results for Common Corn Starch
1
1
15
6
51.53
39.56
1
19
0
0
28.61
17.02
19
2
1
1
40.78
32.94
1
19
3
1
43.17
36.51
4
1
1
19
49.86
41.19
1
19
5
1
47.91
37.38
1
1
19
6
42.81
35.92
1
1
1
39.54
23
30.78
1
43.71
23
2
1
34.77
3
1
1
23
56.69
43.81
23
4
1
1
46.92
37.05
1
1
23
5
44.52
36.97
1
1
59.74
23
6
48.55
27
1
1
1
36.10
27.03
27
2
1
1
42.30
33.15
27
3
1
1
42.76
36.48
27
4
1
1
50.04
39.43
27
5
1
1
45.73
37.78
27
1
1
6
61.17
49.57
31
1
28.61
17.02
1
1
31
2
45.29
34.47
1
1
31
3
44.23
34.73
1
4
1
31
46.70
38.98
1
1
31
5
43.34
36.08
1
1
31
6
49.25
40.34
1
35
1
1
26.34
34.35
2
35
17.02
28.56
1
1
35
3
42.89
36.79
4
1
1
35
44.52
37.23
35
1
1
5
45.99
37.15
35
6
1
1
59.72
48.53
24
1
28.61
17.02
1
25
1
1
39.69
30.82
1
1
1
26
41.19
31.21
1
28
1
1
36.60
29.61
1
29
28.62
17.02
32
1
1
1
33.23
23.50
1
1
33
1
36.40
25.88
34
1
28.40
17.03
36
1
1
1
38.03
29.42
37
1
1
1
39.40
30.50
37.21
17.57
31.47
34.17
38.87
35.68
34.14
29.76
33.07
41.15
34.95
34.77
45.32
26.73
32.36
34.42
37.22
35.65
46.36
17.57
33.26
33.19
36.30
34.54
38.33
25.77
17.56
34.93
34.99
34.85
45.49
17.58
29.75
29.79
29.02
17.58
23.42
25.13
17.50
29.08
29.35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42.77
21.07
35.06
37.95
43.30
40.32
37.62
33.36
37.18
47.22
39.64
38.75
51.20
29.95
35.94
37.88
42.23
39.72
52.37
21.07
37.68
37.38
40.66
37.99
42.64
28.82
21.04
38.20
38.91
39.33
51.25
21.07
33.42
34.06
31.74
21.07
26.72
29.14
20.98
32.18
33.08
400
APPENDIX G:
RECURSIVE RESULTS FOR POTATO STARCH
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
401
File set
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
Table G.1 Recursive Results for Potato
Final Result
Init. Estimate
Comb.
OK?
No.
£s1"£s2
*1
esi-fis2
T1
1
1
1
0
0
1
2
10
0
1
1
3
20
0
0
1
4
30
1
1
1
40
5
0
0
1
50
1
1
6
7
10
1
0
0
8
10
10
0
0
9
10
20
1
1
10
10
30
0
0
11
10
40
0
0
12
10
50
0
0
1
13
20
0
0
14
20
10
0
0
20
1
1
15
20
16
20
30
1
1
17
20
40
0
0
20
50
18
0
0
1
30
1
1
19
1
20
30
10
1
21
30
20
0
0
22
30
30
0
0
23
30
40
1
1
24
30
50
0
0
1
25
40
0
0
40
10
26
1
1
27
40
20
1
1
28
40
30
0
0
29
40
40
1
1
40
50
30
0
0
50
1
31
1
1
32
50
10
1
1
50
33
20
1
1
34
50
30
1
1
50
35
40
1
1
36
50
50
1
1
1
25
2
0
0
2
25
9
0
0
3
25
16
1
1
4
25
23
0
0
5
25
31
0
0
Starch
SSE
1
2
mean
48.48
82.30
60.12
75.62
64.89
81.26
58.52
55.03
90.79
55.70
57.16
54.16
55.31
54.85
192.79
119.92
55.08
55.07
73.26
92.31
53.79
57.92
142.60
51.23
54.53
108.58
126.80
57.12
68.96
56.70
82.64
82.40
93.34
81.82
133.02
181.40
55.40
56.51
176.67
55.40
55.28
85.08
122.79
112.36
121.15
110.91
119.27
87.76
80.35
124.00
81.24
85.69
80.86
81.16
80.96
123.95
110.54
81.28
80.91
112.29
114.26
81.14
83.19
110.40
82.05
86.23
110.25
111.05
81.45
107.14
81.34
113.55
119.24
122.35
128.24
114.75
124.67
76.49
80.82
117.04
83.19
81.65
66.78
102.55
86.24
98.39
87.90
100.26
73.14
67.69
107.39
68.47
71.42
67.51
68.24
67.91
158.37
115.23
68.18
67.99
92.78
103.28
67.47
70.56
126.50
66.64
70.38
109.41
118.93
69.28
88.05
69.02
98.10
100.82
107.84
105.03
123.89
153.04
65.95
68.67
146.86
69.29
68.47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
402
File set
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
Table G.1 Recursive Results for Potato
Final Result
Init. Estimate
Comb.
