# Chemical processing of binary mixtures in a continuous flow microwave discharge reactor

код для вставкиСкачатьINFORM ATION T O USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely afreet reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Com pany 300 North Z eeb R oad. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9204345 H eating characteristics of sim ulated solid foods in a microwave oven Lin, Yah-Hwa Eva, Ph.D. The Pennsylvania State University, 1991 UMI 300N.ZeebRd. Ann Aibor, MI 48106 The Pennsylvania Stace University The Graduate School Department of Food Science HEATING CHARACTERISTICS OF SIMULATED SOLID FOODS IN A MICROWAVE OVEN A Thesis in Food Science by Yah-Hwa Eva Lin Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 1991 We approve the thesis of Yah-Hwa Eva Lin. Date of Signature Ramaswamy C. Anantheswaran Assistant Professor of Food Science Thesis Advisor Chair of Committee ST.(jf> \ X i___ Rbbert B. Beelman Professor of Food Science c Virendra M. Puri Associate Professor of Agricultural Engineering sjix-h/ • &>\ n f t l Food Science C / z z / 1? / Carlos ... Zuritz Assistant Professor of Agricultural Engineering rofessor of Food Science Head of the Department of Food Science Ill ABSTRACT The power distribution and power-load relationship in a Tappan 500 microwave oven were characterized. Sinusoidal power distributions were found inside the oven at different heights during microwave heating. The power absorption by a load heated in the oven correlated with the logarithm of the size of load. Effects of salt content, geometrical configuration, and electrical shielding on product surfaces on the rates of temperature increase (RTR) at different locations in the heated samples were studied using sodium alginate gel. Center heating was observed in cylindrical samples of small diameters. As diameter increased, surface heating predominated. Corner heating and surface heating occurred in large rectangular slabs. When salt concentration was increased, microwave power absorption was higher at the sample surface. Shielding of microwave power at the surface of the sample using aluminum foil allowed the investigation of microwave heating from different directions (radial or axial). It was found that reflected waves from the oven floor were not as uniform as primary waves incident from the top of the oven. A finite element package, TWODEPEP, was used to solve the transient heat transfer equation with absorbed microwave power density as heat source. Microwave power absorbed at any location in a sample during heating was derived as a function of dielectric properties and sample geometry. Convective and evaporative heat losses at the surface of the foods were incorporated as boundary conditions in addition to the initial temperature condition. The model predictions were compared to the experimental results at the 0.01 significance level. The experimental data were significantly different from the predictions mostly in the center regions of the cylindrical samples. The differences between the experimental results and the model predictions were not significant in slabs. Sensitivity analyses of the model on selected parameters were performed. Changes in material properties (thermal diffusivity and dielectric properties) and microwave power output resulted in a significant change on the predicted temperature. V TABLE OF CONTENTS Eage LIST OF T A B L E S ................................................. ix LIST OF FIGURES ................................................. xi LIST OF SYMBOLS ............................................... ACKNOWLEDGEMENTS ............................................. xvii xxi Chapter 1 INTRODUCTION ......................................... 1 Chapter 2 LITERATURE REVIEW ................................... 5 Electromagnetic Waves.. ............................... 7 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.2 Wave Equations............................... Standing W a v e s ............................... Penetration Depth ........................... Transmission, Reflection andRefraction . . . . Power I n t e n s i t y ............................. Field Distribution ........................... 7 10 11 12 13 14 Microwave Cavity OvenSystem .......................... 17 2.2.1 2.2.2 2.2.3 2.3 Magnetron and Power Supply ................... Power Distribution System ................... Oven C a v i t y ................................. 17 19 20 Product Properties ................................... 21 2.3.1 Permittivity and Conductivityof Dielectrics . . 2.3.1.1 2.3.1.2 2.3.1.3 Effect of Composition ............... Temperature Effect ................... Effect of Multiphase Mixture ......... 26 27 28 Transmission Properties ..................... Thermal and Physical Properties ............. Geometrical Configurations ofthe Product . . . Loading E f f e c t ............................... P a c k a g i n g ................................... 29 32 32 34 35 Modeling of Microwave HeatT r a n s f e r .................. 36 2.4.1 2.4.2 36 37 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.4 21 Heat Transfer M o d e l s ......................... Heat and Mass Transfer M o d e l s ............... Vi TABLE OF CONTENTS (continued) Pape 2.4.3 2.5 Chapter 3 3.1 Numerical Techniques ......................... 38 S u m m a r y ............................................. 39 ............................... 44 MATERIALS AND METHODS Characterization of the Microwave O v e n ................ 3.1.1 Empirical measurement of the Power Distribution ........................... 3.1.1.1 3.1.1.2 3.1.2 3.2 3.3 44 45 Microwave Power Measurement .......... Measurement of Power Distribution . . . 45 46 Model Development for the Power Distribution . . 48 Determination of Microwave Power in Relationship to Load i n g ............................................. 49 Derivation of Heat Transfer Model with Microwave Heat G e n e r a t i o n ......................................... 49 Model Development ........................... Equation for Microwave Power Source (#). . . . 50 51 3.3.1 3.3.2 3.3.2.1 3.3.2.2 3.3.3 3.3.4 Cylindrical Coordinate ................ Rectangular Coordinate ................ Boundary Conditions Material Properties 51 54 ......................... ......................... 56 58 3.4 Finite Element Method and TWODEPEP ................... 59 3.5 Experimental Design 64 3.5.1 3.5.2 3.5.3 3.6 Effects of Size and S h a p e ................ Effect of Shielding of P o w e r .............. Effect of Salt Content.................... Temperature Measurement 3.6.1 3.6.2 3.7 3.8 3.9 ................................. .... 66 66 69 ..................... Temperature Measurement by Fluoroptic Probes . . Temperature Measurement by ThermocoupleProbes . Data A n a l y s i s ....................................... Model Verification ................................... Sensitivity Analysis of the Model Prediction ......... 69 70 70 74 75 75 vii TABLE OF CONTENTS (continued) Page Chapter 4 RESULTS AND DISCUSSION ............................... 77 4.1 Experimental Results ................................. 77 4.1.1 Mapping the Power Distribution................ 4.1.1.1 4.1.1.2 4.1.2 4.1.3 4.1.4 Three-Dimensional Presentation of Power Distribution ...................... Multiple Regression Model ............ Effect of Loading on Energy Coupling E f f i c i e n c y ............................. Effect of Salt Concentration.................. Effect of Geometrical Configuration on Temperature Distribution ................ 4.1.4.1 4.1.4.2 Effect of Shape ...................... Effect of S i z e ........................ 4.1.5 Effect of Shielding the Surface Microwave Power 4.1.5.1 4.1.5.2 4.2 Axial Shielding .................... Radial Shielding .................... Mathematical Modeling ............................. 4.2.1 Microwave Power absorption 4.2.1.1 4.2.1.2 4.2.2 4.2.3 4.2.4 Effect of Geometrical Configuration on Microwave Power Absorption . . . . Effect of Sample Temperature on Microwave Power Absorption . . . . Effective Microwave Power Gain .............. Sensitivity Analysis ....................... Model Verification ......................... 4.2.4.1 4.2.4.2 4.2.4.3 CHAPTER 5 .................. CONCLUSIONS Temperature Distribution in Slabs and Cylinders with Thermal Insulation . Temperature Distribution in Cylinders and Slabs with Heat Loss from the B o u n d a r i e s ...................... Summary ........................... 78 78 79 85 86 91 91 97 107 108 112 113 115 115 118 121 122 129 129 133 154 155 vtii TABLE OF CONTENTS (continued) Page B I B L I O G R A P H Y ................................................. Appendix A Appendix B Appendix C 160 FORTRAN PROGRAMS USED IN TWODEPEP FOR PREDICTION OF TEMPERATURE DISTRIBUTION DURING MICROWAVE HEAT I N G ..................................... 165 THERMOCOUPLE TEMPERATURE MEASUREMENT USING 21X MICROLOGGER AND AM32 M U L T I P L E X E R ............ 172 SUMMARY OF THE STATISTICAL ANALYSIS ON THE RESULTS FROM EXPERIMENTAL MEASUREMENTS AND MODEL PREDICTIONS................................. 174 ix LIST OF TABLES Page 3.1 Element Sizes and Time Steps Used In TWODEPEP Programs for Cylinders and Slabs of Different Sizes .................... 62 Experimental Design for the Effect of Size and Shape of the Model Food Systems during Microwave Heating ................ 64 4.1 Multiple Regression Model for Field Distribution............ 82 4.2 Variations of the characteristic parameters of the microwave heating model for sensitivity analysis .................. 124 4.3 Relative Sensitivity Calculated from Sensitivity Analysis . 128 4.4 The Means and Standard Deviations of RTR at Different Locations in Cylindrical Gels of Various Sizes Measured by Thermocouples and Fluoroptic Probes and Compared to the Model Predicted RTR ................................... 143 Sample Program Listings for Rectangular Geometry Used in T W O D E P E P ............................................... 165 Sample Program Listings for Cylindrical Geometry Used in T W O D E P E P ............................................... 168 A-3 Specifications for the Inputs in the TWODEPEP Programs 171 B-2 Program Instructions for Differential Thermocouple Measurement Using 21X with A M 3 2 ......................... 173 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 2 cm Radius Cylindrical Gels and Compared at 0.01 Significance Level ...................... 174 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 3 cm Radius Cylindrical Gels and Compared at 0.01 Significance Level ...................... 175 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius Cylindrical Gels and Compared at 0.01 Significance Level ...................... 176 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 6 cm Radius Cylindrical Gels and Compared at 0.01 Significance L e v e l ...................... 177 3.2 A-l A-2 C-1 C-2 C-3 C-4 . . X LIST 07 TABLES (continued) Page C-5 C-6 C-7 C-8 C-9 C-10 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical Gels with 1% Total Salt and Compared at 0.01 Significance Level ..................................... 179 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical Gels with Axial Shielding and Compared at 0.01 Significance Level ..................................... 180 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical Gels with Shielding on the Top Surface and Compared at 0.01 Significance Level ...................... 181 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical Gels with Shielding on the Bottom Surface and Compared at 0.01 Significance Level ...................... 182 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical Gels with Radial Shielding and Compared at 0.01 Significance Level ..................................... 183 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in Rectangular Gels of Various Sizes Under Two-Dimensional Microwave Heating and Compared at 0.01 Significance Level ..................................... 184 xi LIST OF FIGURES Page 2.1 Dielectric properties as functions of frequency (von Hlppel, 1 9 5 4 ) ..................................................... 24 Discretization of the power measurement points for microwave power distribution in a Tappan 500 microwave o v e n .......... 47 3.2 Shell balance within a cylindrical model 53 3.3 Initial triangulations for cylinders of various sizes . . . . 61 3.4 Geometrical configurations of cylindrical and rectangular gel samples ............................................... 67 Arrangement of microwave shielding using aluminum foil (shaded areas represent surfaces shielded with aluminum f o i l ) ..................................................... 68 Gel sample temperature measurements using thermocouple assemblies and fluoroptic probes with data acquisition from a P C ..................................................... 71 Positioning of thermocouple probes on templates for temperature measurements in (a) cylindrical gels, and (b) rectangular gels of different sizes ........................ 72 Position and the orientation of sample with respect to microwave oven during heating .............................. 73 Measured power distribution in a Tappan 500 microwave oven at three levels (a) z - 2.5 cm, (b) z - 0, and (c) z - -2.5 c m ....................................................... 81 Time-temperature profiles of cylindrical gels of sizes (a) 3 cm radius by 5 cm height, and (b) 4 cm radius by 2.5 cm height measured by fluoroptic probes at different distances from the center at z - 0 l e v e l ............................ 83 Power distribution in a Tappan 500 microwave oven determined by regression model at (a) z - 2.5 cm, (b) z - 0, and (c) z - -2.5 cm l e v e l s ....................................... 84 3.1 3.5 3.6 3.7 3.8 4.1 4.2 4.3 4.4 ................... Effect of loading on the microwave power absorption by cylindrical and rectangular shaped loads .......... 86 xil LIST OF FIGURES (continued) P a pe 4.5 RTR values In 4 cm radius by 2,5 cm height cylindrical gels at different salt concentrations versus radial distance from the center at z - 0 level ............................... 88 Effect of NaCl concentration and temperature on the attenuation factors of aqueous Ionic solutions (Mudgett, 1986) ................................................... 88 The means and standard deviations of RTR In a 4 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels (thermally insulated on the top and bottom surface) ............................. 90 Mean RTR in a 3 cm radius by 5 cm height cylindrical gel at different locations from the center at z - 0 and z - -2.5 cm levels ................................................... 92 Mean RTR in z - -2.5 cm and y axes, to the axes a 6 cm x 6 cm x 5 cm rectangular gel at z - 0, (both measured in the directions parallel to x and z - -2.5 cm measured in diagonal directions ............................................... 93 4.10 Mean RTR in a 4 cm radius by 5 cm height cylindrical gel at different locations from the center at z - 0 and z - -2.5 cm levels ................................................... 95 4.11 Mean RTR in z - -2.5 cm and y axes, to the axes a 8 cm x 8 cm x 5 cm rectangular gel at z - 0, (both measured in the directions parallel to x and z - -2.5 cm measured in diagonal directions ............................................... 96 The means and standard deviations of RTR in a 2 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels ....................... 98 Themeans and standard deviations of RTR in a 3 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels ....................... 99 Themeans and standard deviations of RTR in a 4 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels .................... 100 4.6 4.7 4.8 4.9 4.12 4.13 4.14 x iit LIST OF FIGURES (continued) Page 4.15 4.16 4.17 4.19 4.20 4.21 4.22 101 Themeans and standard deviations of RTR in a 2 cm radius by 5 cm height cylindrical gel measured at different orientations and distances from the center at z - 2.5 cm, z - 0, and z - -2.5 cm three l e v e l s ...................... 102 Themeans and standard deviations of RTR in a 3 cm radius by 5 cm height cylindrical gel measured at different orientations and distances from the center at z - 2.5 cm, z - 0, and z - -2.5 cm three l e v e l s ...................... 103 The means and standard deviations of RTR in a 4 cm radius by 5 cm height cylindrical gel measured at different orientations and distances from the center at z - 2.5 cm, z - 0, and z - -2.5 cm three l e v e l s .................. 104 Themeans and standard deviations of RTR in a 6 cm radius by 5 cm height cylindrical gel measured at different orientations and distances from the center at z - 2.5 cm, z - 0, and z - -2.5 cm three l e v e l s ...................... 105 Themeans and standard deviations of RTRmeasured at different orientations and distance from the center at z 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5 cm height cylindrical gel shielded with aluminum foil on the z - ±1.25 cm s u r f a c e s .............. 109 The means and standard deviations of RTR measured at different orientations and distance from the center at z 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5 cm height cylindrical gel shielded with aluminum foil on the z - -1.25 cm surface................ 110 The means and standard deviations of RTR measured at different orientations and distance from the center at z 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5 cm height cylindrical gel shielded with aluminum foil on the z -1.25 cm s u r f a c e ................. Ill . 4.18 Themeans and standard deviations of RTR in a 6 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three l e v e l s .................... xlv list OF FIGURES (continued) Page 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 The means and standard deviations of RTR measured at different orientations and distance from the center at z 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5 cm height cylindrical gel shielded with aluminum foil on the r - 4 cm surface.................... 114 Two-dimensional microwave power absorbed at z - 0 in a 6 cm x 6 cm x 5 cm slab gel at 3 0 ° C ......................... 117 Two-dimensional microwave power absorbed in a 3 cm radius by 5 cm height cylindrical gel at 3 0 ° C ..................... 118 Power absorption in a 3 cm radius by 2.5 cm height cylindrical gel along the (a) z direction, and (b) r direction at different temperatures ..................... 120 Effect of attenuation factor on the radial distance of concentrated heating from the center in cylinders of various r a d i i ................................................... 122 Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying microwave power output ± 5% and ± 10% 125 Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying the..... attenuation factor ± 30% .... 126 Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying the thermal diffusivity ± 10% and ± 20% 126 Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying the heat transfer coefficient ± 20% and ± 4 0 % ................................................... 127 Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying the difference of relative humidity between sample and ambient air ± 30% and ± 6 0 % .......... 127 (a) Predicted RTR and (b) experimental RTR in a 4 cm radius by 2.5 cm height cylinder with no heat loss from the b o u n d a r i e s ............................................. 130 (a) Predicted RTR and (b) experimental RTR in an 8 cm x 8 cm x 5 cm slab at z - 0 with no heat loss from the boundaries 132 XV LIST OF FIGURES (continued) Page 4.35 4.36 4.37 4.38 4.39 4.40 4.41 4.42 4.43 The RTR a 2 cm, means and standard deviations of predicted and measured at various radial distances from the center (r/R - 0) in cm radius by 2.5 cm height cylindrical gel at z - 1.25 z - 0, and z - -1.25 cm levels ..................... 134 The RTR a 3 cm, means and standard deviations of predicted and measured at various radial distances from the center (r/R - 0) in cm radius by 2.5 cm height cylindrical gel at z - 1.25 z - 0, and z - -1.25 cm levels ..................... 135 The RTR a 4 cm, means and standard deviations of predicted and measured at various radial distances from the center (r/R - 0) in cm radius by 2.5 cm height cylindrical gel at z - 1.25 z - 0, and z - -1.25 cm levels ..................... 136 The RTR a 6 cm, means and standard deviations of predicted and measured at various radial distances from the center (r/R - 0) in cm radius by 2.5 cm height cylindrical gel at z - 1.25 z - 0, and z - -1.25 cm levels ..................... 137 The RTR a 2 z - means and standard deviations of predicted and measured at various radial distances from the center (r/R - 0) in cm radius by 5 cm height cylindrical gel atz - 2.5 cm, 0, and z - -2.5 cm levels ........................... 138 The RTR a 3 z - means and standard deviations of predicted and measured at various radial distances from the center (r/R - 0) in cm radius by 5 cm height cylindrical gel at z - 2,5 cm, 0, and z - -2.5 cm levels ........................... 139 The RTR a 4 z - means and standard deviations o . predicted and measured at various radial distances from the center (r/R - 0) in cm radius by 5 cm height cylindrical gel at z - 2.5 cm, 0, and z - -2.5 cm levels ........................... 140 The RTR a 6 z - means and standard deviations of predicted and measured at various radial distances from the center (r/R - 0) in cm radius by 5 cm height cylindrical gel at z - 2.5 cm, 0, and z - -2.5 cm levels ........................... 141 Model predicted RTR and measured RTR at z - 0 in slabs of (a) and (d) 4 cm x 4 cm x 5 cm, (b) and (e) 6 cm x 6 cm x 5 cm, and (c) and (f) 8 cm x 8 cm x 5 cm, respectively . . . 145 xvi LIST OF FIGURES (continued) Page 4.44 Effectof 1% total salt on the predicted and measured RTR In a 4 cm radius by 2.5 cm height cylindrical gel at different locations at (a) z - 1.25 cm, (b) z - 0, and (c) z - -1.25 cm levels ............................................... 148 4.45 Effect of radial microwave heating on predicted and measured RTR In a 4 cm radius by 2.5 cm height cylindrical gel at different locations at (a) z - 1.25 cm, (b) z - 0, and (c) z 149 - -1.25 cm l e v e l s ............. ......................... 4.46 Effect of microwave heating with the top surface shielded on predicted and measured RTR in a 4 cm radius by 2.5 cm height cylindrical gel at different locations at (a) z - 1.25 cm, (b) z - 0, and (c) z - -1.25 cm levels .................. 151 Effect of microwave heating with the bottom surface shielded on predicted and measured RTR in a 4 cm radius by 2.5 cm height cylindrical gel at different locations at (a) z 1.25 cm, (b) z - 0, and (c) z - -1.25 cm levels ......... 152 Effect of axial microwave heating on the predicted and measured RTR at different locations in a 4 cm radius by 2.5 cm height cylindrical gel at (a) z - 1.25 cm, (b) z - 0, and (c) z - -1.25 cm levels ................................. 153 Circuitry of 21X micrologger with AM32 multiplexer 172 4.47 4.48 B-l .... LIST OF SYMBOLS empirical coupling coefficients element size or surface area surface area at r outside surface area at r - R magnetic flux density in [weber/m2] a material property parameter material property electric flux density in [coulomb/m2] electric field intensity in [volt/mj electric field incident electric field reflected electric field transmitted electric field incident wave reflected wave frequency element forcing function conductance magnetic field in [ampere/m] convective heat transfer coefficient, [W/m2* latent heat of vaporization, [kJ/kg*C] current density in [ampere/m2] xviii LIST OF SYMBOLS (continued) Jt0t8l sum of conduction-current density (oe) and displacement-current density (oee)in time-phase domain k thermal conductivity in [W/m°C] empirical coupling constant k mass transfer coefficient of water vapor in air in [kg/s*m2,kPa] L inductance my mass flow rate of water vapor in [kg/s] OXX material property * dU/dx OXY material property * dU/dy P value of the parameter chosen Pm maximum power generated by the frequency generator P0 surface power coupled by unmatched load [W] Py water vapor pressure in [kPa] Pr" power per unit area at a distance r the r direction in [W/cm2] PR" power per unit surface area at r » E Pz" power per unit area at a distance z the z direction in [W/cm2] P“ volumetric microwave power absorption in [w/m3] P"c absorbed power density at r and z in a cylinder in [W/cm3] P * B h e U -r from the origin in from the center in absorbed power density of the shell 2irArAz in the r direction in [W/cm3] P"ghell.z absorbed power density of the shell 2trArAz in the z direction in [W/cm3] RH equilibrium relative humidity xix LIST OF SYMBOLS (continued) ARH difference of relative humidity between sample and the ambient air RTR rate of temperature rise In [Deg C/s] 5 Poyntlng vector; a vector in the direction of wave propagation At duration of microwave heating time in [s] AT temperature difference between the initial temperature and the final temperature Ta ambient temperature in [Deg C] U unknown variable in question v velocity of the wave in [m/s] Vg volume fraction of suspended phase in the mixture W# electric energy density Wra magnetic energy density X fraction of phase Y model response Greek Symbols a attenuation constant in [cm*1] at thermal diffusivity in [m2/s] /3 phase shift constant Y2 propagation constant 6 angle of a phase difference between the field intensity E and the current density e permittivity of the material in [farad/m] ec complex permittivity of continuous phase LIST OF SYMBOLS (continued) effective permittivity complex permittivity of a mixture permittivity at free space complex permittivity of suspended phase dielectric constant dielectric loss relative dielectric loss a time-shape function, which equals to 0.5 critical angle - sin*1 /e2/«i time coefficient permeability of the material in [henry/m] relative permeability density in [kg/m3] conductivity of the material in [siemens/m] effective conductivity irreversible rate of energy change due to viscous dissipation xxi ACKNOWLEDGEMENTS First, I want to thank Dr. Swamy Anantheswaran for his guidance, wisdom, and consideration throughout the course of my Ph.D. program. I am very thankful to Dr. Virendra M. Puri and Dr. Carlos A. Zuritz for initiating the part on mathematical modeling. I would like to extend my appreciation to Dr. Puri for assisting me with the use of the finite element method and TWODEPEP program. Dr. Robert B. Beelman and Dr. Gregory R. Ziegler are acknowledged for their valuable inputs and the critical reviews of the manuscript. 1 am also thankful to the Departments of Food Science at Penn State and University of Nebraska for granting me the graduate assistantship which made this research possible. 