# Analysis of temperature redistribution in model food during pulsed microwave heating

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ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. ANALYSIS OF TEMPERATURE REDISTRIBUTION IN MODEL FOOD DURING PULSED MICROWAVE HEATING by Huai-Wen Yang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Agricultural Engineering) at the UNIVERSITY OF WISCONSIN-MADISON 2002 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3060540 ___ ® UMI UMI Microform 3060540 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. C o m m ittee’s Page. This page is not to be hand-written except for A dissertation entitled ANALYSIS OF TEMPERATURE REDISTRIBUTION IN MODEL FOOD DURING PULSED MICROWAVE HEATING submitted to the Graduate School of the University of Wisconsin-Madison in partial fulfillment of the requirements for the degree of Doctor of Philosophy by HUAI WEN YANG Date of Final Oral Examination: C om m ittee’s Page. This page is not to be hand-written except for the signatures Month April 18, 2002 &Year Degree to be awarded: December May August 2002 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Approval Signatures of Dissertation Committee U). Signature, Dean of Graduate School R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. ANALYSIS OF TEMPERATURE REDISTRIBUTION IN MODEL FOOD DURING PULSED MICROWAVE HEATING Huai-Wen Yang Under the supervision of Professor Sundaram Gunasekaran At the University of Wisconsin-Madison ABSTRACT Two-percent agar gel was treated as a model food to study the temperature distribution (TD) during continuous and pulsed microwave heating. Microwave power, derived from Maxwell’s equations and/or the Lambert’s law, was applied to a heat transfer model which was solved numerically to predict the TD within the 3.5-cm and 4cm radius agar gel samples. The sample temperatures were measured and compared to the numerical predictions. The TDs predicted based on the Maxwell’s equations were more accurate than those predicted based on the Lambert’s law, especially around the sample edges. This is because, unlike the Lambert’s law, the power solution based on Maxwell’s equations accounts for the standing wave effect. The predicted and measured TDs were more uniform under pulsed than continuous microwave heating. When microwave energy is applied to a food material, the TD within the material depends on the heating duration, pulsing ratio (ratio of total microwave heating time to R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 power-on time under pulsed mode), sample size, and power level. These effects were examined using a 3 x 2 x 2 factorial experimental design. The pulsed and continuous microwave applications were maintained at the same average power based on the oven settings. Analysis of variance indicated that all variables affect the sample TD significantly. The interactions among the experimental parameters were also determined. The results show that pulsed microwave heating is preferred when temperature uniformity is a major concern. Pulsed microwave heating was applied to mashed potato cylinders to determine an optimal set of operating parameters. The effects of sample radius, power level, processing duration and temperature constraints during the microwave application were evaluated. The sample radius and temperature constraints were critical. Depending on the dielectric properties of the sample, the pulsed microwave heating is best suited only over a certain range of sample sizes (about 1-2 times of penetration depth). The simulation model presented in this thesis is suitable for evaluating optimal pulsed microwave heating of other solid foods. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill ACKNOWLEDGEMENTS I would like to sincerely thank my advisor, Professor Sundaram Gunasekaran, for his guidance during the course of this research. His encouragement and constructive criticism made the completion of this study possible. I am also grateful for Professors Richard W. Hartel, John W. Mitchell, Ronald J. Vernon, and Robert J. Witt for their suggestions and serving on my thesis committee. For the financial support, I am indebted to Department of Biological Systems Engineering. Finally, this thesis is dedicated to two persons who are thousands of miles away from the United States, my parents, Mr. and Mrs. H. Y. Yang for their perpetual support over the years. R eproduced w ithperm ission of th e copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ABSTRACT........................................................................................................................ i ACKNOWLEDGEMENTS ....................................................................................iii LIST OF TABLES............................................................................................................ x LIST OF FIGURES........................................................................................................xii CHAPTER 1 INTRODUCTION......................................................................................1 1.1 Development of Microwave O ven................................................................................ I 1.2 Pulsed Microwave Application......................................................................................1 1.3 Objectives........................................................................................................................ 3 1.4 Limitations......................................................................................................................5 1.5 References.......................................................................................................................6 CHAPTER 2 LITERATURE REVIEW....................................................................... 11 2.1 Microwave Frequencies............................................................................................... 11 2.2 Microwave Food Process............................................................................................. 11 2.3 Characteristics of Microwave Heating........................................................................13 2.3.1 Methods o f Heating Food......................................................................................13 2.3.2 Theory o f Microwave Heating.............................................................................. 14 2.3.2.1 Dielectric Properties........................................................................14 23.2.2 Term Derived from Dielectric Properties...................................... 17 2.3.2.3 Basic Mechanisms of Microwave Heating.................................... 18 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 2.4 Microwave Power Formulations.................................................................................20 2.4.1 Lambert’s Law....................................................................................................... 20 2.4.2 Maxwell’s Field Equations................................................................................... 22 2.5 Temperature Modeling.................................................................................................25 2.5.1 Finite-difference Formulations............................................................................25 2.5.1.1 Explicit Formulation.........................................................................27 2.5.1.2 Implicit Formulation.........................................................................28 2.5.2 Temperature Rise Analysis...................................................................................28 2.6 Pulsed Microwave Application...................................................................................29 2.7 References.....................................................................................................................31 CHAPTER 3 GENERAL METHODOLOGY............................................................. 35 3.1 Laboratory Microwave O ven...................................................................................... 35 3.2 Preparation of 2% Agar Gel Sample Cylinders.........................................................35 3.3 Temperature Measurement.......................................................................................... 36 3.4 Average Absorbed Microwave Power........................................................................37 3.5 Average Surface Heat Transfer Coefficient............................................................... 38 3.6 Azimuthal Wave Assumption..................................................................................... 39 3.7 One-dimensional Heat Transfer Assumption............................................................. 40 3.8 References.....................................................................................................................42 CHAPTER 4 TEMPERATURE PROFILES IN A CYLINDRICAL MODEL FOOD DURING PULSED MICROWAVE HEATING.......................................................... 56 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission 4.0 Notations.......................................................................................................................57 4.1 Abstract.........................................................................................................................60 4.2 Introduction...................................................................................................................61 4.3 Mathematical M odel....................................................................................................63 4.3.1 Implicit Finite-difference M odel..........................................................................63 4.3.2 Evaluating Microwave Absorbed Power.............................................................65 4.4 Methods and Materials.................................................................................................67 4.4.1 Microwave Heating Process................................................................................ 67 4.4.2 Data Analysis........................................................................................................ 68 4.5 Results and Discussion.................................................................................................70 4.5.1 Model Validation................................................................................................... 70 4.5.2 Temperature Profiles............................................................................................. 70 4.6 Conclusions...................................................................................................................72 4.7 References.....................................................................................................................73 CHAPTER 5 COMPARISON OF TEM PERATURE PROFILES IN A CYLINDRICAL MODEL FOOD BASED ON MAXWELL’S AND LAM BERT’S L A W ........................................................................................................................................ 87 5.0 Notations.......................................................................................................................88 5.1 Abstract.........................................................................................................................91 5.2 Introduction...................................................................................................................92 5.3 Theory and Analyses....................................................................................................93 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission 5.3.1 Heat Transfer Equation......................................................................................... 93 5.3.2 Wave Propagation..................................................................................................94 5.3.3 Power dissipation...................................................................................................97 5.3.4 Maxwell’s equations fo r a cylinder...................................................................... 98 5.3.5 Analytical Solution fo r Absorbed Power........................................................... 100 5.3.6 Temperature Distribution Prediction.................................................................101 5.4 Methods and Materials................................................................................................ 101 5.4.1 Microwave process...............................................................................................101 5.4.2 Temperature Prediction...................................................................................... 102 5.5 Results and Discussion................................................................................................ 103 5.6 Conclusions..................................................................................................................105 5.7 References....................................................................................................................114 CHAPTER 6 EFFECT OF EXPERIMENTAL PARAMETERS ON TEMPERATURE DISTRIBUTION DURING CONTINUOUS AND PULSED MICROWAVE HEATING........................................................................................... 116 6.0 Notations....................................................................................................................117 6.2 Introduction..................................................................................................................119 6.3 Methods and Materials................................................................................................120 6.3.1 Microwave Heating Processes...........................................................................120 6.3.2 Statistical Analysis...............................................................................................121 6.4 Results and Discussion................................................................................................122 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission 6.4.1 Microwave Absorbed Power.............................................................................. 122 6.4.2 Temperature Distribution (TD) in Samples Under the same average OSP Condition....................................................................................................................... 123 6.4.3 General Linear Model Under.............................................................................124 6.4.4 Effects o f Different Pulsing Ratios on Sample Temperature Distribution Under The Same Average Absorbed Condition......................................................................125 6.5 Conclusions................................................................................................................. 127 6.6 References................................................................................................................... 128 CHAPTER 7 OPTIMIZATION OF PULSED MICROWAVE HEATING.......... 137 7.0 Abstract....................................................................................................................... 138 7.1 Introduction................................................................................................................. 139 7.2 Methods and Materials............................................................................................... 140 7.2.1 Sample Preparation............................................................................................. 140 7.2.2 Physical and Dielectric Properties................................................................... 140 7.2.3 Microwave Heating............................................................................................. 141 7.2.4 Analysis o f Optimal Process.............................................................................. 141 7.3 Results and Discussion............................................................................................... 144 7.4 Conclusions................................................................................................................. 147 7.5 References................................................................................................................... 148 CHAPTER 8 SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK ......................................................................................................................................... 158 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 8.1 Summary..................................................................................................................... 158 8.2 Recommendations for Future Work..........................................................................159 APPENDICES.................................... 160 Appendix A Engineering Equation Solver (EES, F-Chart Software Co., Middleton, WI) program code for calculating temperature profiles in continuous and pulsed microwave heated 2% agar gel cylinders........................................................................161 APPENDIX B.l MATLAB (MathWorks Inc., Natick, MA) program code for the calculating of temperature profile within 4-cm radius agar gel cylinders according to the Maxwell’s equations.................................................................................................. 169 APPENDIX B.2 PREDICTED AND MEASURED TEMPERATURE DISTRIBUTION OF 3.5- AND 4-cm RADIUS 2% AGAR GEL CYLINDERS AT 250-W OVEN SETTING..................................................................................................176 APPENDIX C.l General linear model with time and covariate interaction terms using MiniTab (Minitab Inc., State Park, PA) analysis............................................................183 APPENDIX C.2 General linear model without time and covariate interaction terms involved using MiniTab (Minitab Inc., State Park, PA) analysis (including the Probability Plot of the Residuals and Residuals versus the Fitted Values).................. 186 APPENDIX D Determination o f the experimental error due to 30-s lag during temperature measurement.................................................................................................191 APPENDIX E The evaporation cooling effect during microwave heating............... 196 with perm ission of the copyright owner. Further reproduction prohibited without permission. X LIST OF TABLES Table 1.1 Summary of experimental sample R/H (radius/height) ratios of food used for temperature distribution (TD) simulation studies...........................................................8 Table 2.1 Microwave Food Processing Applications (from Decareau and Peterson, 1986).................................................................................................................................13 Table 2.2 Methods of Heat Processing of Foods (from Heldman and Singh, 1981)........ 14 Table 3.1 Absorbed power and energy-transfer efficiency in the microwave oven at 2.45 GHz..................................................................................................................................43 Table 3.2 Effect of sample placement (1 and 2 cm from the center of turntable in the microwave oven) on the absorbed power at 2.45 GHz-statistical analysis................ 44 Table 3.3 Temperatures at different angular locations in a 3.5-cm agar cylinder..............45 Table 3.4 Effect of angular locations (0, 120 and 240 °) on the measured temperatures in agar gel cylinders-statistical analysis............................................................................46 Table 3.5 Predicted and measured temperatures for 5 x 7 cm (radius x height) agar gel cylinders (from Mudgett, 1986)..................................................................................... 47 Table 3.6 Measured temperatures for 3.5 x 7 cm (radius x height) agar gel cylinders in the microwave oven at the 250-W oven setting........................................................... 48 Table 3.7 Measured temperatures for 4 x 7 cm (radius x height) agar gel cylinders in the microwave oven at the 250-W oven setting................................................................. 48 Table 3.8 Paired /-test for temperatures at mid-plane (MP, depth=3.5 cm) compared to other locations along z-axis........................................................................................... 49 R eproducedw ith permission of the copyright owner. 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XI Table 4.1 Explicit and implicit schemes for heat fluxes (qn and qi+i) for finitedifference analysis...........................................................................................................76 Table 4.2 Dielectric, physical, and thermal properties of 2% agar gela..............................76 Table 4.3 Parameter values of an infinite cylinder used for validation of the implicit finite difference model..............................................................................................................77 Table 5.1 The Chi-square values for Maxwell’s and the Lambert’s predictions compared to the measured temperatures during microwave heating..........................................106 Table 6.1 Analysis of Variance for the Factorial Experiment in a General Linear Model. ........................................................................................................................................ 136 Table 7.1 Thermal and dielectric properties of mashed potatoes with 82.7% moisture content (M, %) and different temperatures (T, °C)..................................................... 