O K?
No.
es1"es2
T1
Es 1"Es 2
T1
25
37
6
0
0
1
7
30
2
1
9
1
1
8
30
30
16
1
9
0
10
30
23
0
1
11
30
31
0
37
1
12
30
1
2
1
1
13
35
1
14
35
9
1
16
1
15
35
1
35
23
1
1
16
17
35
31
35
37
1
1
18
19
40
2
1
1
1
20
40
9
1
21
40
16
1
1
22
40
23
23
40
31
37
1
1
24
40
45
2
1
1
25
26
45
9
1
1
27
45
1
16
1
28
45
23
29
45
31
1
1
30
45
37
1
1
50
2
1
1
31
32
50
9
1
1
33
50
16
1
1
34
50
23
1
1
35
50
31
1
1
36
50
37
1
1
1
30
3
1
1
2
32
1
1
1
3
32
2
1
1
4
32
3
1
1
5
34
1
1
1
6
34
2
1
1
7
34
3
1
1
8
35
36
0
0
9
35
38
0
0
10
35
40
0
0
11
35
42
0
0
12
44
35
1
1
Starch
SSE
1
2
mean
56.29
92.08
123.06
95.33
56.31
120.33
142.24
126.40
89.09
149.93
154.99
51.47
146.09
101.69
87.01
80.44
58.59
54.59
136.58
91.24
83.91
107.71
54.07
82.34
169.22
72.67
147.05
104.32
85.65
96.44
88.66
64.73
84.08
118.01
70.86
92.17
94.18
135.41
57.81
52.28
56.11
55.33
66.66
81.96
111.30
108.30
115.54
82.61
110.02
109.78
105.38
123.62
114.13
114.47
81.61
110.03
107.98
120.30
116.35
81.37
80.87
109.21
118.45
122.65
112.65
81.06
123.18
114.98
104.10
109.65
117.28
131.24
120.49
129.35
116.54
105.50
103.91
92.81
110.32
110.28
107.35
82.75
81.99
80.71
76.66
91.26
69.12
101.69
115.68
105.43
69.46
115.18
126.01
115.89
106.36
132.03
134.73
66.54
128.06
104.84
103.65
98.40
69.98
67.73
122.90
104.84
103.28
110.18
67.57
102.76
142.10
88.39
128.35
110.80
108.45
108.46
109.00
90.64
94.79
110.96
81.83
101.24
102.23
121.38
70.28
67.13
68.41
66.00
78.96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
403
File set
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
Table G.1 Recursive Results for Potato Starch
Final Result
Init. Estimate
Comb.
O <?
NO.