1 want to thank Mr. Soumya Roy in assisting me in this research in many ways. I am also very thankful to all my friends in State College, especially the McClellans, in providing support during the writing of the manuscript. Last but not least, I am grateful to my parents, Mr. Shen, Yun-Jen and Mrs. Shen, Dai Shio-jen, and my husband, Chu-Neng John Lin, for their encouragement and sustaining help that pressed me toward my goal. 1 Chapter 1 INTRODUCTION One of the revolutions In the American kitchen during the twentieth century has been microwave cooking. Due to rapid changes In life styles, 56% of American women are working and 54% of households are dual-lncome families (Messenger, 1987). Today, microwave ovens have become an essential appliance in homes. Microwave ovens are also making their way into work places and fast food restaurants, and are even found beside vending machines. The ownership of microwave ovens In the United States has reached a level of 70%, and the sales of microwave ovens are still increasing (Anonymous, 1989). Oven manufacturers are now producing smaller economy size ovens for use by low income families or as a second microwave oven in a household. the market. This has further stimulated the sales of microwave ovens in In pace with such a strong increase in popularity of microwave ovens, consumers' demands for microwaveable food products are constantly growing. This has created opportunities for food processors and manufacturers to develop food products and packages specifically for microwave heating. However, regardless of the success and the acceptance by the consumers, there are many problems associated with microwave cooking. They can be summarized as follows: 1) unsatisfactory product quality-nonuniform temperature distribution, rubbery or soggy texture in the end products and unacceptable flavor development; 2) concerns about Insufficient microbial destruction due to uneven cooking; and 3) safety hazards such as over-heating of the center in infant formula bottles or the volcano heating effect due to localized superheating in casseroles or high salt/fat sauce products. The uneven microwave radiation within a microwave oven and different rates of microwave power absorptions by various food components cause non-uniform heating and temperature distribution In the food product. This is often recognized as the major shortcoming in microwave food processing. Methods such as stirring, standing time and rotation of the container may help to equalize the temperature distribution, but they actually detract the user from the convenience of microwave cooking. Organoleptic properties, such as good texture and appearance, browning of roasted meats and baked products, and crispness of bread crust, pizza crust and potato chips, are demanded by the customer but often cannot be achieved satisfactorily by microwave heating alone. Knowledge of the temperature distribution or temperature profile during microwave heating is important for the calculation of lethality or time of inactivation of enzymes during pasteurization/blanching or sterilization. Also, a better understanding of the interaction between the food components, packaging system and the microwave during microwave heating will be useful for the design of microwave processes, packaging and product formulation to combat the problems associated with microwave heating of foods. Microwaves propagate as electromagnetic waves. In order to solve problems of nonuniform heating, a thorough understanding of the physical characteristics of the oven design, mechanisms of electromagnetic wave transmission, the effects of geometry, chemical composition and physical structure of the food and container, and the mechanisms by which food products absorb microwave energy and dissipate it as heat is required. In the microwave oven, the microwave field distribution is affected by the transmission properties of waves in the oven cavity. On the other hand, microwave absorption and dissipation by the food products is governed by the dielectric and thermal properties of foods during microwave heating. Based on the complete knowledge of the microwave oven system and the products, the microwave heating of foods can be modeled using the fundamental theories of electromagnetic wave propagation and heat and mass transfer. The solutions to the mathematical models can simulate the time-temperature distribution of a food load being heated in a microwave oven. The best possible improvement in microwave heating can then be obtained by the optimization of all of the variables in the model. If the model can successfully simulate the heating behavior of a specific food product in a microwave oven, product development scientists will then be able to test the effects of different variables, including packaging and food formulation, without having to go through expensive experimental work. The microwave processes can then be utilized more effectively and successfully. The study of the effect of size and shape of food on the temperature distribution can provide information about cold spots and hot spots within foods heated within a container of various shapes. This information will be useful for microwave food package design. Based on this information, the intensity of the microwaves incident on the food can be altered by the insertion of susceptor materials to facilitate improved microwave absorption or by partially blocking the waves by aluminum foil (Keefer, 1986, and Huang, 1987). Thus, the problem of uneven heating of foods can be eliminated by the design of appropriate packaging materials. The study on the effect of physical structure and chemical constituents on the microwave heating characteristics will be useful for product formulation and for developing microwave cooking instructions for microwaveable foods. For example, the heating rates of food with different salt concentrations are drastically different. Chemical composition influences the microwave penetration depth and power absorption directly. The effects of these factors can be simulated by a mathematical model which can take into account all of the possible conditions of the food, microwave heat source and heating environments during microwave heating. 5 Chapter 2 LITERATURE REVIEW Microwaveable foods are becoming increasingly popular in the marketplace. The number of microwave products in the market has been growing rapidly. The 1988 sales of microwaveable foods in the U.S. have been estimated at eight billion dollars, and the sales of microwave ovens have also been reported as breaking the record of the sales of any kitchen appliance (Anonymous, 1987). Nevertheless, there has been only a limited application of microwave processing in the food industry. Microwave energy was first used in food processing in the early 1940's (Decareau, 1984). Since then, attempts have been made to use it in thawing, cooking, reheating, drying, blanching, pasteurization, sterilization, puffing, baking, proofing and in many other unit operations. Though microwave ovens are popular at home, established microwave applications at the industrial level are scarce in the food industry. Food processors were not able to adapt microwave processing to commercial applications due to insufficient knowledge and technical support (Osepechuk, 1984). Due to the many advantages in microwave processing, such as improved product quality, increased yield, savings in operating time and floor space, and cool operation of the plant, there is a growing need for more applications of microwave energy in the food industry. It is necessary to conduct basic research on the effect of different parameters on the microwave heating of food products to provide the food industry with valuable information necessary for the more successful use of microwave processing. Thermal processing is one of the most important unit operations in food processing. Microwave heating can be utilized for thermal processing, by which food is heated to an end point temperature at a desired rate of heating. Most food products prepared from raw ingredients to the stage of ready-to-eat require some form of heating. The amount of heat treatment and the method of heating depends on the final form of the product and the shelf-life requirement. During conventional thermal processing, heat is transferred from a heat source to the product by at least one of the principle modes of heat transfer: conduction, convection and radiation. In conduction, heat or energy is transferred from molecule to molecule by collision, which occurs primarily in solids. In fluids, energy is transferred by the movement of molecules possessing high energy from one point to another which describes the mechanism of convective heat transfer. Heat radiation transfers energy in the form of electromagnetic waves between materials at different temperatures without a need for a medium. Unlike the three mechanisms of energy transport mentioned above for conventional heating, microwave heating results from the interactions of chemical constituents of materials with an electromagnetic field. During conventional heating, heat transfer occurs when there is a temperature gradient, and the rate of heat transfer is dependent on the temperature difference. However, microwaves are absorbed by dielectric food components, such as molecules with dipole moments like water, and conductive ions. The electric energy then dissipated into thermal energy by molecular friction and transferred into the surroundings. Thus food is heated up "instantaneously" along with the absorption of microwave power. When microwaves generate heat within the food, components with different dielectric activities heat up at different rates. The temperature in the food is equilibrated by conduction. Factors influencing microwave heating include: properties of the material, operating frequency, configurations of the power distribution system, source of electromagnetic power and field distribution. These factors are reviewed in the following sections. 2.1 Electromagnetic Waves Electromagnetic waves consist of coupled electric and magnetic waves in the form of accelerated charged particles. The electric wave (E) and the magnetic wave (H) propagate perpendicular to each other. The progress of the wave is accompanied by a flow of energy which is propagating perpendicular to the plane of electric wave and magnetic wave (Strother, 1977). 2.1.1 Wave Equations In 1873 James C. Maxwell, a professor at Cambridge University, England, published the first unified theory of electricity and magnetism and founded the science of electromagnetics (Kraus, 1984). A wave of both E and H transverse to the direction of propagation is called a "Transverse Electromagnetic (TEM) wave." Maxwell described the relationships of electric and magnetic field by the following two equations (Kraus, 1984): V x B - J * |£ (2.1) V x - . - H where E is (2.2) theelectric field in [volt/m], D is theelectric flux density in [coulomb/m2], H is themagnetic field in [ampere/m], B is themagnetic flux density in [weber/m2], and J is the current density in [ampere/m2]. The field components are related by material properties as follows: J - oE, D - eE, and B - pH where o isthe conductivity of the material in [siemens/m], e isthe permittivity of the material in [farad/m], fj isthe permeability of the material in [henry/m]. and Therefore, Eqs. 2.1 and 2.2 become: V x JST - oJT + e j ? <2,3) V x l . - i . |f (2 .4 ) E, H, D, B and J are complex vector quantities of rms (root-mean square) value and e^-t dependency, which are time-harmonic functions. The time- harmonic functions can also be expressed as sinusoidal function of time since e^ut - cos(ot) V x H + j sin(ut). Therefore,Eq. 2.3 becomes - oE0 sin(ot)+ ueEQcos(ut) (2.5) Rewriting Eq. 2.5 In a phasor form, V x H - oE + jueE (2.6) Similarly, Eq. 2.4 can be written as V x E - -ju/iH (2.7) For most materials other than ferromagnetic substances, the value of fj is essentially the same as p0. This holds true for biological materials and agricultural products (Nelson, 1973). Substituting for the complex conductivity a with a ' + jo" and complex permittivity e - e' - Je", Eq. 2.6 becomes V x H - (o'+oe")E + ju(e'-o"/w)E (2.8) or it can be expressed as V x H - o.E + jue.E (2.9) where oe and ee are the effective conductivity and permittivity, respectively. Since V x H - Jtotal (total current density), Jtotai is a sum of a conduction*current density (oa) and a displacement-current density (oea) in time-phase domain. The operator j with the displacement-current density indicates that it leads the conductioncurrent density, which is in phase with the applied field, by 90°. Eqs. 2.6 and 2.7 are the wave equations which relate the space and time variation of the electric field intensity. The solution for the electric field intensity as functions of space and time as a trigonometrical expression is shown below (Kraus, 1984). Ey - E0sin(/3x+ut) + E0sin(/3x-ot) (2.10) 10 Eq. 2.10 represents two waves, one to the left and one to the right, which make a complete transverse sine wave. The trigonometric solution can also be expressed in exponential form as Ey - E0 eJ(Bt^ K> (2.11) The electric field has an imaginary part and a real part of the exponential function. The imaginary part is Ey - E0 Im eK"*’*> - E0 sin(ot-px) (2.12) and the real part is: Ey - E0 Re 2.1.2 - EQ cos(ut-/3x) (2.13) Standing Waves When a wave propagates from one medium to a second medium, a part of the wave is transmitted through the boundary of the two media and continues on in the second medium while the other part of the wave is reflected from the boundary back to medium one. travels in the opposite direction. The reflected wave At the point of reflection, the reflected wave leads the incident wave by a phase shift 6 in time-phase (Kraus, 1984). The reflected wave has the same frequency and the same sinusoidal form as the incident wave. When the incident wave is defined as Ey0 - E0 e ^ <*t+^x) and the reflected wave as Ey1 - E1 e ^ Bt‘^x+4>, then the total electric field is Ey - Eyo + Ey1 - E0 sin(ot+/3x) + E, sin(ut-0x+6). In most situations, the amplitude of reflected wave is smaller than the incident wave, and the general form of the standing wave equation is Ey - /(E0+E,)2 cos20x + (Eq -E^2 sin2/3x sin(ut+/3x) (2.14) 11 The two waves add to each other at some points (constructive) and subtract from each other at other points (destructive). the resultant wave Is like an envelope. The shape of The maximum value of the envelope corresponds to the sum of the amplitudes of the Incident and reflected waves (E0 + Ej), while the minimum corresponds to the difference between the two (EQ - Ej). 2.1.3 Penetration Depth The differential form of the general wave equation can be expressed as follows (Kraus, 1984) — -f (2.15) vm13E v ox2 where y* i-s the propagation constant. The solution of Eq. 2.15 for a wave traveling in the positive x direction is Ey - E0e‘*x. Since y - a + j/3, the solution becomes Ey - E0 e-« e*W* (2.16) where, a is the attenuation constant and j3 is the phase shift constant which determines the wavelength in the material. Eq. 2.16 gives the variation of Ey in both magnitude and phase as a function of x. The electric field attenuates exponentially and is retarded linearly in phase with increasing x. When Eq. 2.16 is rewritten as Ey - EQ e'K/l)e'^x/0>, D is the 1/e depth of penetration, which is equal to 1/i £rr/jo. The penetration depth decreases inversely proportional to the square root of the frequency. Therefore, a high-frequency field penetrates a shorter distance than a 12 low-frequency field In a given medium. This phenomenon Is known as the "skin effect." 2.1.4 Transmission, Reflection and Refraction In the case of a linearly polarized plane wave obliquely Incident on a boundary between two media, the plane wave can be resolved into perpendicular (the E field perpendicular to the plane of incident) and parallel (the E field parallel to the plane of incidence) components. These waves are perpendicular polarized and parallel polarized, respectively (Kraus, 1984). For the perpendicular polarized wave, Snell's law of reflection says that the angle of reflection is equal to the angle of incidence. Also, Snell's law indicates sin(6{) - [q1/q2] sin C©f), where 6( is the angle of refraction and 8( is the angle of incident. For a dielectric medium the index of refraction rj is equal to //irep and Snell's law becomes sin(8t) * J sin(0i) (2.17) >| The critical angle is 6fc - sin'1 V e2/e1• When a wave is incident from the more dense medium onto the less dense medium (e, > e2) at an angle greater than the critical angle, the incident wave is totally internally reflected back into the more dense medium. In the case of parallel polarization, an incident angle exists when the wave is totally transmitted into the second medium (Kraus, 1984). This angle is called the Brewster angle, where 0{B - tan*1 13 / e2/ei• T*1® Brewster angle Is also sometimes called the polarizing angle since a wave composed of both perpendicular and parallel components and incident at the Brewster angle produces a reflected wave with only a perpendicular component. 2.1.5 Power Intensity Consider a planewave traveling with averticalelectric and a horizontal magneticfield H. field E The energy density W# at a point in an electric field is W„ - h eE2 (2.18) The energy density Wn at a point in a magnetic field is WB - h pH2 (2.19) The instantaneous electric and magnetic energy densities expressed as a rate of change of energy in the wave is |f - -V • X xH (2.20) The vector product of the electric and magnetic field is called the Poynting vector (Baden Fuller, 1979). 8 ■ K x B The Poyntingvector (S) is avector in the and isperpendicular to theplane of (2.21) direction of E and Hvectors. wave propagation Since the intrinsic impedance is equal to E/H - /p/e, by rearranging Eq. 2.18 and 2.19 we can arrive at W( - H pH2 - *1 cE2 - tfe. Thus the electric and magnetic energy densities in a plane traveling wave are equal, and the 14 total energy density is the sum of the electric energy density and the magnetic energy density, and therefore W - eE2 - juH2. Although equal amounts of energy are associated with the electric and magnetic fields, in practice most of the measurable effects of electromagnetic waves on a material are due to the electric field energy (Strother, 1977). Considering two plane waves traveling in opposite directions, the instantaneous value of Ey resulting from the two waves is (2 .22 ) Ey - E0 sin (ut + J3x) + E1 sin (ut - /Jx) In the situation of a pure standing wave, E0 - - Ej, the net Poynting vector is zero, and hence no power is transmitted. The electric energy density is We - 2eE02 cos2(ut) sin2(/3x) and the magnetic energy density is Wm - 2/iHQ2 sin2(ot) cos2(/3x). The electric energy density is at maximum when the magnetic density is zero, and vice versa. Furthermore, the points where they are maximum are a quarter wavelength apart. The energy oscillates back and forth from the electric form to the magnetic. The Poynting vector of a pure standing wave is expressed as Sx » ~4EJi0 cos(ut)sin(wt)cos (Px)sin(Px) ■ -4 E% cos(u£)sin(cot)cos(px)sin(Px) and the peak value of the Poynting vector is V e/fj E02. (2.23) Eq. 2.23 shows clearly that Sx is maximum at ut-ir/4. 2.1.6 Field Distribution In the microwave oven, the cavity dimensions are of the same order of magnitude as the wavelength of the microwaves. Thus a number of standing waves will exist inside the microwave oven. Each of these 15 possible standing wave patterns Is referred to as a mode. The mode pattern field Is unique to every microwave oven and depends on the geometrical dimensions and the microwave distribution system. It is also affected by the geometry and composition of the food present In the oven. Calculation of the three-dimensional field distribution inside a microwave oven requires a good knowledge of all the modes present in the microwave oven. The transverse electromagnetic waves (TEH) in the transmission lines are waves with electric and magnetic to the direction of propagation. fields entirely transverse The mode of wave transmission can be identified by the subscripts m and n for TE (transverse electric) and TH (transverse magnetic) waves in a waveguide. The value of m or n indicates the number of half-cycle variations of each field component with respect to z (broad side) and y directions, respectively. Each combination of m and n values represents a different field configuration (Kraus, 1984). To obtain the field configuration and complete information concerning waves in a transmission line, one has to solve the wave equation subject to the appropriate boundary conditions. equation can be developed from Maxwell's equations. A wave The boundary condition applied to the wave equation is that the tangential component of the resultant E must vanish at perfectly conducting walls. The restrictions are that the field components vary harmonically both with time and distance and also attenuate with distance. Then one has to choose the type of higher-order mode of transmission to be analyzed. transverse electric (TE) wave has Ex - 0 while a transverse magnetic A 16 (TM) wave Hx - 0, Next a component can be obtained solution of a scaler-wave equation In one that fits the boundary conditions. This solution is substituted back into the equations for other field components. Thus a set of equations can be obtained which give the variation of each field component with respect to space and time (Kraus, 1984). The particular mode or modes that are actually present the dimension of the system, the method the discontinuity of the structure. dependon of microwave distribution and Each mode of transmission has a particular cutoff wavelength, velocity and impedance. When the frequency is high enough to permit transmission in more than one mode, the resultant field is equal to the sum of the fields of all the modes present. In a waveguide, there is no attenuation of the wave if the guide walls are perfect conducting and the medium is a lossless dielectric. However, if there are any losses incurred by the wave due to imperfect conducting walls or a lossy medium, there is attenuation. Therefore, the propagation factor, y . composes of a real (a) and an imaginary (/3) component. A finite tangential component Et is developed from the electric field due to the finite wall conductivity, and a is not zero due to the lossy dielectric medium. The field decays exponentially with distance in the direction of propagation as given by (Kraus, 1984) P - P0 e'2tfX a is the attenuation factor which can be defined by Eq. 2.25. (2.24) 17 dp ~ 2 Pdx ■ 1 p o w e r lost p a x unit d i stance 2 p o w e r transmitted (2.25) The attenuation in this case is due to an actual power loss into Joule heating. 2.2 Microwave Cavity Oven System The performance of microwave ovens varies between different manufacturers and even between models of the same manufacturer. It is helpful to know the construction and operation of microwave cavity ovens so that the variability in the microwave heating process can be better understood and controlled. 2.2.1 Magnetron and Power Supply The Federal Communications Commission allocates the radio frequencies that can be used for heating as ISM (industrial, scientific, and medical) frequencies (Decareau, 1985). At present, 915 MHz and 2450 MHz are the frequencies used most in food industry and commercial ovens, respectively, in the United States. According to Raytheon, high frequency (2450 MHz) permits better coupling of microwave energy to small loads and allows for the presence of a greater number of modes in the oven, which eliminates hot and cold spots due to the more random heating patterns, while General Electric has claimed that use of 915 MHz provides greater penetration and less thermal runaway in defrosting (Osepchuk, 1984). 18 Most consumer microwave ovens operate at a frequency of 2450 MHz. The magnetron designed for consumer microwave ovens Is different from the magnetrons In the Institutional types of ovens and ovens operating at 915 MHz frequency. There is a variation in the microwave power come- up time in different models of microwave ovens. It takes from 2 s to 4.5 s, depending on the oven manufacturers, for the magnetron in a consumer oven to operate after the power is turned on. This is because, for a magnetron in a consumer oven to generate microwave power, the filament must come to the operating temperature. Normally, it takes about 1.5 s for the filament to come to the operating temperature in a new magnetron (Gerling, 1987). Variations in the lag time for a magnetron to operate after the oven power is on make a great difference in the final temperature distribution within a small product for the same time and power setting. Operation of a magnetron is partly determined by the amount of power reflected back to the magnetron and the phase of the reflected power. This can be measured in relationship to the delivered power as voltage standing wave ratio (VSWR), which is dependant on the exact nature of the impedance in the feed system and the load (Gerling, 1987). The power generated from a magnetron is affected by three factors: line voltage, heat, and load. According to Gerling (1987), a change of ±5% on line voltage can lead to ±5 to ±20 % difference in power supplied. Overheating of the magnetron can cause a reduction of power to the oven by as much as 10%. The size of the load in the cavity determines the amount of power reflected back to magnetron which in turn affects the performance of the magnetron. 19 Because of the reduction of power generation due to the reflected power from the cavity, oven manufacturers have Installed glass or ceramic shelves to absorb some power and reduce the power reflection. The efficiency of microwave power absorption by a product is affected by the load size in the cavity for the same reason. The larger the load size, the lower the power reduction by reflection and cavity loss and, hence, the higher the efficiency of microwave power absorption. 2.2.2 Power Distribution System Waveguide: Transmission lines, which can convey electromagnetic waves only in higher-order modes at high frequencies, are usually called waveguides. Microwaves can be considered to travel down the waveguide in a zigzag fashion with waves being reflected repeatedly between the metal walls of the guide. The waves of higher-order modes have components of E or H in the direction of propagation. The cutoff frequency (or cutoff wavelength), below which the transmission is impossible in the waveguide, occurs when the wavelength is of the same order of magnitude as the dimension of the line (Kraus, 1984). The longest wavelength which can be transmitted in a higher-order mode is twice the width of the spacing of the waveguide. Mode Stirrer: The mode stirrer in a microwave oven rotates when the power is turned on and causes the magnetron frequencies to shift slightly as it rotates, changing the field conditions. It enables different mode patterns to be alternated in the oven during the heating 20 cycle. A combination of a number of different patterns of modes during heating helps to ensure uniform heating. With the mode stirrer, unfavorable coupling between modes is avoided by the time separation of modes. Such coupled modes are considered as degenerative and generally give a diagonal power density pattern as opposed to the regular single mode pattern which parallels the vertical cavity walls (Risman et al., 1987). The mode patterns excited at different times and from different locations are less structured and may be non-symmetrical. This creates an effective field which has maxima and minima at different locations. Rotating food on a turntable in reality is analogous to a mode stirrer. During microwave heating of food, field pattern will change with the heating cycle since dielectric properties of food are temperature dependent. Moving food in the microwave oven will affect the mode patterns and allow food to pass through areas of high and low power density and, thus, improve the heating uniformity (Ohlsson, 1989). 