149 Table 7.2 Power-on (PO) to total processing (TP) time ratios under the ATon (ATd) =20(3) °C and ATon (ATd) =15(3) °C constraints at different final average sample temperature (Tfas, °C).....................................................................................................150 Table 7.3 Microwave pulsing sequences applied to different radius samples and comply the AT0„ (ATd)=20(3) °C criteria...................................................................................150 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. xn LIST OF FIGURES Figure 1.1 Temperature equalization in roast beef sample just after microwave application (0 min) and after a 30-min rest time (30 min) (from Poliak and Foin, I960).................................................................................................................................. 9 Figure 1.2 Two-dimensional microwave power absorbed in a 3-cm radius (r), 5-cm high (z) alginate gel cylinder at 30 °C (from Lin et al. 1995).............................................. 10 Figure 2.1 The electromagnetic spectrum (from Brennan et al., 1969).............................. 12 Figure 3.1 Schematic of the microwave oven with arrows indicating the directions of circulating air.................................................................................................................. 50 Figure 3.2 Pictures of the fans and waveguide in the microwave oven............................. 51 Figure 3.3 Type-T thermocouple, datalogger and cardboard temperature guide used for temperature measurements............................................................................................ 52 Figure 3.4 Absorbed power at 250-W oven setting in the microwave oven by placing the sample at 0, 1 and 2 cm from the center of the turntable.............................................53 Figure 3.5 Cylindrical coordinate system and corresponding unit vectors for the case of electromagnetic radiation incident normal to the surface............................................54 Figure 3.6 Temperature measurement locations in 3.5 x 7 cm (radius x height) agar gel cylinders for validating one-dimensional heat transfer assumption........................... 55 Figure 4.1 Schematic of a typical inner node of a cylindrical object for one-dimensional analysis (Ri=0 along the sample axis)...........................................................................78 Figure 4.2 Power-on/-off periods employed during microwave applications for pulsing ratios (PR) of 1, 2 and 3, with total microwave power on-time of 3 min. The solid R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. xiii segments represent power-on periods, dotted segments represent poweroff periods, and vertical dash segments represent the power on/off transition 79 Figure 4.3 Analytical and implicit finite-difference numerical temperature profiles in an infinite cylinder (see Table 3.3 for parameter values used in analytical approximation)................................................................................................................ 80 Figure 4.4 Comparison of temperature profiles according to implicit fmite-difference model (FD) and that of the temperature-rise (TR) model by Padua (1993), shell thickness increments are 0.1, 0.15, and 0.2 cm........................................................... 81 Figure 4.5 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in radius and 7 cm in height) after 1 min of total microwave power application at different pulsing ratios (PR). The power incident is from the sample outer periphery (the radial distance of 3.5 cm from the centerline)......................................................82 Figure 4.6 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in radius and 7 cm in height) after 2 min of total microwave power application at different pulsing ratios (PR)...........................................................................................83 Figure 4.7 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in radius and 7 cm in height) after 3 min of total microwave power application at different pulsing ratios (PR)...........................................................................................84 Figure 4.8 Measured (M) and finite-difference model predicted (P) temperature profiles at different pulsin j ratios (PR) at a radial distance of 0 cm (center line)..................85 Figure 4.9 Measured (M) and finite-difference model predicted (P) temperature profiles at different pulsing ratios (PR) at a radial distance of 1 cm from the central line... 86 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. XIV Figure 5.1 A cylindrical sample exposed to plane waves normal to the surface with sub-shell sections for numerical modeling..........................................................107 Figure 5.2 Microwave power absorbed in 2 % agar gel cylinders (3.5-cm and 4-cm radius) as a function of radial distance from sample center. The electric field is oriented along the vertical z-axis.o f the cylinder........................................................108 Figure 5.3 Microwave power density absorbed in 2% agar gel cylinders (3.5-cm and 4cm radius) as a function of radial distance from sample center. The electric field is oriented along the vertical z-axis of the cylinder........................................................109 Figure 5.4 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 4-cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 1 (i.e. continuous)..................................................................................................................... 110 Figure 5.5 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 4-cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 3.....111 Figure 5.6 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 3.5-cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 1 (i.e. continuous power)........................................................................................................ 112 Figure 5.7 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 3.5-cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 3...... 113 R eproduced with perm ission of the copyright owner. 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XV Figure 6.1 Temperature distribution in 2 % agar gel cylinders, 4-cm radius, after 1, 2 and 3 min of microwave heating with a same average microwave output power of different pulsing ratios (PR=1 @ 250-W setting and PR=2 @ 500-W setting)............................................................................................................................130 Figure 6.2 Temperature distribution in 2 % agar gel cylinders, 3.5-cm radius, after 1, 2 and 3 min of microwave heating with a same average microwave output power of different pulsing ratios (PR=1 @ 250-W setting and PR= 2 @ 500-W setting) 131 Figure 6.3 Mean sample temperature vs. processing time at two pulsing ratios (PR).... 132 Figure 6.4 Mean sample temperature vs. processing time for samples of different radii (R)...................................................................................................................................133 Figure 6.5 Mean sample temperature vs. pulsing ratio for different sample radii (R).... 134 Figure 6.6 Temperature distribution in 2 % agar gel cylinders, 4-cm radius, after 4 min of heating by using an average microwave absorbed power of 225 W under continuous (Mode A) and pulsed (Mode B) microwave applications.......................................... 135 Figure 7.1 Measured and predicted temperature profiles in a 4-cm radius mashed potato cylinder after 1 min of continuous microwave heating at the 250-W oven setting. 151 Figure 7.2 Predicted temperature profiles in 1.6,2.4,2.8 and 3.2-cm radius mashed potato cylinders after 30 s of microwave heating at the 250-W oven setting. Measured data for 2.4-cm radius sample is also shown...................................................................... 152 Figure 7.3 Average sample temperature profiles in 2.4-cm radius, 7 cm-long, mashed potato cylinders heated by pulsed microwave at the 250-W oven setting. Power-on R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. XVI constraints (A TJ were 20 and 15 °C and power-off temperature difference constraints (ATJ were 5 and 3 °C (i.e. ATon(ATd)=15(5), 15(3), 20(5) and 20(3)).............................................................................................................................. 153 Figure 7.4 Average sample temperature profiles in 2.8-cm radius, 7-cm long, mashed potato cylinders heated by pulsed microwave at the 250-W oven setting. Power-on constraints (A TJ were 20 and 15 °C and power-off temperature difference constraints (ATJ were 5 and 3 °C (i.e. ATon(ATd)=15(5), 15(3), 20(5) and 20(3)). 154 Figure 7.5 Average sample temperature profiles in 3.2-cm radius, 7-cm long, mashed potato cylinders heated by pulsed microwave at the 250-W oven setting. Power-on constraints (A T J were 20 and 15 °C and power-off temperature difference constraints (ATd) were 5 and 3 °C (i.e. ATon(ATd)=15(5), 15(3), 20(5) and 20(3)). 155 Figure 7.6 Temperature distribution in 2.4-, 2.8- and 3.2-cm radius potato cylinders under the ATon(ATd)=20(3) criteria after average sample temperature is about 60 °C at the 250-W oven setting........................................................................................................156 Figure 7.7 Time and average sample temperature of a 3.2-cm mashed potato cylinder with 20 °C power-on constraints and a power-off differential of 3 °C lower than the power-on constraint (i.e. ATon(ATd)=20(3)) at the 250- and 500-W oven settings. 157 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 CHAPTER 1 INTRODUCTION 1.1 Development of Microwave Oven In February 1940, two British scientists, Randal and Boot, developed an electronic magnetron tube that generated large amounts of microwave energy. This equipment enabled the British military to develop smaller and more powerful radar systems to detect German aircrafts during World War n. In September 1940, two other British scientists, Lizard and Cockroft, visited the Massachusetts Institute of Technology (MIT Cambridge, MA), and demonstrated a magnetron microwave generator. Originally, these two scientists planned to have an American company build radar systems for them. In 1945, Dr. Percy Spencer purportedly used microwave radiation to prepare popcorn. This was the first known application of microwave radiation for heating food (Reynolds, 1989). The first commercial microwave oven was introduced into the restaurant and institutional market in 1947. In 1967, Amana Inc. introduced the first “countertop” microwave ovens to the U.S. households (Reynolds, 1989; Happel, 1992), and more than 90 % of the U.S. households owned a microwave oven in 1997 (Liegey, 2001). The Association of Home Appliance Manufacturers (AHAM, 2000) forecasted that the annual sale of home microwave ovens in the U.S. will exceed 12 million units in 2002. 1.2 Pulsed Microwave Application The major reason for infrequent usage of microwave heating by the food industry, which also echoes complaints made by home consumers, is the uneven heating of food R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 2 materials. This is caused by the characteristics of the food materials that absorb microwave energy. The idea of pulsed drying was first investigated by Edholm in 1933 (Hamoy and Radajewski, 1982). In this technique, the convection energy for drying is periodically turned on and off. Intermittent drying improves energy efficiency and product quality (Farkas and Rendik, 1997). Poliak and Foin (1960) measured the temperature distribution (TD) of roast beef immediately after microwave cooking and after a 30-min rest period. The temperature redistribution they observed is presented in Fig. 1.1. This is a good illustration of temperature redistribution within foods during power-off times. The temperature equalization during power-off times occurs due to thermal diffusion (Chamchong and Datta, 1999). The redistribution effect of pulsed microwave heating implies a remedial approach to obtain a uniform temperature profile within a heated food. Microwave heating produces large temperature variations within a food sample due to penetrating nature of microwaves. However, if a proper pulse mode is employed, thermal diffusion can reduce temperature variations during power-off periods. Temperature redistribution can be reliably predicted and explained via numerical models. In this thesis, microwave power within the oven cavity was assumed to be one dimensional, which is often employed to study microwave heating and thawing of foods (Ayappa, et al., 1991; Pangrle et al., 1991; Padua, 1993; Barringer et al., 1995). In reality, microwaves are three-dimensional and can be described as transverse electromagnetic components. The absorbed power in a 6 x 5 cm (diameter x height) alginate gel cylinder at 30°C resulted in more uniform temperature along the sample axis R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. than along the sample radius (Fig. 1.2). This was supported by a two-dimensional finite element analysis of microwave heating reported by Lin et al. (1995). The TD along the sample vertical depth (z-axis) has also been reported to be more uniform than that along the sample radial depth (r-axis) in a 3.5 x 7 cm (radius x height) potato cylinder (Chen et al. 1993). In Table 1.1 different experimental R/H (radius/height) ratios of food samples used for TD simulation studies are summarized, the R/H ratios range from 0.36 to 0.71. These studies assumed and/or reported that the TD along sample vertical depth (z-axis) is not significant compared to that along sample radius. The microwave power solution is either derived from the Lambert’s law or Maxwell’s equations. The Lambert’s law is a simplified solution representing a multitude of boundary conditions. Standing waves, with their attendant nodes and anti-nodes, are not represented in the Lambert’s law. Maxwell’s equations better represent the standing wave effects encountered with microwaves. 1.3 Objectives The overall objective of this thesis was to evaluate the temperature distribution in a model food material during pulsed and continuous microwave heating. The specific objectives of this thesis are listed below under Chapter numbers in which they addressed. The objectives are to: [Chapter 2] • Review literature related to pulsed microwave heating. [Chapter 3] R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 4 • Describe the microwave oven used for the experiments. • Provide the overall experimental procedures. [Chapter 4] • Develop a heat transfer model for predicting temperature distribution in 2% agar gel cylinders heated by continuous and pulsed microwave energy. • Validate the model predictions with experimental data. [Chapter 5] • Investigate internal temperature distributions in 2% agar gel cylinders with different sample radii heated by continuous and pulsed microwave energy. • Predict sample temperature based on the solutions of the Lambert’s law and Maxwell’s equations of microwave power absorbed by the sample. • Compare the measured and predicted temperature distribution for both continuous and pulsed microwave energy. [Chapter 6] • Examine the effect of continuous and pulsed microwave heating at the same average power setting. • Evaluate the effects of microwave processing duration, power mode, power level, and sample size on the temperature distribution within 2% agar gel cylinders. R eproduced with perm ission o fth e copyright owner. Further reproduction prohibited without permission. 5 [Chapter 7] • Evaluate proper sample dimension which pulsed microwave energy can be efficiently employed to create heating in a uniform manner. • Investigate the interior sample temperature of mashed potato cylinders affected by different pulsed modes of microwave energy and temperature constraints. 1.4 Limitations The quantitative results obtained and presented in Chapter 6 are based on data collected using the Labtron 500 (Zwag Inc., Epone, France) microwave oven. The General Linear Model for the statistical analysis is valid only for the range of experimental conditions specified. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 1.5 References AHAM. 2000. Forecasts. Appliance Manufacturer. 48(6): 11. Ayappa, K.G., Davis, H.T., Crapiste, G., Davis, E.A. and Gordon, J. 1991 Microwave heating an evaluation of power formulations. Chem. Engineering Science. 46 : 1005-1016. Barringer, S. A., Davis, E. A., Gordon, J., Ayppa, K. G., and Davis, H. T. 1995. Microwave- heating temperature profiles for thin slabs compared to Maxwell and Lambert law predictions. J. Food Sci. 60(5): 1137-1142. Chamchong, M. and Datta, A. K. 1999. Thawing of Foods in a Microwave Oven: I. Effect of Power Level and Power Cycling. J. Microwave Power and Electromagnetic Energy. 34(1): 9-21. Chen, D. D., Singh, R. K., Haghighi, K. and Nelson, P. E. 1993. Finite Element Analysis of Temperature Distribution in Microwaved Cylindrical Potato Tissue. J. of Food Engineering. 18:351-368. Farkas, I. and Rendik, Z. 1997. Intermittent Thin Layer Com Drying. Drying Technology. 15 (8): 1951-1960. Happel, M.E. 1992. Consumer attitudes + marketing statistics = the consumer equation. Microwave World. 12(2):7-8. Hamoy, A. and Radajewski, W. 1982. Optimization of Grain Drying-With Resting Period. J. Ag. Eng. Research 27(4): 291-308. Liegey, P. R. 2001. Hedonic Quality Adjustment Methods For Microwave Ovens In the U.S. CPI. U.S. Bureau of Labor Statistics. R eproduced with permission of thecopyright owner. Further reproduction prohibited without permission. 7 Lin, Y. E. Anantheswaran, R.C. and Puri, V.M. 1995 Finite Element Analysis of Microwave Heating of Solid Foods. J. of Food Engineering 85-112 Padua, G.W. 1993. Microwave heating of agar gels containing sucrose. J. Food Sci. 58(60): 1426-1428. Pangrel, B.J. Ayappa, K.G. Davis, H.T. Davis, E.A. and Gordan, J. 1991 Microwave thawing of cylinders J. AIChE 37(12): 1789-1800. Poliak, G.A. and Foin, L.C. 1960. Comparative Heating Efficiency of a Microwave and Convection Electric Oven. Food Technology 14:454-457. Reynolds, L.R. 1989. The history of the microwave oven. Microwave World 10(5): 7-11. R eproduced with perm ission of the copyright o w n e r Further reproduction prohibited without permission. 8 Table 1.1 Summary of experimental sample R/H (radius/height) ratios of food used for temperature distribution (TD) simulation studies. Sample R/H ratio Agar gel 0.71 Salted ice NDa Agar gel 0.36-0.42 Potato 0.5 Alginate gel 0.6 a Not described bOne-dimensional cTwo-dimensional d Not significant e Not evaluated Model used 1 Db ID 1D 2 Dc 2D TD along sample axis NSd NEe NE NS NS Reference Mudgett (1986) Pangrle et al. (1991) Padua (1993) Chen et al. (1993) Lin et al. (1995) R eproduced with permission of the copyright owner. Further reproduction prohibited without perm ission. 9 Temperature, 80 40 30 min 0 min 0 0 5 10 Thickness, cm Figure 1.1 Temperature equalization in roast beef sample just after microwave application (0 min) and after a 30-min rest time (30 min) (from Poliak and Foin, 1960). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 ab8 •'€«. I ■so oo IcbJ 0 ■ 00 r fen] so Figure 1.2 Two-dimensional microwave power absorbed in a 3-cm radius (r), 5-cm high (z) alginate gel cylinder at 30 °C (from Lin et al. 1995). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 CHAPTER 2 LITERATURE REVIEW 2.1 Microwave Frequencies Electromagnetic waves are classified by their frequencies. The electromagnetic spectrum is shown in Fig. 2.1. Microwave radiation ranges from 300 MHz to 300 GHz, which lies between radio and infrared waves on the spectrum. Microwave frequencies of 915 MHz, 2.45 GHz, 5.8 GHz, and 24.2 GHz are approved for industrial and scientific applications by the U.S. Federal Communications Commission. Industrial food equipment uses only 915 MHz and 2.45 GHz. At 2.45 GHz, the penetrability is up to 10 cm and has characteristics similar to that of radar waves. At 915 MHz, the penetrability in a food material is up to 30 cm and has characteristics similar to that of radio waves. Most food applications, including cooking and heating in home microwave ovens, are at 2.45 GHz with a few at 915 MHz (Brennan et al., 1969; Clary, 1994; Copson, 1975; Barringer, 1994). 2.2 Microwave Food Process The most common applications of microwave food processing are drying, freeze drying, vacuum drying, baking, pasteurizing, sterilization, cooking, tempering, roasting, blanching and rendering (Decareau, 1985; Decareau and Peterson, 1986). These are summarized in Table 2.1. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1021 COSMIC RAYS 1020 10 19 GAMMA RAYS 10 1* 10 17 10 16 X-RAYS 1015 1014 ULTRA VIOLET 1013 VISIBLE 10 12 10 11 THERMAL (infrared) 10 10 109 RADAR 10* 107 106 mm ^m mm mm mm mm m b ■ MICROWAVE HEATING mm ^m ^m mm mm mm mm m 105 104 103 102 RADIO 101 (Hz or cycle/s) Figure 2.1 The electromagnetic spectrum {from Brennan et al., 1969). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 Table 2.1 Microwave Food Processing Applications (from Decareau and Peterson, 1986). Process Drying Freeze drying Vacuum drying Baking Pasteurizing Sterilization Cooking Tempering Roasting Blanching Rendering Products Pasta, onion, rice, cakes, egg yolk, snack foods, seaweed Meat, vegetables, fruits Orange juice, grains, seeds Bread, Doughnuts Bread, yogurt Pouch-packed foods Bacon, meat patties, sausage, potatoes, chicken Meat, fish, poultry Nuts, cocoa beans, coffee beans Corn, potatoes, fruits Lard, tallow 2.3 Characteristics of Microwave Heating 2.3.1 Methods o f Heating Food There are two methods used to heat food materials: direct and indirect. The major difference between direct and indirect heating is the need for a heat exchanger or medium. Indirect heating requires heat transfer media such as steam, air, water or other exchangers. The major examples of indirect heating are conduction, convection, and thermal radiation. Direct heating and microwave heating do not require any intermediate heat exchangers to transfer the heat into foods and results in a more rapid heating rate (Brennan et al., 1995; Copson, 1975; Decareau, 1985). The methods describing the application of heat to food materials have been summarized in Table 2.2 (Heldman and Singh, 1981). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission 14 Table 2.2 Methods of Heat Processing of Foods (from Heldman and Singh, 1981). Major category Indirect heating Methods By vapor or gas such as steam or air. By liquid such as water or organic heat exchange Liquids. By electricity in resistance and radiation heating system (ohmic heating). Direct heating Using electricity, by dielectric or microwave. 2.3.2 Theory o f Microwave Heating 2.3.2.1 Dielectric Properties Electric field behavior basically classifies materials into three categories: conductors, semiconductors, and insulators. The electrons in the outermost shell within atoms of conductors are loosely held, and tend to migrate easily from one atom to another. Most metals are conductors. The electrons in the outermost shell of insulators (dielectrics) are confined to the specific orbit. Under normal circumstances, they cannot be liberated. Most biological materials, including foods, are dielectrics. The field behavior of semiconductors falls between that of conductors and dielectrics (Cheng, 1992). When a conductor is placed in an external electric field, the free charges move to the surface and the electric field in the interior vanishes. However, the bound charges of a R ep rod uced with perm ission of the copyright owner. Further reproduction prohibited without permission. 15 dielectric in the presence of an external electric field cause small displacements of both positive and negative charges in opposite directions due to the lack o f free charges. These small displacements result in the polarization of a dielectric material. Water molecules, which are polar, possess permanent dipole moments. Under normal conditions, the individual dipoles are randomly oriented. The net dipole moment is zero in the macroscopic view. The application of an electric field will exert torque on the individual dipoles, and partially align them with the field. The fundamental postulate of electrostatics in free space specifies the divergence of E (electric field intensity): ( 2. 1) where p = volume charge density of free charges, and Eo = permittivity of free space. Since a polarized dielectric gives rise to an equivalent volume charge density pp, the divergence must be modified as: V « E = — (p + p ) £0 ( 2 .2 ) where (2.3) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 16 and P = polarization vector. Equation 2.2 could be expressed as: V • (e0E + P) = p or V »(D ) = p (2.4) where D = electric displacement. When the dielectric properties of the media are linear and isotropic, the polarization is directly proportional to the field intensity. The proportionality constant Xe is a dimensionless quantity called susceptibility. The polarization vector in terms of %e is: P=eoX*E (2-5) Substituting equation 2.5 in equation 2.4 yields: £>=eo(l+Xe)£ or =8oK’£=£’£ (2.6) where and k’ = dimensionless quantity known as the relative permittivity or dielectric constant of a medium, and Eo = absolute permittivity or simply permittivity. The dielectric constant s ’ of a material describes its ability to store electrical energy. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 17 With the application of an alternating electric field, the inertia of charged particles tends to prevent particle displacements, and hence causes a frictional damping that leads to power dissipation. Power dissipation in a dielectric is expressed by the loss factor e”, and is included in the imaginary part of the complex permittivity e*: (2.7) where 2.3.2.2 Term Derived from Dielectric Properties The ratio e’Ve’ is the loss tangent, tan 5 = e”/e \ The tan 8, along with dielectric properties, is useful for the calculation of some terms that are of interest in microwave heating. The attenuation constant ((3) is important in determining the penetration depth of microwave in a sample. At higher frequencies, (3 increases, and thus the penetration depth of microwaves in the sample decreases. The reciprocal of (3, known as the penetration depth (Dp), is the distance at which the incident field intensity decreases to i/e of its incident value. The (3 in terms of dielectric properties and tan 5 is: ( 2 .8 ) where c = speed of light a n d /= frequency of radiation used. R eproduced with perm ission of the copyright o w n e r Further reproduction prohibited without permission. 18 The phase constant (a) represents the change in phase of the propagating wave. a 2nf jfc'i/l + itanS)2 +1 = — — c 2 (2.9) The a is related to the wavelength of radiation in the sample (Xs) by: ( 2 . 10) The k s is very important when considering standing wave interactions in the microwave oven. 2.3.2.3 Basic Mechanisms o f Microwave Heating When a material is placed within a coil carrying a high frequency alternating current, it is heated by the current induced in that material. This is analogous to the action within a transformer when the application of a current on the primary side induces a current in the secondary winding (Copson, 1975, Brennan et al., 1969; Copson, 1975). Dielectric heating, on the other hand, heats materials due to power losses within those materials when subjected to alternating current electric fields (Brennan et al., 1969; Decareau, 1985). Copson (1975) also described dielectric heating as the heating of poor dielectric materials such as air, paper, and foods. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 Dielectric heating of substance is generally achieved by placing the sample between two parallel plates (electrodes). The plates are connected to a source of high frequency alternating current (Brennan et al., 1969; Copson, 1975; Decareau, 1985). Heating is brought about by molecular friction due to the rapid orientation of the electric dipoles within the materials (Brennan et al., 1969). The basic concepts of dielectric heating can be applied to microwave heating. However, microwaves, a radiation phenomenon, differ from induction heating between parallel plates in the following ways: 1. Higher energies associated with the higher frequencies used in microwave heating allow for the same energy input to be achieved through the application of a lower voltage (Brennan et al., 1969; Copson, 1975; Decareau, 1985). 2. Loss factors are higher as the frequency becomes higher (Copson, 1975). The electromagnetic field in the microwave region affects the orientation of free water molecules and ionic polarization, and generates the heating in a food product. Dipole rotation of free water molecules is the major force in producing heat within foods, due to the fact that water molecules generally make up the largest portion of raw food materials. Water molecules themselves are randomly polarized in the absence of external forces. In the presence of an electric field, water molecules align themselves in the direction of the field. An alternating current electric field, such as microwaves, causes water molecules to repeatedly change their orientation according to the direction of the field. This is the way dipole rotation generates heat. The rate of rotation corresponds to the frequency of the alternating field. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 The other major mechanism is ionic polarization. The movement of both dissolved anions and cations in solutions can accelerate in response to an electric field. The accelerated ions collide with each other and convert kinetic energy into heat (Brennan et al., 1969; Clary, 1994; Copson, 1975). Food components classified by proximate analysis are moisture, protein, lipids, carbohydrates and ash. Among these components, free water or moisture is associated with dipole rotation, while dissolved ions of ash are associated with ionic polarization. Other components are relatively inert to microwaves. 2.4 Microwave Power Formulations Power formulations for microwave heating are related to the electric field in the oven cavity. When the microwaves contact the sample surface or at interfaces within the sample, some of the radiation is transmitted/absorbed and some of it is reflected. The amount of transmitted and reflected radiation depends on the dielectric properties of materials at both sides of the interface or surface. The electric field intensity entering the sample is a critical parameter to be determined. The electric field intensity and power formulation at a given depth can be determined by using either the Lambert’s law or Maxwell’s field equations. 2.4.1 Lambert’s Law Von Hippel derived the Lambert’s law (1954). It is a semi-infinite solution for electric field intensity and power terms. The Lambert’s law predicts an exponential decay of radiation with increasing depth into the sample. The value of transmitted power must R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 be determined in order to simulate the power term at a given depth within the sample. Different methods for the determination of transmitted power have been proposed. They are: monitoring gas breakdown (MacLathy and Clements, 1980), toner particle melting (Washisa and Fukai, 1980), electrical discharge in fluorescent lights (Bosisio et al., 1975) and relative temperature of a tray of water segmented into cubes (Watanabe et al., 1978). However, these methods do not give an absolute value of power that is needed for model simulation. In 1987, the International Electrotechinal Commission (IEC) accepted a standard method for determining the power rating of a microwave oven. This is based on the assumption that the power absorption of a beaker of water is equal to the energy transmitted into the sample. It is one of the first and the most common methods for measuring the power delivered in a microwave oven. Generally, a beaker containing certain amount of water is placed in the microwave oven and heated. The temperature rise over a certain time is measured. The absorbed power (PabS), which causes the temperature to rise, is calculated by energy balance. Pabs=mCp(AT/At) (2.11) where AT = temperature rise, At = microwave heating time, m = sample mass, and Cp = specific heat capacity of water. Ohlsson and Bengsston (1971), Stuchly and Hamid (1972), Mugett (1986) and Padua (1993) used this standard method to determine the absorbed power. The knowledge of transmitted power flux is beneficial for calculating the power distribution within the sample, P(z), at a given depth (Mudgett, 1986; Padua, 1993). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 P(z) = 2 I0Pe-2pz ( 2 . 12) where Io= transmitted power flux, P = attenuation constant, z = given sample depth. Equation 2.12 is the Lambert’s law which predicts an exponential decay of radiation with increasing depth into the sample. This semi-infinite power formulation does not take the reflected radiation at the exit interface into account. The reflected radiation is evidenced by center focusing in cylinders and causes local hot spots. Even though this produces some errors in model simulation, because of its simplicity, the Lambert’s law is still often used in food research. 2.4.2 Maxwell's Field Equations Power formulation can also be determined by solving Maxwell’s field equations (Cheng, 1992). The equations are more comprehensive than the Lambert’s law, and are capable of modeling the exact electric field intensity at any point within a microwave oven. The knowledge of electric field intensity can be incorporated with the Poynting vector, a power density vector associated with an electromagnetic field. The power dissipated per unit of volume is: P M(Z) = - Re(V • S) = ^(o e0K" E ■Ec (2.13) where Re = real part of a complex number, S = Poynting vector, to = angular velocity, So = permittivity of free space, k” = dielectric loss factor and Ec = conjugate electric field intensity at a given point. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 Equation 2.13 is the local power formulation in unit of power per volume. Jia and Jolly (1992) and Ofoli and Komolprasert (1988) have modeled the exact theoretical electric field at any point within a specific microwave oven. These studies are complex, and applicable only to their specific systems. Ayappa et al. (1991 a and b) developed models according to the solution of Maxwell’s field equations for infinite slab, composite slabs and cylinders. The models were based on two major assumptions about the microwave radiation: (1) it is incident perpendicularly on the sample surface, and (2) it can be treated as uniform plane wave. A plane wave is a one-dimensional spatial dependence wave. A uniform plane wave is a particular solution of Maxwell’s field equations which assumes the same direction, magnitude and phase in infinite planes perpendicular to the direction of propagation. Strictly speaking, a uniform wave does not exist in practice because a source infinite in extent would be required to create it. However, it is a common assumption made by food engineers and others. During modeling with uniform plane wave assumptions according to the solution of Maxwell’s field equations, reflected and transmitted radiation are taken into account. In the case of normal incidence at a plane dielectric boundary, two parameters are very important for modeling. They are reflection coefficient (Ru+i) and transmission coefficient (Tu+i). In terms of intrinsic impedance (q), they are: , /+1 ~ n Hi+1 o. « (2.14) and R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. They can be also be expressed as a function of the attenuation constant (3 and the phase constant a , they are: Jsu-i ( 2 . 16) where \R i m ~ ■ (a, - a l+l)2+CP, - f$M)z (a, +aMy- + U3, +i3l+l)2 S i.m = Tan -i 2(a/+, f l -<*,PM ) (a,2 + p y ) - ( f x M 2 + PM 2) (2.17) ( 2 . 18) and T —\T \a */./+1 —*/./+! F (2.19) where \1 i m “ 4(ar l + P,2) 7 " m ’ («i + « i+i ) + ( A ^ 7 7 U -2 U ) + A +I-) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 *l.M = tan"‘ (off2 + 0 t2 ) + ( a la M + 0, p M ) ( 2 .21) The power dissipated within a sample of single slab geometry (one- dimensional) with radiation is: ( 2 .22 ) Ayappa (1992) presented a more complete derivation of the power formulation. Equation 2.22 is an evidence of the power formulation according to the solution of Maxwell’s field equations calculating reflection at front and back of the sample. The internal standing waves are predicted, and therefore account for focusing and edge heating. 2.5 Temperature Modeling 2.5.1 Finite-difference Formulations Microwave radiation delivers energy into the sample and, thus, heat transfer occurs within the sample. Microwave power is treated as a volumetric internal power generation (P) that can be incorporated with transient (unsteady state) heat transfer. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 d 2T d 2T d 2T p 1 dT T T + T V + -r^ - + — = rdx2 dy2 dz2 k a H dr For a solid, the energy equation is: (2.23) For a cylinder, the energy equation is: d 2T d 2T I d T p I dT - r r - + ^ r ^ + — — + — = ------— dz2 dr2 r dr k a H dr (2.24) where T = temperature, x, y, z and r are length along dimension, k = thermal conductivity, p = power or heat generation, and a H = thermal diffusivity and x = time. Although the partial differential equation can be solved analytically, it is complicated when the heat generation is nonlinear. Alternatively, numerical techniques such as finitedifference, finite-element, or boundary-element methods can be used. The first step in any numerical analysis is to select nodal points in the discretized sample. Each nodal point represents a certain region around that point. The temperature for that nodal point is an average temperature of the region. The heat transfer equation at each nodal point as an exact differential equation is reduced to an approximate algebraic equation. The approximate finite-difference form of the heat equation may be applied to any interior node that is equidistant to the neighboring nodes. The finite-difference equation for a node may also apply conservation of energy to a control volume about the node. Since the actual direction of heat flow (inward or outward) is often unknown, it is convenient to formulate the energy balance by assuming that all the heat flows are into the node. Such a R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 condition is, of course, impossible, but if the rate equation is expressed in a manner consistent with this assumption, the correct form of the finite-difference equation is obtained (Incropera and DeWitt, 1995; Arpaci, 1966). For an unsteady state condition with heat generation, the general form is: E „ + E '= E „ = ^ (2.25) where the subscript “in” = energy inward, “g” = heat generated, and “st” = heat store. Equation 2.25 indicates that time dependent temperature information will be available when solving the unsteady state equation in finite difference form. 2.5.1.1 Explicit Formulation The explicit finite-difference form is used to solve for the unknown nodal temperature for a new time with known nodal temperature at previous time. The explicit method is not conditionally stable because of the numerically-induced oscillations. To prevent such erroneous results, the time interval chosen must be maintained below a certain limit, which depends on the spatial properties of the system. The limit is termed a stability criterion, and is discussed by Incropera and Dewitt (1996). For a given space increment of explicit formulation, it is frequently indicated that an extremely small time step must be chosen. In other words, a very large number of time intervals may be necessary to obtain a reasonable solution. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 28 2.5.1.2 Implicit Formulation An implicit formulation, rather than explicit scheme, evaluates all other nodal temperatures at new time. It provides a backward difference approximation, instead of a forward one obtained by explicit formulation. The new temperature of any given node depends on the new temperature of its adjoining nodes, which are generally unknown. This implies that to determine the unknown nodal temperature at new time, the corresponding nodal equations must be solved simultaneously. The implicit formulation has the important advantage of being unconditionally stable. That is, the solution remains stable for all space and time intervals. The oscillating results induced from the explicit method can then be avoided by using this implicit approach. Since a larger time interval may be used with the implicit method, computation time may often be reduced with little loss of accuracy. Nevertheless, a sufficiently small time interval must be chosen to ensure maximum accuracy (Incropera and DeWitt, 1996). Ohlsson and Bengtsson (1971) developed microwave-heating profiles using a finite-difference technique. Kirk and Holmes (1975) predicted the temperature profile of a high moisture substance (1 % ion-agar gel in water) at any given time by using the finite-difference method. Nykvist et al. (1976) investigated microwave beef roasting using the finite-difference method by considering the case of non-perpendicular wave incidents. 2.5.2 Temperature Rise Analysis Thermal conductivity of a biological material is small compared to other material like metals. In the case of large volumetric heat generation like microwave heating of R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. food materials, the thermal conduction may be neglected for simplicity. The energy balance equation (equation 2.11) of this type of analysis is termed the temperature rise method. Padua (1993) proposed a temperature analysis for 2 % agar gel cylinders using the Lambert’s law of power formulation. Barringer et al. (1995) proposed a temperature analysis for 2 % agar gel using the Lambert’s law, Maxwell’s field equations, and a combined power formulation for slab geometry with microwave radiation from one side of the slab. Barringer (1994) used temperature rise method to analyze microwave heating of emulsions. The emulsion system, either oil-in-water or water-in-oil, is generally a liquid or semi-liquid, and it is very difficult to measure the temperature at given point within the sample because a convection effect occurs. The bulk (average) temperature rise is a better alternative to obtain some additional information for microwave application. 2.6 Pulsed Microwave Application Despite the many seemingly excellent reasons for adopting microwave energy, the non-uniform microwave heating of foods is recognized as a major drawback to microwave heating. Ohlsson and Riman (1978) conducted a study on the non-uniform temperature distribution in meat and potato cylinders; central heating effects (local hot and cold spots) were found in small sample cylinders. One of the approaches to improve R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 the temperature uniformity in samples heated by microwave is pulsed (intermittent) operation. Pulsed microwave drying operation can also reduce energy loss through flowing air, which is often evidenced in continuous microwave application. That is, pulsed microwave application results in a lower energy requirement. However, the total heating or drying time increases. In spite of the time factor, pulsed microwave application is still worthwhile from product quality and energy cost consideration. Shivhare et al. (1992 a, b, and c) studied continuous and intermittent microwave drying of com with various duty cycles. Yongsawatdigul and Gunasekaran (1996 a and b) investigated microwave drying of cranberries with at different vacuum and power levels, and duty cycles. The drying efficiency improved under pulsed application compared to continuous application. Tulasidas et al. (1994) evaluated pulsed microwave drying of grapes in a single mode resonant cavity. The cost (energy consumption) of pulsed application was substantially lower than that of continuous application. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 2.7 References Arpaci, V. S. 1966. Conduction Heat Transfer. Addison-Wesley Publishing Company, Reading, MA. Ayappa, K.G. 1992. Analysis o f microwave heating o f dielectric material. Ph. D. Dissertation, University of Minnesota, St Paul, MN Ayappa, K.G., Davis, H.T., Crapiste, G., Davis, E.A. and Gordon, J. 1991a. Microwave heating an evaluation of power formulations. Chem. Engineering Science. 