1
T1
Es 1"Es 2
e s1_es2
11
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
1
2
3
36
36
36
37
37
37
37
37
38
38
38
39
39
40
40
40
40
40
42
42
42
42
42
42
42
42
44
44
44
45
45
45
45
45
46
46
46
48
48
48
34
34
34
1
2
3
36
38
40
42
44
1
2
3
1
2
3
36
38
42
44
1
2
3
36
38
40
42
44
1
2
3
36
38
40
42
44
1
2
3
1
2
3
43
44
45
0
1
1
1
0
0
1
0
1
0
1
1
1
1
1
1
1
0
1
0
1
1
0
0
1
1
0
1
1
1
1
0
1
1
1
0
1
1
0
1
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
57.48
94.20
92.87
141.41
67.35
58.04
141.06
55.56
88.46
68.53
129.41
72.07
93.11
92.74
158.36
140.70
120.35
64.80
77.51
57.27
136.68
183.47
50.57
57.60
158.84
122.59
56.16
90.91
134.97
162.17
137.73
68.80
164.35
160.51
89.19
53.78
130.25
93.83
65.46
128.08
56.41
66.79
65.31
SSE
2
mean
66.57
111.75
113.67
110.00
89.06
80.88
109.35
82.49
106.24
80.98
103.83
117.21
113.73
115.01
115.29
108.49
110.22
88.43
102.96
80.64
107.38
120.31
89.00
81.94
117.67
110.43
79.91
117.90
107.36
117.93
109.56
87.61
116.73
118.03
121.21
80.51
106.73
118.69
69.00
106.32
80.54
84.09
88.61
62.02
102.98
103.27
125.70
78.21
69.46
125.20
69.02
97.35
74.75
116.62
94.64
103.42
103.87
136.83
124.59
115.29
76.62
90.24
68.95
122.03
151.89
69.78
69.77
138.26
116.51
68.04
104.40
121.16
140.05
123.65
78.21
140.54
139.27
105.20
67.14
118.49
106.26
67.23
117.20
68.48
75.44
76.96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
404
File set
4
4
4
4
4
5
5
5
5
5
Table G.1 Recursive Results for Potato Starch
Final Result
Init. Estimate
Comb.
0 K?
No.
1
£S1-E82
ti
Ss1-Es2
T1
4
5
6
7
8
1
2
3
4
5
35
35
36
36
36
33
33
33
34
35
43
45
43
44
45
44
45
46
46
46
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
55.31
62.79
55.66
51.30
63.13
55.11
62.10
52.59
55.06
59.97
SSE
2
mean
80.66
74.70
81.22
82.81
88.32
82.12
87.52
88.75
80.83
80.80
67.98
68.75
68.44
67.06
75.72
68.61
74.81
70.67
67.95
70.39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX H
COMPARING THE DIELECTRIC PROPERTIES
OF STARCH WITH THOSE OF WATER
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
406
(a) e/ heating
starch
w a te r
Frequency (Hz)
Temperature (°C)
(b) e/ cooling
starch
w a te r
Frequency (Hz)
Temperature (°C)
Figure H .l Comparison of the relative dielectric permittivity of waxy maize starch suspention and water during the
a) heating and b) cooling cycles.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
407
a) 8r" heating
starch
water
Frequency (Hz)
Temperature (°C)
(b) e," cooling
starch
water
Frequency (Hz)
Temperature (°C)
Figure H.2 Comparison of the relative dielectric loss of waxy maize starch suspention and water during the a)
heating and b) cooling cycles.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
408
(a)
e/
h e a tin g
starch
water
Frequency (Hz)
Temperature (°C)
(b) e/ cooling
starch
water
Frequency (Hz)
Temperature (°C)
Figure H.3 Comparison of the relative dielectric permittivity of common corn starch suspention and water during
the a) heating and b) cooling cycles.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
409
a) e/'heating
starch
water
Frequency (Hz)
Temperature (°C)
(b) e/' cooling
starch
water
Frequency (Hz)
Temperature (°C)
Figure H.4 Comparison of the relative dielectric loss of common corn starch suspention and water during the a)
heating and b) cooling cycles.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
410
(a) e/heating
starch
water
Frequency (H z)
Temperature (°C)
(b) s,' cooling
starch
water
Frequency (Hz)
Temperature (°C)
Figure H.5 Comparison of the relative dielectric permittivity of potato starch suspention and water during the a)
heating and b) cooling cycles.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
411
a) e/' heating
starch
water
Frequency (Hz)
Temperature (°C)
(b) e/' cooling
starch
water
Frequency (H z)
Temperature (°C)
Figure H.6 Comparison of the relative dielectric loss of potato starch suspention and water during the a) heating
and b) cooling cycles.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX I
PLOTS O F D IELECTRIC DATA AND FITTED CURVES FO R
STARCH SUSPENSIONS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
413
(a) s/
80
heating
75
70
40
10
135
Frequency (Hz)
Temperature (°C)
80
10°
(b) 6,'
cooling
65
45
10°
Frequency (Hz)
Temperature ( C)
Colored curves = measured data;
80
10°
black curves = fitted data.