2.2.3 Oven Cavity The dimensions of the oven cavity are generally several times greater than the wavelength at 2450 MHz in free space. Multiple reflections of microwaves in the microwave oven result in a number of possible three-dimensional standing wave patterns. According Watanabe et al. (1978), the power density distribution in a domestic small sized microwave oven contained over ten propagating modes and many non propagating modes. In general, the field in a typical microwave oven 21 can be described as a mixture of about 20 to 30 modes. Thus, the microwave oven is often referred to as a multimode cavity; the number of modes Increases rapidly as the cavity size Increases (Lorenson, 1990). The field distribution will change if the microwave generation and distribution system is changed. Also, putting a container of food into the oven changes the reflections and will, therefore, modify the mixture of modes in the oven. 2.3 Product Properties The mechanisms by which the constitutive parameters govern the conversion of electromagnetic energy to heat are the flow of free charge carriers and the polarization effects. These constitutive properties are, in general, functions of geometry, frequency, and temperature. 2.3.1 Permittivity and Conductivity of Dielectrics For a dielectric material, the electrons are bound at the equilibrium positions and cannot be detached by the application of electric fields; and there is no migration of charges in a dielectric under the electric field. Under normal conditions, the negatively charged electron cloud surrounds the nucleus symmetrically. When a dielectric is placed in an electric field, the electron cloud becomes slightly displaced or asymmetrical with respect to the nuclei. field is removed, the atoms and molecules return to their normal unpolarized state and the dipoles disappear. Once the 22 field is removed, the atoms and molecules return to their normal unpolarized state and the dipoles disappear. The major component of most foods is water. Hater is a dipole molecule which will become polarized under an applied field. If the electric field oscillates along the length of the water dipole, it sets up a vibration along the bond. When the electric field is perpendicular to the bond length, there will be a torque exerted on the molecule which will set it into rotation. electric dipole moment. The polarization of a dielectric produces an For the energy to dissipate as heat, it must be transferred into molecular vibration against its surroundings rather than remain as absorbed internal vibration (Curnutte, 1980). The constitutive parameters, permittivity, permeability, and conductivity are important parameters in describing the interaction of electromagnetic waves with the material. the capacitance C per unit length. The permittivity e is equal to The permittivity e is a complex quantity of great importance in the microwave heating of foods. The complex form of permittivity e - e' - je" has real and imaginary parts, both of which are frequency dependent. The real part of the complex permittivity e' is the dielectric constant. the dielectric loss. The imaginary part en is The permeability p is equal to the Inductance L per unit length; and, the conductivity a is equal to the conductance G per unit length. According to Eq. 2.6, the space rate of change of H equals the sum of the conduction-current and displacement-current densities. will be three possible conditions: (1) oc » o. There o, (2) oe » o, or (3) oc « When the displacement current is much greater than the conduction current, as in the case of condition (1), the medium behaves like a dielectric. If o - 0, the medium is a perfect, lossless dielectric. On the other hand, when the conduction current is much greater than the displacement current, as in condition (3), the medium may be classified as a conductor. In conductors, current results from movement of free electrons because of conductivity o. In dielectrics, the bound charges are predominant; therefore, the current is normally considered from the displacement jueE. Since both conductivity and permittivity are functions of frequency, frequency is an important factor in determining whether a medium acts like a dielectric or like a conductor. Frequency also determines how microwave power is absorbed by a dielectric and dissipated as heat, since the dielectric properties are functions of frequency. As frequencies increase from low values, the polar molecules can follow the change in the direction of the electric field up to a critical point. At that point, the dipole motion can no longer keep up with the changing field; as a result, the dielectric constant drops while the dielectric loss increases with increasing frequencies in this region. The energy is absorbed asa result of phase lag between the dipole rotation and the field. the These dispersion and relaxation behaviors, in relation tothe frequency change, were modeled by von Hippel (1954) as shown inFig. 2.1. The energy lost from the wave, as a result of attenuation, is dissipated in the propagating medium as heat. The real part of Eq. 2.8, also referred to as effective conductivity oe - (o'+ue")E, represents a dissipative current in phase with E; and (ue'-o")E, or the effective permittivity ee, represents a reactive current out of phase with E 24 (Nelson, 1973). The effective conductivity o0 and the effective permittivity e# can also be measured experimentally as dielectric loss and dielectric constant, respectively. These empirical constants imply the capacity of a material to interact with the field (dielectric constant) and the ability of a material to convert the absorbed energy to thermal energy (dielectric loss) (Curnutte, 1980); a" generally turns out to be negligible and, in this situation, ee is still referred to as the dielectric constant, which decreases with increasing frequency as seen in Fig. 2.1. Dielectric constant is associated with the stored- energy density e#E2 - e'E2, while dielectric loss oe (includes both the o' and the e" components) is considered to be the total loss current associated with the energy dissipated as heat, oeE2 - (o'+ue")E2 (Vermeulen and Chute, 1987). ar'T Km-K* to r 100 CUT Fig. 2.1 Dielectric properties as functions of frequency (von Hippel, 1954) 25 The ability of a material to absorb microwave energy and dissipate as heat can be expressed as dissipation factor, which is normally known as the loss tangent (Nelson, 1973). The loss tangent is defined as tan 6 - a ^/na '. 6 is the angle of a phase difference which exists between the field intensity E and the current density; it is called the dielectric loss angle. For a nonconducting dielectric material the conductivity a is zero, then the loss tangent can be expressed as tan 6 - «€"/»«' “ «"/«'. (2.26) The inductive losses in biological materials are normally considered zero; i.e., the relative permeability fir is assumed to be unity. The dielectric constant of a material is associated with the capacitive loss which does not affect the heating rate of a material; whereas the dielectric loss, which is related to the resistive loss, determines the voltage coupling of the material and power absorption levels. The variation of power absorption with dielectric loss can be shown as (Mudgett and Nash, 1980) P" - 2jrfe0E2ep" (2.27) where P" is the volumetric microwave power absorption [w/m3J, f is the frequency, e0 is the permittivity at free space, E is the electric field er" is the dielectric loss. intensity, and The phases of different intrinsic impedance in a heterogeneous system may be arranged in series or in parallel analogous to electrical 26 circuits (Mudgett and Nash, 1980). The phase which intercepts more power will heat at a faster rate than the phase that absorbs less power. 2.3.1.1 Effect of Composition The composition of food affects the dielectric properties which, in turn, affect the heating characteristics of food. Uater accounts for much of the primary absorption of microwaves in biological materials. Dielectric properties of food systems, as functions of frequencies and temperatures, can be predicted based on the activities of water molecules, conductive ions and inert solid contents. The model can be useful for product formulation of composite food systems to reduce the phase differential heating effect. Dielectric properties of biological materials have been studied extensively by Risman and Bengtsson (1971), Mudgett et al. (1974, 1977, 1979 and 1980), and Ohlsson and Bengtsson (1975). Most of these works are related to the measurement and modeling of dielectric properties in various food products. It has been found that the dielectric behavior of food components is mainly affected by moisture (especially unbound water) and ash content (dissociated salt ions in the aqueous solution). In general, the higher the moisture content, the higher the dielectric constant. Dielectric loss increases with increasing moisture content to a point, then it decreases with higher moisture contents (Schiffmann, 1986). ensure that the majority of microwaves are being absorbed by foods, materials of the mlcrowaveable containers are designed with low To 27 dielectric loss; therefore, the containers do not absorb much microwave energy. Dissolved salts or ions bind water molecules and depress the dielectric constant to a level lower than that of pure water. The dielectric loss of aqueous ionic solutions is elevated above that of pure water due to conductive and/or electrophoretic migration of free ions in the applied field. In the aqueous mixtures, however, both the dielectric constant and dielectric loss are depressed by insoluble and immiscible constituents such as lipids, proteins, and carbohydrates. (Mudgett, 1985). 2.3.1.2 Temperature Effect Vibration of a dipole is not only dependent on the frequency and intermolecular bindings, but is also temperature dependent. Foods at high temperatures absorb less microwave energy than at low temperatures. At high temperatures, thermal movements are more intense, and energy necessary for overcoming the inter-molecular bonds is less; therefore, the dielectric heating decreases with the increasing temperature (Ohlsson, 1989). Debye predicted the dielectric behavior of water as functions of frequency and temperature. Increasing the temperature depressed both e ' and e" of free water at fixed frequency and shifted the region of dispersion to higher critical frequency (Mudgett, 1985). In aqueous ionic solution, hydrated ions try to move in the direction of the oscillating electric field. During their movement, the ions transfer energy randomly to adjacent water molecules. Water 28 molecules are more mobile and less tightly bound to the Ions at higher temperatures. Thus, Ions absorb and dissipate energy freely, and the conductive heating Increases with the Increasing temperature (Ohlsson, 1989). The dipole component of dielectric loss decreases as the temperature Is Increased, while the conductive dielectric loss increases as the temperature is Increased. Thus, the total dielectric loss of a food initially decreases as the initial temperature Increases, when it is dominated by the dipole loss effect. The total dielectric loss then increases as the temperature increases further, when the conductive effect dominates (Mudgett, 1985). 2.3.1.3 Effect of multiphase mixture Mudgett et al. (1974) modeled the effect of structure on the dielectric properties of a material by using a combination ofthe Maxwell and Rayleigh models. This model was originally proposed for the prediction of complex conductivity of randomly distributed spheroidal particles in a continuous phase by Fricke (1955). Mudgett adapted the model for the prediction of the complex permittivity for a non*interactive two-phase mixture: . _ ecl€t ii+xvt) +eji-v.)x] e0U+V,)+€#(l-V,) where «m is complex permittivity of a mixture, ec is complex permittivity of continuous phase, eg is complex permittivity of suspended phase, (2 28) 29 X is fraction of phase, Vg is volume fraction of suspended phase in the mixture. Thus, the dielectric properties of a non-interactive, two-phase mixture may be predicted from the proportion and material properties of each phase if the particle shape and orientation are known. The density of a product has an effect on its dielectric constant. Since the dielectric constant of air is very low, the presence of air will reduce the material's dielectric constant. The dielectric constant of a material increases almost in a linear fashion with density (Schiffmann, 1986). The study by Rzepecka and Pereira (1974) on dried milk and dried whey powder showed that the dielectric permittivity (e) is linearly correlated with bulk density under constant moisture content. Nelson (1980) also found that the dielectric constant of fresh fruits and vegetables correlated with moisture content and tissue density. 2.3.2 Transmission Properties Vhen a wave is propagating in an unbounded medium of permeability and permittivity e, the velocity of the wave is v - 1/7pe, or the phase velocity is - ■“ ■4m ■f <*•**> The index of refraction rj is defined as the reciprocal of the relative phase velocity of the wave in the medium with respect to the velocity of 30 light, or rj - ^/ir«r. For nonferrous media, fir is nearly unity, so that The refraction of microwave energy is proportional to the square root of the dielectric constant. much greater than that of air. Most foods have a dielectric constant The refracted wave in food travels in a direction close to the surface normal. Consequently, once the microwave energy is transmitted into the food, there is very little possibility for it to travel close to the interface between the food and the air or escape out of food (Ohlsson, 1990). The relative intrinsic impedance of a dielectric material, as defined in Eq. 2.30, determines the level of power transfer from the field in terms of reflection and transmission at the boundaries. (2.30) j*7n The relationship of power incidence and transmission is expressed by Snell's law as indicated in Eq. 2.17. The relative reflected and transmitted power with respect to the total power incident may be estimated from two coefficients, the reflectivity coefficient and the transmissibility coefficient, which are defined as follows: (2.32) Et 1* T^CQSflt T^cosflj where E{ is the incident electric field, Ep is the reflected electric field and E( is the transmitted electric field (2.33) 31 At the boundary of two media with extremely different dielectric properties, the reflected wave is nearly as large as the incident wave. The field is essentially a standing wave. Whereas at the boundary of two media with similar dielectric properties, the reflected wave is small and the field is a traveling wave. There are three types of impedance: intrinsic, characteristic, and wave. The characteristic impedance is basically related to electricity. The intrinsic impedance and the wave impedance are field or wave quantities involving ratios of electric field Intrinsic impedance refers to the ratio tomagnetic field. of H for a plane (TEH) wave in an unbounded medium. complex the phaserfieldsE and For a medium with and e, the intrinsic impedance is calculated as (2.34) For low conductivity dielectric material, substituting y in Eq. 2.34 with jut/jue, the intrinsic impedance becomes (von Hippel, 1954) (2.35) The wave impedance refers to the ratio of an electric field component to a magnetic field component at the same point for the same wave. For a TEM wave the wave impedance is the same as the intrinsic impedance; but for higher-order modes, there can be as many wave impedances as there are combinations of electric and magnetic field components. The transverse-wave impedance is a function of the intrinsic impedance of the medium, and is also the dimensions of the 32 waveguide. As Che dimensions become very large compared to the wavelength, Che transverse-wave Impedance approaches Che intrinsic impedance of Che medium (Kraus, 1984). 2.3.3 Thermal and Physical Properties In solid food materials, the rate of heat conduction is much slower than the rate of microwave heating. Uneven microwave absorption and slow heat conduction result in temperature distributions in food during microwave heating. The only remedy is to heat the food product at a lower power level (Ohlsson, 1983). The microwave absorption by fats and oils is very low compared to that of high moisture foods. their low specific heat. But fats and oils heat well because of It takes less time to heat fats and oils to a certain temperature than water under the same mass and the same heat flux. 2.3.4 Geometrical Configurations of the Product Kritikos and Schwan (1975) indicated that the microwave energy absorbed by a large spherical biological tissue at high frequencies is similar to that absorbed by an infinite slab. In this situation, heating is described as a skin phenomena and microwave energy decays exponentially into the sphere. Kritikos and Schwan (1975) also characterized the maximum microwave heating in the spherical geometry over a wide spectrum of frequencies in terms of quasi-physical optic 33 region and resonance region. When the radius of the sphere Is longer than the wavelength Inside the sphere (high frequency region), the field converges and Is concentrated In the half wavelength region (quaslphyslcal optic region). When the wavelength exceeds the radius (low frequency region), the electric and magnetic modes resonate inside the sphere and nonuniform heating occurs. The effective gain of the power at a certain distance from the center in one dimension can be calculated by the product of the ratio of the surface area to the inside area and the attenuation effect at that area as shown in Eq. 2.36 (Kritikos and Schwan, 1975). Ap 9Min where PR" . 2 * ° --------- . p" e-2.(ii-r) isthe power per unit surface area at r - R, Pr" isthe power per unit surface area at r, Ar isthe outside surface area at r - R and Ar (2.36) At is the surface area at r. If the value of the effective gain in power is greater than one, then a concentrated heating effect will occur inside the sphere. Ohlsson and Risman (1978) studied temperature distribution due to microwave heating in spheres and cylinders of different diameters. found that the concentrated heating effects are more pronounced for spheres than for cylinders. Conductive heat transfer proceeds at a slower rate than the concentrated heating effect resulting from the geometrical configuration. The study of the effect of geometrical They 34 configuration will help to eliminate the problem of nonuniform heating by the proper design of the shape and size of a product. 2.3.5 Loading Effect The efficiency of microwave power absorption is affected by the load condition in the oven. efficiency." This is sometimes referred to as "coupling Energy coupling efficiency is dependent on the impedance characteristics of the microwave generator (Mudgett, 1986). Power coupling by a loaded microwave cavity is affected by product impedance characteristics which may vary considerably during heating. An acceptable load impedance is one which couples sufficient power from the magnetron to prevent excessive anode heating, and yet, does not load the magnetron so heavily that it falls to oscillate at the correct frequencies, shifting to another mode (Mudgett, 1985). Furthermore, when foods with different impedance characteristics are heated together in a microwave oven, there will be a competition for the microwave power between them (Ohlsson, 1983). The coupling efficiency can be estimated empirically by a correlation based on liquid and solid calorimetry. The true power incident at the dielectric surface can be estimated from the maximum power generated from the magnetron by using either of the two equations given by Mudgett (1986) as follows: P0 - PB [l-exp(-aV)] P ■ 0 * Km + Vb (2.37) (2.38) 35 where P0 is the power coupled by unmatched load, Pm Is the maximum power generated by the frequency generator, Kg is the empirical coupling constant, a,b are empirical coupling coefficients. The coupling coefficients are unique for each heating apparatus. They can be determined by calorlmetrlc measurement. The energy coupling efficiency (Ey) is determined as: E v - -^2 x 100% 2.3.6 (2.39) Packaging Since basic microwave oven performance cannot be changed, food processors and manufacturers can only manipulate the ingredients and the packaging system to improve the quality attributes of microwaveable foods. Factors such as shape and size of food product, covering material, and cooking directions can significantly influence the performance of microwave cooking. It is very important for food companies to conduct thorough testing and evaluation to see how the designed package performs under a variety of conditions in different microwave ovens. 36 2.4 Modeling of Microwave Heat Transfer Mathematical modeling can play an Important role in the design and optimization of the product and process parameters in order to obtain the best possible improvement in heating. 2.4.1 Heat Transfer Models The general model takes into account conductive heat transfer, internal microwave heat generation, and surface cooling due to convective and evaporative heat losses at the boundaries. Therefore, initial temperature, ambient air temperature, and water vapor pressure at the surface of the product and in the air are needed to describe initial and boundary conditions. Ohlsson and Bengtsson (1971) developed microwave heating profiles by computer simulation using finite difference technique and showed good agreement with experimental time-temperature data collected in meat products during microwave heating. He also showed the influence of thermal conductivity of the material and the surface heat transfer coefficient on heat transfer during microwave processing. Mass transfer of water vapor from the food surface occurred simultaneously during microwave heating. Surface cooling effects were found due to both the convective heat transfer and evaporative cooling. Kirk and Holmes (1975) modeled temperature profiles in model foods being heated in a microwave field. The model predicted the temperature profile of high moisture content substances at any given time and 37 thickness of the sample by using the finite difference method. The profiles agreed well with the experimentally measured temperature profiles in gels of 1% ion-agar in water. Ofoli and Komolprasert (1988) derived a mathematical model in which the conductive heat transfer was considered negligible in comparison with the internal heat generation due to absorption of microwave power. The temperature increase predicted by this model was purely a function of electric field intensity and time. The center heating or edge and corner heating phenomena which occur normally in microwave heating was not predicted by this model. 2.4.2 Heat and Mass Transfer Models Lyons et al. (1972) have done experiments to measure the radial and axial changes in temperature, moisture, and pressure with respect to time in porous cotton yarn cylinders during microwave heating. During microwave heating, pressure may build up due to the accumulation of super-heated steam. Pressure increases significantly as the temperature rises to the boiling point, and results in explosions and volcano effects in eggs and sauces cooked in the microwave oven. The phase change may cause distortion on the product. A water-laden sandstone model system was used by Wei and Davis (1985) to predict heat and mass transfer phenomena in porous materials during microwave drying. data. Their model agreed well with experimental The model also predicted local moisture concentration, gas densities, and pressure gradient with respect to time. During microwave 38 heating, the center of sandstone Is hotter than the surface; air flows slowly toward the center and water vapor migrates toward the surface as liquid vaporizes on its way out. Both heat and mass transfer must be considered when modeling microwave thawing or dehydration. Thermal and dielectric properties vary drastically with the phase change during thawing. much work reported in these areas. There is not Taoukis et al.(1987) used a modified Isotherm migration method to develop a mathematical model to predict the thawing time and temperature profiles of cylindrical meat products during microwave processing. Finite difference method was used to solve the heat and mass transfer equations under the moving boundary condition. The model and experimental results compared well at low microwave power at 2450 MHz. Low frequencies, 300 MHz and 915 MHz, were found to thaw the frozen meat with more efficient energy consumption as compared to 2450 MHz. 2.4.3 Numerical Techniques Numerical models can also be used to predict heat transfer and for optimization of microwave packaging and food formulation. The analysis can be carried out by dividing the food load and container into discrete elements. The nonuniform heating effect and the differential heating effect of multi‘Component meals or layered products can also be studied by knowing their dielectric, thermal, and physical properties. So far, the development of computational methods and techniques for analysis of the multi-mode cavity has been very limited. A three 39 dimensional, finite element method makes it possible to numerically calculate the absorbed power density in every discretized volume element. Ohlsson and Risman (1978) developed a finite element program to investigate the n'onuniform temperature distribution in model foods of simple geometries. The microwave power density at a given location in the sample was modeled for a cylinder and a sphere. normal plane wave incident at the surface. The model assumed TM and TE incident waves in the x, y, and z axes were considered for sphere, whereas only horizontal (TE wave in y direction) and vertical (TM wave in z direction) were considered in the cylindrical model. Uithin the sizes and shapes of meat and potato products studied, the model simulated temperature distribution very well. Center heating effects were found in spheres of diameters less than 6 cm and in cylinders of diameters less than 3.5 cm for the particular material studied. Swami (1982) developed a finite difference model to describe microwave heating of high moisture foods in cylindrical and rectangular shapes. The volumetric power for each finite shell was derived from the surface power assuming a 45° or 90° incident angle. Convective and evaporative surface heat loss were considered in the boundary conditions. Dielectric properties of the gel samples at various sodium chloride concentrations were predicted by Hasted-Debye equations. The model predicted temperature distribution in good agreement with the experimental measurement for gel samples of high moisture and different salt concentrations under two-dimensional and three-dimensional microwave heating. The finite element method is more versatile and has several advantages over the finite difference technique. It can handle irregular geometries, analyze nonhomogeneous and nonisotropic food products and is generally more accurate (Segerlind, 1984). The finite element method (FEM) is a procedure for solving problems by dividing the domain of interest into small, basic elements. The FEM is derived using either variational methods or Galerkin's method for solving operator equations. The starting point for this method is the partial differential equations describing the process. The solution is approximated by writing it as a linear combination of known functions with unknown coefficients. This approximation function is then substituted into the original partial differential equations, and both sides of the equation are multiplied by the full set of approximation functions. Finally, integrating over the solution region results in a matrix equation, which is solved for the unknown coefficients. provides the approximate solution. This The accuracy of a finite element analysis depends both on the type of approximation function used in each element and on the number of elements used. The sizes of the elements must be chosen to be a small fraction of the minimum wavelength in the material. According to Lorenson (1990), the size of the discretized element should be less than about 3 mm. Many FEM software packages are available for used on mainframes, workstations, and PCs. They leave the engineers free to focus on the design issues instead of algorithms (Cendes, 1989). Finite element analysis was used to solve the Maxwell equation for the three dimensional microwave field of a microwave heating furnace 41 system (Tejika et al. 1987). In this case, the configuration and the operation of the microwave transmission system and the cavity must be clearly defined. The major drawback in using numerical methods for calculating the power density In a microwave oven is the difficulty in clearly defining the mode pattern. Full specification of the mode pattern may be impossible because of the theoretical complexity of the microwave propagation within the oven and because of the interference of the field pattern from various load conditions in the cavity (Risman et al., 1987). 