46 : 1005-1016. Ayappa, K.G., Davis, H.T., Davis, E.A. and Gordon, J. 1991b. Analysis of microwave heating of material with temperature-dependent proties. AIChE J 37(3): 313-322. Barringer, S.A. 1994. Experimental and predictive heating rates o f microwaved food systems. Ph. D. Dissertation, University of Minnesota, St Paul, MN. Barringer, S.A., Davis E. A., Gordon, J., Ayappa, K. G. and Davis, H. T. 1995. Microwave- heating temperature profiles for thin slabs compared to Maxwell and Lambert Law prediction. J. Food Science. 60(5): 1137-114. Brennan, J.G., Butters, J.R., Cowell, N.D., and Lily, A.E.V. 1969. Food Engineering Operation. Elservier Publishing Company Limited, New York, NY Bosisio, R. G., Nachman, M. and Nobert, R. 1975. A simple method for determining the electric field distribution along a microwave applicator. J. Microwave Power 11(1): 3-24. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 Cheng, D. K. 1992. Field and Wave Electromagnetics. Addison-Wesley Publishing Company, Reading, MA Clary, D.C. 1994. Application o f microwave vacuum and liquid media dehydration fo r the production o f dried grapes. Ph. D. Dissertation, Michigan State University. Copson, D. A. 1975. Microwave Heating. The AVI Publishing Company, Inc., Westport, CT. Decareau, R.V. 1985. Microwave Application in Food Processing Industry. Academic Press, Inc., Orlando, FL Decareau, R.V., and Peterson, R.A. 1986. Microwave processing and Engineering. Ellis Horwood Ltd, Chichester, England. Heldman, D.R., and Singh, R.P. 1981. Food Processing Engineering. The AVI Publishing, Westport CT. Incropera, F. P., and DeWitt, D.P. 1996. Introduction to Heat Transfer. John Wiley & Sons, Inc., New York, NY. Jia, X. and Jolly, P. 1992. Simulation of microwave field and power distribution in a cavity by three-dimensional finite element method. J. Microwave Power and Eletromagnetic Energy 27(10): 11-22. Kirk, D. and Holmes, A. W. 1975. The heating of foodstuffs in a microwave oven. J. FoodTechnol., 10: 375-384. MacLatchy, C.S. and Clements, R.M. 1980. A simple technique for measuring high microwave electric field strengths. J. Microwave Power 15(1): 7-14. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Mudgett, R. E. 1986. Microwave propertiesand heating characteristics of foods. Food technol. 40(6): 84-93. Nykvist W. E., and Decareau R. V. 1976. Microwave meat Roasting. J. Microwave Power 11(10): 3-24. Ofoli, R. Y., and Komolprasert, V. 1988. On the thermal modeling of foods in an electromagnetic field. J. Food Process. Preserv. 12: 219-241. Ohlsson, T. and Bengtsson, N. 1971. Microwave heating profile in Foods- a comparison between heating and computer simulation. Microwave Energy Application New Letter 6: 3-8. Ohlsson, T,and risman, P.O. 1978. Temperature distrubution of microwave heatingspheres and cylinders. J. Microwave Power 13(4): 303-309. Padua, G. W. 1993. Microwave heating of agar gels containing sucrose. J. Food Science 58(60): 1426-1428 Shivhare, U.S., Raghavan, G.S. and Bosisio, R.G. 1992a. Microwave drying of com I. Equilibrium moisture content. Transactions of the American Society of Agricultural Engineers. 35(3): 947-950. Shivhare, U.S., Raghavan, G.S. and Bosisio, R.G. 1992b. Microwave drying of com II. Constant power, continuous operation. Transactions of the American Society of Agricultural Engineers. 35(3): 951-957. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Shivhare, U.S., Raghavan, G.S., Bosisio, R.G., and Mujumdar, A.S. 1992c. Microwave drying of com HI. Constant power, intermittent operation. Transactions of the American Society of Agricultural Engineers. 35(3): 959-962. Stuchly, S.S. and Hamid, M.A. K. 1972. Physical Parameters in microwave heating processes. J. Microwave Power 7(2): 117-137. Tulasidas, T.N., Raghavan, G.S.V., Hudra, T. Gariepy, Y, and Akyel, C. 1994. Microwave drying of grapes in a single mode resonant cavity with pulsed power. Paper No. 94-6547, present at Ann. Mtg. Of the American Society of Agricultural Engineers. von Hippel, A. R. 1954. Dielectric and Waves. The MIT Press. Cambridge, MA Washisu, S. and Fukai, 1980. A simple method for indication of the electric field distribution in a microwave oven. J Microwave Power 15(1): 59-61. Watanabe, M., Suzuki, M. and Ohkawa, S. 1978. Analysis of power density distribution in a microwave oven. J. Microwave Power 13(2): 173-181 Yongsawatdigul, J. and Gunasekaran, S. 1996a. Microwave drying of cranberries: part I. Energy and Efficiency. J. Food Processing and Preservation 20: 121-143. Yongsawatdigul, J. and Gunasekaran, S. 1996b. Microwave drying of cranberries: part II. Quality evaluation. J. Food Processing and Preservation 20: 145-156. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 CHAPTER 3 GENERAL METHODOLOGY 3.1 Laboratory Microwave Oven A laboratory microwave oven (Labotron 500, Zwag Inc., Epone, France) was used for all the experiments. The microwave oven operates at 2.45 GHz and has two continuous output power settings, 250 and 500 W. The oven cavity is 33 x 22 x 35 cm (width x height x depth) and houses a 25 x 3.5 cm (diameter x height) turntable that rotates at 15 rpm. A of 6.5 x 6.5-cm waveguide (microwave energy source) is positioned on the left cavity wall. The edges of the waveguide are 12.5 cm from the bottom and 6 cm from the front edge of the cavity. There are two mode fans in the oven cavity. The horizontal mode fan, 6.5 x 6.5 cm, is positioned right across the waveguide on the cavity wall; the vertical mode fan, 5 x 5 cm, is on the top wall, 18 cm from the front and 8.5 cm from the left edge. The third fan, 6 cm in diameter, is an air outlet circulator. Its center was positioned at 15 cm from the bottom and 23 cm from the left edge on the rear wall (Figs. 3.1 and 3.2). 3.2 Preparation of 2% Agar Gel Sample Cylinders Agar gel samples were prepared by dissolving 40 g (2%) of agar powder (Bacto Agar, Difco Inc., Detroit, MI) in 1,960-mL of warm (~ 40°C) distilled water in a 2,000mL pyrex glass beaker. The agar-water mixture was heated until agar powder was totally dissolved and the gel solution was clear. It was then poured into 600-mL or 400-mL pyrex glass beakers and cooled to room temperature into solid sample cylinders. The radii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 of samples prepared in 600-mL and 400-mL beakers are 4 and 3.5 cm, respectively. The sample cylinders were stored at 4 °C for 16 h to ensure uniform initial sample temperature for microwave heating. The plastic film wrap (Saran Warp, The Dow Chemical Company, Indianapolis, IN) was used to cover the beakers during the heating, cooling and storage to prevent moisture loss. 3.3 Temperature Measurement For each heating experiment, one agar gel cylinder was placed at the center of the turntable in the microwave oven. Temperatures were measured across the horizontal mid plane at the radial distances of 0, I, 2, and 3 cm for 3.5-cm radius samples and 0, 1, 2, 3 and 4 cm for 4-cm radius samples. The sample cylinder was removed from the microwave oven after every minute of microwave heating, and temperature measurements were made. A type-T thermocouple probe (Omega Engineering Inc., Stamford, CT) connected to a datalogger (Model 34970A, Hewlett Packard, Beaverton, OR, see Fig. 3.3) was used to measure the temperature. A 0.5-cm thick cardboard with holes drilled 1 cm from each other along the radius (Fig. 3.3) was placed on top of the agar gel cylinder matching the sample edge. The thermocouple probe was then inserted into the sample via the holes in the cardboard until it reached the mid-plane. Single thermocouple was used for all measurements. The readings were taken starting from the sample center (0 cm) and moving outward. The temperature at each location was measured in triplicates using three samples. The average and the standard deviation of the three temperature measurements were calculated to represent the temperature distribution Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 in the sample cylinder. For each sample, all temperature measurements were taken within 30 s after the heating process was completed and the sample was removed from the microwave oven. The temperature variation at each location due to the 30-s delay during the measurements was determined (Appendix D) not to affect the temperatures measured significantly (P >0.01, Table D). 3.4 Average Absorbed Microwave Power The average absorbed power (PabS) in the microwave oven was determined experimentally. The 250- and 500-W oven settings were employed to heat distilled water in different volumes for 180 s. The efficiency of energy transfer between the microwave oven and the food sample is related to the sample’s dielectric properties. The efficiency is also related to the nature and volumetric load of the sample (Mudgett, 1986). The average temperature rise (ATav), the difference between the temperatures before and after microwave heating, were measured. The temperatures were measured after stirring the heated water for 10 s using a hand mixer (KitchenAid, KTM-7) at the lowest speed (250 rpm) to ensure temperature uniformity. The distilled water can be lost during the microwave heating due to the moisture evaporation. The effect of evaporative cooling was determined by measuring the difference in mass of distilled water (Aw) before and after microwave heating; Aw multiplied by latent heat of water (hfg=2,338 U/kg at 27 °C and at atmospheric pressure) then divided by the total heating time (t) is the power correction term for the evaporative cooling. The average temperature rise (ATav) can be calculated from the following energy balance relationship: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 p V C A T + h f &w - ' ■ t * (3-1), where p = density of distilled water (1,070 kg.m'3), V = volume of the distilled water, Cp = specific heat capacity of distilled water (4.18 kJ.kg'l.°C'1), ATav= average temperature rise (°C) and t = microwave heating period (180 s). Three replications of temperature rise measurements were performed for each microwave power setting. The variation in the absorbed microwave power at the center and in the vicinity of the turntable was also determined. The absorbed power at the 250-W oven setting in the microwave oven was measured by placing the samples at I and 2 cm from the center of the turntable in triplicates and compared to the absorbed power measured at the center (Fig. 3.4). The absorbed powers and the energy-transfer efficiencies in the microwave oven for all experimental conditions are listed in Table 3.1. The variation in absorbed power at the range between the center and at 2 cm from center was (3.7 W) and was not statistically significant (P >0.01, Table 3.2). The energy-transfer efficiency (ratio of absorbed power to oven setting) increased as the sample volume increased. 3.5 Average Surface Heat Transfer Coefficient The average surface heat transfer coefficient was determined by measuring the temperature of a cylindrical aluminum block, 3.5 x 7 cm (radius x height). The aluminum block was cooled to 4°C and placed at the center of the microwave oven and allowed to warm by the ambient temperature in the oven. The temperature change (at center of the aluminum block) was recorded every 10 s using a fiberoptic sensing system (MetricCor, Model 1400, Woodivelle, WA). The average surface heat transfer coefficient was R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. calculated according to Rizvi and Mittal (1992). Since the thermal conductivity of the aluminum block is very high (i.e. Biot number « 1), it was assumed that a lumped capacitance method could be used. The experimental parameters are: air temperature = 23°C, initial temperature of the aluminum block = 4°C, surface area = 0.0194 m2, mass of the block = 0.72 kg, and specific heat capacity = 0.930 kJ/kg."C. The measured average surface heat transfer coefficient was 41.7 ± 0.56 W .m '^ C '1(mean ± standard deviation). 3.6 Azimuthal Wave Assumption In this research, microwave radiation was assumed to be incident normal to the cylinder surface (Bowman, 1988; Pangrle et al., 1991). For propagation of waves in polar (cylindrical) coordinates, the electric E,(r) and the magnetic H<t>(r) components lie along the cylindrical surface of uniform intensity varying only in the direction of wave propagation along the radial r-axis (azimuthal condition, Fig. 3.5). The azimuthal microwave heating also means that the sample temperatures at the same radial distance but at different angular locations should be the same. To verify this, 3.5-cm radius agar gel sample cylinders were heated in the microwave oven for 30 s at the 250-W oven setting. The temperatures at every 120° angular location (0°, 120° and 240°) in the same sample were measured centimeter-wise in triplicates. At the sample center, only one angular location is available. The resulting temperature measurements with respect to the angular variation are shown in Table 3.3. The temperature variation at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 different spatial angles is not significant (P >0.01, Table 3.4). Therefore, the assumption of azimuthal wave behavior is acceptable. 3.7 One-dimensional Heat Transfer Assumption The heat transfer equation was solved by finite-difference approximation, with intemal-heat-generation terms expressed as volumetric heating rate based on the levels of microwave power absorption in successive shell volumes. The transient finite-difference model used in this thesis is one-dimensional, temperature variation was considered only along r-axis, i.e. the predicted temperature of each incremental shell represents the average temperature of the entire shell. Mudgett (1986) used a one-dimensional model to predict the temperatures within agar gel cylinders ( 5 x 7 cm, radius x height) heated by microwaves (750 W, 1.5 min). The predicted and measured temperatures at different vertical (z-axis) and radial (r-axis) locations (Table 3.5) indicated that one-dimensional heat transfer assumption is reasonably valid. Temperature variation along the z-axis in 3.5 x 7 and 4 x 7 cm (radius x height) agar gel cylinders (used as sample in Chapters 4, 5 and 6) after being heated in the microwave oven used in this thesis was evaluated. The heating duration was 3 min at the 250-W oven setting and the temperature distribution in the agar gel cylinders was measured (method described in Section 3.3). The measured points are shown in Fig. 3.6. The temperature distributions in the 3.5- and 4-cm radius sample cylinder are shown in Tables 3.6 and 3.7 respectively. Each set of temperatures measured at the same horizontal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 plane along z-axis was compared to that at the mid-plane (depth = 3.5 cm) using a paired /-test. The temperature sets at all horizontal planes are not significantly different from that at the mid-plane (P >0.01, Table 3.8). The temperatures at 6.5-cm depth are the highest because the bottom of the agar gel cylinder is in direct contact with the turntable. The dielectric boundary causes the reflected and transmitted radiation. The reflection and transmission coefficients are related to the power dissipation in terms of dielectric properties of both materials across the boundary and described in Section 2.4.2. The reflected waves cause higher power dissipation near the contact surface of the agar gel cylinder. However, the results indicated that the mid-plane temperatures can represent the average temperatures along the z-axis (P > 0.01, Table 3.8). Therefore, one-dimensional heat transfer assumption is statistically acceptable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 3.8 References Bowman, F. 1988. Introduction to Bessel functions. Dover Publications Inc., New York, NY. Mudgett, R. E. 1986. Microwave properties and heating characteristics of foods. Food technol. 40(6): 84-93. Pangrel, B.J. Ayappa, K.G. Davis, H.T. Davis, E.A. and Gordan, J. 1991 Microwave thawing of cylinders J. AIChE 37(12): 1789-1800. Rizvi, S.S.H. and Mittal, G.S. 1992. Experimental Methods in Food Engineering. AVI Book Co., New York, NY. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 Table 3.1 Absorbed power and energy'transfer efficiency in the microwave oven at 2.45 GHz. Oven setting (W) 250 500 Sample radius (cm) 1.6 2.4 2.8 3.2 3.5 4.0 3.2 3.5 4.0 Sample volume (cm3) 56 127 160 225 269 352 225 269 352 Absorbed power (W) 110.2 ±4.07 130.7 ± 3.90 144.4 ± 4.63 167.7 ±3.12 223.5 ± 3.70 233.9 ± 2.89 279.9 ± 4.03 331.2 ±3.61 344.1 ±3.12 Energy-transfer Efficiency (%) 44.1 52.3 57.6 67.1 89.4 93.6 55.8 66.2 68.8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 Table 3.2 Effect of sample placement (1 and 2 cm from the center of turntable in the microwave oven) on the absorbed power at 2.45 GHz-statistical analysis. Source of variation Placement Replication Error Total Degrees of Freedom 2 2 4 8 F-ratio 2.21 3.09 P-value 0.226 0.309 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 Table 3.3 Temperatures at different angular locations in a 3.5-cm agar cylinder. Location from center 0° 120° 240° 0 14.9 ± 0.61°C -- -- 1 8.7 ± 0.17°C 8.4 ± 0.25°C 8.2 ±0 .1°C 2 10.5 ± 0.38°C 10.1 ±0.35°C 10.3 ± 0 .3 1°C 3 11.7 ±0.40°C 12.6 ±0.25°C 12.8 ± 0.40°C (cm) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.4 Effect of angular locations (0,120 and 240 °) on the measured temperatures in agar gel cylinders-statistical analysis. Source of variation Radial location Replication Angular location Error Total Degrees of Freedom 3 2 2 28 35 F-ratio 596.67 2.52 2.02 P-value <0.01 0.098 0.151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 Table 3.5 Predicted and measured temperatures for 5 x 7 cm (radius x height) agar gel cylinders (from Mudgett, 1986). Temperature (°C) along r-axis Depth (cm) 2 cm (Midpoint) 45 4 cm (Surface) 49 Predicted All 0 cm (Center) 57 Measured 1 57 45 44 2 63 47 46 3 62 48 50 4 60 45 50 5 60 46 46 6 55 44 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 Table 3.6 Measured temperatures for 3.5 x 7 cm (radius x height) agar gei cylinders in the microwave oven at the 250-W oven setting. Average Temperature (°C) along r-axis 1 cm 2 cm Depth (cm) All 45.3 ±3.14 35.4± 3.49 33.1 ±3.50 32.9 ± 2.77 0.5 40.7 ± 1.27 31.5± 0.35 29.6 ±0.65 31.4 ±0.78 2.0 43.7 ± 1.01 34.2± 0.45 31.7 ±0.87 30.8 ±0.40 3.5 46.4 ± 1.32 34.0± 1.44 31.5 ±0.25 31.9 ±0.78 5.0 46.6 ±0.61 35.7± 0.46 33.6 ± 0.50 32.4 ± 0.46 6.5 49.1 ±0.9 41.4± 0.76 39.3 ±0.32 38.1 ±0.25 Measured 0 cm 3 cm Table 3.7 Measured temperatures for 4 x 7 cm (radius x height) agar gel cylinders in the microwave oven at the 250-W oven setting. Average Measured Depth (cm) All 0.5 2.0 3.5 5.0 6.5 0 cm 35.9 ± 2.59 32.8 ±0.67 33.3 ±0.80 35.3 ± 1.25 36.3 ± 1.00 39.5 ± 1.13 Temperature (°C) along r-axis 1 cm 2 cm 3 cm 23.2 ± 3.20 20.2 ± 1.42 20.8 ± 1.21 21.7 ± 1.44 24.2 ±1.15 27.5 ±0.75 24.2 ±3.19 21.0 ± 1.01 21.9 ± 1.17 23.5 ±0.85 25.3 ± 0.90 28.4 ± 1.16 26.5 ±2.41 24.5 ±0.55 24.5 ± 0.87 25.7 ± 1.23 27.4 ± 1.21 29.7 ± 1.05 4 cm 30.9 ± 1.96 31.3 ±0.47 30.3 ±0.85 31.6 ± 1.07 30.1 ± 1.21 31.7 ±0.85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.8 Paired Mest for temperatures at mid-plane (MP, depth=3.5 cm) compared to other locations along z-axis. 3.5-cm radius sample 4-cm radius sample Depth (cm) r-value P r-value P MP vs. average 1.24 0.323 1.42 0.229 MP vs. 0.5 MP vs. 2.0 MP vs. 5.0 MP vs. 6.5 2.37 1.20 2.46 5.17 0.098 0.455 0.091 0.014 3.83 3.60 1.62 3.87 0.019 0.023 0.181 0.018 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 Figure 3.1 Schematic of the microwave oven with arrows indicating the directions of circulating air. Vertical mode fan Air outlet fan Horizontal mode fan Waveguide 0 Turntable Top view of turntable Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.2 Pictures of the fans and waveguide in the microwave oven. 19 f r Vertical mode fan (on oven ceiling) Air outlet fan (on back wall) itiunrnt Horizontal mode fan (on right wall) Waveguide (on left wall) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 52 Figure 3.3 Type-T thermocouple, datalogger and cardboard temperature guide used for temperature measurements. g a e w r ■■■ Type-T thermocouple and datalogger 3.5-cm radius marked cardboard 4-cm radius marked cardboard Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 250 240 U 1 230 2 TJ 2w o CO A 220 < 210 200 40 0.5 1 1.5 2 2.5 Distance from center of turntable (cm) Figure 3.4 Absorbed power at 250-W oven setting in the microwave oven by placing the sample at 0,1 and 2 cm from the center of the turntable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 Figure 3.5 Cylindrical coordinate system and corresponding unit vectors for the case of electromagnetic radiation incident normal to the surface. Z X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 Figure 3.6 Temperature measurement locations in 3.5 x 7 cm (radius x height) agar gel cylinders for validating one-dimensional heat transfer assumption. u Top view 7 cm i r r •— » "T T cm •— Mid-plane »— A T ^ 3.5 cm *- O' Q 0 A 5 cm 0 6. 5 cm ~9----♦----•----9~ 9------ * ------♦ - Turntable Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 cm •— 56 C H A PT E R 4 TEMPERATURE PROFILES IN A CYLINDRICAL MODEL FOOD DURING PULSED MICROWAVE HEATING* * This chapter has been published in Journal of Food Science: Yang, H.W. and S. Gunasekaran. 2001. Temperature Profiles in a Cylindrical Model Food During Pulsed Microwave Heating. J. of Food Sci. 66(7): 998-1004. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.0 Notations As: surface area exposed to ambient air, m2 Bp Biot number C: coefficient of the analytical solution for an infinite cylinder Cp: specific heat capasity, kJ/kg.°C dR: increment of between two nodal points, m F0: Fourier number h: heat transfer coefficient, W/m2.°C Jo: zero order Bessel function of first kind J.: First order Bessel function of first kind k: thermal conductivity, W/m.