Figure 1.1 Comparison of the fitted curve with the measured data for the relative dielectric permittivity during the
a) heating and b) cooling cycles for waxy maize starch suspention.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
414
a) e,"
80
heating
75
70
65
10
10s
35
Frequency (Hz)
60
Temperature (°C)
80
10°
(b) e,"
80
cooling
75
70
65
45
10°
Frequency (Hz)
60
Temperature (°C)
Colored curves = measured data;
80
10°
black curves = fitted data
Figure 1.2 Comparison of the fitted curve with the measured data for the relative dielectric loss during the
a) heating and b) cooling cycles for waxy maize starch suspention.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
415
(a )
H r'
heating
10
Frequency (Hz)
Temperature ( C)
80
10
(b) e/
80
cooling
75
70
65
10"
Frequency (Hz)
Temperature ( C)
Colored curves = measured data;
80
10
black curves = fitted data
Figure 1.3 Comparison of the fitted curve with the measured data for the relative dielectric permittivity during the
a) heating and b) cooling cycles for common corn starch suspention.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
416
a) e/'
heating
Frequency (Hz)
Temperature ( C)
80
10°
(b) e/'
80
cooling
75
70
10
Frequency (Hz)
Temperature (°C)
Colored curves = measured data;
80
10°
black curves = fitted data
Figure 1.4 Comparison of the fitted curve with the measured data for the relative dielectric loss during the
a) heating and b) cooling cycles for common corn starch suspention.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
417
80
(a) £/
heating
75
70
65
45
10’
Frequency (Hz)
Temperature ( C)
80
10°
80
(b) e/
cooling
■
75
70
65
10’
Frequency (Hz)
Temperature (°C)
Colored curves = measured data;
80
10°
black curves = fitted data
Figure 1.5 Comparison of the fitted curve with the measured data for the relative dielectric permittivity during the
a) heating and b) cooling cycles for potato starch suspention.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
418
a) s/
heating
Frequency (Hz)
60
Temperature (°C)
80
10°
(b) Er"
80
cooling
75
70
65
60
155
150
945
140
10
35
Frequency (Hz)
60
Temperature (°C)
Colored curves = measured data;
'30
80
10°
black curves = fitted data
Figure 1.6 Comparison of the fitted curve with the measured data for the relative dielectric loss during the
a) heating and b) cooling cycles for potato starch suspention.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■
419
80
(a) <*'
heating
75
■70
■65
10’
Frequency (Hz)
Temperature ( C)
80
10°
80
(b) 6/
cooling
75
70
65
140
10’
135
Frequency (Hz)
Temperature ( C)
80
10°
Colored curves = measured data; black curves = fitted data
Figure 1.7 Comparison of the fitted curve with the measured data for the relative dielectric permittivity during the
a) heating and b) cooling cycles for the waxy maize starch suspention with sodium chloride.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
420
a) £r"
80
heating
75
70
s" 175
10’
Frequency (Hz)
60
Temperature (°C)
80
35
10°
(b) £/'
cooling
65
60
s'' 175
10’
Frequency (Hz)
60
Temperature (°C)
80
10°
Colored curves = measured data; black curves = fitted data
Figure 1.8 Comparison of the fitted curve with the measured data for the relative dielectric loss during the
a) heating and b) cooling cycles for the waxy maize starch suspention with sodium chloride.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
421
APPENDIX J:
COORDINATES OF MICROWAVE OVEN
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
422
Table J .l Locations o f microwave oven features and sample points in terms o f cell numbers
in the XFDTD model w ith 1.22-mm cell size
M icrow ave Oven
Feature
Plane
front o f oven
back o f oven
le ft w all
right w all
top o f oven
floor o f oven
Point
Location
1
back
left
bottom
back
right
bottom
front
right
bottom
front
left
bottom
back
left
top
back
right
top
front
right
top
front
left
top
top
center
middle o f
water
2
3
4
5
6
7
8
9
10
x = 303
x= 1
y= l
y = 283
z = 179
z = 11
Coordinates
(x,y,z)
(26, 36,36)
(26,253, 36)
x=26 is 25 cells from back o f oven