2.5 Summary The uneven temperature distribution in food during microwave heating is affected by the microwave field distribution and by the characteristics of the food. The electromagnetic field inside the oven can be described by Maxwell equations, but are difficult to solve analytically. It is necessary to develop a technique to measure the microwave field distribution. Thus far, methods of measurement for the field distribution inside a microwave oven were found in literature to be mostly descriptive. Technique to measure microwave oven field distribution quantitatively has to be developed. Parameters of the food product to be heated in a microwave oven are as important as the field distribution in determining the temperature distribution in a food. Factors such as geometrical configuration, chemical composition, and packaging materials may interact with the electric field and, in turn, influence the microwave 42 power absorption by the product. The dielectric properties and transmission properties of a material are functions of microwave frequency, temperature, and chemical/physical properties. Therefore, heating characteristics of food during microwave processing is a complex phenomenon. A comprehensive experimental design may help to elucidate the complex system and identify the most critical factors. A mathematical model can incorporate the hypothesis of the mechanisms behind the microwave heating behavior of a food and be verified by experiment. Most of the mathematical models for the microwave heating process, found in the current literature, were solved by the finite difference technique. Though FEM is a fairly new numerical method, with its flexibility in handling irregular geometrical configurations and material properties, it would be the best numerical method to solve the microwave heating process in foods. Therefore, the objectives of this research were: 1. to characterize patterns of microwave power field distribution of the microwave ovens studied; 2. to study the effect of geometrical configurations of model food systems on the temperature profiles during microwave heating; 3. to study the effect of dielectric properties of food on microwave heating due to the change of composition; and 4. to develop a heat transfer model and to use a finite element technique to predict temperature distribution in foods of cylindrical and rectangular shapes during microwave heating. This research was designed to fulfill the objectives stated. Is hoped that more elaborate modeling can be developed In the future based on this research to simulate the dynamic interaction occurring during microwave processing. It 44 Chapter 3 MATERIALS AND METHODS This research was conducted by two approaches— theoretical and experimental. A microwave heat transfer model and a method to measure the temperature distribution of high moisture foods were developed. The experimental results were used to verify the temperature prediction by the model. This chapter provides a detailed description of the derivation of microwave power absorption function at different locations in the food and the temperature measurement technique used in this research. 3.1 Characterization of the Microwave Oven Household microwave ovens vary greatly in the design of microwave power generation and distribution systems, cavity size, and type. Variations also can be found In accessories such as turntables and temperature control features. The configurations of the oven, together with the interaction between microwaves and the load, determine the mode pattern for that oven. The field distribution can be obtained by solving the Maxwell equations if the multi-mode pattern can be clearly defined. The mixing of the electric field by the mode-stirrer and the continuous interaction of the field with food load during microwave heating make it very difficult to compute the solution of the electric 45 field by Maxwell equations; therefore, an empirical technique was developed to characterize the oven power distribution. 3.1.1 Empirical measurement of the Power Distribution To determine the microwave power distribution quantitatively, the microwave power at any location in the oven needs to be measured independently. Presence of more than one load in the oven during the power measurement will dis'tort the mode pattern. The procedures for the determination of microwave power distribution were described as follows. 3.1.1.1 Microwave Power Measurement A Tappan 500 (Model 562077) microwave oven operating at 2450 MHz was used in this research. The line voltage on the power supply was monitored and maintained by the Powerstat (Model 3PN126, Superior Electric Co. Bristol, CT) at 115 V. Power output was determined calorimetrically according to the standard procedure of the International Electrotechnical Committee (IEC) (Schiffmann, 1987). Deionized water weighing 1000 g and equilibrated at a temperature of 5°C below room temperature, was heated in the microwave oven at full power. Heating was continued for a period of time until the final temperature of the water reached about 5 °C above room temperature. Temperatures of the water before and after heating were measured by using an Omega type K thermocouple probe (Omega, Inc., Stamford, CT). Water was thoroughly 46 mixed with a spatula before temperature measurement. The microwave power absorbed was calculated by Eq. 3.1. . mCp& T (3>1) At where m is the mass of the water, [kg] Cp is the specific heat of water, [4.184 kJ/kg°C], AT is the temperature difference between the initial temperature and the final temperature, and At is the duration of microwave heating time. The time needed for the magnetron to reach a steady state was determined using a Soar ME-530 digital multimeter. The time period for the filament in the magnetron to warm up to the operating temperature is the come-up time of a magnetron. When the magnetron was operating at its full power, line voltage dropped from 115 V to about 112.8 V. come-up time was recorded by a stop watch. This The magnetron come-up time was subtracted from the time setting of microwave heating to give the net time of duration of the microwave heating. 3.1.1.2 Measurement of Power Distribution Power distribution was measured by determining the rate of heating of water in a plastic cup at different locations and at different heights of the oven cavity. The dimension of the microwave oven cavity was measured to be 29.5 cm x 28.5 cm x 16.5 cm. The horizontal plane of the oven cavity was divided into grids of 3 cm x 3 cm and spaced 1.25 cm 47 vercically as illusCraCed in Fig. 3.1. To standardize the operation of the oven for each test, a 1000 mL of water was heated in a glass beaker 5 min before the first test of the day. Microwave power was also measured periodically according to the standard procedure to make sure that Che oven performance had not changed. I OVEN CAVITY (0 ,0 ,0) Fig. 3,1 Discretization of the power measurement points for microwave power distribution in a Tappan 500 microwave oven A small polyethylene cup (4 cm radius by 3 cm height) was used to contain the water for microwave power measurement at each junction of Che grid shown in Fig. 3.1. According to Ohlsson (1989), small cylinders of dielectrical materials will exert little interference to the microwave field. In each test, 100 g of deionized water at about 15°C ± 3°C was heated in the oven at each of the 245 junctions on the grid for 20 s at full power setting with Che line voltage maintained at 115 V. Temperature measurements at points located at elevated levels 48 were conducted by elevating the cup with styrofoam blocks, which are transparent to microwaves. Before the temperature measurement, water was thoroughly mixed with a plastic spatula to get an average temperature reading. Temperature measurement by a thermocouple probe was done within approximately 8 s after the microwave heating. Complete randomized experimental design was used and three replications were performed, 3.1.2 Model Development for the Power Distribution The temperature increase at different locations in the oven cavity was calculated from the initial and final temperatures obtained from the power distribution measurement. Temperature increase values of four replications at all the locations were analyzed by Statistic Analysis System (SAS Institute Inc., 1985) on an IBM 3090. Stepwise procedure was used to find the polynomial model that would best fit the experimental data by maximizing the coefficient of determination (R2). Temperature increase, as a dependent variable, was expressed as a polynomial function of the coordinates (x, y and z) of the oven cavity up to the eighth order. The origin of the coordinates of the grid was set at (6.75 cm, 2.25 cm, 5.5 cm) of the oven as shown in Fig. 3.1. The ratio of the temperature increase from the regression model to the mean temperature increase for the region, where the load was located, was used as an index of power distribution at any location in the mathematical model. 49 3.2 Determination of Microwave Power in Relationahip to Loading Microwave power absorbed by the load inside the oven is affected by the nature of impedance of the load, such as the size, the geometrical configuration, and the chemical composition of the load. Two sets of experiments were conducted to determine the relationship of microwave power to the nature of loading. Microwave power absorbed by deionized water ranging from 25 g to 2500 g in four different sized cylindrical containers and rectangular containers at the center of the oven were determined calorimetrically. The time of microwave heating for each sample was determined according to the size of the load so that the final temperature would not exceed the room temperature by 10°C. The microwave power absorption was calculated from Eq. 3.1, 3.3 Derivation of Heat Transfer Model with Microwave Heat Generation The transient heat transfer equation consists of heat conduction, rate of gain of volumetric thermal energy, and microwave power absorption. Cylindrical and rectangular coordinate systems were used in this study for the prediction of temperature distribution in cylindrical and rectangular shaped samples using the finite element method. The model was developed based on the assumptions that the model food system used to simulate the food product was homogenous and its properties were isotropic. 50 3.3.1 Model Development For an Incompressible gel undergoing a constantpressure process, the thermalenergy equation can be described as: p cpj ¥ - V-(JcVT) + Voix + * (3.2) where the V u :t term Is the irreversible rate of energy change due to viscous dissipation, and can be assumed to be zero when considering the gel as a solid (i> - 0). The thermal energy equation for the cylindrical coordinate system thus becomes: pcjj - 1 3 r p 3t (r^ | T ) + i a z dz dz z 30 r 30 dz a (ic| r ) + 4 ( 3 .3) dz In the first approach, material properties and the normal incident microwave power were assumed constant and symmetrical; therefore, there is no circumferential variation in the thermal and physical properties and microwave power. The terms with d/dO are equal to zero; therefore, the governing equation reduces to: ar i a , a t * £<*£> * * (3-‘ > The heat conduction equation in a cube shaped material (slab) was derived based on the same procedure and is expressed as shown below: 51 (3.5) 3.3.2 Equation for Microwave Power Source (ft) The microwave power source term, ft, represents the microwave power absorption density. For use in the FEM, ft at any location must be specified as a differential equation for use by the finite element package. Such an equation has not been reported in literature and, hence, was derived as part of this research. The exact forms of absorbed power density terms for cylindrical coordinates and rectangular coordinates can be obtained by shell balance as described below. The microwave power attenuates exponentially from the surface of incident following Lambert's law (von Hippel, 1954). The electric field strength, expressed as a function of distance from the surface of a material, is Ex - E0e*sx. Since power is proportional to the square of the electric intensity, the power unabsorbed at a distance from the surface is (3.6) where PQ is the surface power in [W]. 3.3.2.1 Cylindrical Coordinate It was assumed that the incident power at the surface of the material was uniform and normal to the surface. The surface power per 52 unit surface area is F" - P0/A>l where A# is the total surface area of the product. Therefore, for a cylinder Pc" - P^^trlUL+R)),where R Is the radius of the cylinder and L is the height of the cylinder. The total power incident in the radial direction is equal to PR - 2jrRLPc". Substituting PQ by PR in Eq. 3.6, the power propagating in the radial direction from the surface of a cylinder is equal to Pr - PRe‘2a<“'r> - 2trRLPc"e'2a(R'r> (3.7) Dividing both sides of Eq. 3.7 by 2TrrL, the radial surface area at any given r can be expressed as p" . J ± _ Px 2^1 „ 2*RLPJ /e-“ '”-'' 2M L p ri'R) e -2,{*-r> (3.8) r where Pr" is the power per unit area at a distance r from the origin in the r direction [W/cm2]. Consider the power incident at the ends of a cylinder axially, PL/2 - jtR2Pc" and therefore, p!> „ J k * nR* = *X*P, c0 '3''*~') . nR2 (3-9) c where Pz" is the power per unit area at a distance z from the center in the z direction [W/cm2). By doing shell balance on the cylinder as seen in Fig. 3.2, the power absorbed per unit volume by the finite shell in the radial direction is equal to the difference of the power at r+Ar and the power reaching at r divided by the volume of the shell; therefore, 53 rjH _ Pi*i,r i 2 i t ( r + A x ) A z > - P t ( 2 n r A z ) = 2vzAiAz (3 .10 ) where P"ghen.r is the absorbed power density of the shell 2trArAz in the r direction [W/cm3]. The shell balance of the power absorbed between z+Az and z per unit volume of the shell in the axial direction becomes p'',Ag(2xxAr) - P % (2wrAr) (3.11) 2xrArAz where P'ghgit.g Is the absorbed power density of the shell 2irArAz in the z direction [W/cm3]. z - Lz <k L ♦r z- O z - - Lz Fig. 3.2 Shell balance within a cylindrical model 54 The totalpowerabsorbed quantities inEq. 3.10 by the shellequals the sum and Eq. 3.11. Taking limits, as ofthe Ar and Az approach zero, the total power absorbed by an infinitesimal volume at r and z becomes: „/// _ dPr+dP, _ dip" . BPfJ c ~dv ~ zs r (3. 12) Tz where P"c is the absorbed power density at r and z in a cylinder [W/cra3]. Differentiating Pr" and Pzn in Eq. 3.8 and 3.9 against r and z respectively, they become 2*1 - <*-!■) + 2aRp,c 3r r* 2*1 dz (3.13) -r - <3 1 4 > Substituting Eq. 3.13 and 3.14 into Eq. 3.12, power absorbed per volume in a cylinder becomes _ dip" c ” - 3.3.2.2 rdr , dp'j _ P'J , dP? dz 2 a R P ° 0 -2m(M-ri ~ z dP? dr dz (3,15) + 2 a p f ' e im{i ' a) Rectangular Coordinate For a rectangle with dimensions L, x 1^ x 1^, the power per unit surface area is P8" - P ^ 2 (L1L^+LjL^+L^L^). direction is Surface power along x and y 55 Pl. . P l ,, L*L >P° 2 (LjLj+LjLj+LjLj) ' ■= L^ p o 2 [ L lL 2*LiL i*LiLi) ‘ respectively; therefore, microwave power propagating in the direction of x and y from the surface is: Px “ Pl,© a respectively. and y -a»ti -*) -a<<h -y> 2 , Py ■ Pr,e 2 , a To obtain the expression for unit surface power in the x direction, divide the Px and Py by their surface areas,LgL^ and LjL^, respectively: p" - - p?e'a*<:£"x) (3 -16) £2£j p" - - l z - . P " e ~ 2 m i ^ * ~ y) ( 3 . 1 7 ) LjLj Therefore, the power absorbed per unit volume becomes p/// = U y L ^ U x - ^ y L ^ ) + (AxL3PytAy-AxL3Py) * " A xAyLj (3.18) „ P'Ux-LP" + PyUy-APy Ax Ay Taking limits as Ax and Ay approach zero, Eq. 3.18 becomes a derivative of absorbed power density in x and y direction in the unit of [W/cm3]. • art dx _ ap1 ; dy « f = 2aPite _a « (3.19) (~-x) -2*{h-y) 2 ♦ e 2 ] There is a considerable amount of power transmitted from the surface in the negative directions of the x and y axes if the size of 56 the product is on the order of magnitude of the wavelength in the material. Therefore, the absorbed power density equations for cylinders and rectangles become: (3.20) 4>(x,y> (3.21) Also, because the permittivity of the material heated in the microwave is significantly higher than that of the ambient air, internal reflection should be included. In the case of the first internal reflection, additional power can be added to the source surface power per unit surface area as, P" + P"e'2al-. For multiple (n times) internal reflection as in the case of a small sized product, P"-P"(l + e‘2<lL + e-4«L+....+e-2n«Lj jn this study, only the first internal power reflection term was considered. 3.3.3 Boundary Conditions The boundary conditions for the heat transfer problem in this study are of three kinds. They are: (1) no heat flux (i.e. thermal insulation), (2) convective heat loss when the surface exchanges heat with the surroundings, and (3) evaporative heat loss at the surface 57 where moisture exchange with the surroundings took place (i.e. when the surface was not covered with materials such as Saran wrap). (1) Insulated boundary: q - 0 (3.22) (2) Convective heat loss: q - h (T - Ta) (3.23) (3) Evaporative heat loss: q « where (3.24) ih y A H y /A h is the convective heat transfer coefficient, [W/m2>C], Ta is the ambient temperature, niy is the mass flow rate of water vapor, [kg/s], A H y is the latent heat of vaporization, [kJ/kg*C] and A is the surface area. The surface heat transfer coefficient, h, was assumed to be 37.05 W/m2,C for slab and 39.44 W/m2*C for cylinder (Swami, 1982). The values for mv and AHy as functions of temperature were evaluated from Swami (1982) and Heldman and Singh as follows: ( 1 9 8 1 ) n i y / A - k y ( T ) P y ( T ) k y ( T ) - k y + A H y ( T ) - 2 5 0 1 P V ( T ) - 1 . 1 [ ( T 7 . 8 8 6 2 7 - 3 ( R H - R H a ) / 1 ) / ( T 2 . 3 9 a + 2 7 3 0 0 ( 3 . 2 5 ) ) ] 1 *5 ( 3 . 2 6 ) ( 3 . 2 7 ) ( 3 . 2 8 ) T e ( 0 - W 9 8 T ) where ky is the mass transfer coefficient of water vapor in air, [kg/s'm2-kPa], Py is the water vapor pressure in [kPa],and RH is the equilibrium relative humidity. The mass transfer coefficient is a function of temperature due to the temperature dependency of diffusivity of water vapor toair (Bird al., 1960). et The equilibrium relative humidity of the sample was assumed to be 99% and the RHa of ambient air was assumed to be 60%. Foods may 58 have water activity lower than 99%, and the humidity in the oven cavity may also vary depending on the amount of water vapor driven out of the food during microwave heating. The fan inside the oven ventilated the air during heating and, therefore, kept the relative humidity of ambient air at a relative constant during microwave heating. 3.3.4 Material Properties Thermal conductivity of the gel was assumed to be constant within the temperature range in this research (Swami, 1982), though it may vary considerably with temperature in the case of thawing or freeze-drying. Density of gel was determined from weighing 100 mL gel in a graduated cylinder before the gel was solidified. Specific heat of the gel was determined from the Dickerson equation Cp - 1.675 + 0.025 w (Singh and Heldman, 1984). Density, p - 1010 kg/m3 Thermal conductivity, k - 0.8374 W/m°C Specific heat, Cp - 4.12 kJ/kg°C Dielectric loss of the ionic solution increases because of the increased ionic conductivity, since o - o«r" (Nelson, 1973). Dielectric properties of sodium chloride solutions at different concentrations have been measured by Mudgett (1986) as a function of temperature. Dielectric properties of the gel in this research were found to be similar to that of aqueous ionic solutions based on selected determination of dielectric loss and dielectric constant of the gel samples by an outside laboratory. In order to determine the equivalent 59 sodium chloride concentration for the gel, the conductance of the gel samples was measured by a conductivity bridge (Model FM-70CB, Sybron/Barnstead). Standard curves were obtained by measuring conductance of NaCl solution of 0, 0.05 M, and 0.1 M concentrations from 20°C to 80°C. The conductance of gel measured from 23°C to 77°C was compared with the standard curves, and the equivalent NaCl concentration was thus determined. NaCl concentration. The gel was found to be equivalent to the 0.05 M Dielectric properties data from Mudgett (1986) for 0.05M NaCl solution were used for the gel in this study. From these data, dielectric properties were expressed as functions of temperature as shown in Eq. 3.29 and Eq. 3.30. 3.4 er' - 81.79 - 0.299 T (3.29) er" - 22.6 - 0.378 T +0.00293 T2 (3.30) Finite Element Method and TWODEPEP TWODEPEP, a commercial finite element software package available on the Penn State IBM 3090-ES mainframe was used. The software is in FORTRAN and distributed by the International Mathematical and Statistical Libraries (IMSL) Inc. TWODEPEP can solve transient heat conduction equation with internal heat generation. TWODEPEP solves the parabolic (time-dependent) partial differential equation of heat conduction with heat generation in twodimensional and axisymmetric regions. It uses six-node (and higher order) triangular elements, with the quadratic basis shape function, and the Crank-Nicolson scheme to discretize time domain (IMSL, 1984). The 60 initial triangulation was made with only enough triangles to define the region, where the parameter NTF defines the number of final triangles desired. A function D3EST(x,y) was used to refine the triangulation by setting the appropriate density of final triangles for the region of interest. Fig. 3.3 shows the initial triangulation of all the geometries studied. Table 3.1 shows the element sizes and time steps of cylindrical and slab geometries determined by minimizing the percent error between the FEM prediction and the experimental results. The element sizes and time steps were examined to be within the constraint of the following equation (Puri, 1989): At < (3.31) 2l\ 9D(1 -0) where At is the time step, 1 is the time coefficient, which equals to Cl or pCp, A is the element size, D is the material property or thermal conductivity, 0 is a time-shape function, which equals to 0.5. k, In every case, the algebraic equations are solved by Newton's method and the linear system (which must be solved to do a Newton iteration) is solved directly by Gaussian elimination (IMSL, 1984). reverse Cuthill-McKee algorithm and a special bandwidth reduction algorithm are used by TWODEPEP to number the nodes and to give this linear system a banded structure. The frequency of updating the The 61 R**2cm H*»2.5cm R“3cra H"2.5cm R=4cm H“2.5cm R“3cm R-4cm H“5cm R=6cm H**2.5cm 1 t— R=2cm H-5cm H“5cm R*6cra H»5cm * Drawing not according to scale Fig. 3.3 Initial triangulations for cylinders of various sizes 62 Table 3.1 Element Sizes and Time Steps Used in TWODEPEP Programs for Cylinders and Slabs of Different Sizes Geometry* 2 cm rad. X 2.5 cm 2 cm rad. X 5 cm 3 cm rad. X 2.5 cm 3 cm rad. X 5 cm 4 cm rad. X 2.5 cm 4 cm rad. X 5 cm 6 cm rad. X 2.5 cm Element Sizes2 [cm2} Time Step3 [s] 0.469-0.625 5 0.312 5 0.469-1.406 5 0.938 5 0.938-1.562 5 1.000 5 0.469-1.406 5 1.000 5 4 cm x 4 cm x 5 cm 0.160 1 6 cm x 6 cm x 5 cm 0.360 1 8 cm x 8 cm x 5 cm 0.640 1 6 cm rad. X 5 cm * Dimensions of cylinders and slabs in cm. 2 Numbers show the range of final triangle element sizes used. 3 Element size (A) and time step (At) relationship falls in the criteria described in Eq. 3.31. 63 Jacobian matrix was determined adaptively. The general equation used in TWODEPEP is; CHi.y.u.t)™ .-3.oxx(*.y.$:.fy .u.C) * (3 32) * « where Cl is a material property parameter, U is the dependent variable, OXX is the material property * dU/dx, OXY is the material property * dU/dy and Fl is the element forcing function. For heat dissipation problems, Cl equals pCp; the material property in OXX and OXY is the thermal conductivity, k; and Fl is the rate of volumetric heat generation. Slab; ♦ £<*§> * •<*■» <3-33) >* *<*•*> (3.34) Cylinder: I? “ ^ (JrS ) + * <r,Z> where pCp can be a constant or can be expressed as functions of x, y, T, and t. When TWODEPEP is used to solve axisymmetric problem, the (k/r)dT/dr term in the cylindrical heat conduction equation was added to the element forcing function. The r-axis can be replaced by the x-axis 64 and Che z-axis by Che y-axis, and Chus Che problem of cylindrical specimen can be solved using Che CarCesian coordinate syscem. For slabs, the region of inCerest is one quarter of Che horizontal slice through the center of the slab because of the symmetry in the x and y direction. plane passing through r-0. cylindrical sample. The region of interest in the cylinder is any Radial symmetry was assumed in the These regions were divided initially into eight triangles, and the number of final triangles varies according to the geometry and size of the specimen. The TWODEPEP programs for slab and cylinder and the specifications of the input statement are shown in the Appendix A. The solutions of temperature and heat flux at final node locations at each time step (NOUT) were produced as an output by TUODEPEP. The scalar and vector fields for a specific time step were also generated as part of the graphical output. 