°C P: power generation from microwave, W/m3 P(x): power dissipated at depth x, W/m3 Po: incident power or power at the surface, W/m2 Pp power generation with volume V j, W Pto tal- total microwave energy absorbed by the heated body, W PR: pulsing ratio of microwave application Q i-i: heat flux from an inner nodal point, W/m2 Q i+ l: heat flux from an outer nodal point, W/m2 r: radial distance, m R: radius of the sample, m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 Rj: t: radial distance o f a nodal point, m processing time, s toff: microwave power-off time, s ton: microwave power-on time, s T: temperature, °C Tajr: temperature of ambient air, °C Tj: predicted temperature at nodal point i, present time, °C Tj.t: predicted temperature at the inner nodal point i-1, present time, °C Ti+j: predicted temperature at the outer nodal point i+1, present time, °C Tint: uniform initial temperature, °C T„,i: predicted temperature at nodal point i, at new (next) time step, °C T„,j.i: predicted temperature at the inner nodal point i-1, at new (next) time step, °C Tn.j+i: predicted temperature at the outer nodal point i+1, at new (next) time step, °C V: total volume of the sample, m3 V;: volume of sub-shell i for numerical model, m3 x: distance from sample surface to the center, m Z: height of the sample, m z: axial distance, m an: thermal diffusivity, m2/s (3: attenuation constant X2: chi square (statistical table) value 5: loss angle, rad R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 ATav- average temperature rise in the sample, °C n: series term involved in analytical heat transfer solution for an infinite cylinder K’: dielectric constant K”: dielectric loss factor Xq: incident wavelength, m u: angle in cylindrical coordinates, rad P: density of the sample, kg/m3 dx: time increment, s & eigenvalue or positive root of the transcendental equation R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 4.1 Abstract Agar gel cylinders (2 %) were heated by pulsed and continuous microwave energy. The total microwave application time of three minutes was maintained for all experiments. Sample temperature distribution (TD) was measured at various radial distances along the mid-plane as a function of heating time and heating mode. A local hot spot was observed around the sample center during the continuous microwave application. This hot spot was less significant during pulsed microwave applications, especially when longer intermittent power-off times were employed. An implicit finitedifference simulation based on the Lambert’s law was used to predict TD in the sample during microwave heating. The predicted TD matched the experimental values statistically. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 4.2 Introduction The prediction of temperature distribution (TD) in food and biological materials heated by microwave radiation is critical for developing food processes and medical treatments. Regarding medical treatments, Ho et al. (1971) reported microwave-heating patterns of human limbs (cylindrical models) associated with human arms and thighs. In their study, the limb tissue was exposed to a direct contact energy source in which the waveguide and transmission lines were coupled through small apertures. They found differences in the heating patterns of the tissue cylinders when aperture size was different. Kritikos and Schwan (1975) reported microwave-heating patterns of spheres representing human and animal heads of various sizes (radii). Regarding food heating process using microwave energy, it is important to simulate the time-dependent TD and design the process related to product quality and microbial safety (Mudgett, 1986). Finite-difference approximations have been used to obtain reasonable predictions of TD in the sample during microwave heating. Ohlsson and Bengtsson (1971) offered a one-dimensional numerical solution for a finite slab to approximate the temperature profiles in meat blocks heated by microwave radiation. Nykvist and Decareau (1976) developed a two-dimensional model for cylinders representing meat roasts. Padua (1993) developed a temperature rise model for agar gel cylinders containing sucrose in terms of dielectric properties and total absorbed power. Barringer et al. (1995) reported another one-dimensional model representing thin slabs for temperature prediction of agar gels in terms of dielectric properties and various formulations of microwave absorbed power. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 One of the major drawbacks concerning microwave-heated foods is the nonuniform TD. Ohlsson and Riman (1978) studied the non-uniform TD inside meat and potato cylinders. Local hot spots at the sample center were observed in their study. Pulsed (intermittent) microwave energy has been reported to result in lower energy requirement and more uniform TD in food materials compared to continuous microwave energy. Shivhare et al. (1992 a, b, and c) studied com drying using pulsed microwave energy. They reported that the pulsed microwave energy is more efficient than conventional hot air drying. Yongsawatdigul and Gunasekaran (1996 a and b) investigated the pulsed microwave drying of cranberries. They found that a pulsed application (30-s power-on, 150-s power-off) under vacuum (5.33 kPa) resulted in maximum drying efficiency. However, in these studies temperature uniformity in the heated sample were not considered. The objectives of this Chapter were to: • Develop and verify a finite-difference heat transfer model in an implicit form that is capable of simulating TD in microwave heated model food cylinders (2% agar gel). • Compare sample TD under the heating of continuous and pulsed microwave energy using the heat transfer model. • Evaluate the temperature equalization effect via thermal conduction during pulsed microwave heating. • Validate the model predicted TD using experimental data and a statistical model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 4.3 Mathematical Model 4.3.1 Implicit Finite-dijference Model One-dimensional finite-difference models have been used under the assumption that the incident microwave radiation is normal to the material surface (Padua, 1993; Barringer et al., 1995; Pangrle et al., 1991). For a cylinder, the unsteady state (transient) differential equation can be solved considering term-by-term difference approximation of the differential equation: BZT 1 BT + dr~ r dr BZT P BT ------T k a Hdt (4-1) + ~ ^~ r + ~ i Bz where T = temperature, r = radial distance, z = axial distance, P = power generation, k = thermal conductivity, t = time, and an = thermal diffusivity. In the case of one-dimensional heat transfer, the third term of equation 4.1 on the left drops out and it becomes a 2nd-order partial differential equation (Incropera and DeWitt, 1996). Arpaci (1966) illustrated a finite-difference (FD) formulation for cylindrical geometry. For the cylindrical geometry corresponding to a typical inner point, the first law of thermodynamics yields: V't^ ' dT‘ = - y ) 2 , ! Z + « ■ * '( R ' + Y )27e* p ~ p' ( 4 '2 ) where Ri = distance from the sample center, V; = volume between Rj±dR/2, Pj = power generation in Vj, p = density of the material, Cp= specific heat capacity, t = time increment, dTj = temperature difference at nodal point i between present time (t) and new Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 time step (t+At) and Pc = power correction term due to the evaporation cooling (see Appendix E for the evaluation of Pc). The heat flux terms, and qj+i. are related to temperature by Fourier’s law of conduction. Fig. 4.1 shows the schematic of a typical inner node (nodal point) of a cylindrical object for finite-difference simulation. In an explicit finite-difference scheme, the temperature of any node at t+At can be calculated from the knowledge of temperature at the same and neighboring nodes for the preceding time t. This method suffers from limitations on the selection of At to ensure compatibility with stability requirements. In an implicit finite-difference scheme, the temperature of neighboring nodes relates to new (next) time. For the implicit method, the temperature of each node at new time step (t+At) depends on new temperatures of its adjoining nodes, which are generally unknown. Hence, to determine the unknown temperature at (t+At), the corresponding nodal equations must be solved simultaneously. The marching solution would then involve simultaneously solving the nodal equation at each time t= At, 2A t,... until the desired final time is reached. Compared to the explicit method, the implicit formulation has the advantage of being unconditionally stable, that is, it remains stable for all space and time intervals (Incropera and DeWitt, 1996). Thus, the implicit method was used to evaluate the TD in the model food, 2% agar gel cylinders. The heat flux terms for the explicit and implicit schemes are listed in Table 4.1. For a typical boundary nodal point, the second term on the right side in equation 4.2 is substituted by the boundary condition: hAs(Ti-Tajr), where h = average surface heat transfer coefficient, As = surface area at the boundary and Tajr= temperature of ambient air. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 4.3.2 Evaluating Microwave Absorbed Power The average temperature rise in a material within time t depends on the total microwave energy absorbed by the heated material (Pabs) (Padua, 1993; Barringer et al., 1995). The energy balance gives: A7* “v (4.3) VpCp where ATav = average temperature rise, Pabs = total absorbed power (evaluated using the method described in Section 3.4), t = heating time and V = sample volume, p = sample density and Cp = sample specific heat capacity. If one-dimensional simulation is considered and the incident radiation is assumed to be normal to the surface, the power dissipated at a certain sample depth is given by the exponential decay of the incident power along that direction. The power term was derived by von Hippel (1954) and is often referred as the Lambert’s law: P(x) = P0e '2* (4.4) 231 (—)(Vl + tan25 - l ) 2 (4.5) where ' and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 K tan S = — k (4.6) ' where x = depth or distance from surface along the radial axis (in the reverse direction of radial axis), P(x) = power dissipated at the depth x, PQ= incident power or power at the surface, (5 = attenuation constant in terms of dielectric properties, incident wavelength (Xo), k = dielectric constant and ’ k” = loss factor. To calculate the incident power (P0), the total absorbed power (Pabs) is expressed as the volume integral of the P(x) function. For a cylindrical sample in an azimuthal wave Field: ZZnR Pat, = j P ( x ) d V = j j j P 0e - iPxdxdudz (4.7) 0 0 0 where Z and R are the height and radius of the test sample, respectively. Integrating between limits and solving for Po gives: PpVCAT (48) The following expression can be used to calculate power absorbed by each annular shell (Pj) in the sample (Padua, 1993). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 P0e '2^dxdudz = ^ — {e'2Pxi ~ e 2Px' ) 0 O r, (4.9) P where the subscripts “ 1” and “2” refer to the outer and inner peripheries of the shell. In the case of pulsed microwave application, the power term for each node (Pj) is applied as a heat generation term in equation 4.2 during power-on periods and is zero for those time intervals during which the microwave energy source was not powered. This means that during every power-off period, there is no heat generation within the sample, and there is presumably only thermal conduction occurring in the sample and convective heat transfer across the boundary between the sample and ambient air. 4.4 Methods and Materials Agar gel cylinders, 3.5 cm in radius, were prepared (method described in Section 3.2). The sample TDs were measured (method described in Section 3.3). The microwave absorbed power in the microwave oven (PabS) was evaluated (method described in Section 3.4). The power absorbed by each cylindrical shell (Pj) during each microwave power-on period was calculated using equations 4.8 and 4.9. The average surface heat transfer coefficient in the microwave oven was determined (method described in Section 3.5). 4.4.1 Microwave Heating Process The microwave oven (described in Section 3.1) was used as the microwave energy source. Microwave power-on and -off times were adjusted to obtain different Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 pulsing ratios (duty cycles) as follows (Fig. 4.2): 1 (continuous), 2 (30-s power-on, 30-s power- off), and 3 (20-s power-on, 40-s power-off). The pulsing ratio, PR, is defined as: p R _ (to* + toff'> (4.10) where ton and toff = duration the microwave power is on and off per duty cycle, respectively. The total microwave power-on time was maintained 3 min for each PR. However, the total heating (process) times were 3 ,6 and 9 min respectively for PR=1, 2 and 3. 4.4.2 Data Analysis The physical, dielectric, and thermal properties of 2% -agar gel reported by Barringer et al. (1995) were used for calculation (Table 4.2). The attenuation constant (P) was calculated using equation 4.5. The implicit transient finite-difference simulation was used to predict the sample TDs heated by both continuous and pulsed microwave energy. The convection effect at the surface was also considered. In the case of an infinite cylinder, which is initially at a uniform temperature and experiences a change in convective boundary condition with one-dimensional transient heat conduction, the exact solution is given by (Incropera and DeWitt, 1996): T.-T (4.11) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 where F 0 = a Mt , ^ ^ „ —£y-, the coefficient Cn is Cn R~ 2 ■/,(£,) C, y2o(C,)+y2.(C,) and the discrete values (eigenvalues) of £n are positive roots of the transcendental equation which are related by the Biot number ( B j ) . The quantities J! and J0 are Bessel functions of the first kind. For F0 > 0.2, the series analytical solution (equation 4.11) can be approximated by single term: (4.12) The implicit finite-difference (FD) scheme was validated first by considering a non-power generation case (i.e. P=0) and comparing the model evaluation to the analytical solution of equation 4.12 for an example situation with parameter values listed in Table 4.3. The effect of shell interval on the proposed implicit FD model including microwave power was then validated by comparing the results to the temperature rise model of Padua (1993) under the same conditions: continuous microwave heating for 15 s, sample radius = 3 cm, sample height = 8.6 cm, initial temperature = 23°C, power level = 1500 W and ATav = 4°C. The predicted internal temperature profiles were calculated using Engineering Equation Solver (Klein and Alvarado, 1996). The X test was used to accept or reject the hypothesis that measured and predicted temperatures are the same, at P = 0.01 and degrees of freedom (n-1) = 35, where n = 3 PRs x 3 time intervals x 4 measured locations (Bender et al., 1981). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 4.5 Results and Discussion 4.5.1 Model Validation The analytical and numerical solutions for the infinite cylinder case matched very well, though the analytical solutions were slightly lower than the corresponding numerical values (Fig. 4.3). This is because only the first term of the series solution was used (i.e. used only ^ instead of £n=i,2 , 3 .... ~) in equation (4.11) for the analytical solutions. The temperature predictions of the implicit FD model also agreed well with the solution of the temperature-rise model of Padua (1993). Different shell increments used in FD model (0.1,0.15, and 0.2 cm) did not affect the predictions significantly (Fig. 4.4). These results indicated that the implicit FD model was unaffected by the increment chosen. The nodal increment of 0.1 cm was chosen for ongoing simulation. 4.5.2 Temperature Profiles The measured sample TDs after the total microwave applications of 1, 2 and 3 min with PR = 1, 2 and 3 are shown in Figs. 4.5,4.6 and 4.7, respectively. The center (R = 0 cm) temperature of all samples was the highest among temperatures at all locations. This local hot spot became significant within a short time of continuous microwave heating (PR = 1). The measured temperature profiles showed a difference of 9.1°C between radial distances of 0 and 1 cm at the end of 1 min with PR = 1. As the continuous microwave processing time increased, the local hot spot became more significant. The measured sample temperature profiles showed a difference of 13.4°C between radial distances of 0 and 1 cm after 3 min with PR = 1. The uneven TD within Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 continuous microwave heated food materials is not desirable for both process safety and quality. Under pulsed microwave applications (PR = 2 and 3), the unevenness of temperature distribution decreased substantially. After 3 min of a total microwave application (TMA) with PR = 2 and 3, the differences between temperatures at R=0 and I cm were 8.0 and 6.3°C, respectively. These results indicate that pulsed microwave application is preferable to continuous application in avoiding the development of local hot spots. The higher the pulsing ratio, the better temperature uniformity should be expected during microwave heating. The convective heat transfer affects temperature distribution within the sample during heating. The air temperature inside the microwave oven was around 22 °C. Once the sample surface temperature is higher than the ambient temperature, air convection becomes a cooling effect. The measured and FD predicted temperature profiles for different pulsing ratios at sample radial distances of 0 and 1 cm are shown in Figs. 4.8 and 4.9, respectively. The predicted TD at the sample center (Fig. 4.8) indicated that the temperature decreased during the power-off periods. This implies that the direction of heat flux was outward because of the lower neighboring temperature. The predicted TD at a radial distance of 1 cm shows that the temperature held steady during the power-off period. The pulsed applications for either PR = 2 or PR = 3 minimized the sample center temperatures significantly. The conductive temperature equalization during the power-off times is a major reason for a more uniform TD during pulsed heating compared to continuous Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 microwave heating. The chi square value, x2, was determined for the difference between predicted and measured temperatures. After 3 min of total microwave application, the predicted temperatures for PR = 1, 2 and 3 were not significantly different (P >0.01) from the corresponding measured temperatures (calculated x2 = 1185 and x2 = 50.7 at P = 0.01 with degrees of freedom = 35, Bender et al., 1981). 4.6 Conclusions Temperature uniformity in 2% -agar gel cylinders was improved during pulsed microwave application compared to continuous application. The local hot spot observed at the sample center during continuous microwave application was minimized during pulsed applications. The longer the power-off period (i.e. the higher the pulsing ratio), the better the expected temperature uniformity. The conductive temperature equalization during the power-off periods leads to better temperature uniformity in samples heated by pulsed microwave energy. The predicted temperature distribution based on one dimensional finite-difference heat transfer simulation in the sample matched with the corresponding measured temperature detebution statistically. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.7 References Arpaci, V. S. 1966. Conduction Heat Transfer, Addison-Wesley Publishing Co., Reading, MA. Barringer, S. A., Davis, E. A., Gordon, J., Ayppa, K. G., and Davis, H. T. 1995. Microwave- heating temperature profiles for thin slabs compared to Maxwell and Lambert law predictions. J. Food Sci. 60(5): 1137-1142. Bender, F.B., Douglass, L.W., and Kramer, A. 1981. Statistical Methods fo r Food and Agriculture, Food Products Press Inc., Binghamton, New York. Ho, H.S., Guy, A.W., Sigelman, R.A., and Lehmann, J.F. 1971. Microwave heating of simulated human limbs by aperture sources. IEEE Trans. Microwave Theory & Techniques 19(2): 224-231. Incropera, F. P. and DeWitt, D.P. 1996. Introduction to Heat Transfer, 3rd Ed. John Wiley & Sons Inc., New York, NY. Klein, S. A. and Alvarado, F. L. 1996. Engineering Equation Solver fo r Microsoft Windows Operating Systems. F-Chart Software Co., Middleton, WI. Kritikos, H.N. and Schwan, H.P. 1975. The distribution of heating potential inside lossy spheres. IEEE Trans. Biomed. Eng. 22(6): 457-463. Mudgett, R. E. 1986. Microwave properties and heating characteristics of foods. Food Technol. 40(6): 84-93. Nykvist W. E. and Decareau R. V. 1976. Microwave meat Roasting. J. Microwave Power 11(10): 3-24. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 Ohlsson, T. and Bengtsson, N. 1971. Microwave heating profiles in foods- a comparison between heating and computer simulation. Microw. Energy Appl. Newsletter 6: 38. Ohlsson, T. and Risman, P.O. 1978. Temperature distribution of microwave heatingspheres and cylinders. J. Microwave Power 13(4): 303-309. Padua, G.W . 1993. M icrowave heating of agar gels containing sucrose. J. Food Sci. 58(60): 1426-1428. Pangrle, B.J., Ayappa, K.G., Davis, H.T., Davis, E.A. and Gordon, J. 1991. M icrowave thawing of cylinders. Journal AIChE 37(12): 1789-1800. Shivhare, U.S., Raghavan, G.S., and Bosisio, R.G. 1992a. Microwave drying of com I. Equilibrium moisture content. Trans, of the ASAE 35(3): 947-950. Shivhare, U.S., Raghavan, G.S., and Bosisio, R.G. 1992b. Microwave drying of com II. Constant power, continuous operation. Trans, of the ASAE 35(3): 951-957. Shivhare, U.S., Raghavan, G.S., Bosisio, R.G., and Mujumdar, A.S. 1992c. Microwave drying of com HI. Constant power, intermittent operation. Trans, of the ASAE 35(3): 959-962. von Hippel, A. R. 1954. Dielectric and Waves. The M IT Press. Cambridge, MA. Yongsawatdigul, J. and Gunasekaran, S. 1996a. Microwave drying of cranberries: Part I. Energy and efficiency. J. Food Proc. and Pres. 20: 121-143. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 Yongsawatdigul, J. and Gunasekaran, S. 1996b. Microwave drying of cranberries: Part II. Quality evaluation. J. Food Proc. and Pres. 20: 145-156. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1 Explicit and implicit schemes for heat fluxes (qM and qi+i) for finitedifference analysis Explicit scheme qi.1=k(Ti.r Ti)/dR Implicit scheme q j. i=K).5k(Tj. i-Tj+Tn,j.i-Tn,i)/dR qi+l=k(Ti+l-Ti)/dR qi+l=0.5k(Ti+1-Ti+Tn,i+i-Tn.i)/dR Table 4.2 Dielectric, physical, and thermal properties of 2% agar gela Property Specific heat capacity, Cp (kJ/kg.