y =36 was chosen to avoid back “ bump”
z =36 is 25 cells above oven floo r
y=253 is 30 cells from right w all to avoid back
“ bump”
(278,258, 36)
x=278 is 25 cells from front w all
y=258 is 25 cells from right w all
(278,26, 36)
y=26 is 25 cells from le ft w all
(26, 36, 154)
z=154 is 25 cells from top
(26,253, 154)
(278,258, 154)
(278,26,154)
(153, 142, 154)
(153,142, 50)
at the top o f the oven above the center o f the
water in the bowl
geometric center o f water
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX K:
FINITE DIFFERENCE TIME DOMAIN MODELING
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
424
(a) 30°C
10 °
10*
10
20
30
40
50
10
20
30
40
50
10
20
30
t (ns)
40
50
60
10
20
30
40
50
60
20
30
40
50
60
10
20
30
t (ns)
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
20
30
t (ns)
40
50
60
10c
r c
10*
10
10c
fn T ^ ~
10*
(b) 40°C
10£
10
lifer:
0
10
10
1---10 °
-
J f t ir c 0
(C)
50°C
10c
10*
10c
10*
10
10c
10
10
Figure K.1 Electric field amplitudes at 10 sample points versus real oven time with water as the load at a) 30°C, b)
40°C, c) 50°C.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
425
(a) 60°C
10
10'
20
30
40
50
60
20
30
40
50
60
10
20
30
t (ns)
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
10
20
40
50
60
10
20
40
50
60
10
20
30
40
50
60
10
20
30
t (ns)
40
50
60
10
10
*o
10
10°
-
0
10
---------
(b) 70°C
10
10'
10
10 '
10
10
10 '
(c) 75°C
10c
10*
10
10'
10
10
10
Figure K.2 Electric field amplitudes at 10 sample points versus real oven time with water as the load at a) 60°C, b)
70°C, c) 75°C.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
426
(a) 30°C
10
10
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
t (ns)
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
t (ns)
40
50
10
20
30
40
50
20
30
40
50
20
30
t (ns)
40
50
10l
■o
u.
10
10
10
(b) 40°C
10
10'
10
10'
U.
10
10
10'
(c) 50°C
10c
10
60
■o
10
—
10'
10
10
--
10°
0
10
60
Figure K.3 Electric field amplitudes at 10 sample points versus real oven time with 3% waxy maize starch
suspension as the load at a) 30°C, b) 40°C, c) 50°C.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
427
(a) 60°C
10
10
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
t (ns)
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
t (ns)
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
t (ns)
40
50
60
10c
10
10
10
10'
(b) 70°C
10
10
10c
u.
10
10
10c
10
(C)
75°C
10c
10‘
■o
a
10c
■o
10'
10
10
10'
Figure K.4 Electric field amplitudes at 10 sample points versus real oven time with 3% waxy maize starch
suspension as the load at a) 60°C, b) 70°C, c) 75°C.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
428
(a) 30°C
10
(b) 40°C
10
Figure K.5 Electric field magnitudes in the entire oven at the end o f the calculation with water as the load at a) 30°C,
b) 40°C. Scale given in V/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
429
(a) 50°C
Figure K.6 Electric field magnitudes in the entire oven at the end o f the calculation with water as the load at a) 50°C,
b) 60°C. Scale given in V/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
430
Figure K.7 Electric field magnitudes in the entire oven at the end of the calculation with water as the load at a) 70°C,
b) 75°C. Scale given in V/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
431
Figure K.8 Electric field magnitudes in the entire oven at the end o f the calculation with 3% waxy maize starch
suspension as the load at a) 30°C, b) 40°C. Scale given in V/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
432
(a) 50°C
(b) 60°C
Figure K.9 Electric field magnitudes in the entire oven at the end o f the calculation with 3% waxy maize starch
suspension as the load at a) 50°C, b) 60°C. Scale given in V/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
433
(a) 70°C
Figure K.10 Electric field magnitudes in the entire oven at the end o f the calculation with 3% waxy maize starch
suspension as the load at a) 70°C, b) 75°C. Scale given in V/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
434
a)
(b)
(e) 60°C
30°C
I
(c)
40°C
(0 70°c
I
t
L
■'
..