3.5 Experimental Design To verify the time-temperature profile simulated by TWODEPEP, a series of experiments were designed using sodium alginate gel as a model food system. The gel simulates a high moisture solid food product. The advantage of using model food systems is that they are highly reproducible; and their thermal, physical, as well as electrical properties are well defined as a function of temperature. A thermally irreversible sodium alginate gel, Hanucol DM (Kelco, San Diego, CA) was used. The formulation of the gel is listed below: 65 Hanucol DM gel 1.5% CaCl-2H20 0 .2% Hexa meta sodium phosphate 0.12% Citric acid 0 .2% Water 97.98% All of the dry ingredients, except citric acid, were mixed together, and then vigorously mixed into heated water in small quantities. When the whole mixture reached 70-80°C, the dissolved citric acid solution was added in with gentle mixing. If gel crumbs were not dissolved properly, the gel was blended in a high speed blender to break up the crumbs before citric acid was added. The gel was poured into ready made sample containers and allowed to set over night until it equilibrated with the ambient temperature (25°C). Sample containers were made with materials transparent to microwave. Rigid plastic films Scotch T-501 (3M, St. Paul, MN) and styrofoam blocks were used to make rectangular and cylindrical containers of different sizes. The experimental design and the configurations of the samples are shown in Table 3.2 and Fig. 3.4. Heating time of each sample was determined according to the size of the sample such that the final temperature after heating would not exceed room temperature by 10°C. 66 Table 3.2 Experimental Design for the Effect of Size and Shape of the Model Food Systems during Microwave Heating Cylinder Radius Slab height Length Width Height [cm] [cm] [cm] [cm] [cm] 3 5 6 6 5 4 5 8 8 5 3.5.1 Effects of Size and Shape Rectangular and cylindrical samples of various sizes were prepared by Manucol DM gel. Rectangular samples of 4 cm, 6 cm and 8 cm widths and lengths and 5 cm height and cylindrical samples of 2 cm, 3 cm, 4 cm and 6 cm radii and 2.5 cm and 5 cm heights were the dimensions of samples used in the study of effect of size and shape. Cylindrical gel samples were heated from all the directions in a Tappan 500 microwave oven for different time Intervals according to the size of the sample. Rectangular gel samples were also heated at different time intervals, but the gels were microwave shielded on the top and bottom surfaces by aluminum foil in the z direction during heating. 3.5.2 Effect of Shielding of Power Double layers of aluminum foil were used to shield microwaves from different directions of the oven by covering different faces of the cylindrical samples (4 cm radius x 2.5 cm height). Aluminum foil is a good conductor; therefore, microwaves cannot transmit through and are, 67 Z= 0 z = — L, x Cylinder z=0 z= Slab Fig. 3.4 Geometrical configurations of cylindrical and rectangular gel samples 68 instead, totally reflected. Blocking the microwave with aluminum foil on the axial ends of the cylinder was equivalent to radial microwave heating, while shielding in the radial direction simulated axial microwave heating. The arrangement of this experiment is shown in Fig. 3.5. Radial Shielding Top Surface Shielded Fig. 3.5 Axial Shielding Bottom Surface Shielded Arrangement of microwave shielding using aluminum foil (shaded areas represent surfaces shielded with aluminum foil) 69 3.5.3 Effect of Salt Content Dielectric properties, as affected by the chemical composition, were studied by adding sodium chloride to the gel solution during sample preparation. Unbound ions such as sodium chloride will hydrate with free water molecules thus depressing the dielectric constant. Dielectric loss, however, is increased due to electrophoretic behavior of the free ions. Uithout additional salt added, the total salt used in the gel formulation was 0.32% (w/w). According to Liu (1990), the dielectric properties of sodium chloride solution change drastically when salt concentration increases up to 2% (w/w), therefore, two levels of salt concentration, 1% (w/w) and 2% (w/w), were used in this study. NaCl powder was added to the desired total salt concentration in making the gel. The NaCl mole equivalent concentrations of these gel samples were determined from the conductance measurement as described in Section 3.3.4. 3.6 Temperature Measurement To verify the time temperature history of the product during microwave heating as predicted by the mathematical model, gel samples were heated in the microwave oven for different time periods. The temperature measurements were performed using both fluoroptic probes during the heating and thermocouple probes before and after microwave heating. 70 3.6.1 Temperature Measurement by Fluoroptlc Probes Four fluoroptlc probes (Luxtron Thermometry System 750, Mountain View, CA) were used to measure sample temperatures during the microwave heating, because they do not Interfere with the electromagnetic field In the oven. Temperatures at different locations relative to the center of the gel, and on the gel surface, were measured. to an IBM-PC via an RS 232 serial port. The unit was Interfaced Time-temperature data were collected using a BASIC program on PC for data analyses. The maximum number of channels for temperature measurement was limited to four fluoroptlc probes. Considering the variation of temperature with location due to factors such as geometry and field distribution, thermocouple probes were constructed to monitor temperature over a wider number of locations in the sample. 3.6.2 Temperature Measurement by Thermocouple Probes Jigs to hold thirty-seven type K thermocouple probes to measure temperatures in rectangular and cylindrical samples at different locations were constructed as shown in Fig. 3.6. The arrangements of thermocouple probes for temperature measurements at different locations of the rectangular and cylindrical samples are shown in Fig. 3.7 (a) and (b). During microwave heating, samples were positioned at the center of the microwave oven as shown in Fig. 3.8. Temperatures at the locations shown in Fig. 3.7 (a) and (b) were measured before and after microwave 71 □ A. B. C. D. E. F. G. H. □□ □ Luxtron fluoroptlc temperature measurement system Microwave oven cavity Gel sample Styrofoam block Thermocouple probe assembly AM32 multiplexer 2IX micrologger Microcomputer data acquisition network Fig. 3.6 Gel sample temperature measurements using thermocouple assemblies and fluoroptlc probes with data acquisition from a PC 72 tBACK 6 CB 4 CB 3 CB 2 CB RIGHT left •4- I FRONT tBACK L=8 c m L=6 c m L=4 c m RICHT LEFT t FRONT Fig. .7 Positioning of thermocouple probes on templates for temperature measurements in (a) cylindrical gels, and (b) rectangular gels of different sizes 73 Center of Sam ple @ Center of Oven Cavity Gel Sam ple Oven JBa ckf)Ve t Oven Front Fig. 3.8 heating. Position and the orientation of sample with respect to microwave oven during heating Probes were aligned according to the orientation of the sample inside the oven as indicated in Fig. 3.8. All temperature data were collected by a data acquisition system consisting of a Campbell 21X Micrologger, an AM32 Multiplexer and a SM192 Storage Module (Campbell Scientific, Logan, UT). The time-temperature data measured by thermocouple probes were stored in the storage module. At the end of measurement, data in the storage module were downloaded unto a floppy disk using PC 208 Datalogger Support Software (Campbell Scientific, Logan, UT). The schematic diagram of the circuitry and data acquisition program are illustrated in Appendix B. The measurement always began from the top level (z - Lz) of the gel sample since the top surface was subjected to surface cooling due to convective and evaporative heat loss. Then the probes were brought down to the center level (z - 0) and then the bottom level (z - -Lz) for the temperature measurement at these levels. Each run/level took 5 a for execution time. Therefore, the 74 time used to complete three runs for temperature measurement In a sample at three levels was around 20 s. 3.7 Data Analysis Temperature measurements were taken at all the locations In each experimental unit In four replications. The Initial and final temperatures, and the heating time obtained from the measurement, were used to calculate the rate of temperature rise, RTR, which is defined by Eq. 3.35. RTR - (3.35) The RTR at different locations for all the experiments were calculated according to Eq. 3.35. The effect of initial temperature from different experiments was normalized, to enable the temperature data measured from thermocouples and fluoroptlc probes to be compared with the model prediction. The mean and standard deviation of the RTR were evaluated for each point of measurement, and they are listed in Appendix C. The mean RTR, based on the study of different factors, were plotted against sample geometrical locations from the center point at Lj, 0, and -Lj— three levels along the z direction. The RTR data within a sample, measured at different orientations from the center of the oven toward the right, front, left and back sides of the oven, were also presented to show the 75 effect of field distribution. The layout of this orientation during temperature measurement Is Illustrated In Fig. 3.8. 3.8 Model Verification Data predicted from the model were compared with the means of the experimental results by t-test. The statistical analysis verifies whether the model predictions of the RTR values at different locations in the sample were significantly different from the experimental results (testing H0: RTRpredicted - R T R ^ ^ ) at 0.01 significance level. Response surfaces from model predictions were plotted in threedimensional graphics by the G3GRID procedure in SAS/GRAFH1CS software by Statistic Analysis System (SAS, 1985). The graphs plotted a continuous surface of temperature distribution by interpolation and extrapolation between the measured points using a cubic spline process. The slowest heating points during microwave heating can be located from these graphs. 3.9 Sensitivity Analysis of the Model Prediction Sensitivity analysis is a technique to examine the extent of variability due to different material properties on the response variables model prediction. Sensitivity analysis for each of the material properties and system conditions was performed. The key parameters included thermal diffusivity, attenuation factor as calculated from dielectric constant and dielectric loss of the material, 76 microwave power output, convective heat loss as affected by the heat transfer coefficient, and evaporative heat loss due to the humidity gradient between the food and ambient air in the oven. This was done by running the base simulation and then varying the parameters with different percentage changes according to the realistic situation encountered. The variability of the response due to the changes in the parameters reflects the sensitivity of the model. The relative sensitivity is an index to determine the influence of a parameter on the model prediction. It was calculated as the ratio of the variability of the model response to the percent change of the parameter as shown in Eq. 3.36, AX R e l at i v e S e n s i t i v i t y * (3.36) ~P where Y is the model response and P is the value of the parameter chosen. 77 Chapter 4 RESULTS AND DISCUSSION Results of mathematical modeling and the experimental data for the heating characteristics of model foods during microwave heating were analyzed and compared based on the rate of the temperature rise (RTR). Microwave heating of foods is affected by the oven parameters and the properties of the food product. The latter consists of material properties and geometrical configuration, while the oven parameters include the field distribution and the power-load relationship. Material properties and oven characteristics interact with each other and cause different microwave heating behavior of the food product under different conditions. In discussion of the results, the results from measurements and the finite element model prediction are presented in sections 4.1 and 4.2. The finite element model predictions were then compared with the means of the experimental measurements. 4.1 Experimental Results All of the experiments were conducted based on four replications in order to obtain information on the range of experimental error. means and the standard deviations of sample means are presented. The 78 4.1.1 Mapping the Power Distribution Since the dimensions of the microwave oven cavity are of the same order of magnitude as the wavelength of the microwaves, a number of standing waves exist inside the microwave oven. The development of the field distribution depends on the oven type and the geometry and composition of the food inside the oven. Due to the lack of information about the mode patterns that exist in the test oven (Tappan 500), the field distribution cannot be solved mathematically. instead. The field distribution was determined empirically Different oven power mapping procedures have been proposed in the literature (Wilhelm and Satterlee, 1973; Washisu and Fukal, 1980 and Schiffmann, 1987). The use of egg white, pancake batter or tonic powder as described in the literature does not determine the field distribution quantitatively. Presence of a large amount of sample or numerous individual samples may also interfere with the development of mode patterns. The method used in this study results in minimal field interference and gives quantitative data on the power distribution. 4.1.1.1 Three-Dimensional Presentation of Power Distribution Temperature difference between the initial and final temperatures (AT) in the three dimensional oven cavity were plotted using the G3GR1D procedure in SAS/GRAPHICS. The three-dimensional plots are shown in Fig. 4.1. The origin represents the front left corner of the oven. is proportional to microwave power absorbed at each location. AT The power 79 absorbed is a function of the microwave field intensity at that particular position. Therefore, AT at different positions of the cavity was used to describe the power distribution within the microwave cavity. The results showed that power distribution patterns are of sinusoidal shape and are composed of high and low nodes. The field Intensity of a level close to the cavity ceiling is higher than that at lower level. This could be due to microwaves that are directly incident from the waveguide and mode stirrer at that location. At the lower level, fewer direct incident and more reflected waves from the side walls and floor strike the sample. The wave may also be attenuated depending on the conductivity of the oven walls and the humidity within the cavity. A.1.1.2 Multiple Regression Model According to the general form of the standing wave equation (Eq. 2.14), the standing wave is a sine and cosine function of the coordinates. In the development of the regression model for the power distribution, these trigonometric functions and the combinations of coordinates up to the eighth order were included. Models with different orders of polynomial functions were tested by the STEPWISE procedure in the Statistical Analysis System (SAS, 1982) to select terms which yielded the maximum R2. In determining the number of terms to retain in the model, a compromise was made between increasing R2 and decreasing the number of terms. A polynomial of the eighth order with trigonometric functions (43 terms in total) was chosen, which yielded an R2 of 83.8%. The entire model and the estimates of parameters are shown in Table 4.1. The coefficient of variation for the standard error is 2.65% of the overall mean from the analysis of variance. The F value from the test of variance is 37.93, which indicates that the variance due to the power distribution is significant. The three dimensional standing wave patterns developed quickly in the oven cavity after the magnetron was turned on. This was seen in the straight line relationship of time-temperature profile in each power measurement at different locations. A time-temperature profile obtained by Luxtron temperature measurement system is shown in Fig. 4.2 to illustrate the time-temperature relationship in a gel at four different locations during microwave heating. All of the temperature profiles were found to be linear with respect to time, hence, RTR was used as a way to describe the microwave heating. Fig. 4.3 shows the three-dimensional graphs of the predicted temperature difference at the same horizontal levels as in Fig. 4.1 within the region of the sample location in the oven. These graphs are based on the multiple regression model characterized the power intensity pattern in the Tappan 500 oven. The power distribution regression model was thus used in the microwave power absorption term in the model to predict the relative power intensity at different locations in the oven. Based on the dimension of the sample, an average value was first calculated from the regression model at the locations r - 0, 1 c m for every n/6 and z - ± Lz within the domain. R Then the ratio of the temperature difference calculated from the power distribution model at a particular location Inside the sample to the average value is used as an index of the power intensity at that point within the sample. The Fig. 4.1 Measured power distribution in a Tappan 500 microwave oven at three levels (a) z - 2.5 cm, (b) z - 0, and (c) z - -2.5 cm. 82 Table 4.1 Multiple Regression Model for Field Distribution Parameter Estimate Intercept Y XY YZ X2Z Y4 Y3Z YZ3 x 2y 2 X2Z2 X5 X4Z y 3z 2 y 2z 3 x 2y 2z XY5 18.033 -2.953 0.167 0.952 -0.044 1.128E-3 -2.673E-3 -3.562E-2 1.690E-3 3.324E-3 -3.502E-5 4.359E-4 -1.180E-3 4.206E-3 -2.241E-3 -2.479E-6 6.545E-4 -4.890E-5 -4.617E-4 8.554E-4 -1.952E-7 7.854E-7 -7.971E-5 9.144E-6 -3.790E-6 -1.601E-5 3.224E-6 3.296E-5 4.464E-6 -1.050E-4 2.326E-5 1.025E-6 8.792E-9 2.189E-7 3.123E-6 6.416E-7 -1.543E-6 -7.285E-8 -1.843E-6 3.200E-6 -4.784E-1 1.223E-1 2.485E-1 1.658E-1 xz5 X4Z2 x 2y z 3 x 2y 2z 2 Y7 Y*Z XZ6 YZ6 X2Y5 y 2z 5 x 4y z 2 x 2y ^z x y 3z 3 x 2y 3z 2 x 2y 2z 3 z8 x 7y X2y6 XYZ6 X3Z5 x 2y 5z X*YZ XY2Z5 X^Z2 COS X COS X * SIN Y SIN Y * COS Z SIN Z * COS Y PR > |T| 0.0 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0007 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0412 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0463 0.0346 0.0002 83 (a) 3 cm rad.x 5 cm ht. o r=0 * ■ 1 60 2 . ' r=2R/3 r=R/3 r=R 0 20 40 Time, s (b) 4 cm rad. x 2.5 cm ht. 80 o $ | 6°- r=R/4 IQ 6 r=0 • r=R/2 40 r=R 20 0 Fig. 4.2 20 Time, s 40 Time-temperature profiles of cylindrical gels of sizes (a) 3 cm radius by 5 cm height, and (b) 4 cm radius by 2.5 cm height measured by fluoroptic probes at different distances from the center at z - 0 level o • and r"**** 85 microwave absorption term was modified by this ratio as shown in the finite element program in the Appendix A. 4.1.2 Effect of Loading on Energy Coupling Efficiency The efficiency of microwave power absorption is affected by the load within the oven. The microwave power absorption may vary depending on the material properties, total mass and geometrical configuration of the product load. Fig. 4.4 shows the effect of loading on the microwave power absorption by different amounts of rectangular and cylindrical loads placed at the center of the oven. The best-fit equations for the loading effect of rectangular and cylindrical loads were determined to be exponential as shown in Eq. 4.1 and 4.2. Rectangle: PBbs “ 50<27 + 56•6 ln<wt> R2-91.5% (4.1) R2-95.4% (4.2) Cylinder: P«bs “ 40-1 + 59•8 ln<wt> where wt is the weight of the test water, [g]. These two equations were used in the mathematical model to determine the power absorbed by samples of different sizes and shapes. For small size loads, the microwaves penetrate through the load without much absorption. Since the load size is small, temperature increases at a faster rate and the dielectric properties change rapidly during the heating cycle. As explained in section 2.3, when temperature increases, dielectric constant and dielectric loss decrease. The loss tangent or dissipation factor decreases with the increasing temperature. Therefore, the efficiency of microwave absorption is lowered at higher 86 temperatures. This may partially explain the lower energy coupling efficiency from smaller loads, and the Increased power absorption by increasing load size. When the load size Increased to a point that most of the microwave power was absorbed, further increase In the size of the load did not Increase the coupling efficiency. Cylinder Rectangle 600 500 4 0 0 §300 o £ 3200 100 0 500 1000 1500 2000 2500 0 Load, g Fig. 4.4 4.1.3 500 1000 1500 2000 2500 Load, g Effect of loading on the microwave power absorption by cylindrical and rectangular shaped loads Effect of Salt Concentration Temperature distribution in the 4 cm radius by 2.5 cm height cylinders with total added salt concentration ranging from 0.32% (w/w) to 2% (w/w) was measured as described in sections 3.5.3 and 3.6. The 0.32% of salt according to the conductivity measurement described In Section 3.5.3 is equivalent to 0.05 H NaCl concentration. Whereas 1% 87 and 2% total salt in the gel are equivalent to 0.172 M NaCl and 0.369 M NaCl, respectively. RTR values at different radial locations and at half the cylinder height (z - 0) were calculated and are shown in Pig. 4.5. The curves in Fig. 4.5 are the means from four different experiments. Surface heating phenomena was observed in all of the samples at all salt levels. concentration increased. At r/R < 1/2, the RTR decreased when salt In the region close to the surface of the cylinder, the higher the salt level, the steeper the RTR with respect to radial distance from the center (r/R). This can be rationalized by the concept of penetration depth as influenced by the dielectric properties of the material. The higher the salt content, the more microwave power is attenuated or the shorter the penetration depth. Hence, more microwave power is absorbed closer to the surface of the sample. Hudgett (1986) investigated the effects of salt concentration and temperature on the dielectric constant and loss of aqueous sodium chloride solution at 2450 MHz. The attenuation factors, a, calculated from dielectric properties are plotted against temperature in Fig. 4.6. The graph supports the results for this part of the study. the salt content, the higher the attenuation factor. The higher At low temperature region (below 25°C), the attenuation factor of salt solutions decreased with increasing temperature. Beyond 50°C, the attenuation factor of salt solutions increased with increasing temperature. However, the attenuation factor decreased with increasing temperature for deionized water (zero salt content). The degree to which the dielectric constant was depressed by salt concentration was not as large as the increase of the dielectric loss. temperature. However, they are both affected by the change in This combined effect can be seen more readily in the Q 1/4 1/2 3/4 1 r/R Fig. 4.5 RTR values In 4 cm radius by 2.5 cm height cylindrical gels ae different salt concentrations versus radial distance from the center at z - 0 level 2 RT R , Deg C/s 1 5 1 0.5 □ 0 25 SO 75 100 Temperature, Deg C Fig. 4.6 Effect of NaCl concentration and temperature on the attenuation Jactors of aqeous ionic solutions (Mudgett, 1986) 89 attenuation factor that changes with the salt concentration and temperature. Because of the increased microwave attenuation at higher temperatures and at high salt concentration, the sample containing 2% salt (0.369 M NaCl equivalent) had a significant drop in RTR near the surface. The sample with 1% salt (0.172 H NaCl equivalent) attenuated less compared to the sample with 2% salt, therefore implying more microwave absorption and higher temperature increase than the sample with 2% salt. The 0.32% of salt (0.05 M NaCl equivalent) is the amount of salt (calcium chloride and sodium phosphate) required for gelling of the Manucol gel and this may be considered as bound ions. They contributed to the least amount of deviation in dielectric properties from pure water. Therefore, less microwave power was absorbed at the surface, which then implied that more power was available at a distance away from the surface. As shown in Fig. 4.5, the rate of temperature increase at the surface was slowest for 0.32% salt samples, yet the same sample at the center had the highest RTR compared with other samples. This phenomenon may also explain how microwave power is coupled differently by materials of the same geometrical configuration but different dielectric properties. Temperature distributions within these samples were different, but the total power absorbed in all the samples of different salt content was approximately the same (Liu, 1990). Samples containing 2% salt did not show higher RTR at the surface than samples containing 1% salt, and this may be due to excessive heat loss from surfaces of higher temperatures. 90 LEFT - RIGHT BACK - FRONT z=1.25 an 1.0 0.5 0.0 RTR[°C/s] z=0 z=0 1.0 0.5 0.0 RTR [*C/s] 1.5 z=—1.25 an z=—1.25 an 1.0 0.5 0.0 1 - 1 /2 0 r/R Fig. 4.7 1 /2 1 1 - 1 /2 0 1 /2 1 r/R The means and standard deviations of RTR in a 4 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels (thermally insulated on the top and bottom surface) The distributions of mean RTR and standard deviation of 4 cm radius by 2.5 cm height cylinder at three levels (z - 1.25 cm, 0 and 1.25 cm) are shown in Fig. 4.7. The graphs show significant variabilityin the temperature distribution comparing the back-front orientation (the set of figures on the left column) to the left-right orientation (the set of figures on the right column) inside the microwave oven, which can be accounted for by the mode pattern in the cavity of the microwave oven. 4.1.4 Effect of Geometrical Configuration on Temperature Distribution The effect of geometrical configuration on the temperature distribution of materials heated using microwaves has been studied by different research groups (Ho et al., 1971; Ohlsson and Risman, 1978 and Swami, 1982). In addition to the material properties and power distribution, geometrical configuration is the major factor affecting the temperature distribution in a food heated within a microwave oven. 4.1.4.1 Effect of Shape Rectangular and cylindrical shapes were considered in this study. Cylindrical samples of 3 cm radius by 5 cm height and rectangular samples of 6 cm length and width and 5 cm height are of comparable overall size. Fig. 4.8 shows the mean cylinder RTR and standard deviation of the mean at different locations away from the center at z 0 and z - -2.5 cm levels. Fig. 4.9 shows the mean slab RTR and standard 92 2.0 0.5 0.0 Distance to the center, an -2.5 cm 2.0 (0 0.0 3 2 1 0 1 2 3 Distance to the center, cm Fig. 4.8 Mean RTR in a 3 cm radius by 5 cm height cylindrical gel at different locations from the center at z - 0 and z - -2.5 cm levels 93 z - 0 .5 to o 00 0) 0 n oz tz 0.5 0.0 3 - 2 - 1 0 1 2 3 Distance to the center, cm z - -2.5 cm to 5 u 60 <U o 0 n 0 0.5 0.0 3 - 2 - 1 0 1 2 3 Distance to the center, cm z - -2.5 cm (diagonal) —to 0 . 