°C) Thermal conductivity, k (W/m.°C) Density, p (kg/m3) Dielectric Constant, k Dielectric loss, k” a Data From Barringer et al. (1995) ' Value 4.2 0.60 1070 73.6 11.5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 Table 4.3 Parameter values of an Infinite cylinder used for validation of the implicit finite difference model. Parameter Value Heat transfer coefficient, h 40 0.03a Radius, R (m) 0 .T Height, Z (m) Thermal conductivity, k (W/m.°C) 0.6 2 Biot num ber5 1.34 Coefficient, Ct 1.6° Eigenvalue, Ci 4 Initial uniform temperature, Tjnt 60 Ambient temperature, Ta;r( °C) 100 T im e, s 1.5* 10'5 Thermal diffusivity, an (m2/s) 0.3d Fourier number, F0 a Chosen to designate an infinite cylinder Z/R>10 b Bi=hR/k c Table value from Incorpera and DeWitt (1996) d Chosen to designate conditions can be approximated Fo>0.2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 Sample axis Node i dR=0.1cm Heat flu x from node i+1 to node i Heat flux from node i-1 to node i Figure 4.1 Schematic of a typical inner node of a cylindrical object for one dimensional analysis (R,=0 along the sample axis). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 ON PR=3 OFF n PR=2 OFF ON PR=1 0 1 2 3 4 5 6 7 8 9 Tim e, min Figure 4.2 Power-on/-ofT periods employed during microwave applications for pulsing ratios (PR) of 1,2 and 3, with total microwave power on-time of 3 min. The solid segments represent power-on periods, dotted segments represent power-off periods, and vertical dash segments represent the power on/off transition. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 80 Temperature (°C) 50 40 30 20 approximate analytical solution 10 numerical solution 0 0 0.5 2 1.5 Radial distance from center (cm) 1 2.5 3 Figure 4.3 Analytical and implicit finite-difference numerical temperature profiles in an infinite cylinder (see Table 3.3 for parameter values used in analytical approximation). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 60 — - A 55 -{ FD increment=0.1 c m FD in c re m e n ts . 15 cm 50 Temperature (°C) TR FD increm ent=0.2 cm 45 i 30 25 - 50 4 2 0 Radius distance from center (cm) Figure 4.4 Comparison of temperature profiles according to implicit finitedifference model (FD) and that of the temperature-rise (TR) model by Padua (1993), shell thickness increments are 0.1,0.15, and 0.2 cm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 30 PR=1 Temperature (°C) 25 PR=2 PR=3 20 15 10 5 0 1 2 3 4 Radial distance from center (cm) Figure 4.5 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in radius and 7 cm in height) after 1 min of total microwave power application at different pulsing ratios (PR). The power incident is from the sample outer periphery (the radial distance of 3.5 cm from the center line). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 35 PR=1 Temperature (°C) PR=2 PR=3 30 25 20 15 0 0.5 1 1.5 2 2.5 3 3.5 Radial distance from center (cm) Figure 4.6 Measured temperature profiles in cylindrical agar gel samples (3.5 cm radius and 7 cm in height) after 2 min of total microwave power application at different pulsing ratios (PR). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 50 PR=1 Temperature (°C) 45 PR =2 PR =3 40 35 30 25 0 0.5 1 1.5 2 2.5 3 3.5 Radial distance from center (cm) Figure 4.7 Measured temperature profiles in cylindrical agar gel samples (3.5 cm in radius and 7 cm in height) after 3 min of total microwave power application at different pulsing ratios (PR). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 60 50 40 9 £3 2 30 2L E PR=1 20 — - P R -2 PR=3 10 • PR=1 ■ PR=2 • PR=3 0 0 100 200 300 400 500 6 00 Time (s) Figure 4.8 Measured (M) and finite*difference model predicted (P) temperature profiles at different pulsing ratios (PR) at a radial distance of 0 cm (center line). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 40 35 30 (0 o) a jn ie ja d u ia i 25 20 PR=1 (P) PR =2 (P) 15 PR=3 (P) PR=1 (M) 10 PR = 2(M ) PR = 3(M ) 5 0 0 100 200 300 400 500 600 Time (s) Figure 4.9 Measured (M) and finite-difference model predicted (P) temperature profiles at different pulsing ratios (PR) at a radial distance of 1 cm from the central line. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 CHAPTER 5 COMPARISON OF TEMPERATURE PROFILES IN A CYLINDRICAL MODEL FOOD BASED ON MAXWELL’S AND LAMBERT’S LAW Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.0 Notations As = surface area, m2 B = magnetic induction, Wb. m '2 c = velocity of radiation, m.s'1 ci, c2, c3 and c4: parameters involved in analytical solution of Maxwell’s equations C,,C2 and C3: parameters involved in the evaluation of electric field intensity Cp = specific heat capacity, J.g'l.°C'1 D = electric displacement, C.m'2 E = electric field intensity, V.m'1 Ec = conjugate electric field intensity, V.m’1 / = frequency of incident radiation, Hz h = heat transfer coefficient, W .m'VC*1 H = magnetic field intensity, A .m '1 Hc = conjugate magnetic field intensity, A.m'1 J = current flux A.m'2 Jo = Bessel function of first kind in zero order Ji = Bessel function of first kind in first order k = thermal conductivity, W .m'l.° C l p = microwave source power term, W .m 2 P = dimensionless microwave source power term r = radial distance of the sample, m r* = dimensionless radial distance of the sample Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 R = radius, m Re=real part of a complex number Im=imaginary part of a complex number S = Poynting vector, W.m'2 t = time, s T = sample temperature, °C Ts = surface temperature of the sample, °C To. = ambient temperature, °C u = dimensionless electric field intensity v = dimensionless real field component w = dimensionless imaginary field component Y0 = Bessel function of second kind in zero order Yi = Bessel function of second kind in first order Greek letters a = wave number, m '1 (Xo = free space wave number, m '1 P = attenuation constant, m 1 %: parameter involved in analytical solution of Maxwell’s equations 5 = Kroncker delta Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 e = permittivity, F.m '1 £o = free space permittivity, F.m'1 e’ = dielectric constant, F.m '1 £ = dielectric loss factor, F.m '1 k= relative permittivity k’ = k” relative dielectric constant = relative dielectric loss factor Xm= wave length in sample, m (A= permeability, H.m '1 |io = free space permeability, H.m'1 a = electric conductivity, mho, m '1 a) = angular frequency, rad.s'1 V= parameter involved in analytical solution of Maxwell’s equations Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1 Abstract Two-percent agar gel cylinders were heated using pulsed and continuous microwave energy. Temperature distribution (TD) inside the sample was measured and compared with numerical predictions based on the Lambert’s law and Maxwell’s equations'. The Maxwell’s equations account for the standing wave effect inside the sample, the Lambert’s law does not. The results show that the predictions based on the Maxwell’s equations are statistically more accurate than those based on the Lambert’s law, especially around the sample edge. The measured TDs and the corresponding predictions using both models indicate better temperature uniformity in the agar gel cylinders under pulsed microwave heating than under continuous microwave heating. 1 Also referred as “Maxwell’s field equations” R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 5.2 Introduction Heating food materials by electromagnetic radiation is widely used in commercial, industrial, and household applications. These applications rely on the internal heat generation due to interaction between the material and electromagnetic radiation. The propagation fields based on Maxwell’s equations are associated with a power flux, namely, Poynting vector of harmonic fields (Ayappa et al., 1991; Chen, 1992). The oscillating power distribution within the material is useful for predicting the locations of hot spots (Fu and Metaxas, 1992). The oscillation power absorbed by thin slabs of food is related to dielectric properties. This is the key to several patents claimed by a research group at Pillsbury Corp., Minneapolis, MN (Atwell et al.,1990, 1992; Peschek et al., 1991a,1991b and 1992). The basic principle of these patents is that the heating rate of a given food layer is a combined function of the layer’s dielectric properties, thickness and thermal mass. In a sphere or a cylinder, overheating frequently occurs at the focal points in the sample during continuous microwave application (Ohlsson and Risman, 1978). Pangrle et al. (1991) proposed cylindrical models to study phase changes of frozen brine (salted water) thawed by microwave energy. In Chapter 4, the Lambert’s law was used to describe the electric field behavior. However, the Lambert’s law does not account for node/anti-node formation (Barringer et al., 1995). In this Chapter, Maxwell’s equations, related to space and time dependence, were solved to determine the absorbed microwave Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 power. The absorbed power was applied intermittently to simulate the temperature distribution in 2 % agar gel cylinders heated by pulsed microwave energy. The objectives of this Chapter were to: • Investigate the temperature distribution (TD) in 2% agar gel cylinders heated by continuous and pulsed microwave energy. • Predict the sample TD by solving the Lambert’s law and Maxwell’s equations of microwave energy absorbed by the sample. • Compare the measured and predicted TD in the sample under continuous and pulsed microwave heating. 5.3 Theory and Analyses 5.3.1 Heat Transfer Equation The heat equation in a material is: St pCp — = V •(kVT) + p(r,t) at (5.1) where p, Cp and k are density, specific heat capacity, and thermal conductivity, respectively. In the case of microwave heating, the absorbed power, p(r, t) is determined from real and imaginary components of the electromagnetic field. The transient one dimensional heat equation is: P C ” TOt~ = ~r dr T ( k r lor T )+ p ( r ’° The boundary conditions for the heat equation are: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (52) 94 (5.3) dr and (5.4) The initial condition is: T(t=0) = Tj fo rO < r< R . (5.5) The absorbed microwave power is obtained by solving the Maxwell’s equations. 5.3.2 Wave Propagation A propagating electromagnetic wave is composed of oscillating electric (£) and magnetic (B) Field components. Maxwell’s equations describing their space and time variations are: and (5.6) where E and H are the electric and magnetic fields, J = current flux, D = electric displacement and B = magnetic induction. The constitutive relations relating J, D and B, to E and H are: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 J = a((o)E(t), D = £(co)E(t) and B = fi( 0))H (t) (5.7) where E=E*e'l(ut and H=HVIUJt. Alternatively e',aican be used to express the time dependence. Equations 5.6 and 5.7 yield: V x £ * =ia)ii((o)H' (5.8) and V x H * = \a((o)-i(DE((o)]p' = -icoe' E ' , (5.9) where the complex dielectric constant, e\ is defined as: e ’ (co) = £(cu) + = £' ((D) + i£" ((D) (5.10) (0 The material’s ability to store and dissipate electric energy is represented by the real and imaginary parts of £*, e’=Re(e*) and e”= Im(E*) respectively, and e” accounts for energy losses through dissipation. The electric conductivity a(co), dielectric constant e((d), and magnetic permeability p((0 ) are generally complex functions of frequency of radiation, u). Neglecting magnetic effects, the magnetic permeability |n(co) is approximated by its value po in free space. The time derivative of a and £ can also be neglected, since the time scale of electromagnetic propagation is smaller than the time scales for thermal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 diffusion. With these assumptions and the condition of electroneutrality of the medium, which implies V- (e*E) =0, equations 5.8 and 5.9 can be combined to give: ~ ) + V 2E ' + kl2E m (5.11) where kt2 =a>2pi0e0(fc'+iK") (5.12) The relative dielectric constant k ’ and the relative dielectric loss K '= e’/ e Q and K"=£"/e0 k” are: (5.13) To simplify the notations, the superscript “stars” on E and H will be dropped from this point on. The propagation constant ki is a complex quantity: kt = a + iP, (5.14) where a and (3 are related to the dielectric properties of the material and frequency of radiation by: and where tan<5=— . (5.17) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 where c is the speed of light and a) is replaced by I n f , f is the frequency of radiation. The phase constant a is related to the wavelength of radiation in the material (Am) by: which in free space reduces to Ao=cI f . The attenuation constant P is a rate constant for the decay of the incident field of intensity E 0. For instance, in a semi-infinite sample the interior field obeys the equation: £ = £„<?■*. (5.19) 5.3.3 Power dissipation The power flux associated with a propagating electromagnetic wave is represented by the Poynting vector (S) and the time average flux for harmonic fields: S= ^E xH c (5.20) The power dissipated per unit volume is: p(r) = —Re(V- S) =±(oe0K"(E £ r ). (5.21) Therefore, given the electric field intensity in the material the local power dissipated is calculated by equation 5.21. The subscript “c” is the conjugate field intensity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 5.3.4 Maxwell's equations for a cylinder Microwaves are assumed to be incident on a cylinder as shown in Fig. 5.1. For propagation of an azimuthal condition in polar coordinates, the electric and magnetic components orient along a cylindrical surface of a uniform intensity varying only in the direction of propagation along r-axis (Section 3.6). The wave equation, Equation 5.9 is: d ' E 1 dE r-H — 1 - k f E = 0, d r 2 r dr (5.22) fo r 0 < r< R Maxwell’s equations are rendered dimensionless by substituting the relationships: r E r* = — and u = — R E0 (5.23) Noting that u = v+iw, equation 5.22 yields to the following two equations that are solved with their appropriate boundary conditions: (5.24) and dr*2 r* dr* + ynv + %v=0, (5.25) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where \j/ = R2a)2Ho£oK’ and x= R2a)2Ho£oK”. The boundary conditions at r* = 0 are: dv _ dw dr* dr* (5.26) while boundary conditions at r* = 1 are: dv dr * + C .V + C-, w = c. (5.27) and dw + c,w - c 2v = c 4, dr * (5.28) where c | —Ret q J l ( a 0R ) J 0(ccQR ) + Y l(cc0R)Y0(ccQR ) J 02( a 0R) + Y02( a 0R) c-, = n [ j 02{ a 0R) + y02( a 0Ji)]’ * J 0' ( a 0R) + Y0' ( a 0R) and J 0(ct0R) n J 0- ( a 0R) + Y0- ( a 0R) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 where J and Y are Bessel functions, subscripts 0 and 1 are zero- and first-order of Bessel functions and Oo = free space wave number. Eo is the intensity of the incident field related to power term Pqby: „ c e nEo~ / >o = - ^ - r - . n a 0R (5.29) The power term as a function of r* is calculated from v and w: P(r*,t) = R 2Q)£0K"E0(v2 + w 2) fo r 0 < /? * < !. (5.30) 5.3.5 Analytical Solution for Absorbed Power For the case of constant properties, the coupled equations 5.24 and 5.25 can be solved with the help o f : C ____________________________ C 3 + / C 4_____________________________ 1 - (C, + iC, )/, (C2 + iC3) + (c,+ ic2)JQ(C2 + /C3) ’ (531) where C, + iC3 = yj\}f + i%. The analytical solutions of the coupled equations are: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 v(r*) = Re{C, J 0[(C, + i'C, )r *]} (5.32) and w(r*) = Im{C| y 0[(Cl + iC 2)r * J (5.33) 5.3.6 Temperature Distribution Prediction Applying the analytical solutions of the electromagnetic equations to the heat equation (equation 5.2) by a finite-difference heat transfer method described in Chapter 4, the interior temperature profile of a microwave-heated sample can be predicted. For pulsed microwave heating, the absorbed power = 0 in the heat transfer equation during power-off periods. 5.4 Methods and Materials 5.4.1 Microwave process Two-percent agar gel cylinders, 3.5 and 4 cm in radius, were prepared (method described in Section 3.2). The microwave oven operating at 250-W oven setting was used as the heating source. Microwave pulsing ratios (duty cycles) of 1 (continuous), 2 (30-s power-on, 30-s power-off) and 3 (20-s power-on, 40-s power-off) were applied. The pulsing ratio, PR, is defined as: pp _ + 1off ) (5.34) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 where to„ and W = duration of the microwave power-on and -off per duty cycle, respectively. The total microwave power-on time for every pulsing ratio was 3 min in the experiments. The TD in the samples was measured (method described in Section 3.3). The microwave power absorbed by the samples (Pabs) and average surface heat transfer coefficient in the microwave oven were determined (method described in Section 3.4 and 3.5, respectively). 5.4.2 Temperature Prediction The physical and thermal properties (Table 4.2) of agar gel reported by Barringer et al. (1995) were used for the calculation. The finite-difference model proposed in Section 4.3.1 was used to solve the heat transfer equation numerically to predict the TD in the agar gel cylinders. The convection effect inside the microwave oven (ambient air effect that contributes to the heat transfer boundary conditions of the agar gel cylinders) was also included in the model. The absorbed microwave power applied to the heat transfer equation was based on either Maxwell’s equations (Section 5.3.2 and 5.3.3) or the Lambert’s law (Section 4.3.2). Maxwell’s absorbed power was solved using the MATLAB software (MathWorks Inc., Natick, MA). The X2-test was used to accept or reject the hypothesis that measured and predicted temperatures were the same, at P = 0.01 (Bender et al., 1981). __ v r X ““ (Tmeasured - T* predicted' _ •, J sc n c \ predicted Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 5.5 Results and Discussion Maxwell’s and Lambert’s microwave power distributions in the 3.5- and 4-cm radius agar gel cylinders as a function of radial distance are shown in Fig. 5.2. The field components were assumed to orient along the z-axis (this assumption was verified to be valid in Section 3.6). The Maxwell’s power distributions present oscillating patterns because of the space and time dependence of electromagnetic fields. Due to the node/anti-node effect of waves is not considered, the Lambert’s power distributions decay exponentially from the sample surface to the center. The exclusion of standing wave effect also caused the Lambert’s power at the outer portion of the sample along the r-axis being significantly larger than the Maxwell’s power. The power distribution in a 3.5-cm radius sample cylinder was generally larger than that in a 4.0-cm radius sample cylinder at the same radial distance. Theoretically, for smaller cylindrical samples, their oscillating power patterns are stronger due to the penetrating nature of the microwaves (Fu and Metaxas, 1992). In the case of a cylindrical body, the annular incremental volume (V j) along the r- axis converges as a function of 1/r2 from the sample surface to the center (the same sample height). However, the microwave power (P) decreases exponentially along the same direction (surface to center). Due to the combined effect of power and incremental volume (P/Vj), the volumetric power (power density) represents the center focusing effect of microwave energy along the r-axis (Fig. 5.3). The existence of the focusing volumetric power at the sample center depends on the sample critical length. For a very large sample, the center focusing of microwave power density may not be observed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 The measured and predicted sample TDs at the end of 1- and 3- min heating by microwave energy with PR = 1 (continuous) and PR = 3 are shown as Figs. 5.4 to 5.7 (2min results shown in Appendix B.2). The results of PR = 2 (Appendix B.2) followed the same trend as PR = 3, only less significant. For all the microwave processes, the largest temperature difference between any two consecutive measured points occurred at R = 0 and R = 1 cm. As the duration of microwave heating increased, the difference increased. The pulsed microwave heating resulted in better temperature uniformity and minimized the central focusing effect. For the continuous microwave heating process, the sample temperature profiles predicted by the Maxwell’s equations represented an oscillating pattern and agreed with the absorbed power pattern (Fig. 5.3). For pulsed microwave heating, the sample temperature profiles based on Maxwell’s prediction represent smooth curves. These results are expected because thermal diffusion occurred during the power-off periods of pulsed microwave applications and equalized the oscillating temperatures established during microwave power-on periods. The measured temperatures after 1-, 2- and 3-min of total microwave applications (continuous and pulsed) were compared to the predicted temperatures based on Maxwell’s equations as well as that based on the Lambert’s law using the x 2-test (Table 5.1). The x 2 = 45.6 at P = 0.01 with degrees of freedom = 26 (Bender et al. 1981). The predicted TD based on either Maxwell’s or Lambert’s power is not significantly different from the measured TD (P > 0.01). However, the Lambert’s prediction resulted in overestimation of temperatures near the sample surface. The calculated x 2 increased as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 microwave heating time increased for the Lambert’s predictions. The calculated x2 values were fairly consistent as a function of time for the Maxwell’s predictions. These statistical results indicate that Maxwell’s prediction is more accurate than the Lambert’s for long heating times. 