i-
•i
i
(d)
50°C
(f) 75°C
I
Figure K.11 Electric field magnitudes in the bowl at the end o f the calculation with water as the load: a) bowl
position in oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, f) 70°C, and g) 75°C. Scale given in V/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
435
(a)
(b)
(e) 60°C
30°C
I
I
(c)
40°C
(f) 70°C
(d)
50°C
(f) 75°C
I
I
Figure K.12 Electric field magnitudes in the bowl at the end o f the calculation with 3% waxy maize suspension as
the load: a) bowl position in oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, f) 70°C, and g) 75°C. Scale in V/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
436
(a)
f
(e) 60°C
(b) 30°C
|
f■
e
\f
•6
5
•5
(f) 70°C
(c) 40°C
e
i g '
f•6
(d) 50°C
JSf
1
1
S
•6
•
(g) 75°C
•5
>
i0
'
•6
I
Figure K.13 SARs in the bowl at the end of the calculation with water as the load: a) bowl position in oven, b) 30°C,
c) 40°C, d) 50°C, e) 60°C, f) 70°C, and g) 75°C. Scale given in W/kg.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
437
(a)
(b)
(e) 60°C
30°C
I
•(
6
•5
|
(c)
I
(f) 70°C
40°C
H
'
•I
I
(d)
50°C
(g) 75°C
Figure K.14 SAR s in the bowl at the end of the calculation with 3% waxy maize suspension as the load: a) bowl
position in oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, f) 70°C, and g) 75°C. Scale given in W/kg.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
438
(a)
(b)
(e) 60°C
30°C
I
•(
|
(c)
■
(f) 70°C
40°C
I
«
5
\< 4
(d)
50°C
(g) 75°C
x
-i
I
I
Figure K.15 1-g SARs in the bowl at the end o f the calculation with water as the load: a) bowl position in oven, b)
30°C, c) 40°C, d) 50°C, e) 60°C, I) 70°C, and g) 75°C. Scale given in W/kg.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure K.16 1-g SARs in the bowl at the end of the calculation with 3% waxy maize suspension as the load: a) bowl
position in oven, b) 30°C, c) 40°C, d) 50°C, e) 60°C, I) 70°C, and g) 75°C. Scale given in W/kg.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Vita for Johnny Casasnovas
A year after Johnny Casasnovas was bom in Chicago, Illinois, his parents decided to take him and
his siblings back to their native Dominican Republic. He received twelve years o f prim ary and secondary
education from the Christian Brothers at the Dom inican School o f La Salle, and then enrolled in the
Universidad Nacional Pedro Henriquez Urefla (UNPHU) in 1984. A fter completing the standard five years
o f coursework in chemical engineering and a required extra semester o f graduate level w ork in food
science, he received his B.S. in chemical engineering in 1990. He chose to study food science for an
additional semester at UNPHU, before returning later that year to the United States to continue his graduate
studies at Penn State University. During the summer o f 1991, he researched the dynamic measurement o f
starch granule swelling during gelatinization under the direction o f Dr. Gregory Ziegler. W orking under
the direction o f Dr. Ramaswamy Anantheswaran, Johnny developed an interest in microwave heating and
computer modeling. This led him to develop an interactive software program to teach thermal processing
to undergraduate students in Food Science. In 1994, after completing his thesis on the thermal processing
o f food packaging waste using microwave heating, Johnny received his M . S. in food science. He
immediately enrolled in the Ph. D. program, continuing under the guidance o f Dr. Anantheswaran and
researching the microwave properties o f food components. During his years at Penn State, Johnny was the
grateful recipient o f the follow ing Penn State awards: the W illiam B. Rosskam M em orial Scholarship, the
John E. Hetrick Scholarship, and the M in o rity Scholars Award. When not working in his laboratory, he
was very active in the Penn State Catholic Community and the Penn State Latin American Student
Association. Johnny le ft Penn State in 1998 to accept a position as a research scientist w ith K raft Foods
Research and Development. He is the author of:
Casasnovas, J. and Anantheswaran, R.C. 1997. E l uso de microondas para el calentam iento de
alim entos. Indotdcnica, Institute Dominicano de Tecnologla Industrial, Dominican Repub. 8(1-3):47-51.
Casasnovas, J. and Anantheswaran, R.C. 1995. A n interactive software to sim ulate therm al processing
o f foods. Int. J. o f Engineering Education 1l(l):6 7 -7 7 .
Casasnovas, J. and Anantheswaran, R.C. 1994. T herm al processing o f food packaging waste using
m icrowave heating. J. Microwave Power and Electromagnetic Energy 29(3): 171-179.
Casasnovas, J. 1994. Therm al processing o f food packaging waste using m icrowave heating. Masters
Thesis. The Pennsylvania State University, U niversity Park, PA.
Ziegler, G.R., Thompson, D.B., and Casasnovas, J. 1993. Dynamic measurement o f starch granule
sw elling during gelatinization. Cereal Chem. 70(3):247-251.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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