60 01 n o4 & 0.5 0.0 -4.25 -2 0 2 4.25 Distance to the center, cm ' Fig. 4.9 Mean RTR in a 6 cm x 6 cm x 5 cm rectangular gel at z - 0, z - -2.5 cm (both measured in the directions parallel to x and y axes, and z - -2.5 cm measured in diagonal directions to the axes 94 deviation of the mean at different locations measured from the center at (a) z - 0, (b) z - -2.5 cm and (c) diagonal directions of the slab at z - -2.5 cm, respectively. Center heating effect was observed at the middle height (z - 0) of the cylindrical sample. In the cylindrical sample, RTR increases toward the center of the sample. At the same height within the slab, temperature distribution was fairly uniform throughout the entire cross section. However, at the bottom level (z - -2.5 cm) of both the cylindrical and slab samples, the heating behavior changed drastically. At this level in the cylinder, surface heating, or edge heating, was significant; center heating still existed but was less apparent compared to the edge heating effect. The slabs at the bottom level also showed significant edge heating effect. In both of the cases, edge heating was evident from r/R - 1/3 to the surface of the sample. Similar results were found for cylinders of 4 cm radius by 5 cm height and slabs of 8 cm x 8 cm by 5 cm height as shown in Figs. 4.10 and 4.11, respectively. The edge heating phenomenon may be explained by the fact that the microwave power is transmitted into the sample from different directions (r and z directions in the cylinder, and x, y and z directions in the slab). The power from the surface is focused at the edges before it is attenuated. The edge heating effect is intensified at the corners of a slab where microwaves come from all three directions. This is shown in Figs. 4.9 and 4.11 for slabs of 6 cm x 6 cm x 5 cm and 8 cm x 8 cm x 5 cm along their diagonals. 95 2 — 0 Cfl 0.5 0.0 Distance to the center, cm z - -2.5 cm in 1 * 0 o * i 0.5 0.0 4 2 0 2 Distance to the center, 4 cm Fig. 4.10 Mean RTR in a 4 cm radius by 5 cm height cylindrical gel at different locations from the center at z - 0 and z - -2.5 cm levels 96 RTR [° z - 0 c/r t 1. 0 0.5 I- 0.0 — I----- 1 -4 I -2 I----- 1 I- '"— I----- 1----- 1 0 2 I■ 4 •2.5cm 1.0 0.5 0.0 RTR [*C/s] 1.5 4 2 2 4 0 Distance to the center, cm z - -2.5 cm (diagonal) Distance to the center, cm Fig. 4.11 Mean RTR in a 8 cm x 8 cm x 5 cm rectangular gel at z - 0, z - -2.5 cm (both measured in the directions parallel to x and y axes, and z - -2.5 cm measured in diagonal directions to the axes 97 4.1.4.2 Effect of Size Cylindrical gels were used to study the effect of size on temperature distribution during microwave heating. Four radii (2 cm, 3 cm, 4 cm and 6 cm) and two heights (2.5 cm and 5 cm) were studied. Figs. 4.12 to 4.15 show the RTR for 2.5 cm height cylinders, whereas Figs. 4.16 to 4.19 show the results for 5 cm cylinders. As expected, the smaller the sample size, the faster the average rate of temperature rise. The average RTR decreased with increasing radius and increasing height. For example, temperature at the center of 2 cm radius by 2.5 cm height cylinder rose at a rate of 3.5 °C every second; the temperature increased at a rate of 1.8 °C per second at the center of a 3 cm radius by 2.5 cm height cylinder while the average RTR for a 3 cm radius by 5 cm height cylinder was 0.8 °C. The curve in each figure shows the RTR within the sample at different radial directions with respect to the axes of the oven (parallel to the x or y axes). These figures also show significant effect of power distribution on the RTR results. In samples of larger sizes, it was observed from the figures that the RTR in the directions of front and right with respect to the oven had higher value than did the RTR in the back and left directions in the oven. It may be due to the specific power distribution present in this type of oven. This is evident from Fig. 4.1. At z - 2.5 cm and z - -2.5 cm levels, the absorbed power at areas 3 to 4 cm away from the center of the oven is very high. The rate of temperature increase was lower in the case of the 4 cm radius cylinders RTR[*C/s] BACK - FRONT LEFT - RIGHT 4 z=l.25 an z=l.25 an z=0 z=0 3 2 1 0 3 2 1 0 RTR [°C/s] 4 z=-l.25 an z=-1.25 an 3 2 1 0 1 0 r/R Fig. 4.12 1 /2 1 1 0 1 r/R The means and standard deviations of RTR in a 2 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z «■ 0, and z - -1.25 cm three levels 99 RTR[° C/ s ] 2 BACK - FRONT LEFT - RIGHT z=l.25 an z=l.25 an 1 0 RTR[°C/s] 2 z=0 z=0 1 0 RTR[°C/s] z=-l.25 an z=-1.25 an 1 0 -1 -2/3 -1/3 0 1/3 2/3 r/R Fig. 4.13 1 1 -2/3 -1/3 0 1/3 2/3 r/R 1 The means and standard deviations of RTR in a 3 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels 100 RTR[° C/s] BACK - FRONT LEFT - RIGHT z=1.25 cm z=l.25 cm 1.0 0.5 0.0 RTR[°C/s] z=0 2=0 1.0 0.5 0.0 RTR[°C/s] z=-l.25 cm z=-l.25 cm 1.0 0.5 0.0 - 1/2 1 /2 r/R Fig. 4 .14 1 - 1 /2 0 r/R 1 /2 1 The means and standard deviations of RTR in a 4 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels RTR[°C/s] BACK - FRONT LEFT - RIGHT z=1.25 on z=l.25 an 101 1.0 0.5 0.01 RTR[°C/s] z=0 z=0 1.0 0.5 0.0 z=-l.25 an z=-l.25 an 1.0 0.5 0.0 1 Fig. 4.15 - 1 /2 0 r/R 1/2 1 1 -1/2 0 r/R 1/2 1 The means and standard deviations of RTR in a 6 cm radius by 2.5 cm height cylindrical gel measured at different orientations and distances from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels C/s] BACK - FRONT LEFT - RIGHT z=2.5 cm z=2.5 cm 102 4 3 2 1 0 m C/s] 4 z=0 z=0 3 2 1 0 m 4 C/s] z— 2 •5 cm 3 2 1 0 -1 - 1/2 0 r/R 4.16 1/2 1 - 1 /2 0 r/R 1 The means and standard deviations of RTR in a 2 cm radius by 5 cm height cylindrical gel measured at different orientations and distances from the center at z - 2.5 cm, z - 0, and z - *2.5 cm three levels RTR[*C/s] 2.0 BACK - FRONT LEFT - RIGHT z=2.5 an z=2.5 cm 103 1.0 0.5 0.0 RTR[*C/s] 2.0 RTR[°C/s] 2.0 z=0 z=-2.5 cm -1 -2/3 -1/3 0 1/3 2/3 r/R Fig. 4.17 z=-2.5 cm 1 1 -2/3 -1/3 0 1/3 2/3 r/R 1 The means and standard deviations of RTR in a 3 cm radius by 5 cm height cylindrical gel measured at different orientations and distances from the center at z - 2.5 cm, z - 0, and z - -2.5 cm three levels 104 BACK - FRONT LEFT - RIGHT z=2.5 cm z=2.5 cm 1.0 0.5 0.0 RTR[°C/s] 1.5 z=0 z=0 z-~2.5 cm z=-2.5 cm 1.0 0.5 0.0 RTR[°C/s] 1.5 1.0 0.5 0.0 1 Fig. 4.18 -1/2 0 r/R 1/2 1 1 -1/2 0 r/R 1/2 1 The means and standard deviations of RTR In a 4 cm radius by 5 cm height cylindrical gel measured at different orientations and distances from the center at z - 2.5 cm, z - 0, and z - -2.5 cm three levels RTR[°C/s] BACK - FRONT 1.5 LEFT - RIGHT z=2.5 cm 105 z=2.5 cm 0.5 • 0.0 RTR[0C/s] 1*5 z=0 z=0 1.0 0.5 0.0 RTR[° C/s] z=-2.5 cm z=-2.5 cm 0.5 0.0 -1/2 1 r/R Fig. 4.19 -1/2 0 1/2 1 r/R The means and standard deviations of RTR in a 6 cm radius by 5 cm height cylindrical gel measured at different orientations and distances from the center at z - 2.5 cm, z - 0, and z - -2.5 cm three levels 106 as shown In Figs. 4.7 and 4.14, as compared to the samples of smaller radii, the sample had a more uniform temperature distribution.Therefore, 4 cm radius by 2.5 cm height cylindrical gel samples were used in studying other aspects of microwave heating. Edge heating at z - 1^ level became significant in geometries of 3 cm radius and larger due to the reasons discussed in section 4.1.4.1. The RTR curves at the middle level (z - 0) in cylinders of 2 cm and 3 cm radius show that RTR peaked at the center of the cylinder for both 2.5 cm and 5 cm height cylindrical gels. the center to the surface. The RTR decreased from For cylinders of larger radii (4 cm and 6 cm), the center heating phenomenon was not observed; instead, temperatures at surface areas were higher than the center. The sharp decrease of the RTR close to the surface of the cylinders of 2.5 cm height was partly due to the surface heat loss from convection, because of higher surface temperatures as compared to the samples of 5 cm height. The other possibility for this pattern of RTR is the unique power distribution at z - 1.25 cm and z - -1.25 cm. However, similar RTR patterns were observed in a 4 cm radius by 2.5 cm height cylinder that was shielded by aluminum foil at the z - 1.25 cm (Fig. 4.22). Therefore, the power distribution at z - 1.25 cm (top surface) had little effect on the RTR patterns in the 4 cm radius by 2.5 cm height cylinder, but the RTR patterns were mostly affected by the waves coming in from the lower part of the oven as seen in Figs. 4.7, 4.14, 4.15, 4.18, and 4.19 at z - -1.25 cm or z - -2.5 cm level. As for cylinders of 5 cm height (Figs. 4.16 to 4.19), since the power attenuated from the z - ± 2.5 cm surfaces toward the center, the influence of the power distribution on the surface power at z - ± 2.5 cm 107 levels may not be as noticeable at center level (z - 0) of the gel. However, the effect of nonuniform power distribution at the z - -2.5 cm level on the bottom surface is very distinct. These observations were based on the variations of the RTR curves at different orientations with respect to the oven cavity at z - 0 and z - -2.5 cm in Figs. 4.16 to 4.19. Fig. 4.1 shows that the variability of the power distribution at z - 2.5 cm and z » 0 levels is less than that from the z - -2.5 cm level. Since the top surface was uncovered, there were evaporative and convective heat losses during the heating cycle and during the temperature measurement by thermocouple probes. This probably is another reason why the distinct patterns due to power distribution in cylinders of 5 cm height at z - 2.5 cm level were not as evident as those at the z - -2.5 cm level. 4.1.5 Effect of Shielding the Surface Microwave Power Nonuniform microwave field distribution within the microwave oven results in a nonuniform temperature distribution within the food. The power intensity within the oven varied with respect to both vertical and horizontal directions. Examination of this variation can be conducted by electrically shielding different sides of the gel samples with aluminum foil. The dimensions of the cylindrical gel samples used in this part of the study were 4 cm radius by 2.5 cm height. 108 4.1.5.1 Axial Shielding By shielding microwave power incident on the top and bottom of the sample (axial direction), the sample is subject to microwave radiation merely in the radial direction. The RTR by radial microwave heating of gel samples of 4 cm radius by 2.5 cm height cylinders at three levels are shown in Fig. 4.20. They are compared to the data from the samples without microwave shielding in Fig. 4.7. The radial microwave heating resulted in a more uniform temperature distribution at all radial locations irrespective of the orientation relative to the oven cavity. As expected, the variations of RTR in the z direction were minimal because of the microwave shielding in the z direction. Surface heating was predominant, and there was a moderate center heating. Comparing Fig. 4.7 and Fig. 4.20, the microwave power absorption apparently was higher when there was no shielding on the top and bottom surfaces. Without shielding, the variations of the RTR at different orientation with respect to the oven were greater. This is attributed to the uneven vertical field intensity in the microwave oven. An attempt was made to verify the induced temperature variations due to the presence of vertical power distribution. Separate studies were conducted with gel cylinders of 4 cm radius by 2.5 cm height with shielding just at the bottom (z - -1.25 cm) and at the top (z - 1.25 cm). Results from the two experiments are shown in Figs. 4.21 and 4.22. It is evident that the variability of the temperature distribution at radial locations with respect to orientation in the oven is higher when the top of the gel was shielded by the aluminum foil. This indicates that the microwaves primarily incident from the top of the RTRfC/s] BACK - FROMT z=l.25 cm LEFT - RIGHT 109 z=l.25 cm 0.5 0.0 RTR[° C/s ] z=0 0.5 0.0 RTR[° C/s ] z=-l.25 cm z=-l.25 cm 0.5 0.0 -1/4 r/R r/R Fig. 4.20 The means and standard deviations of RTR measured at different orientations and distance from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5 cm height cylindrical gel shielded with aluminum foil on the z ±1,25 cm surfaces RTR[°C/s] BACK - FRONT LEFT - RIGHT z=l.25 cm z=l.25 cm z=0 z=0 z=-l.25 cm z=-l.25 cm .5 1.0 0.5 0.0 RTR[°C/s] .5 .0 0.5 0.0 RTR[°C/s] .5 .0 .5 0.0 1 -1/2 0 r/R 1/2 1 1 -1/2 0 1/2 1 r/R Fig. 4.21 The means and standard deviations of RTR measured at different orientations and distance from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5 cm height cylindrical gel shielded with aluminum foil on the z - 1.25 cm surface RTR[°C/s] BACK - FRONT LEFT - RIGHT z=l.25 cm z=l.25 cm z=0 z=0 0.5 0.0 RTR[°C/s] 0.5 0.0 RTR[°C/s] z=-l.25 cm 1.0 0.5 0.0 1 -1/2 0 r/R 1/2 1 1 - 1 /2 0 r/R 1/2 1 Fig. 4.22 The means and standard deviations of RTR measured at different orientations and distance from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5 cm height cylindrical gel shielded with aluminum foil on the z 1.25 cm surface 112 oven are more uniform than the secondary waves reflecting from the oven floor. Comparing Figs. 4.21 and 4.22 with Fig. 4.7 (where there was no shielding), it can be seen that the temperature distributions in Fig. 4.22 are similar to those of Fig. 4.7. This is because of the presence of nonuniform secondary reflected waves from the lower level of the oven. The temperature distributions in Fig. 4.21 at z - 1.25 cm look similar to Fig. 4.7 at the same level since the primary waves incident at the top of the oven are more uniform. Blocking microwave power at the bottom of the sample made less microwave power available to the bottom surface of the gel, and similarly less power was available to the top surface when the gel was shielded at the top. Hence, temperatures behind a surface that was not shielded was higher. However, when the bottom surface was shielded, the RTR values at the top surface level of the gel were not found to be much higher than those at the bottom level. This could be due to the heat loss from the top surface during the thermocouple temperature measurement. This was verified by the significantly higher RTR values calculated from the fluoroptic temperature measurement during microwave heating than the RTR values from the thermocouple temperature measurement. 4.1.5.2 Radial Shielding As expected from the discussion in the previous section, when samples were shielded radially, the axial microwave heating resulted in a nonuniform temperature distribution in the gel. The effect of uneven horizontal power distribution is readily demonstrated by results in Fig. 113 4.23. The microwave power coupling during axial heating seemed to be higher than with both radial and axial heating (without radial microwave shielding). This may be due to the electrical conducting environment provided by the aluminum foil which causes the induction of the secondary field in the cylinder known as near field phenomenon (Risman et al. 1987). For most of the points away from the center, the RTR rose and then dropped at a distance close to the surface. This is due to the shading effect from the shielding of the radial microwave power from the surface. It was also noticed that there was a consistently low RTR toward the right of the oven at all levels, which could be due to the weak field distribution in that area of the oven cavity. Thermocouple probes used in the experiment were examined for calibration to verify that this was not an artifact. 4.2 Mathematical Modeling In this section, results from mathematical modeling of microwave power absorption and temperature distribution in the model food systems under different conditions are discussed. The finite element model was used in this study. Variability in the predication due to changes of estimated parameters was tested by sensitivity analysis, and the model validation was done by comparing the predicted and measured values. RTR[°C/s] BACK - FRONT LEFT - RIGHT z=l.25 cm z=l.25 cm 0.5 0.0 RTR[°C/s] z=0 z=0 0.5 0.0 RTR[°C/s] z=-l.25 cm z=-l.25 cm 1. 0 0.5 0.0 1 - 1 /2 0 r/R 1 /2 1 - 1 /2 0 r/R 1 /2 Fig. 4.23 The means and standard deviations of RTR measured at different orientations and distance from the center at z - 1.25 cm, z - 0, and z - -1.25 cm three levels in a 4 cm radius by 2.5 cm height cylindrical gel shielded with aluminum foil on the r - 4 cm surface 115 4.2.1 Microwave Power absorption The critical step In mathematical modeling of microwave heating lies in the formulation of the term for internal microwave power generation. In the differential equation for heat transfer this term was derived as the localized microwave power absorption per unit volume. Four crucial variables in the microwave power absorption terms are dielectric constant, dielectric loss of the element, location of the element and total microwave power absorbed by the load. Dielectric constant and loss can be lumped into attenuation factor as shown in Eq. 4.3. (4.3) O where a is the attenuation factor in [cm'1]; er* and er" are the dielectric constant and the dielectric loss, respectively. 4.2.1.1 Effect of Geometrical Configuration on Microwave Power Absorption The absorbed microwave power density for cylinders and slabs are given by Eqs. 3.20 and 3.21, respectively. They are functions of the location and the attenuation factor which in turn is a function of the dielectric properties of the material. 116 The effect of geometrical configuration on microwave power absorption can be understood by Eqs. 3.20 and 3.21. For rectangular geometry, the surface microwave power attenuates exponentially into the food while in the case of a cylinder, the wave attenuates exponentially from the surface but also concentrates at regions close to the center. This is because of the 1/r term in the power absorption equation. In two-dimensional microwave heating of a slab, any location inside the food is subject to the microwaves from the positive and negative x and y directions. This is also true for the cylinder where microwaves are incident from both the positive and negative radial and axial directions. The microwave absorption of a cylinder in the z direction attenuated exponentially (similar to a slab) as shown in Eq. 3.20. Since there is no attenuation of the microwave power at the surface, the microwave power is maximum at this location. The microwave power at the line where two surfaces meet, that is at the edge or corner of the product, is the sum of the power from both directions. Microwave power at an edge or a corner is thus much higher than that at other locations upon the surface. Microwave power absorbed by a sample with rectangular geometry is the result of attenuation of surface microwave power from x and y directions as shown in Fig. 4.24. In this figure, microwave power incident at the surfaces of a 6 cm x 6 cm x 5 cm slab perpendicular to both x and y directions at 30°C constant temperature was calculated by Eqs. 3.20 and 3.21, High microwave power distribution at corners is due to the net absorption of microwave power from both of the surfaces at -3 Fig. 4.24 the corner. 00 Two-dimensional microwave power absorbed ac z - 0 in a 6 cm x 6 cm x 5 cm slab gel at 30°C As microwave power travels into the slab, the absorbed power density decreases exponentially as shown in Fig. 4.24. Fig. 4.25 shows the absorbed microwave power distribution in a 3 cm radius by 5 cm height cylinder. Absorbed power attenuated from both the radial (sides) and axial surfaces (top and bottom) toward the center. Edge heating effect discussed earlier can be found along the circumferences at both ends of the cylinder (z - ± 2.5 cm). However, because of the decreasing control volume as microwave energy approaches the center along the radial direction, power density increased dramatically close to the center. This is also indicated by the equation for radial microwave power density absorption (Eq. 3.20). An increase in the denominator r would cause the calculated microwave power absorption to increase drastically at regions close to the center. 118 12 9 6 3 a 3 •3 00 Fig. 4.25 Two-dimensional microwave power absorbed In a 3 cm radius by 5 cm height cylindrical gel ac 30°C 4.2.1.2 Effect of Sample Temperature on Microwave Power Absorption Figs. 4.24 and 4.25 were both calculated from the microwave power absorption terms In Eqs. 3.20 and 3.21 based on a constant temperature (30°C) to show the differences in microwave power absorption between a slab and a cylinder. Since temperature affects the material dielectric properties, it will affect the microwave power distribution during heating. Effects of temperature and distance (from the surface) on the microwave power absorption in a cylinder (3 cm radius by 5 cm height) in the axial and radial direction are shown in Figs. 4.26(a) and (b), respectively. 119 From the conductivity measurements It was determined that the gel samples used have an equivalent NaCl concentration of 0.05 M. Referring to Fig. 4.6, the attenuation factor varies with respect to temperature; it first decreases with temperature from 0°C to 50°C and then increases after 50°C. Looking at Fig. 4.26(a), the power density at the surface varies similar to the variation of attenuation factor with respect to temperature as shown in Fig. 4.6. they attenuate exponentially. As microwaves travel into the food, At temperature regions where the attenuation factor is high, microwave power absorbed is lower than that at room temperatures (low attenuation factor). Therefore, in Fig. 4.26(a) the surface microwave power decreases toward the center (z-0) at a faster rate up to 50°C. Then at higher temperatures (low attenuation factor), the decrease of the microwave power absorption is at a slower rate. In Fig. 4.26(b) the effect of temperature on the microwave power absorption due to the temperature dependency of the attenuation factor is more evident. The power absorption term in the radial direction in a cylinder as seen in Eq. 3.20 has an r term in the denominator. The microwave power absorption becomes very high as r approaches zero (radial center). At these temperatures, when the attenuation factor is low, there is more power available toward the center. This is indicated by the high microwave power at a temperature range from 45°C to 75°C. di<eC dltec cioti * /c i» aV>® 16^ Fig, 26 Power 3 cmz direct J-*calabsorption gel alongin thea (a) *4<f€erent temperatures 4 .cylindrical 20 _ Bdirection at diffe- 121 It also shows that power absorption at the center becomes less efficient as temperature increases because of the high attenuation factor. At the latter stages of heating, the microwave absorption is more of a surface phenomenon because of the increased attenuation factor toward the surface. The microwave power absorption decreases from the surface up to half the radius and then increases. This would indicate the location of the slowest heating point in the sample to be at half the radius from the surface of a cylinder. 4.2.2 Effective Microwave Power Gain The volumetric concentration with the increase in microwave power absorption at areas close to the center of the cylinder was defined by Kritikos and Schwan (1975) in Eq. 2.36. Fig. 4.27. This is also illustrated in When the effective gain of microwave power becomes greater than one, the concentration heating effect occurs. The distance from the radial center where this phenomenon occurred was calculated for cylinders with radii ranging from 2 cm to 6 cm and with attenuation factors of 0.3, 0.4 and 0.5. Fig. 4.27 showed that this concentrated heating effect is predominant in cylinders of smaller radii and at low attenuation factors. For instance, center heating effect occurs at a cylinder of 2 cm radius over two thirds of the area from the center when a equals to 0.3 while it occurs less than one thirds of the area if a equals to 0.5. As the radius increases, this center heating effect becomes insignificant because heating occurs in a very narrow region close to the center of the cylinder. 122 1.4 2 3 4 5 6 CyIinder radlI, cm V<. : EZJO.3 E30.4 E S I 0.5 Fig- 4.27 Effect of attenuation factor on the radial distance of concentrated heating from the center in cylinders of various radii 4.2.3 Sensitivity Analysis Listings of TUODEPEP finite element programs used for cylinders and slabs are listed in the Appendix A. selected in the sensitivity analysis. Five parameters used were They are thermal diffusivity (a( — k/pCp), convective heat transfer coefficient, evaporative heat loss, microwave surface power and attenuation factor. Two of these parameters, i.e. thermal diffusivity and attenuation factor, are 123 material properties. Others are related to the environment and operating conditions. The parameters were varied in ranges according to realistic situations. They are listed in Table 4.2. The justification of the selected percent variation in the parameters is as follows. Thermal diffusivity lumps together thermal conductivity, density and specific heat of the material. Thermal diffusivity of high moisture foods varies within the range of ± 20% from the value used in the model (Choi and Okos, 1983). The convective heat transfer coefficient as used by Swami (1982) may vary due to the degree of ventilation caused by the fan during microwave heating in different ovens. The heat transfer coefficient of the moving air may change from 11.3 to 55 W/m2oC . The relative humidity of the ambient air may vary significantly from day to day. Since the water activity of gel is very high and close to that of water, fluctuation in the relative humidity in the air may account for about ± 60% from the assumed 39% difference in the relative humidity between the sample and the air. According to Gerling (1987), variability of the voltage from the power supply may cause ± 10% change in the power output from the magnetron. The variations in the attenuation factor were derived based on the attenuation factor of water and 0.1 M NaCl solution. Cylinders of sizes 2 cm and 4 cm radii by 5 cm height were used for sensitivity analysis. They represented a small size cylinder (2 cm radius) which has pronounced center heating and a larger size cylinder (4 cm radius) with little center heating. The changes of temperature 124 Table 4.2 Variations of the characteristic parameters of the microwave heating model for sensitivity analysis Parameters1 1 at hc Values Variation “t 2.011E-3 fcm2/®] ± 20 % hc 3.944E-3 (w/cm2oC) ± 40 % ARH 39 « ± 60 % Po 40.1+59.81n(wt) [w] ± 10 % a f(er\ ± 30 % er") [cm’1J is thermal diffusivity, k/pCp, is convective heat transfer coefficient, ARH is the difference of relative humidity between sample and the ambient air, P0 a is the total microwave power absorbed, is the attenuation factor calculated from Eq. 4.3. 125 prediction due to the variation of the various parameters in cylinder of 2 cm radius by 5 cm height were significant. The temperature distribution as a result of a change in the various parameters was simulated by the finite element program. The time-temperature relationships at one point in the cylinder due to changes of parameters are shown in Figs. 4.28 to 4.32. Temperature variations due to the change of the material's thermal diffusivity, attenuation factor and microwave power output were compared at the center point of the cylinder. 2 can rad. x 5 cm ht. r=0 4 cm rad. x 5 cm ht. z=0 r=0 z=0 40 80 ■ 10% 10% 40 5 10 Time, s Fig. 4.28 15 20 10 20 30 Time, s Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying microwave power output ± 5% and ± 10% 126 T [°C] T [°C] 100 2cm rad. x 5cm ht. r-0 z-0 4cm rad. x 5cm ht. -30: 80- -3 0 % 40 30% 60 40- Time , 8 Fig. 4.29 Time, s Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying the attenuation factor ± 30% T [°C] T [°C] 100 2cm rad. x 5cm ht r-0 4cm rad. x 5cm ht z-0 r-0 80 z-0 40 60- 20% 10% 20% 10% 30 -10% -20% 20 5 10 15 Time, s 20 10 20 30 Time, s Fig. 4.30 Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying the thermal diffusivity ± 10% and ± 20% T [°C] 127 T [«C] 100 2 cm rad- x 5 cm ht. r=2 cm 4 cm rad. x 5 cm ht. z=2.5 on r=4 cm z=2.5 cm 40 80 's -40% ♦ - 20% » 0 * 20% x 40% 60 40 30 i i i1V "i— i — i — i — i— i— i — i— i — r 10 15 20 20 20 10 Time, s Fig. 4.31 20 Time, s 30 Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying the heat transfer coefficient ± 20% and ± 40% T [°C] T [°C] 100 2 an rad. x 5 an ht r=2 an 4 an rad. x 5 cm ht z=2.5 an r=4 cm z=2.5 cm 40 80 60 30 40 20 20 5 10 15 Time, s Fig. 4.32 20 10 20 Time, s 30 Sensitivity analysis for 2 cm and 4 cm radii cylinders of 5 cm height by varying the difference of relative humidity between sample and ambient air ± 30% and ± 60% 128 The effects of changes of the boundary conditions (heat transfer coefficient and relative humidity) on temperature predictions were evaluated at r - R and z - 2.5 cm surface. The relative sensitivity was calculated as the ratio of the change in temperature with respect to the change of parameters to the original value as shown in Eq. 3.36. The results of this part are presented in Table 4.3. Table 4.3 Relative Sensitivity Calculated from Sensitivity Analysis1. 2 cm rad. x 5 cm 4 cm rad. x 5 cm “t 53.4 32.2 hc 1.4 1.8 ARH 4.7 7.2 Po 59.8 30.3 40.0 10.9 Parameters a 'Calculated based on Eq. 3.36 expressed in percentage. The effects of ± 40% change in heat transfer coefficient and ± 60% change in relative humidity were not as significant as compared to the variations in other parameters. This is due to the fact that these parameters are only affecting the boundary condition. The model was equally sensitive to changes in thermal diffusivity and power absorption. Variations in attenuation factor also lead to considerable changes in temperature predicted by the finite element model for 2 cm radius by 5 cm height cylinder. 129 4.2.4 Model Verification The temperature distribution during microwave heating of cylindrical and rectangular gels under different conditions were predicted by the finite element program TWODEPEP. The results of model predictions were verified using experimental measurements. Comparisons of the predicted RTR and the measured RTR were performed at 0.01 significance level by t-test. 4.2.4.1 Temperature Distribution in Slabs and Cylinders with Thermal Insulation Cylinder Fig. 4.33 shows (a) the predicted RTR in a half cylinder of 4 cm radius and 2.5 cm height using the finite element program and (b) the RTR distribution obtained from the experimental results, respectively. Duration of heating was 30 s in a microwave oven. When the magnetron come-up time (3 s) was subtracted, the exact heating time was 27 s. The element size of 1.562 cm2 and 5 s time step were determined for the finite element program that resulted in the least percent error when compared with the predicted RTR with the experimental RTR. Assuming uniform surface microwave power (uniform field distribution all around the cylinder), the temperature distribution within the cylinder has radial and axial symmetry. The diagrams show from the center to the bottom part of cross sectional area of a cylinder. The boundary and initial conditions were set as 25 °C initial temperature and no heat loss from the boundaries (thermally insulated). 130 (a) Predicted 1.0 u 60 01 Q 0.5 I 0.0 0.0 -0.625'J'Ujv z [cm] r [cm] -1.25 (b) Experimental u> o 1. 0 60 0) <=> (A 0.5 0.0 -0.625 z [cm] -1.25 Fig. 4. 33 (a) Predicted RTR and (b) experimental RTR in a 4 cm radius by 2.5 cm height cylinder with no heat loss from the boundaries 131 Center heating was observed in the predicted RTR. The variation of RTR in the z direction indicated the attenuation of bottom surface power from z - -1.25 cm and then concentrated at the center. Fig. 4.33(b) shows the asymmetrical variation of RTR when compared to the predicted RTR, which reflected the nonuniform microwave power distribution in the area where the sample was heated. Because of the imperfect insulation on the radial surface, the experimental RTR decreased at r - ±4 cm surface areas due to convective heat loss. Slab Figs. 4.34(a) and 4.34(b) show the RTR values from the finite element program prediction and experimental measurements in slabs of 8 cm x 8 cm x 5 cm. The initial and boundary conditions are 25 °C initial temperature and no heat losses at the boundaries. of microwave heating was 35 s. cm2 while the time step was 1 s. The duration The element size was determined as 0.64 The microwave radiations from x and y directions only were taken into account due to the two-dimension limitation of the finite element program. The figures show RTR results over the cross sectional area of the slab at z - 0 level. distribution of RTR was observed in both figures. Symmetrical Again, the effect of microwave power distribution is evident in the experimental result shown in Fig. 4.34(b). Predicted Edge heating and corner heating effects are shown in both the RTR prediction and the experimental RTR. The predicted RTR values were shown higher than the measured RTR in many locations at surface of the gel, which may be caused from imperfect insulation and heat loss during temperature measurements. 132 (a) Predicted u> o 00 4> 0 p? 0.5 1 0.0 y [cm] x [cm] (b) Experimental ID U t>0 4) P . 0.5 I 0.0 y x [cm] [cm] -4 -4 Fig. 4. 34 (a) Predicted RTR and (b) experimental RTR in an 8 cm x 8 cm x 5 cm slab at z - 0 with no heat loss from the boundaries 133 4.2.4.2 Temperature Distribution in Cylinders and Slabs with Heat Loss from the Boundaries Cylinder RTR as predicted by the model for cylinders of different sizes are plotted in Figs. 4.35 to 4.42. The model considered convective heat loss radially, and convective and evaporative heat loss on the top surface of the gel. For the most part, the predicted temperature distributions in cylinders of different sizes were comparable with the results from the experimental data. Results from finite element model predictions and the experimental measurements were compared at 0.01 significance level and are listed in Appendix C. At most of the locations the model predictions were not significantly different from the experimental results. Locations where model predictions were significantly different from the experimental results were at the center point or at regions close to the center line of the cylinder ( r - 0 ) , especially at the bottom level (z - -Lz) of the cylinder. It was found that the model predictions at the center points were mostly higher than the experimental results, while lower than the experimental results at regions near by the center line. This is possibly due to the time delay from when the samples were taken out of the microwave oven and the temperature measurements were taken. 134 4.5 z=l.25 can co CJ 3.0. i 0.0 4.5 z=0 m 3.0' o S 0.0 4.5 z=-l.25 cm to o * 0.0 r/R measured Fig. 4.35 — predicted The means and standard deviations of predicted and measured RTR at various radial distances from the center (r/R - 0) in a 2 cm radius by 2.5 cm height cylindrical gel at z 1.25 cm, z - 0, and z - -1.25 cm levels 135 3.0 z=l.25 cm § rj> 2.0 a 0.0 3.0 z=0 1.0 0.0 3.0 z=-l.25 cm (0 u 2.0 S 1.0- 0.0 r/R measured Fig. 4.36 |— predicted The means and standard deviations of predicted and measured RTR at various radial distances from the center (r/R - 0) in a 3 cm radius by 2.5 cm height cylindrical gel at z 1.25 cm, z - 0, and z - -1.25 cm levels 136 z=l.25 cm 0.5- 0.0 1.5 z=0 0.0 1.5 z=-l.25 cm 1. 0 0.5 0.0 1/2 3/4 r/R * measured Fig. 4. 1— predicted The means and standard deviations of predicted and measured RTR at various radial distances from the center (r/R - 0) in a 4 cm radius by 2.5 cm height cylindrical gel at z 1.25 cm, z - 0, and z - -1.25 cm levels 137 z=l.25 an (0 £ 0.5- 0.0 1.5 z=0 (0 0.0 1.5 z=-l.25 an CO I 0.5- 0.0 |— Fig. 4.38 measured r/R *— |— predicted The means and standard deviations of predicted and measured RTR at various radial distances from the center (r/R - 0) in a 6 cm radius by 2.5 cm height cylindrical gel at z 1.25 cm, z - 0, and z - -1.25 cm levels 138 4.5 z=2.5 an 3.0 0.0 4.5 z=0 S 0.0 4.5 z=-2.5 an $ 0.0 1 /2 r/R measured Fig. 4. — I— predicted The means and standard deviations of predicted and measured RTR at various radial distances from the center (r/R - 0) in a 2 cm radius by 5 cm height cylindrical gel at z - 2.5 cm, z - 0, and z - -2.5 cm levels 139 3.0 z=2.5 an 0.0 3.0 2=0 * ° 2.0 0.0 3.0 z=-2.5 an § $ 0.0 2/3 r/R measured Fig. 4.40 — f— predicted The means and standard deviations of predicted and measured RTR at various radial distances from the center (r/R - 0) in a 3 cm radius by 5 cm height cylindrical gel at z - 2.5 cm, z - 0, and z - -2.5 cm levels 140 z-2.5 cm § a 0.5 0.0 1.5 z=0 (0 \ a 0.0 1.5 z=-2.5 cm & 0.5 0.0 1/4 1 /2 r/R measured Fig. 4.41 3/4 — 1 predicted The means and standard deviations of predicted and measured RTR at various radial distances from the center (r/R - 0) in a 4 cm radius by 5 cm height cylindrical gel at z - 2.5 cm, z - 0, and z - -2.5 cm levels 141 .5 z=2.5 an .0 .5 0.0 1.5 z=0 I 0.0 1.5 z=-2.5 an $ 0.5 - 0.0 r/R measured Fig. 4.42 predicted The means and standard deviations of predicted and measured RTR at various radial distances from the center (r/R - 0) in a 6 cm radius by 5 cm height cylindrical gel at z - 2.5 cm, z - 0, and z - -2.5 cm levels 142 The field distribution was incorporated in the power absorption term. However, this did not effectively improve the prediction by the model. This may result from the fact that the technique used to determine the field distribution cannot accurately predict the local field intensity in the presence of a load, and thus the mode patterns within the microwave oven cannot be accurately described. It is also possible that the regression model tends to smooth the high and low nodes as was observed by comparing the experimental data (Fig. 4.1) with data from the regression model (Fig. 4.2). In all cases, except 6 cm radius cylinders, center heating phenomenon was observed. As was discussed in the previous section, the area where center heating occurred decreases as the radius increases. The temperature increase predicted from cylinders of 5 cm height is about half of that from cylinders of 2.5 cm height. Vertical temperature distribution is consistently uniform in cylinders of 2.5 cm height. For 5 cm high cylinders, the temperature predicted at the center plane (z - 0) is lower than the temperatures at the top and bottom surfaces. This was found in all the 5 cm high cylinders. Table 4.4 compares the calculated RTR from model prediction with the thermocouple measurement and the fluoroptic probe measurement. Limited by the number of channels for the temperature measurement by fluoroptic probes, the locations which gave rise to large temperature changes were selected. The predicted RTR values were compared with results of experimental measurements at 0.01 significance level. Most of the disagreement between the model prediction and the experimental results came from samples of smaller diameters where center heating was 143 Table 4.4 The Means and Standard Deviations of RTR In Cylindrical Gels of Various Sizes Measured by Thermocouples and Fluoroptic Probes and Compared to the Model Predicted RTR r [cm] z [cm] Measured RTR Thermocouple Fluoroptic Predicted RTR 2 cm rad, x 2.5 cm ht. 0 1.25 1.25 2 0 0.00 0 -1.25 1.65110.311 1.22110.200 3.48810.307 2.51110.335 2.01110.649 1.47410.357 3.63210.500 1.34210.397 3.087 1.615 3.187 3.307 2 cm rad. x 5 cm ht. 0 2.50 2 2.50 0 0.00 0 -2.50 1.15810.128 1.49611.894 2.39010.352 1.25510.177 1.30410.340 1.89410.320 3.22510.699 0.78410.169 2.947 0.810 3.079 2.645 3 cm rad. x 2.5 cm ht. 0 1.25 3 1.25 0 0.00 3 -1.25 0.80610.124 1.02110.100 1.75210.134 1.75910.173 0.82110.180 1.35610.208 1.97110.594 2.12210.383 1.494 0.826 2.110 0.863 3 cm rad. x 5 cm ht. 0 2.50 3 2.50 0 0.00 3 -2.50 0.44110.091 0.95810.285 0.84110.119 0.90110.081 0.51310.134 1.60410.476 0.78210.212 1.10710.178 1.004 0.519 0.900 0.569 4 cm rad. x 2.5 cm ht. 0 1.25 3 1.25 4 1.25 4 -1.25 0.57510.047 0.64310.063 0.65810.089 1.38210.173 0.72610.120 0.59710.063 0.58210.097 1.49010.410 0.703 0.499 0.592 0.635 4 cm rad. x 5 cm ht. 0 2.50 3 2.50 4 2.50 4 -2.50 0.41310.078 0.43010.084 0.76710.190 0.81610.057 0.50010.030 0.33910.071 1.03210.277 0.69110.147 0.449 0.333 0.408 0.445 6 cm rad. x 2.5 cm ht. 5 1.25 6 1.25 5 0.00 6 -1.25 0.37810.034 0.33810.047 0.41110.029 0.34810.042 0.34310.074 0.39910.091 0.68210.120 0.73210.082 0.302 0.364 0.337 0.410 6 cm rad. x 5 cm ht. 2.50 5 2.50 6 6 0.00 -2.50 6 0.19110.007 0.18710.005 0.13410.003 0.19010.017 0.163+0.041 0.42310.600 0.25110.035 0.43010.081 0.195 0.251 0.198 0.282 144 predominant. The model predictions at the center of the cylinders were higher for cylindrical samples with smaller diameters. The predicted temperatures at the center of the top and bottom surfaces of the cylinder were higher than those experimentally determined in all the samples. For the most part, the discrepancies at the center of the cylinder were less than those from the top center and the bottom center of the cylinder. At areas close to the center (r - 0) at three levels in the z direction, the predicted RTR values of most of the cylinders were found significantly different from the experimental results. The RTR calculated from experimental temperature measurement with fluoroptic probes showed similar trends as those from the thermocouple measurement. Deviation of the model prediction from experimental results may be from the inability to sufficiently incorporate the interaction of microwave power with the gel sample. Apart from the over-estimation of the center temperature in cylinders of small radii, the model predicted the trend in temperature distribution with respect to different locations in the gel reasonably well. The predicted temperature distribution for the slower heating region were in close agreementwith the experimental results as shown in Table 4.4. This will assurea safe estimation of heating time for microwave thermal processing. Slab Fig. 4.43 shows the RTR from the model simulation and experimental results for slabs of 3 sizes with microwave incident in the x and y directions. The slab geometry is essentially a three- dimensional geometry, but TWODEPEP cannot handle three-dimensional problems. Because of the limitation of TWODEPEP in solving only two (b)predict: 6an x 6cm x 5cm (e)measure: 6cm x 6cm x 5cm (c)predict: 8cm x 8cm x 5cm (f)measure: 8cm x 8cm x 5cm Fie. 4.43 Model predicted RTR and measured RTR at z - 0 in slabs of (a) and (d) 4 cm x 4 cm x 5 cm, (b) and (e) 6 cm x 6 cm x cm, and (c) a n d (f) 8 cm x 8 cm x 5 cm, respectively 146 dimensional conditions, the experiments were conducted by shielding gel slabs with aluminum foil on the top and bottom surface. The figure shows RTR at the central cross sectional layer of the slab. predictions were close to the experimental results. The model This was also verified by the comparison of the predicted RTR with experimental RTR atO.Ol significance level as shown in Appendix C-10. However, the field distribution effect incorporated in the power source term did not improve the model prediction significantly. For all the slabs, the model predicted an RTR at the center to be lower than the experimental results. As seen in Fig. 4.43(d), the RTR in a slab of 4 cm x 4 cm x 5 cm was unique, and this was not predicted by the model. The center of the slab in this case was not the slowest heating point as predicted by the model. This phenomenon is similar to the center heating of the cylindrical gel. It is possible that the microwave resonance effect causes the generation of internal mode patterns in the small gel sample. In order to understand this further, the true mechanism of microwave coupling needs to be further elucidated. Gel with 1 % Total Salt Content Temperature distribution in a 4 cm radius by 2.5 cm height cylindrical gel containing 1 % total salt concentration during microwave heating was predicted using TWODEPEP. Since the NaCl Molar equivalent concentration for 1% total salt content in gel was determined to be 0.172 M, the attenuation factor for NaCl solution at 0.172 M 147 (Interpolated from Fig. 4.5) was used in the model. A multiple linear regression model was developed to describe the change In attenuation factor as a function of temperature from 0°C to 100°C as shown In Eq. 4.4. a - 0.758 - 0.00619T + 0.00012T2 R2-99.4 « (4.4) This function was used in the program, and the RTR was calculated based on predicted temperature data. The predicted RTR at different locations In the 4 cm radius by 2.5 cm height cylinder were plotted with respect to radial distance as shown in Fig. 4.44 and compared with the experimental RTR. The model predictions are not significantly different from the experimental results at 0.01 significance level except for the points at r - 0 and r - 1 cm at z - 0 and z - 1.25 cm two levels within the cylinder as shown in Appendix C-5. Radial Microwave Heating To simulate radial microwave heating, the source term in the z direction was eliminated, leaving just the radial microwave absorption term in the program. The RTR values based on the predicted data were compared with the RTR values from the experimental results of radial microwave heating in a 4 cm radius by 2.5 cm height cylinder. This is shown in Fig. 4.45. The model predictions appeared to be generally lower than the experimental results except at the center region. Microwave heating of 4 cm radius by 2.5 cm height cylinder with electrical shielding from just the top surface was simulated by a finite element model. The microwave power coming from the z - 1.25 cm surface 148 z=l .25 can (0 a S 0.0 z=0 m o $ 0.0 1.5 z=-l.25 cm 1. 0 0.5 0.0 1/4 r/R measured - i - predicted Fig. 4.44 Effect of 1% total salt on the predicted and measured RTR in a 4 cm radius by 2.5 cm height cylindrical gel at different locations at (a) z - 1.25 cm, (b) z - 0, and (c) z - -1.25 cm levels 149 1.5 z=l.25 an (0 'u 1.0 g 0.5 0.0 1.5 z=0 5 g 0.5 0.0 1.5 S z=-l.25 an 1.0- g 0.0 0 1/4 v — +— 1/2 3/4 1 r/R— t4— predicted measrued Fig. 4.45 Effect of radial microwave heating on predicted and measured RTR In a 4 cm radius by 2.5 cm height cylindrical gel at different locations at (a) z - 1.25 cm, (b) z - 0, and (c) z - -1.25 cm levels 150 was eliminated from Che power source term. The RTR results were plotted against the radial distance from the center as shown In Fig. 4.46. Like the radial microwave heating simulation, the temperature predictions at the center of the cylinder were all higher than the experimental results but were lower at other locations at level z - 0. Appendix C-7 shows the comparison of measured RTR and predicted RTR at all the locations in the cylinder. The microwave power incident from the z - -1.25 cm was removed from the power source term to predict the microwave heating of a 4 cm radius by 2.5 cm height cylinder with bottom surface shielded with aluminum foil. The predictions were compared with the experimental results as shown in Fig. 4.47 and Appendix C-8. RTR values calculated from temperature predictions at center areas of z « 1.25 cm and z - 0 levels were higher than the experimental results. Axial Microwave Heating When the radial microwave heat source was taken away from the microwave power absorption term, the model simulated axial microwave heating. The prediction was compared with the RTR experimental results by shielding at the 4 cm radius by 2.5 cm height cylindrical surface. The data are plotted in Fig. 4.48 with respect to radial distance from the center. At z - 0 and z - -1.25 cm levels the model predicted lower RTR as compared to the experimental results. Discrepancy between the experimental and predicted RTR may be due to induced internal electric field inside the aluminum foil circumference which was not incorporated in the mathematlc model. 151 1.5 z=l.25 cm (0 1.0 $ 0.5 0.0 1.5 z=0 0.0 z=-l.25 cm s s 174 172 374" r/R — measured — |— predicted Fig. 4.46 Effect of microwave hearing with the top surface shielded on predicted and measured RTR In a 4 cm radius by 2.5 cm height cylindrical gel at different locations at (a) z - 1.25 cm, (b) z 0, and (c) z - -1.25 cm levels 152 1.5 z=1.25 cm 1.0 0.5 0.0 1.5 z=0 $ 0.0 1.5 z=-l.25 cm (0 0.5- 0.0 1/2 3/4 r/R measured -- 1— predicted Fig. 4.47 Effect of microwave heating with the bottom surface shielded on predicted and measured RTR In a 4 cm radius by 2.5 cm height cylindrical gel at different locations at (a) z - 1.25 cm, (b) z 0, and (c) z - -1.25 cm levels 153 1.5 z=l.25 an (0 ^ 1.0 s 0.5 0.0 z=0 § 1.0 - ir | 0.5! 0.0 1.5 co o 0.5 0.0 0 1 /2 r/R measured — J— 1/4 |— 3/4 1 predicted Fig. 4.48 Effect of axial microwave heating on the predicted and measured RTR at different locations in a 4 cm radius by 2.5 cm height cylindrical gel at (a) z - 1.25 cm, (b) z - 0, and (c) z - -1.25 cm levels 154 4.2.4.3 Summary Considering the many variables during microwave heating, the model simulation of microwave heating compared favorably with the experimental results for gels of different salt concentration, geometry and size. In many cases, especially with cylindrical geometry, the center temperature predicted by the model was higher than that from the experiments. The model predicted lower temperature distribution in cylindrical samples which were microwave-shielded in the radial direction when compared to the experimental results. A thorough understanding on the interaction of microwaves with metals is needed in order to improve the model by inputting the functions that better describe the phenomenon. The asymmetrical temperature distribution observed during experimental measurements was not effectively predicted by the model. A better method to assess the microwave power distribution will enable simulation closer to the experimental results. Since microwave power distribution varies from oven to oven, it is almost impossible to specify this factor clearly without over complicating the model. It can be concluded that this mathematical model can be used to simulate the temperature distribution for samples of different material properties and geometrical configurations under different microwave heating conditions within a reasonably small range of deviation. 155 CHAPTER 5 CONCLUSIONS The heating characteristics of foods In a microwave oven depend on the oven parameters and the physical and dielectric properties of the food. This research was aimed at understanding the nonuniform heating of food in a microwave oven by means of experimental measurement and mathematical modeling the microwave heating process based on the interaction between the microwaves and the food material. In many research activities related to microwave heating of dielectric materials, temperature measurements were often performed at few locations on the sample and the factor of nonuniform microwave power distribution was not investigated. In this research, temperature measurements were carried out three-dimensionally in samples to better understand the whole picture of temperature distribution in a high moisture food during microwave heating. In order to characterize the microwave oven, the patterns of microwave power distribution at different locations in the oven were determined. The relationships between the microwave power absorption and the loading in cylindrical and rectangular containers were also studied. Microwave power distribution in a Tappan 500 microwave oven used in this study is composed of high and low nodes. The patterns are different from one level to another in the vertical direction. The power intensity increases with increasing height within the oven cavity. The absorption of microwave power with different sizes of load follows a 156 logarithmic function. The power-load relationship is also dependent on the shape of the container because of the effect of microwave power coupling efficiency. A model food system made of sodium alginate gel was used in studying the effects of salt content, geometrical configuration, and shielding of microwaves (at different surfaces of the sample) on temperature distribution of high moisture foods during microwave heating. Surface heating was predominant in gels of high salt content because of the increased attenuation due to the elevation of dielectric loss and depression of dielectric constant from increased conductivity. Cylindrical and rectangular shaped gels of different sizes were studied. Center heating was predominant in samples with smaller diameters (4 cm and 6 cm). Surface heating was observed in cylinders of diameters from 8 cm and larger. The center heating or surface heating can be determined by effective power gain of a cylinder with respect to the distance from the center. In small cylinders, the absorbed microwave power density becomes very large (due to decreasing volume) as microwaves are transmitted towards the center of the cylinder. In a slab, microwave power attenuated exponentially from the surface towards the center. Therefore, the slowest heating point was located at the center of the slab. Corner and edge heating occurred in the slab due to incident microwave power from two or three surfaces. Shielding of the microwave incident power in the axial and radial surfaces on a cylinder resulted in microwave heating from the radial direction and axial direction, respectively. The temperature increase at the surface adjacent to the aluminum foil (used for shielding) was 157 lower. Shielding the microwaves coming In from the bottom part of the oven eliminated the nonuniform reflected microwaves from the bottom of the oven. Shielding at the radial surface resulted In an uneven temperature distribution much higher than that from other different configurations. A mathematical model was developed to simulate transient heat transfer In a food during microwave heating. Boundary conditions were Included in the model to consider the surface heat loss due to convective and/or evaporative heat loss. The volumetric microwave power absorption term with respect to distance from surface was derived for cylindrical and rectangular shaped products. In the power absorption term, internal power reflection and the microwave power from the positive and negative direction of the coordinates were included. The magnetron come-up-time was determined for Tappan 500 microwave oven and was incorporated in the program. The TUODEPEP finite element program was used to calculate the temperature distribution in different locations and time step within the model food during microwave heating in different conditions. The model prediction agreed well with the experimental results, except for the samples of small sizes. In small size samples, the model generally predicted higher center temperature than the measured experimental values. The model was able to characterize the heating behavior of the food but was not able to totally describe the nonuniform temperature distribution due to the uneven field distribution in the oven. The changes in temperature predictions by the model due to variations in selected parameters were tested. These parameters 158 Included the material's dielectric properties, thermal diffusivity, convective heat transfer coefficient, ambient relative humidity and microwave oven power output. The sensitivity analysis indicated that the model prediction will be subjected to a larger amount of variation when the material properties change. Change in power output level influenced the model prediction significantly. However, it was found that model prediction will not be affected by the variations in the two boundary conditions, i.e. changes in convective heat transfer coefficient or relative humidity. This study was limited to the conductive heat transfer of high moisture solid foods during microwave heating. Microwave heating of the liquid foods and the effect of salt concentration and viscosity on the natural convection heating has been investigated (Liu, 1990). More work should be conducted in the area of convective/conductive heat transfer and mass transfer modeling for foods with different physical properties and chemical compositions during microwave heating. Physical properties of materials such as density, porosity, heterogeneity (e.g. liquid with particulates) affect the dielectric and transmissive properties, which in turn influence the microwave power coupling and wave propagation inside the food. Heat transfer and moisture migration occur simultaneously during microwave heating. Modeling of heat and mass (moisture) transfer should consider the phase change and the resulting pressure changes, if a food is not porous enough to let the moist air diffuse out freely. Moisture migration measurement techniques need to be developed for conducting experiments to verify the mass transfer models. A better technique to determine the mode pattern within the oven cavity during microwave heating needs to be developed. From the empirical determination of the mode pattern, analytical solutions for the electric field may be obtained, which can be incorporated into the microwave power source term. This can lead to a better prediction of the temperature distribution as affected by the product parameters and oven performance characteristics. 160 BIBLIOGRAPHY Anonymous. 1989. A whole new wave. Prepared Foods New Products Annual. 159(8): 147. Anonymous. 1987. The new (micro)'wave: heating up the market. Frozen Food Report. May/June, 19. Baden Fuller, A. J. 1979. 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Thesis, University of Massachusetts, Amherst, MA. Taoukis, P., Davis, E. A. Davis, H. T., Gordon, J. and Talmon, Y. 1987. Mathematical modeling of microwave thawing by the modified isotherm migration method. J. of Food Science. 52(2): 455-463. Tejika, Y . , Komatsu, F., Masaki, T., Yoshida, N. and Fukai, I. 1987. Three-dimensional analysis of a microwave heating furnace system. J. Microwave Power. 23(3): 107-114. Vermeulen, F. E. and Chute, F.S. 1987. On the classification of processes using electric and magnetic fields to heat materials. Microwave Power Energy and Electromagnetics. 23(3): 187-191. J. von Hippel, A. R. (Ed.). 1954. Macroscopic approach. In "Dielectrics and Waves," p. 3. M.I.T. Press, Cambridge, MA. Washisu, S. and Fukai, I. 1980. A simple method for indicating the electric field distribution in a microwave oven. J. Microwave Power. 15(1): 59-61. Watanabe, M . , Suzuki, M. and Ohkawa, S. 1978. Analysis of power density distribution in microwave ovens. J. Microwave Power. 13(2): 173-181. 164 Wei, C. K. and Davis, E. A. 1985. Heat and mass transfer in water laden sandstone: microwave heating. Am. Inst. Chem. Engr. J. 31(5): 842-848. Wilhelm, M. and Satterlee, L. D. 1973. A 3-dimensional method for mapping microwave ovens. Microwave Energy Appl. Newsl. 4(5): 3-5. 165 Appendix A FORTRAN PROGRAMS USED IN TWODEPEP FOR PREDICTION OF TEMPERATURE DISTRIBUTION DURING MICROWAVE HEATING A-l Sample Program Listings for Rectangular Geometry Used In TWODEPEP //V03XXXXX JOB // EXEC TWODEPEP //TWODEPEP.INPUT DD * 1 50 1 8.37D-3*UX OXX OXY 8.37D-3*UY Cl 1.01D0*4.12D0 FI FLD(X,Y)*PHEE(U,T,X,Y) UO 25. DO ARC-1 GB1 -3.705D-3*(U-28.D0) ARC-2 -3.705D-3*(U-28.D0) GB1 ARC-3 -3.705D-3*(U-28.D0) GB1 NX 4 NY 8 45 NOUT MUR 6 SYMMETRY 1 2 NUPDT 0.5D0 ALPHA DT 1.D0 45. DO TF 1.D0 D3EST VXY 0.DO,-4.DO, 4.DO,-4.DO, 4.DO,0.DO, 4.DO,4.DO, 0.DO,4.DO, 0.DO,0.DO, 2.D0.-2.DO, 2.DO,2.DO VXY IABC 1,2,7, 2,3,7, 3,4,8, 4,5,8, 5,6,8, 6,1,7, 3,6,7, 6,3,8 I 1,2,2,3,4,4,0,0 ADD. DOUBLE PRECISION FUNCTION FLD(X,Y) IMPLICIT REAL*8 (A-H.O-Z) SUM-0.DO COUNT-O.DO A-4.D0 B-4.D0 C-5.D0 M-2*A+1 N-2*B+1 DO 10 1-1, M XI— (A+l. DO)+1+9. DO 166 30 20 10 DO 20 J-l, N X2--(B+l.D0)+J+9.DO Z-5.D0 Q-18.033D0+X2*<-2.953D0)+X2*X1*0.167D0+X2*Z*0.952D0+ * Xl*Xl*Z*(-4.358D-2)+DCOS(Xl)*(-4.784D-l)+ * DCOS(Xl)*DSIN(X2)*1.223D-l+DSIN(X2)*DCOS(Z)*2.485D-l+ * DSIN(Z)*DCOS(X2)*0.1658D0+(X2**4)*1,128D-3+ * (X2**3)*Z*(-2.673D-3)+X2*(Z**3)*(-3.562D-2)+ * X2*X2*X1*X1*1.69D-3+Xl*Xl*Z*Z*3.324D-3+ * (Xl**5)*(-3.502D-05)+(Xl**4)*Z*4.359D-4+ * (X2**3)*Z*Z*(-1.18D-3)+X2*X2*(Z**3)*4.206D-3+ * X2*X2*Xl*Xl*Z*(-2.241D-3)+(X2**5)*Xl*(-2.479D-06)+ * X1*(Z**5)*6.545D-4+(Xl**4)*Z*Z*(-4.89D-05)+ * X2*X1*X1*(Z**3)*(-4.617D-4)+X2*X2*Xl*Xl*Z*Z*8.554D-4+ * <X2**7)*(-1.952D-07)+(X2**6)*Z*7.854D-07+ * Xl*(Z**6)*(-7.971D-05)+X2*(Z**6)*9.144D-06+ * (X2**5)*X1*X1*(-3.79D-06)+X2*X2*(Z**5)*(-1.601D-05)+ * X2*(X1**4)*Z*Z*3.224D-06+(X2**4)*X1*X1*Z*3.296D-05+ * (X2**3)*Xl*(Z**3)*(4.464D-06)* (1.05D-4)*(X2**3)*X1*X1*Z*Z+ * X2*X2*X1*X1*(Z**3)*2.326D-05+(Z**8)*1.02 5D-06+ * X2*(X1**7)*8.792D-09+(X2**6)*X1*X1*2.189D-07+ * X2*<X1**6)*Z*(-7.285D-08)+X2*Xl*(Z**6)*3.123D-06+ * (Xl**3)*(Z**5)*(6.416D-07)+(X2**5)*X1*X1*Z* * (-1.543D-06)+X2*X2*Xl*(Z**5)*(-1.843D-06)+ * (X2**4)*X1*X1*Z*Z*(3.2D-06) SUM-SUM+Q COUNT-COUNT+1.DO CONTINUE CONTINUE CONTINUE QM - SUM/COUNT Xl-(-l.DO)*X + 9.DO X2=Y + 9.DO Q-18.033D0+X2*(■2.953D0)+X2*X1*0.167D0+X2*Z*0.952D0+ * X1*X1*Z*(-4.358D-2)+DC0S(Xl)*(-4.784D-1)+ * DC0S(X1)*DSIN(X2)*1,223D-l+DSIN(X2)*DCOS(Z)*2.485D-1+ * DSIN(Z)*DCOS(X2)*0.1658D0+(X2**4)*1.128D-3+ * (X2**3)*Z*(-2.673D-3)+X2*(Z**3)*(-3.562D-2)+ * X2*X2*X1*X1*1.69D*3+Xl*Xl*Z*Z*3.324D-3+ * (Xl**5)*(-3.502D-05)+(Xl**4)*Z*4.359D-4+ * (X2**3)*Z*Z*(-1.18D-3)+X2*X2*(Z**3)*4.206D-3+ * X2*X2*X1*X1*Z*(-2.241D-3)+(X2**5)*X1*(-2.479D-06)+ * X1*(Z**5)*6.545D-4+(Xl**4)*Z*Z*(-4.89D-05)+ * X2*X1*X1*(Z**3)*(-4.617D-4)+X2*X2*Xl*Xl*Z*Z*8.554D-4+ * (X2**7)*(-l.952D-07)+(X2**6)*Z*7.854D-07+ * Xl*(Z**6)*(-7.971D-05)+X2*(Z**6)*9.144D-06+ * (X2**5)*X1*X1*(-3.79D-06)+X2+X2*(Z**5)*(-1.601D-05)+ * X2*(Xl**4)*Z*Z*3.224D-06+(X2**4)*X1*X1*Z*3.296D-05+ * (X2**3)*Xl*(Z**3)*(4.464D-06)* (1.05D-4)*(X2**3)*X1*X1*Z*Z+ * X2*X2*X1*X1*(Z**3)*2.326D-05+(Z**8)*l.025D-06+ * * * * * X2*(X1**7)*8.792D-09+(X2**6)*Xl*Xl*2.189D-07+ X2*(X1**6)*Z*(-7.285D-08)+X2*Xl*(Z**6)*3.123D-06+ (Xl**3)*(Z**5)*(6.416D-07)+(X2**5)*X1*X1*Z* (-1.543D-06)+X2*X2*Xl*(Z**5)*(-1.843D-06)+ (X2**4)*X1*X1*Z*Z*(3.2D-06) FLD - Q/QM RETURN END DOUBLE PRECISION FUNCTION PHEE(U,T,X,Y) IMPLICIT REAL*8 (A-H.O-Z) X1-4.D0 X2-4.D0 X3-5.DO WT-2.D0*X1*2.DO*X2*X3 P0-50.27D0+56.6DO*DLOG(WT) PS-PO/(2.DO*(2.D0*Xl*X3+2.D0*X2*X3)) DC-81.7935D0-0.299D0*U DL-22.6D0-0.378D0*U+0.00293D0*U*U ATU-2.D0*3.142D0*DSQRT(DC*(DSQRT(1.D0+ * (DL/DC)**2)-1.D0)/2.D0)/12.245D0 C— 2.D0*ATU EPX-DEXP(C*(XI-X))+DEXP(C*(Xl+X)) EPY-DEXP(C*(X2-Y))+DEXP(C*(X2+Y)) Pl-2.DO*ATU*PS*(EPX+EPY) P2-2.DO*ATU*PS*(DEXP(C*2.D0*X1)*EPX+ * DEXP(C*2.D0*X2)*EPY) PWR-P1+P2 TL^3.D0 IF (T.LT.TL) PHEE-0.DO IF (T.GE.TL) PHEE-PWR RETURN END DOUBLE PRECISION FUNCTION TT(U) IMPLICIT REAL*8 (A-H.O-Z) TT-(U+273.D0)**1.5D0 IF(U.GE.105.DO)TT-7 349.16D0 RETURN END DOUBLE PRECISION FUNCTION VP(U) IMPLICIT REAL*8 (A-H.O-Z) VP-1.1779D0*DEXP(.04498D0*U) IF(U.GE.105,DO)VP-273.13D0 RETURN END DOUBLE PRECISION FUNCTION H(U) IMPLICIT REAL*8 (A-H.O-Z) H-2501.86DO-2.39D0*U IF(U.GE.100.D0)H-2250.91D0 RETURN END 168 A-2 Sample Program Listings for Cylindrical Geometry Used In TWODEPEP //V04XXXXX JOB // EXEC TWODEPEP //TWODEPEP.INPUT DD * 1 20 1 OXX 8.37D-3*UX OXY 8.37D-3*UY 1.01D0*4.12D0 Cl FI 8.37D-3*UX/X+FLD(X,Y)*PHEE(U,T ,X ,Y) UO 25. DO ARC-1 -2.7D-4*(U-28.D0)/2.5D0 GB1 ARC-2 -3.944D-3*(U-28.D0) GB1 ARC-3 -3.944D-3*(U-28.D0)-(1.251D-9)*TT(U)*VP(U)*H(U) GB1 NX 4 NY 2 NOUT 1 MWR 6 NUPDT 2 ALPHA 0.5D0 DT 5.DO TF 35. DO D3EST 1.D0 0.D0.-2.5D0, 4.D0.-2.5D0, 4.DO,0.DO, 4.D0,2.5D0, VXY 0.D0.2.5D0, 0.DO,0.DO, 2.0D0,-1.25D0, 2.0D0.1.25D0 VXY IABC 1,2,7, 2,3,7, 3,4,8, 4,5,8, 5,6,8, 6,1,7, 3,6,7, IABC 6,3,8 I 1,2,2,3,4,4,0,0 ADD. DOUBLE PRECISION FUNCTION FLD(X.Y) IMPLICIT REAL*8 (A-H.O-Z) SUM-0.DO COUNT-O.DO R-4.D0 ZL-5.OD0 ZH-ZL/2.DO M-R+l DO 10 1-1, 12 F-(I-1.)/6.D0 A-F*3.14159D0 DO 20 J-l, 3 Z-(J-2.D0)*ZH+5.DO DO 30 K-l, M RX-K-1.D0 Xl-RX*DCOS(A)+9.DO X2-RX*DSIN(A)+9.D0 169 Q-18.033D0+X2*(-2.953D0)+X2*X1*0.167D0+X2*Z*0.952D0+ X1*X1*Z*(-4.358D-2)+DC0S<Xl)*(-4.784D-1)+ DCOS(XI)*DSIN(X2)*1.223D-1+DSIN(X2)*DCOS(Z)*2.485D-1+ DSIN(Z)*DCOS(X2)*0.1658D0+(X2**4)*1.128D-3+ (X2**3)*Z*(-2.673D-3)+X2*(Z**3)*(-3.562D-2)+ X2*X2*X1*X1*1.69D-3+Xl*Xl*Z*Z*3.324D-3+ (Xl**5)*(-3.502D-05)+(Xl**4)*Z*4.359D-4+ (X2**3)*Z*Z*(-1.18D-3)+X2*X2*(Z**3)*4.206D-3+ X2*X2*X1*X1*Z*(-2.241D-3)+(X2**5)*Xl*(-2.479D-06)+ XI*(Z**5)*6.545D-4+(Xl**4)*Z*Z*(-4.89D-05)+ X2*X1*X1*(Z**3)*(-4.617D-4)+X2*X2*Xl*Xl*Z*Z*8.554D-4+ (X2**7)*(-1.952D-07) KX2**6)*Z*7.854D-07+ X1*(Z**6)*(-7.971D-0a)+X2*(Z**6)*9.144D-06+ (X2**5)*Xl*Xl*(-3.79D-06)+X2*X2*(Z**5)*(-1.601D-05)+ X2*(X1**4)*Z*Z*3.224D-06+(X2**4)*Xl*Xl*Z*3.296D-05+ (X2**3)*Xl*(Z**3)*(4.464D-06)(1.05D-4)*(X2**3)*X1*X1*Z*Z+ X2*X2*X1*X1*(Z**3)*2.326D-05+(Z**8)*l.025D-06+ X2*(XI**7)*8.792D-09+(X2**6)*X1*X1*2.18 9D-07+ X2*(X1**6)*Z*(-7.285D-08)+X2*Xl*(Z**6)*3.123D-06+ (Xl**3)*(Z**5)*(6.416D-07)+ (X2**5)*X1*X1*Z* (-1.543D-06)+X2*X2*X1*(Z**5)*(-1.843D-06)+ (X2**4)*X1*X1*Z*Z*(3.2D-06) SUM-SUM+Q COUNT—COUNT+1.DO CONTINUE CONTINUE CONTINUE QM - SUM/COUNT X1-X*DCOS(3.14159DO*1,ODO) + 9.DO X2-X*DSIN(3.14159D0*1.0D0) + 9.DO Z-Y+5.DO Q-18.033D0+X2*(-2.953D0)+X2*Xl*0.167D0+X2*Z*0.952DO+ * X1*X1*Z*(-4.358D-2)+DC0S(Xl)*(-4.784D-1)+ * DC0S(Xl)*DSIN(X2)*l.223D-l+DSIN(X2)*DCOS(Z)*2.485D-1+ * DSIN(Z)*DCOS(X2)*0.1658D0+(X2**4)*1.128D-3+ * (X2**3)*Z*(-2.673D-3)+X2*(Z**3)*(-3.562D-2)+ * X2*X2*X1*X1*1.69D-3+Xl*Xl*Z*Z*3.324D-3+ * (Xl**5)*(-3.502D-05)+(Xl**4)*Z*4.359D-4+ * (X2**3)*Z*Z*(-1.18D-3)+X2*X2*(Z**3)*4.206D-3+ * X2*X2*X1*X1*Z*(-2.241D-3)+(X2**5)*Xl*(-2.4790-06)+ * X1*(Z**5)*6.545D-4+(Xl**4)*Z*Z*(-4.89D-05)+ * X2*X1*X1*(Z**3)*(-4.617D-4)+X2*X2*Xl*Xl*Z*Z*8.554D-4+ * (X2**7)*(-l.952D-07)+(X2**6)*Z*7.854D-07+ * X1*(Z**6)*(-7.971D-05)+X2*(Z**6)*9.144D-06+ * (X2**5)*X1*X1*(-3.79D-06)+X2*X2*(Z**5)*(-1.601D-05)+ * X2*(Xl**4)*Z*Z*3.224D-06+(X2**4)*X1*X1*Z*3.296D-05+ * (X2**3)*Xl*(Z**3)*(4.464D-06)* (1.05D-4)*(X2**3)*X1*X1*Z*Z+ * X2*X2*X1*X1*(Z**3)*2.326D-05+(Z**8)*1.025D-06+ * X2*(Xl**7)*8.792D-09+(X2**6)*Xl*Xl*2.189D-07+ * X2*(X1**6)*Z*(-7.285D-08)+X2*Xl*(Z**6)*3.123D-06+ * * * * * * * * * * * * * * * * * * * * * * 30 20 10 * (Xl**3)*(Z**5)*(6.416D-07)+(X2**5)*Xl*Xl*Z* * (-1.543D-06)+X2*X2*Xl*(Z**5)*(-1.843D-06)+ * (X2**4)*Xl*Xl*Z*Z*(3.2D-06) FLD - Q/QM RETURN END DOUBLE PRECISION FUNCTION PHEE(U,T,X,Y) IMPLICIT REAL*8 (A-H.O-Z) R-4.D0 D-2.D0*R Z-5.0D0 ZH-Z/2.DO WT-(R**2)*3.14L6D0*Z P0-40.1DO+59.8DO*DLOG(WT) PS-PO/(6.2832D0*(R*Z+(R**2))) DC-81.7935D0-0.299D0*U DL-22.6DO-0.378D0*U+0.00293D0*U*U ATU-2.DO*3.142D0*DSQRT(DC*(DSQRT(1.D0+ * (DL**2/DC**2))-1.D0)/2.D0)/12.245DO C— 2.D0*ATU EPX-DEXP(C*(R-X))+DEXP(C*(R+X)) EPY-DEXP(C*(ZH-Y))+DEXP(C*(ZH+Y)) Pl-2.D0*ATU*PS*(R*EPX/X+EPY) P2-2.DO*ATU*PS*(DEXP(C*D)*R*EPX/X+DEXP(C*Z)*EPY) PWR-P1+P2 TI^3.DO IF (T.LT.TL) PHEE-0.DO IF (T.GE.TL) PHEE-PWR RETURN END DOUBLE PRECISION FUNCTION TT(U) IMPLICIT REAL*8 (A-H.O-Z) TT-(U+273.D0)**1.5D0 IF(U.GE.105.D0)TT-7349.16D0 RETURN END DOUBLE PRECISION FUNCTION VP(U) IMPLICIT REAL*8 (A-H.O-Z) VP-1.1779D0*DEXP(.04498D0*U) IF(U.GT.105.DO)VP-132.5067D0 RETURN END DOUBLE PRECISION FUNCTION H(U) IMPLICIT REAL*8 (A-H.O-Z) H-2501.86DO-2.39D0*U IF(U.GE.105.D0)H-2250.91D0 RETURN END 171 A-3 NEQ NTF ND1H OXX OXY U Cl FI UO ARC GB1 NX NY NOUT MWR NUPDT ALPHA DT TF D3EST VXY IABC I ADD FLD PHEE A,B,C R ZL PO PS DC.DL ATU Pi P2 PWR TL TT VP H Specifications for the Inputs in the TWODEPEP Programs Number of differential equation(s) used Number of final triangles desired Stiffness matrix storage reservation 3T/3x 3T/3y Temperature pCp Force functions including heat source terms Initial temperature Boundary site Free boundary condition as a heat flux Number of output points in xdirection Number of output points in ydirection Number of output with time Output logical unit number Number of iterations (time steps) Numerical method used to solve the equation Time step Final time Method of discretization of elements Coordinates of the initial triangle vertexes Numbering the nodes on each element Identification of the arc associated with each element Subfunctions defined after this line Polynomial power distribution function Microwave power absorption function Width, length and height of a rectangle Radius of a cylinder Height of a cylinder Total microwave power absorbed by the sample Surface microwave power intensity Dielectric constant and loss Attenuation factor Primary microwave power absorbed Secondary microwave power absorbed Microwave power absorbed per unit volume Magnetron come-up-time Mass transer coefficient temperature correction function Water vapor pressure as a function of temperature Latent heat of vaporization as a function of temperature 172 Appendix B THERMOCOUPLE TEMPERATURE MEASUREMENT USING 2IX MICROLOGGER AND AM32 MULTIPLEXER B-l Circuitry of 21X Micrologger with AM32 Multiplexer 2IX AM32 Analog In 1 Com n Ground Coin L Digital Port 1 Switch Ex. 1 Reserve Clock + 12 V + 12 V Ground Ground 173 B-2 Program Instructions for Differential Thermocouple Measurement Using 2IX with AM32 *A: Memory allocation (Default-28) Inst.# Para.# 1 2 3 4 5 6 7 8 9 10 11 P 1 P 1 2 P 1 2 P 2 3 4 P 1 2 3 4 5 6 7 8 P P 1 P 1 2 3 4 5 6 7 8 P 1 P 1 P 1 2 Entry t 17 1 20 1 1 87 0 n 22 1 0 5000 14 1 1 1 1 1 2C 1 0 95 20 0 14 m 1 2 1 1 n+2 1 0 86 10 77 11 70 n+m+1 1 Description Execution interval Measure panel temperature Store ref. temp, in Loc.#l Enable AM32 to set port Set port high (5V) Port #1 (Analog I/O #1) Loop Delay Count (n £ 32) Excite channel #1 Delay excitation in 0.01s 0 s delay after excitation Excitation range mV Measure differential TC at AM32 Repetition Range code ± 5 mV 1st measurement @ channel #1 TC type (3 for type K) Reference temperature loc. Indexed destinated 1st input loc. Multiplier Offset End Set port (Reset AM32) Set port low (0V) Differential TC measurement at 21X Repetition Range code Measure 1st TC channel TC type Reference temp. loc. Destinate input loc. Multiplier Offset Output along data collection Set flag 0 to 1 (Output) Time instruction Hr:min:sec Sample Repetitions Loc. of 1st input data 174 Appendix C SUMMARY OF THE STATISTICAL ANALYSIS ON THE RESULTS FROM EXPERIMENTAL MEASUREMENTS AND MODEL PREDICTIONS C-l Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 2 cm Radius Cylindrical Gels and Compared at 0.01 Significance Level r[cm] z[cm] Experimental RTR STD Predicted RTR STD ^calc 0 1 2 1.25 1.25 1.25 1.651 1.177 1.317 0.311 0.211 0.257 3.087 1.095 1.615 0.004 0.010 0.016 9.233 * 0.773 2.313 0 1 2 0 0 0 3.488 2.844 1.203 0.307 0.421 0.258 3.187 1.030 1.578 0.008 0.016 0.022 1.957 8.606 * 2.899 0 1 2 -1.25 -1.25 -1.25 2.511 1.577 2.049 0.335 0.117 0.336 3.307 1.199 1.800 0.013 0.028 0.039 4.749 6.265 * 1.478 0 1 2 2.5 2.5 2.5 1.158 0.971 1.224 0.128 0.287 0.202 2.947 0.989 0.810 0.003 0.005 0.008 27.947 * 0.124 4.092 0 1 2 0 0 0 2.390 1.192 0.749 0.352 0.229 0.076 3.079 0.850 0.615 0.007 0.009 0.013 3.915 2.971 3.453 0 1 2 -2.5 -2.5 -2.5 1.255 0.817 1.140 0.177 0.024 0.167 2.645 0.519 1.670 0.010 0.021 0.033 15.678 * 18.891 * 6.233 * 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) 3> ^0.01,3 ” 5 -8^l 175 C-2 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 3 cm Radius Cylindrical Gels and Compared at 0.01 Significance Level r[cm] z[cm] Experimental RTR STD 0 1 2 3 1.25 1.25 1.25 1.25 0.806 0.868 0.735 0.785 0.124 0.087 0.163 0.222 1.494 0.638 0.830 0.826 0.003 0.006 0.010 0.017 11.090 * 5.241 1.169 0.372 0 1 2 3 0 0 0 0 1.752 1.211 1.271 0.803 0.134 0.135 0.281 0.273 2.110 0.411 0.717 0.792 0.006 0.009 0.015 0.018 5.341 11.795 * 3.939 0.073 0 1 2 3 -1.25 -1.25 -1.25 -1.25 1.214 0.941 0.982 1.290 0.062 0.039 0.174 0.306 1.516 0.762 0.865 0.863 0.009 0.017 0.025 0.021 9.635 * 8.332 * 1.333 2.785 0 1 2 3 2.5 2.5 2.5 2.5 0.441 0.504 0.733 0.675 0.091 0.044 0.312 0.196 1.004 0.481 0.488 0.519 0.002 0.003 0.004 0.006 12.365 * 1.049 1.568 1.583 0 1 2 3 0 0 0 0 0.841 0.620 0.420 0.361 0.119 0.076 0.101 0.102 0.900 0.366 0.381 0.396 0.004 0.005 0.008 0.010 0.995 6.689 * 0.783 0.692 0 1 2 3 -2.5 -2.5 -2.5 -2.5 0.436 0.351 0.542 0.763 0.037 0.020 0.123 0.182 1.034 0.548 0.557 0.569 0.006 0.012 0.021 0.026 31.864 * 16.674 * 0.252 2.114 Predicted RTR STD **calc 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) 3 - ^0.01,3 “ 5 841 176 C-3 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius Cylindrical Gels and Compared at 0.01 Significance Level r[cm] z[cm) Experimental RTR STD 0 1 2 3 4 1.25 1.25 1.25 1.25 1.25 0.575 0.474 0.501 0.491 0.552 0.047 0.020 0.122 0.145 0.147 0.703 0.421 0.408 0.499 0.592 0.002 0.004 0.007 0.012 0.021 5.429 5.248 * 1.513 0.111 0.533 0 1 2 3 4 0 0 0 0 0 0.716 0.549 0.555 0.832 0.571 0.180 0.051 0.097 0.255 0.236 0.906 0.431 0.453 0.490 0.574 0.003 0.005 0.009 0.012 0.016 2.115 4.613 2.097 2.687 0.019 0 1 2 3 4 -1.25 -1.25 -1.25 -1.25 -1.25 0.604 0.605 0.564 0.668 0.897 0.073 0.058 0.095 0.163 0.426 0.729 0.521 0.490 0.598 0.635 0.005 0.011 0.015 0.014 0.008 3.403 2.873 1.530 0.851 1.228 0 1 2 3 4 2.5 2.5 2.5 2.5 2.5 0.413 0.361 0.299 0.302 0.423 0.078 0.023 0.036 0.089 0.171 0.449 0.289 0.296 0.333 0.408 0.001 0.002 0.003 0.005 0.011 0.911 6.106 * 0.164 0.701 0.181 0 1 2 3 4 0 0 0 0 0 0.371 0.273 0.261 0.267 0.209 0.036 0.025 0.051 0.060 0.036 0.329 0.193 0.191 0.254 0.324 0.002 0.003 0.004 0.006 0.010 2.359 6.502 * 2.742 0.412 6.193 * 0 1 2 3 4 -2.5 -2.5 -2.5 -2.5 -2.5 0.285 0.257 0.284 0.494 0.731 0.010 0.019 0.058 0.236 0.349 0.481 0.345 0.345 0.389 0.445 0.003 0.008 0.014 0.016 0.017 37.109 * 8.730 * 2.082 0.886 1.632 Predicted RTR STD ^alc 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) ^.01,3 “ 177 C-4 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 6 cm Radius Cylindrical Gels and Compared at 0.01 Significance Level r [cm] z[cm] Experimental RTR STD 0 1 2 3 4 5 6 1.25 1.25 1.25 1.25 1.25 1.25 1.25 0.204 0.203 0.202 0.201 0.269 0.396 0.409 0.014 0.009 0.017 0.026 0.031 0.026 0.062 0.249 0.218 0.208 0.219 0.251 0.302 0.364 0.002 0.002 0.004 0.007 0.011 0.013 0.011 6.449 * 3.352 0.655 1.338 1.087 6.423 * 1.441 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0.203 0.217 0.230 0.244 0.354 0.510 0.414 0.006 0.011 0.023 0.034 0.048 0.079 0.071 0.276 0,270 0.191 0.247 0.300 0.337 0.386 0.001 0.003 0.004 0.006 0.010 0.015 0.016 24.087 * 8.971 * 3.379 0.183 2.183 4.312 0.787 0 1 2 3 4 5 6 -1.25 -1.25 -1.25 -1.25 -1.25 -1.25 -1.25 0.189 0.202 0.214 0.227 0.276 0.453 0.540 0.012 0.010 0.021 0.031 0.019 0.139 0.197 0.284 0.286 0.291 0.300 0.313 0.388 0.410 0.003 0.006 0.007 0.007 0.005 0.012 0.020 15.343 * 14.066 * 6.968 * 4.590 3.764 0.935 1.310 Predicted RTR STD ^calc 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) 178 C-4 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 6 cm Radius Cylindrical Gels and Compared at 0.01 Significance Level (continued) r[cm] z [cm] 0 1 2 3 4 5 6 2.5 2.5 2.5 2.5 2.5 2.5 2.5 0.206 0.193 0.180 0.167 0.147 0.230 0.274 0.042 0.020 0.039 0.059 0.033 0.092 0.120 0.206 0.126 0.131 0.155 0.155 0.195 0.251 0.001 0.001 0.001 0.003 0.005 0.007 0.008 0.003 6.867 * 2.492 0.390 0.497 0.755 0.372 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0.114 0.116 0.119 0.121 0.165 0.192 0.206 0.009 0.011 0.023 0.034 0.047 0.047 0.045 0.093 0.067 0.067 0.070 0.096 0.154 0.198 0.000 0.001 0.001 0.002 0.004 0.007 0.010 4.615 8.502 * 4.477 2.966 2.904 1.586 0.344 0 1 2 3 4 5 6 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 0.119 0.126 0.133 0.140 0.182 0.370 0.470 0.018 0.010 0.020 0.031 0.071 0.218 0.312 0.218 0.170 0.172 0.181 0.198 0.245 0.282 0.002 0.004 0.006 0.008 0.007 0.008 0.011 11.025 * 8.019 * 3.664 2.597 0.464 1.141 1.202 Experimental RTR STD Predicted RTR STD Cealc 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) " 5 *841 179 C-5 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical Gels with 1% Total Salt and Compared at 0.01 Significance Level r [cm] z[cm] Experimental RTR STD Predicted RTR STD 0 1 2 3 4 1.25 1.25 1.25 1.25 1.25 0.331 0.392 0.432 0.554 0.785 0.047 0.054 0.090 0.165 0.245 0.602 0.608 0.622 0.788 1.121 0.001 0.005 0.009 0.020 0.042 11.635 * 7.915 * 4.217 2.826 2.704 0 1 2 3 4 0 0 0 0 0 0.399 0.478 0.523 0.831 0.940 0.054 0.072 0.119 0.307 0.379 0.220 0.185 0.280 0.355 0.727 0.000 0.002 0.005 0.008 0.023 6.588 * 8.173 * 4.096 3.093 1.120 0 1 2 3 4 -1.25 -1.25 -1.25 -1.25 -1.25 0.598 0.728 0.720 0.991 1.164 0.087 0.123 0.155 0.344 0.498 0.581 0.578 0.597 0.759 1.085 0.003 0.012 0.017 0.018 0.019 0.394 2.410 1.574 1.344 0.317 *"calc 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) 180 C-6 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical Gels with Axial Shielding and Compared at 0.01 Significance Level Predicted RTR STD r[cm] z(cm] Experimental RTR STD 0 1 2 3 4 1.25 1.25 1.25 1.25 1.25 0.247 0.174 0.177 0.283 0.509 0.036 0.037 0.019 0.069 0.075 0.358 0.113 0.106 0.218 0.322 0.001 0.001 0.002 0.005 0.011 6.182 * 3.361 7.573 * 1.879 4.957 0 1 2 3 4 0 0 0 0 0 0.426 0.286 0.291 0.469 0.613 0.009 0.019 0.041 0.104 0.112 0.516 0.141 0.130 0.197 0.304 0.002 0.002 0.003 0.005 0.008 19.542 * 15.498 * 7.834 * 5.247 5.497 0 1 2 3 4 -1.25 -1.25 -1.25 -1.25 -1.25 0.286 0.186 0.179 0.259 0.487 0.022 0.010 0.030 0.043 0.114 0.334 0.106 0.103 0.210 0.312 0.003 0.003 0.004 0.005 0.004 4.312 15.214 * 4.984 2.263 3.074 *calc 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) 3 - *0.01,3 ” 5 *8^1 181 C-7 Means and Standard Deviations of Measured and Predicted RTR at Different Locations In 4 cm Radius by 2.5 cm Height Cylindrical Gels with Shielding on the Top Surface and Compared at 0.01 Significance Level r[cmj z[cm] Experimental RTR STD 0 1 2 3 4 1.25 1.25 1.25 1.25 1.25 0.312 0.298 0.355 0.432 0.344 0.069 0.062 0.058 0.059 0.065 0.414 0.174 0.164 0.276 0.368 0.001 0.002 0.003 0.007 0.013 2.947 3.987 6.587 * 5.209 0.724 0 1 2 3 4 0 0 0 0 0 0.544 0.534 0.593 0.746 0.508 0.063 0.077 0.058 0.115 0.110 0.699 0.286 0.281 0.337 0.431 0.002 0.004 0.006 0.008 0.012 4.910 6.419 * 10.741 * 7.109 * 1.397 0 1 2 3 4 -1.25 -1.25 -1.25 -1.25 -1.25 0.621 0.704 0.733 0.738 0.843 0.103 0.079 0.103 0.126 0.272 0.718 0.446 0.433 0.538 0.609 0.005 0.009 0.013 0.012 0.008 1.884 6.490 * 5.801 3.154 1.720 Predicted RTR STD ^calc 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) 3 ‘ * 0 .01,3 “ 5 * 8 41 182 C-8 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical Gels with Shielding on the Bottom Surface and Compared at 0.01 Significance Level r[cm] z[cm] Experimental RTR STD 0 1 2 3 4 1.25 1.25 1.25 1.25 1.25 0.450 0.411 0.383 0.424 0.682 0.015 0.055 0.040 0.075 0.168 0.762 0.469 0.451 0.558 0.629 0.002 0.004 0.007 0.013 0.021 41.410 * 2.130 3.366 3.514 0.620 0 1 2 3 4 0 0 0 0 0 0.567 0.435 0.446 0.610 0.671 0.045 0.034 0.052 0.121 0.162 0.700 0.286 0.282 0.337 0.431 0.002 0.004 0.006 0.008 0.012 5.900 * 8.765 * 6.277 * 4.505 2.962 0 1 2 3 4 -1.25 -1.25 -1.25 -1.25 -1.25 0.376 0.320 0.295 0.362 0.484 0.039 0.022 0.028 0.050 0.139 0.385 0.166 0.158 0.266 0.356 0.003 0.004 0.005 0.006 0.004 0.456 13.552 * 9.762 * 3.805 1.848 Predicted RTR STD ^calc 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) 183 C-9 Means and Standard Deviations of Measured and Predicted RTR at Different Locations in 4 cm Radius by 2.5 cm Height Cylindrical Gels with Radial Shielding and Compared at 0.01 Significance Level r [cm] z[cm] Experimental RTR STD 0 1 2 3 4 1.25 1.25 1.25 1.25 1.25 0.393 0.485 0.597 0.571 0.372 0.061 0.076 0.085 0.136 0.017 0.411 0.410 0.404 0.406 0.385 0.001 0.003 0.006 0.009 0.013 0 1 2 3 4 0 0 0 0 0 0.573 0.687 0.733 0.663 0.302 0.011 0.113 0.155 0.166 0.021 0.293 0.278 0.299 0.283 0.279 0.001 0.004 0.006 0.007 0.008 50.721 * 7.267 * 5.607 4.581 2.089 0 1 2 3 4 -1.25 -1.25 -1.25 -1.25 -1.25 0.774 0.975 0.938 0.798 0.383 0.027 0.178 0.223 0.244 0.066 0.393 0.391 0.387 0.391 0.373 0.002 0.008 0.011 0.009 0.005 28.166 * 6.571 * 4.943 3.326 0.294 Predicted RTR STD Ccalc 0.603 1.987 4.545 2.418 1.148 1. Number of replications is 4 2. Calculated t values are significant at 0,01 level (*) 184 C-10 Means and Standard Deviations of Measured and Predicted RTR at Different Locations In Rectangular Gels of Various Sizes Under Two-Dimensional Microwave Heating and Compared at 0.01 Significance Level x [cm] y [cm] Experimental RTR STD Predicted RTR STD ^alc 4 cm x 4 cm x 5 cm 0 1 2 1.114 0.871 1.264 0.216 0.100 0.185 0.616 0.824 1.191 0.001 0.013 0.024 4.613 0.931 0.771 0.283 0,355 0.499 0.612 0.023 0.035 0.079 0.129 0.195 0.268 0.524 0.805 0.004 0.005 0.012 0.020 5.785 4.962 0.614 2.956 0.098 0.125 0.213 0.362 0.506 0.016 0.012 0.030 0.042 0.040 0.066 0.092 0.188 0.402 0.634 0.010 0.002 0.004 0.012 0.020 3.407 5.318 1.607 1.858 5.730 0 1 2 6 cm x 6 cm x 5 cm 0 1 2 3 0 1 2 3 8 cm x 8 cm x 5 cm 0 1 2 3 4 0 1 2 3 4 1. Number of replications is 4 2. Calculated t values are significant at 0.01 level (*) 3 * ^ . 01,3 “ 5 -841 VITA Yah-Hwa was born on January 26, 1958, in Taipei, Taiwan, the Republic of China. In June, 1976, she graduated from Taipei First Girls' High and enrolled in the Department of Food Science at National Chung-Hsing University in Taichung, Taiwan. In 1980, Yah-Hwa obtained her B.S. Degree in Food Science and worked in Wei-Chuan Food Company and National Taiwan University untill she came to the United State in 1981. She started her Master's program with the Department of Food Science and Technology at Texas A & M University, and then completed an M.S. Degree in the Department of Agricultural Engineering specializing in Food Engineering. She received her Master’s Degree in 1985 and entered a Doctoral program in the Department of Food Science and Technology at University of Nebraska. She has been actively working in the area of microwave heating of foods and other food engineering related research projects and was awarded the Widman Distinguished Graduate Assistant Award in 1987. In 1988, she transferred to the Department of Food Science at Penn State and continued the research on microwave heating. During her graduate study, she published three papers and five abstracts. She is currently a member of the Institute of Food Technologists, the American Association of Cereal Chemists, Sigma XI (the Scientific Research Society), Phi Tau Sigma (the Honor Society for Food Science), and Gamma Sigma Delta (the Honor Society of Agriculture).

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