5.6 Conclusions Pulsed microwave heating improves the temperature uniformity in heated samples. The local hot spot observed at the sample center during continuous microwave application was substantially minimized by pulsed microwave applications. The predicted temperature distribution in the sample based on either the Lambert’s law or Maxwell’s equations was statistically accurate as compared to the measured temperature distribution. The power formulation based on Maxwell’s equations is more accurate than that based on the Lambert’s law, especially for longer heating time. The Lambert’s law is based on the assumption that the sample critical length is semi-infinite and does not consider standing wave effect, therefore, leads to the overestimation of temperatures near sample surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 Table 5.1 The Chi-square values for Maxwell’s and the Lambert’s predictions compared to the measured temperatures during microwave heating Microwave processing time (min) 1 2 3 X2of Maxwell’s prediction 5.56 3.86 4.24 %2 of Lambert’s prediction 7.25 10.01 16.06 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 Incident microwaves normal to the surface Top view of cylindrical sample Figure 5.1 A cylindrical sample exposed to plane waves normal to the surface with sub-shell sections for numerical modeling. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 25 4 cm, Maxwell's 4 cm, Lambert's 20 3.5 cm, Maxwell's © § Q. 15 T3 (D X3 W O CO n as © 10 > as 3.5 cm, Lambert's % 5 0 0 1 2 3 4 5 Radial distance from center (cm) Figure 5.2 Microwave power absorbed in 2 % agar gel cylinders (3.5-cm and 4-cm radius) as a function of radial distance from sample center. The electric field is oriented along the vertical z-axis of the cylinder. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 14 Absorbed microwave density (MW/m3) 4 cm, Maxwell's 12 4 cm, Lambert's 3.5 cm, Maxwell's 10 3.5 cm, Lambert's 8 6 4 2 0 0 1 2 3 4 5 Radial distance from center (cm) Figure 5.3 Microwave power density absorbed in 2% agar gel cylinders (3.5-cm and 4-cm radius) as a function of radial distance from sample center. The electric field is oriented along the vertical z-axis of the cylinder. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 60 MP, 1 min LP, 1 min MP, 3 min - - - LP, 3 min Temperature (°C) 50 40 30 20 <>>' 10 0 0 1 2 3 4 5 Radial distance from center (cm) Figure 5.4 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 4-cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 1 (i.e. continuous). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 111 60 MP, 1 min LP, 1 min MP, 3 min - - - LP, 3 min Temperature (°C) 50 40 30 20 0 2 3 4 Radial distance from center (cm) 1 5 Figure 5.5 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 4-cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L12 60 MP, 1 min Temperature (°C) MP, 3 min ■ - -L P , 1 min M, 1 min " LP, 3 min M, 3 min 40 0 1 2 3 Radial distance from the center (cm) 4 Figure 5.6 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 3.5cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 1 (i.e. continuous power). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 MP, 1 m in ................. LP, 1 min MP, 1 m in - • M, 1 min ■ LP, 3 min SO 40 s s 3 - * - 30 20 .I- 0 1 2 3 4 Radial distance from the center (cm) Figure 5.7 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 3.5cm radius 2 % agar gel cylinders after 1 and 3 min of microwave heating with a pulsing ratio of 3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 5.7 References Atwell, W.H., Peschek, P., Krawjecki, M. and Anderson, G. 1990. U.S. patent 4926020 May 15. Atwell, W.H., Peschek, P., Krawjecki, M. and Anderson, G. 1992. U.S. patent 5101084 May 15. Ayappa, K.G., Davis, H.T., Crapiste, G., Davis, E.A. and Gordon, J., 1991a, Microwave heating an evaluation of power formulations. Chem. Engineering Science. 46 : 1005-1016. Barringer, S.A., Davis E. A., Gordon, J., Ayappa, K. G. and Davis, H. T. 1995 Microwave- heating temperature profiles for thin slabs compared to Maxwell and Lambert Law prediction. J. Food Science. 60(5): 1137-1142. Bender, F.B., Douglass, L.W., and Kramer, A. 1981. Statistical Methods for Food and Agriculture, Food Products Press Inc., Binghamton, New York. Cheng, D. K., 1992. Field and Wave Electromagnetics. Addison-Wesley Publishing Company. Reading, MA. Fu, W. and Metaxas, A. 1992 A mathematical derivation of power penetration depth for thin lossy materials. J. Microwave Power 27(2): 217-222. Ohlsson, T. and Risman, P.O. 1978. Temperature distrubution of microwave heatingspheres and cylinders. J. Microwave Power 13(4): 303-309. Pangrel, B.J. Ayappa, K.G. Davis, H.T. Davis, E.A. and Gordan, J. 1991 Microwave thawing of cylinders J. AIChE 37(12): 1789-1800. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Peschek, P., Atwell, W.H., Krawjecki, M. and Anderson, G. 1991a. U.S. patent 4988841 January 29. Peschek, P., Atwell, W.H., Krawjecki, M. and Anderson, G. 1991b. U.S. patent 5008507 April 16. Peschek, P., Atwell, W.H., Krawjecki, M. and Anderson, G. 1992. U.S. patent 5140121 August 18. Rizvi, S.S.H. and Mittal, G.S. 1992. Experimental Methods in Food Engineering. AVI Book Co., New York, NY. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 CHAPTER 6 EFFECT OF EXPERIMENTAL PARAMETERS ON TEMPERATURE DISTRIBUTION DURING CONTINUOUS AND PULSED MICROWAVE HEATING Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 6.0 Notations a = Effect of processing time (3 = Effect of pulsing ratio Y= Effect of radius (sample size) C = Constant of general linear model a, ^Coefficients of general linear model Pabs = Microwave absorbed power, W PR = Pulsing ratio t = Time, s toff = Duration of the microwave power off per duty cycle, s ton = Duration of the microwave power on per duty cycle, s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 6.1 Abstract The temperature distribution (TD) in two-percent agar gel cylinders heated by the same average microwave power level was evaluated. The same average power level based on the oven settings (OSP) and the absorbed power (AP) was applied to the sample. A 3 x 2 x 2 factorial design was used to evaluate the effect of different experimental variables: (A) heating time (1,2 and 3 min), (B) microwave application (continuous vs. pulsed by employing the same average OSP) and (C) sample size (3.5and 4-cm radius). Analysis of variance under the same average OSP condition indicate that all variables affected sample TD significantly. Overall, the pulsed microwave treatment resulted in a more uniform TD than the continuous treatment under the same average OSP condition. The interactions among variables under the same average OSP condition were also observed; the results showed that pulsed microwave heating should be preferred to the continuous microwave heating when temperature uniformity in the sample is critical. The agar gel cylinders were also heated under the same average AP condition and present different TDs when different microwave pulsing ratios were employed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 6.2 Introduction Microwave energy can enhance interior heating of foods and other materials. Such heating mechanism of electromagnetic radiation depends on the dielectric properties and dimensions of the food material (Zhang and Datta, 1999). The power level applies to foods is also very important (Decareau and Peterson, 1986). Tulasidas et al. (1994) reported the drying of Thompson seedless grapes using pulsed microwave energy at different power levels. They concluded that the quality of dried grapes (raisins) was highly acceptable in all pulsed microwave duty cycles. Shivhare et al. (1991) studied the effects of pulsed microwave and hot air drying characteristics of com. They reported that the use of pulsed microwaves for com drying resulted in reducing the heat loss through exhaust air and actual time that grains was exposed to microwaves. They also indicated that the magnitude of microwave power and the pulsing period affected the drying rate and product quality. Yonsawstdigul and Gunasekaran (1995) investigated pulsed microwave-vacuum drying of cranberries. Comparing continuous and pulsed modes at microwave output setting of 250-W, they concluded that the pulsed microwave energy was more efficient than the continuous mode. Shorter duty cycles provided more favorable drying efficiency in the pulsed mode. Yang and Gunasekaran (2001) proposed a model for predicting the interior temperature distribution (TD) during pulsed microwave heating based on the Lambert’s law. The uneven interior TD during continuous microwave heating was dramatically reduced by pulsed microwave heating. However, the effect of the sample dimension and the microwave power level were not investigated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 The objectives of this part of research were to: • Examine the effects of sample size, pulsing ratio and microwave processing duration on the sample temperature distribution by employing the same average microwave power level based on the oven settings (OSP). • Evaluate the effect of the same average microwave power level based on the absorbed power (AP) on the sample temperature distribution by employing different pulsing ratios. 6.3 Methods and Materials Two-percent agar gel cylinders were prepared (method described in Section 3.2). The sample cylinders were 3.5 x 7 cm and 4 x 7 cm, radius x height (i.e. volume of the later was about 30 % more than the former). The sample temperature distributions (TDs) were measured (method described in Section 3.3). The absorbed power in the microwave oven (Pabs) at two continuous microwave power output settings, 250 and 500 W, was determined (method described in Section 3.4). 6.3.1 Microwave Heating Processes Two microwave heating processes of the same duty cycle of 60s (to„ + toff) under the same average OSP condition (the same average microwave power level based on the oven settings) were studied: 1. Pulsed process at the 500-W oven setting using a pulsing ratio of 2 (PR = 2, defined in Section 4.4.1) with 30-s tmand 30-s toff. 2. Continuous microwave process at the 250-W oven setting (PR = I). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 These two microwave processes were selected to maintain under the same average OSP condition. The total heating time for both the same average OSP processes was 3 min. The sample TDs were measured after 1,2 and 3 min of total heating time (note: the sampie TDs presented in Chapters 4 and 5 were measured based on the total microwave power-on duration). Two microwave modes, continuous and pulsed, each under the same average absorbed power level (AP) o f 225 W were applied to 4-cm radius agar gel cylinders: A. Continuous microwave power (PR=1, 250-W oven setting, 225-W absorbed power) for 4 min. B. Pulsed microwave power (PR=1.47, l63/77s power-on/off, 500-W oven setting, 331-W absorbed power) for 4 min. The sample temperatures were measured (method described in Section 3.3) at the end of the 4-min processing duration. The measured TDs were also compared to the predicted temperatures based on the Maxwell’s model described in Chapter 5. 6.3.2 Statistical Analysis Analysis of variance (ANOVA) was performed to evaluate the effect of experimental variables on sample temperature under the same average OSP condition. The three main effects, heating time (a), pulsing ratio based on the same average OSP condition (|3) and sample radius (y) were treated as fixed factors, i.e. levels of each factor were discrete (non-continuous). The microwave energy was applied to the samples in a non-continuous manner during the pulsed treatment, therefore, the heating time was Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 considered to be a discrete factor. Each measured temperature was treated as a response in a general linear model (GLM) for correlation analysis. The radial distance was treated as a co-variance (continuous factor). The model was also used to evaluate a 3 x 2 x 2 factorial experimental test: (1) heating time (1,2 and 3 min), (2) microwave application (continuous vs. pulsed) and (3) sample size (3.5- and 4-cm radius). The significant analysis included (1) main effects (a, (3 and y), (2) primary interactions (a*(3, a*y and P*y) and (3) the second-order interaction (a*p*y) (Brender et al., 1981; Fienberg, 1981). MINITAB 13.0 (Minitab Inc., State Park, PA) was used for the statistical analysis. 6.4 Results and Discussion 6.4.1 Microwave Absorbed Power The absorbed powers were 225 ± 3.7 W and 331 ± 3.6 W at the oven settings of 250 W and 500 W, respectively, for a 3.5-cm radius sample. 234 ± 2.9 W and 344 ± 3.1 W at the oven settings of 250 W and 500 W, respectively, for a 4-cm radius sample (Section 3.4). This indicated that heating at the 250-W oven setting under the continuous mode (PR =1) should result in a higher sample temperature than heating at the 500-W oven setting with PR= 2. The power delivered (PD) by the microwave oven and the power absorbed (Pabs) by the sample are not the same. The Pabs/PD ratio is often referred as the energy-transfer efficiency and it depends on the sample load volume (Table 3.1) and the sample’s ability to store and dissipate the delivered microwave energy. The energy-transfer efficiency can range from 18 to 100% at different load volumes in a home microwave oven; only water samples of 1000 mL or more satisfies a 100 % efficiency (Mudgett, 1986). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 6.4.2 Temperature Distribution (TD) in Samples Under the same average OSP Condition Figs. 6.1 and 6.2 present the TD in 3.5- and 4.0-cm radius agar gel cylinders after 1-, 2- and 3-min heating by both microwave processes (PR = 1 and 2 under the same average OSP condition). The TD at different pulsing ratios resulted in a small difference at the end of one minute of microwave heating. As the microwave heating duration increased, the continuous microwave heating (PR=l) resulted in higher interior sample temperatures. The largest temperature difference occurred at the sample center when it was heated using PR=1 and 2. For continuous microwave heating (PR=1), temperature at the sample center was higher than at other locations (radial distances of 1,2, 3 and/or 4 cm). The TD obtained under pulsed microwave heating remained considerably flat, i.e. TD was fairly uniform. As the microwave heating time increased, the temperature profiles in the samples heated by continuous microwave energy exhibited increases in temperature variation between sample center and surface. The TD in the 3.5-cm radius sample was more uneven compared to the 4-cm radius sample when heated by continuous microwave power. This can be related to the penetrating and decaying nature of microwave energy (Mudgett, 1986; Cheng, 1992). As the penetration depth in the 2 % agar gel (2.6 cm) and microwave power remain the same, the absorbed power along the radial axis in a 3.5-cm radius agar gel cylinder is greater than in a 4-cm radius agar cylinder (Figs. 5.2 and 5.3). In the 3.5-cm radius sample, there was more focusing power at the radial center. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 6.4.3 General Linear Model Under Figs. 6.1 and 6.2 show that the measured temperatures (responses) are nonlinearly correlated to the radial distance (RD) under the same average OSP condition. Therefore, a quadratic term of radial distance (RD2) was included to fit GLM. The analysis of variance (ANOVA) including all the interaction terms (a*RD2, P*RD2 and y*RD2) was programmed (Appendix C.l) using MINITAB 13.0 (Minitab Inc., State Park, PA). The interaction of a*RD 2 was not significant (P=0.062) and was dropped for a second trial. The second model is: T = C + axRD + a zRD 2 + a^qRD + axf}RD + asyRD + a6(JRD2 + a ^R D 2 The second trial appeared to be a good model because: 1. All sources of variation included in the GLM were significant (P <0.01, tvalues in Appendix C.2). 2. Residuals were generally scattered randomly about zero. There were no special features or patterns, such as non-constant variance (residual plot, i.e. residuals vs. fitted values, Fig. C .l, Appendix C.2). 3. The responses (temperatures) are normally distributed because the points come close to form a straight line in the normal probability plot of residuals (Fig. C.2, Appendix C.2). According to the second trial, all main effects (F-values of a, (3 and y, Table 6.1) on sample temperatures are significant (P < 0.01) and each main effect significantly interacts with the other two main effects because all primary interactions (F-values of a*P, a*y and P*y, Table 6.1) were significant (P < 0.01). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 The interaction plots (Figs. 6.3,6.4 and 6.5) demonstrate the primary interaction effects on the response (temperature). The relative strength of the effects was also compared. The effect of pulsing ratio vs. processing time is demonstrated in Fig. 6.3. The mean temperature (Y-axis) is the average of measured temperatures at all locations under that specific condition. As processing time increased, the temperature for PR=2 is significantly lower than that for PR=1. This is mainly due to the power absorbed under the experimental conditions (as described in Section 6.4.1). As the processing time increased, the heating rate of a 3.5-cm radius sample was greater than that of the 4-cm radius (Fig. 6.4). The effect of pulsing ratio is dependent on the sample radius (Fig. 6.5). For the 3.5-cm radius sample, the sample temperature increased greatly as PR changed from 1 to 2. Even though the total microwave power output was at the same average OSP, for the 4-cm radius sample, the temperature was relatively unaffected by the pulsing ratio. The microwave power is a decaying function penetrating into the sample from the surface. The decaying function is significantly related to the sample radius. 6.4.4 Effects o f Different Pulsing Ratios on Sample Temperature Distribution Under The Same Average Absorbed Condition Fig. 6.6 shows the predicted and measured TD after the agar gel cylinder heated by the two microwave modes under the same average absorbed power condition The measured center temperature of the gel cylinder heated by Mode A and B were 42.7 and 34.2 °C respectively. The sample center temperature heated by continuous microwave (Mode A) was higher than heated by pulsed microwave (Mode B). The temperatures at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the mid-annular section along the sample radius were approximately the same for both Modes. During the power-off period, the temperatures near the surface tended to be lower due to convective cooling. Pulsed application of microwave energy can produce more uniform sample TD than the continuous application under the same average absorbed power (AP) condition. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ill 6.5 Conclusions The pulsed microwave heating resulted in more uniform TD in samples than continuous microwave heating at the same average microwave output power based on the oven settings (OSP). There was a significant interaction between microwave power in the different pulsing ratios under the same average OSP condition and sample radius due to the non-continuous application of microwave radiation and the penetrating nature of microwaves. Even though the same average absorbed microwave power was delivered into the samples of the same dimensions, the pulsed application resulted in different sample temperature distribution as compared to the continuous application. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 6.6 References Brender, F. E., Douglass, L. W. and Kramer, A. 1981. Statistical methods for food and agriculture. Food Products Press, New York, NY. Cheng, D. K., 1992. Field and Wave Electromagnetics. Addison-Wesley Puclishing Co., Reading, MA. Decareau, R. V., and Peterson, R. A., 1985. Microwave Processing and Engineering. Ellis Horwood Ltd., Chichester, Engilshed. Fienberg, S. E. 1981. The analysis of cross-classified categorical data. MIT Press Cambrige, MA. Mudgett, R. E., 1986. Microwave properties and heating characteristics of foods. Food Technol. 40(6): 3-24. Shivhare, U. S., Raghavan, G.S.V., Bosisio R. G., and Mujumdar, A. S. 1991. Drying of com in a pulsed microwave field. Paper No. 91-3009. ASAE International Summer Meeting, Albuquerque, NM. June 23-26. Tulasidas, T. N., Raghavan, G. S. V., Kudra, T., Gariepy, Y. and Akyel, C. 1994. Microwave drying of grapes in a single mode resonant with pulsed power. Paper No. 94-6547. ASAE International Winter Meeting, Atlanta, GA. December 13-16. Yang, H. W. and Gunasekaran, S. 2001. Temperature Profiles in a Cylindrical Model Food During Pulsed Microwave Heating. J. Food Science 66(7): 998-1004. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 Yongsawatdigul, J. and Gunasekaran, S. 1996. Microwave-vacuum drying of cranberries: Part I. Energy use and efficiency. J. Food Processing and Preservation 20(2): 121144. Zhang, H. and Datta, A. K. 1999. Distributions of heating potentials inside spherical shaped foods in electromagnetic fields. Abstract No. 75-1. IFT Book of abstracts IFT annual meeting, Chicago, IL, July 24-28. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 50 P R = 1,1 min P R = 1,2 min P R = 1,3 min PR=2,1 min - - o - - P R s2 ,2 min Temperature (°C) PR =2,3m in 0 1 2 3 4 5 Radial distance from center (cm) Figure 6.1 Temperature distribution in 2 % agar gel cylinders, 4-cm radius, after 1, 2 and 3 min of microwave heating with a same average microwave output power of different pulsing ratios (PR=1 @ 250-W setting and PR=2 @ 500-W setting). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 50 P R = 1,1 min PR=2, 1 min P R = 1,2 min • - o- • PR=2,2 min PR=1,3min - • «* - PR=2,3 min 45 Temperature (°C) 40 35 ^ 30 25 *1 20 15 10 5 0 0 1 2 3 4 Radial distance from the center (cm) Figure 6.2 Temperature distribution in 2 % agar gel cylinders, 3.5-cm radius, after 1,2 and 3 min of microwave heating with a same average microwave output power of different pulsing ratios (PR=1 @ 250-W setting and PR= 2 @ 500-W setting). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 PR=1 PR=2 30 3 k- 2L E 2 «»e S 10 0 1 2 3 4 Processing time (min) Figure 6.3 Mean sample temperature vs. processing time at two pulsing ratios (PR). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 40 R=3.5 cm — R=4.0 cm Mean temperature (°C) 30 20 10 0 1 2 3 4 Processing time (min) Figure 6.4 Mean sample temperature vs. processing time for samples of different radii (R). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 30 Mean temperature (°C) 25 20 15 0 2 1 3 Pulsing ratio Figure 6.5 Mean sample temperature vs. pulsing ratio for different sample radii (R). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 50 Temperature (°C) a M ode A, Predicted - - - M ode B, Predicted M ode A, M easured o Mode B, M easured 40 0 1 2 3 4 5 Radial distance from center (cm) Figure 6.6 Temperature distribution in 2 % agar gel cylinders, 4-cm radius, after 4 min of heating by using an average microwave absorbed power of 225 W under continuous (Mode A) and pulsed (Mode B) microwave applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 Table 6.1 Analysis of Variance for the Factorial Experiment in a General Linear Model. Source of variation Sum of squares Radial distance (RD) RD2 a: Time P: Pulsing ratio y: Sample radius a*p a*y P*y a*3*y a*Radial distance P*Radial distance y*Radial distance P* RD2 y*RD 2 Error Total 184.86 568.36 7380.09 1041.20 865.64 267.53 277.67 75.26 26.12 167.89 89.79 100.21 113.10 96.15 427.39 11681.25 Degrees of freedom 1 1 2 1 1 2 2 1 2 2 1 1 I 1 142 161 Mean square 802.47 606.03 1906.26 824.78 732.27 140.86 99.26 77.11 13.06 83.94 170.15 154.45 113.10 96.15 3.10 F 266.6 201.3 633.3 274.0 243.3 46.80 32.98 25.62 4.34 27.89 56.53 51.31 37.58 31.95 P 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.015 0.000 0.000 0.000 0.000 0.000 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 CHAPTER 7 OPTIMIZATION O F PULSED MICROWAVE HEATING Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 7.0 Abstract A simulation model was used to optimize pulsed microwave heating of precooked mashed potato cylinders of 82.7% moisture content. The experimental variables were: (1) sample radius: 2.4, 2.8 and 3.2 cm (or 1.5,1.75 and 2.0 time of the penetration depth of microwave radiation), (2) microwave power-on temperature constraints (ATon): at 20 and 15 °C, (3) power-off temperature constraints (ATd): at 5 and 3 °C lower than ATon, (4) total processing time (< 1000 s) and (5) average sample temperature (70 °C). The evaluation showed that the samples of 2.4- to 2.8-cm radius were heated uniformly and efficiently. The ATon is very critical for optimum pulsed microwave heating; ATon= 20 °C is a better choice than ATon =15 °C. ATd affects the total processing time for a large sample more significantly than for a small sample. The total processing time depends on both ATon and ATd as well as sample radius. In the case of 2.4-cm radius samples (1.5 times of penetration depth) with ATon(ATd )= 20(3) °C , the pulsed microwave energy can heat the sample to an average temperature of 60 °C in 336 s. This was the most efficient process among all evaluated conditions with respect to total processing time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 7.1 Introduction Temperature uniformity is achieved in foods during pulsed microwave heating due to thermal equalization via conduction from hot to cold region during power-off periods (Yang and Gunasekaran, 2001). However, to optimize the pulsed microwave heating process, certain parameters should be constrained. Penetration of microwave energy inside a material is a function of dielectric properties, which can alter the temperature distribution (TD) within the sample. As the sample size increases, the sample regions away from the surface are not heated satisfactorily due to decaying microwave energy as it propagates into the sample. Poliak and Foin (1960) reported, in a microwave-heated beef cylinder (radius = 6 cm, penetration depth = 2.1 cm), the temperature at the center was lower than at the surface. For a small sample (relative to microwave penetration depth), the focusing effect of microwave energy accumulates as a function of time and cause overheating at the sample center. The pulsed heating is especially suitable for such cases. Therefore, sample dimension and heating time should be optimized to prevent over and under heating during pulsed microwave application. The objective of this study was to evaluate the effect of different levels of processing parameters on temperature uniformity in a microwave heated food. Precooked mashed potato was used as the sample food material. The parameters studied were: • Sample dimension, • Sample temperature distribution, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 • Processing time, and • Power level. 7.2 Methods and Materials 7.2.1 Sample Preparation Mashed potato flakes (Idaho Spuds, The Pillsbury Company, Minneapolis, MN) were purchased from a local supermarket. Mashed potato was prepared according to the manufacturer’s information with some exceptions (no butter and salt added). 1800 mL of distilled water was brought to a rolling boil and 370 g of potato flakes were stirred in with a hand mixer (KitchenAid KHM-7) at the lowest speed (250 rpm), to a uniform consistency. 7.2.2 Physical and Dielectric Properties Sample moisture content was determined in triplicate by an infrared moisture analyzer (U 16 Mettler Toledo Inc, Switzerland) with a set temperature of 160 °C. About 2 g of sample was placed in the moisture analyzer. The sample mass was measured by a built-in digital balance capable of reading to 4 decimal places during the infrared heating. The sample moisture content was automatically calculated by the moisture analyzer. The measurement was terminated, if after 3 consecutive 10 min intervals, the moisture content varied less than ±0.2 %. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 Thermal and dielectric properties of the mashed potato were determined using moisture content dependent models available in the literature. The models and corresponding property values are summarized in Table 7.1. 7.2.3 Microwave Heating The mashed potato samples were transferred into 4-cm radius pyrex glass beakers to a height of 7 cm and covered with waxed paper to prevent moisture loss. They were stored in a refrigerator at 4 °C for at least 16 h to ensure sample temperature uniformity before microwave heating experiments. The sample along with the glass beaker was then placed at the center of turntable in the microwave oven cavity and heated for 1 min at 250-W oven setting. The sample temperature distribution (TD) was measured (as described in Section 3.3) and compared to the model (Maxwell’s as described in Chapter 5) predicted TD (Fig. 7.1). The power correction term due to the evaporative cooling was not considered in the numerical models because the samples were not exposed to air. 7.2.4 Analysis of Optimal Process The measured and predicted TDs agreed, and the temperature generally decreased from the surface to center. This is because the penetration depth (Dp in Equation 7.1) of microwaves is 1.6 cm for the mashed potato sample (82.7 % moisture content, 4 °C). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 where A0 is the incident wavelength (12.24 cm), k” is the dielectric loss factor (20.4), 5 is the loss angle (0.298 rad). The pulsed microwave heating is most effective for samples with hot spots around their centers, i.e. when the center temperature of the sample tends to be higher than that of the surrounding material. In such cases, the equalization of thermal energy due to conduction during microwave power-off periods tends to result in a more uniform sample temperature. Therefore, to study the optimization of pulsed microwave heating using mashed potato as the example food, the sample radius should be less than 4 cm (2.5 Dp) and comparable to Dp. The 3.5- and 4-cm radius agar gel cylinders (used in Chapter 5) are only appropriate to demonstrate the effect of pulsed microwaving for foods with similar Dp (Note: for 2% agar gel, Dp = 2.8 cm). Therefore, sample radii of 1.6, 2.4, 2.8 and 3.2 cm (or 1,1.5, 1.75 and 2 Dp) were considered. All samples were 7-cm high. The corresponding sample volumes and absorbed power are listed in Table 3.1. Thirty seconds of continuous microwave heating was simulated (using the Maxwell’s model as described in Chapter 5). The simulation results are shown in Fig. 7.2. The center temperature in the 1.6-cm radius sample was the highest and is an excellent candidate for illustrating the temperature equalization under pulsed microwave heating, but was considered too small to be practical (volume = 56 cm3). Thus, 2.4-, 2.8- and 3.2-cm radius samples were considered for further analysis using the following criteria: 1. Initial sample temperature is 4 °C; R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 2. Microwave energy, 250-W oven setting, is applied continuously until the maximum temperature difference between any two locations (0.1 cm apart for every two consecutive locations) in the sample (ATon) just exceeds 15 and 20 °C; 3. This is followed by a power-off period until the temperature difference during the power-off period (ATd) is reduced by 3 and 5 °C (i.e. maximum temperature difference at any two locations in the sample is 12 and 10 °C for ATon=15 °C, and 17 and 15 °C for ATo„=20°C); 4. The total processing time should be less than 1000 s; 5. Maximum sample temperature anywhere should not exceed 70 °C, because the predictive dielectric property models available in the literature (Table 7.1) are valid only between 0 and 70 °C; 6. 500-W oven setting can be used alternatively to satisfy some constraints, if necessary. The simulation model based on Maxwell’s equations was modified as follows to evaluate the effect of different parameters: 1. Automatically calculate the sample average, maximum and minimum temperatures. 2. Each power-on period continues until the maximum and minimum sample temperature differential exceeds the set criterion, then followed by a poweroff period. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 3. Each power-off period continues until the interior temperature differential decreases to the set criterion. 4. At the end of each power-off period, the dielectric properties are evaluated according to the average sample temperature. 5. Conditional loops repeat the computations until power-on or power-off temperature differentials or time limit (1000 s) is reached. 7.3 Results and Discussion The results of the optimization analysis using pulsed microwave power at the 250W oven setting are shown in Figs. 7.3, 7.4 and 7.5 for 2.4-, 2.8- and 3.2-cm radius samples, respectively. At the beginning, the heating rate is the greatest because microwave energy is applied continuously. The time of continuous power application (done at the beginning of the simulation) for ATon = 20 °C is longer than ATon = 15 °C and resulted in higher average sample temperatures. When ATon = 20 °C, the total processing time was shorter than for ATon = 15 °C. The power-on (PO) to total processing (TP) time ratios under the ATon =20 °C and ATon =15 °C conditions with ATj=5 °C at the same average sample temperature were calculated (Table 7.2). The PO/TP ratios are higher for ATon = 20 °C than for ATon = 15 °C condition. This is because of longer PO and shorter TP for ATon =20 °C. The penetrating nature of microwaves causes an uneven power distribution in the material. As the heating time progresses, non-uniform TD in the sample becomes more pronounced. It can be resolved by setting proper power-on temperature constraint. In the case of ATon R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 =15 °C, it obviously takes less time to heat the sample from the starting point of the cycle than it does to heat the same sample to ATon =20 °C. However, the energy needed to heat the sample under the ATon =20 °C constraint is greater than that under the AT0ff =15 °C constraint. The temperature gradient within the sample, which is the driving force for sample temperature equalization, is greater under the ATon =20 °C condition. Therefore, under the ATon =20 °C constraint, the on-off cycle is shorter than under the ATon =15 °C constraint. The power-off temperature differences (ATd) were 5 and 3 °C. Generally, the larger ATd, the longer the total processing time needed to achieve the same average sample temperature. For the case of ATon = 20 °C and sample radius= 2.4 cm (1.5 Dp), there is only small difference between ATd= 5 and 3 °C constraints with respect to the time-dependent average sample temperature curves (Fig. 7.3). As the sample radius increased, the difference became larger (Figs. 7.4 and 7.5), and was dependent on the ATon constraint. For sample radius of 2.4 cm (1.5 Dp) with ATon(ATd) = 20(3) °C constraints, the final sample average temperature (Tfsa) can reach around 60 °C in 336 s. For sample radii of 2.8 and 3.2 cm (1.75 and 2 Dp), all other parameters remaining the same, Tfsa = 60 °C was achieved at 449 s and 997 s respectively. Fig. 7.6 shows the sample TD after the microwave heating under ATon(ATd) =20(3) °C constraints. Under the ATon(ATd) = 15(3) °C constraints, it takes too long to heat the sample to the average temperature of 60 °C (443 s and 586 s for 1.5-DP and 1.75-DPradius samples, respectively). Under the ATon(ATd) =15(3) °C constraints, the average temperature of a 3.2-cm (2-Dp) radius R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 sample did not reach 60 °C within the preset 1000 s maximum processing time constraint. All simulation results showed that the sample can be heated to an average temperature of about 30 °C. However, the larger the sample radius, the longer the difference in total processing time between the ATon (ATd) = 15(5) and 20(5) °C constraints. The microwave pulsing sequences are consistent for 2.4- and 2.8-cm (1.5- and 1.75-DP) radius samples, the sequences that can be applied to the samples to obtain a final average temperature of 60 °C and satisfy AT0„ (ATd) = 20(3) °C criteria are listed in Table 7.3. For the 3.2-cm (2Dp) radius sample, longer power-off periods are needed due to the fact that temperature is high at the surface than at the center (Fig. 7.6). Therefore, pulsed microwave application is not as beneficial for the 3.2-cm radius samples as it is for the 2.4- and 2.8-cm radius samples. Accordingly, it may be concluded that pulsed microwave application is most effective when the critical sample size (radius) is less than 2 Dp. The use of 500-W oven setting was also simulated for the 3.2-cm radius sample and the result was compared to that heated at the 250-W oven setting under ATon(ATd) = 20(3) criterion (Fig. 7.7). The sample average temperature of 60 °C was reached in a shorter total processing time. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 7.4 Conclusions The pulsed microwave heating is most effective when the sample radius is < 2 Dp. Maximum microwave power-on and -o ff temperature constraints are very critical for the optimal application of pulsed microwave heating. Power-on temperature constraint produces suitable temperature gradient. Power-off temperature constraint allows the temperature equalization to occur. The power-off temperature constraint affects the total processing time as the sample radius increases. The most efficient process among all the cases is the heating of 2.4-cm (1.5-DP) radius precooked mashed potato sample under the ATon(ATd)=20(3) constraint. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.5 References Poliak, G.A. and Foin, L.C. 1960. Comparative Heating Efficiency of a Microwave and Convection Electric Oven. Food Technology 14: 454-457. Mohsenin, N.N. 1980. Thermal Properties o f Foods and Agricultural Materials. Gordan and Breach Science Publishers, Inc. New York, New York. Calay, R.K., Newborough, M., Probert, D. and Calay, P.. S. 1995. Predictive Equations for the Dielectric Properties of Foods. International Journal of Food Science and technology 29: 699-713. Tinga, W.R. and Nelson, S.O. 1973. Dielectric Properties of Materials for Microwave Processing-Tabulated. Journal of Microwave Power. 8(1): 24-65. Yang, H. W. and Gunasekaran, S. 2001. Temperature Profiles in a Cylindrical Model Food During Pulsed Microwave Heating. J. Food Science 66(7): 998-1004. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 Table 7.1 Thermal and dielectric properties of mashed potatoes with 82.7% moisture content (M, %) and different temperatures (T, °C) Parameter Thermal conductivity, W/m2.°C Specific heat capacity, J/g.°C Dielectric constant Model 0.00493M+0.148a Temperature (°C) 33.3M+833.3b Value 0.55 3577 2.14-0.104T+0.808MC 4 10 20 30 40 50 60 70 68.4 67.8 66.7 65.7 64.6 63.6 62.6 61.5 4 10 20 30 40 50 60 70 20.4 20.0 19.4 18.7 18.1 17.5 16.8 16.2 Dielectric loss factor a from Sweat (1974) b from Mohsenin (1980) c from Calay et al. (1995) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 Table 7.2 Power-on (PO) to total processing (TP) time ratios under the AToa (ATd) =20(3) °C and ATon (ATd) =15(3) °C constraints at different final average sample temperature (Tfas, °C). Radius (cm) PO/PT at ATon : Tfas =20: 60 2.4 0.88 2.8 0.66 3.2 0.43 PO/PT at ATon : Tfas =15: 30 0.67 0.47 0.22 Table 7 3 Microwave pulsing sequences applied to different radius samples and comply the ATon (ATd)=20(3) °C criteria. Radius (cm) 2.4 2.8 (Time interval): Pulsing ratio: Number of duty cycles (0-92): 1: 1 (0-108): 1: 1 (92-336): 1.17 : 7 (108-449): 2.2:10 Reproduced^with perm ission of the copyright owner. Further reproduction prohibited without permission. 151 25 Predicted Measured Temperature (°C) 20 15 10 5 0 0 1 2 3 Radial distance from center (cm) 4 Figure 7.1 Measured and predicted temperature profiles in a 4-cm radius mashed potato cylinder after 1 min of continuous microwave heating at the 250W oven setting. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 45 40 R=1.6 cm R=2.4 cm ■ R=2.8 cm ■ “ *R=3.2 cm O R=2.4 cm M easured 35 0 30 O ^ 1 25 * 2Q» 20 E £ 15 10 5 0 0 1 2 3 4 Radial distance from center (cm) Figure 7.2 Predicted temperature profiles in 1.6,2.4,2.8 and 3.2-cm radius mashed potato cylinders after 30 s of microwave heating at the 250-W oven setting. Measured data for 2.4-cm radius sample is also shown. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 Average sample temperature (°C) 70 60 50 40 30 20(3) 20 20(5) 15(3) 10 15(5) 0 0 200 400 600 800 1000 Time (s) Figure 7.3 Average sample temperature profiles in 2.4-cm radius, 7 cm-long, mashed potato cylinders heated by pulsed microwave at the 250-W oven setting. Power-on constraints (AT,,,) were 20 and 15 °C and power-off temperature difference constraints (ATd) were 5 and 3 °C (i.e. AT„n(ATd)=15(5), 15(3), 20(5) and 20(3)). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 Average sample temperature (°C) 70 60 50 40 30 -20(3) 20(5) •15(3) •15(5) 20 10 0 0 200 400 600 800 1000 Time (s) Figure 7.4 Average sample temperature profiles in 2.8-cm radius, 7-cm long, mashed potato cylinders heated by pulsed microwave at the 250-W oven setting. Power-on constraints (ATon) were 20 and 15 °C and power-off temperature difference constraints (ATd) were 5 and 3 °C (i.e. ATon(ATd)= 15(5), 15(3), 20(5) and 20(3)). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 155 Average sample temperature (°C) 70 -i -20(3) 20(5) ■15(3) 15(5) 60 50 40 30 10 0 200 600 400 800 1000 Tim e (s) Figure 7.5 Average sample temperature profiles in 3.2-cm radius, 7-cm long, mashed potato cylinders heated by pulsed microwave at the 250-W oven setting. Power-on constraints (ATon) were 20 and 15 °C and power-off temperature difference constraints (ATd) were 5 and 3 °C (i.e. ATon(ATd)=15(5), 15(3), 20(5) and 20(3)). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 80 70 Temperature (°C) 60 50 40 30 -2.4 cm 2.8 cm *3.2 cm 20 2.4 cm Measured 10 0 0 0.8 1.6 2.4 3.2 Radial distance from center (cm) Figure 7.6 Temperature distribution in 2.4-, 2.8- and 3.2-cm radius potato cylinders under the ATon(ATd)=20(3) criteria after average sample temperature is about 60 °C at the 250-W oven setting. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 70 Average sample temperature (°C) 250 500 40 0 200 400 600 800 1000 Time (s) Figure 7.7 Time and average sample temperature of a 3.2-cm mashed potato cylinder with 20 °C power-on constraints and a power-off differential of 3 °C lower than the power-on constraint (i.e. ATon(ATd)=20(3)) at the 250and 500-W oven settings. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 CHAPTER 8 SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK 8.1 Summary Temperature distribution in 2 % agar gel cylinders was studied experimentally and numerically. The following conclusions can be drawn are: • Pulsed microwave heating results in more uniform temperature distribution than continuous microwave heating. • Pulsed microwave heating can be employed to minimize the overheating of the sample center. • Simulation of temperature distribution using absorbed power based on Maxwell’s equations is more accurate than that based on the Lambert’s law. • At the same average output power, the effect of sample size, pulsing ratio, and power level are statistically significant. Regarding the optimization of pulsed microwave application to the re-heating of mashed potato cylinders, the following conclusions can be drawn: • Sample radius, power-on and power-off temperature constraints, power level and processing time are important parameters to be considered. • The optimal process depends on the sample radius related to microwave penetration depth for the sample being heated. • The optimal process is also affected by power-on and power-off temperature constraints and power level. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 8.2 Recommendations for Future Work The following suggestions are for future research regarding the pulsed microwave heating: • Numerical simulation of temperature distribution including phase change in frozen food. • Mass and heat transfer simulation for long duration considering moisture and temperature simultaneously. • Wider scope of optimization applications to different foods with processing parameter constrained. • Numerical simulation of temperature distribution using different geometry such as slabs and spheres. • Optimization of additional samples of varying dielectric properties of different geometry. • ‘Runaway’ heating in frozen foods. R eproduced with perm ission of t h e W i g h t owner. Further reproduction prohibited without permission APPENDICES R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 Appendix A Engineering Equation Solver (EES, F-Chart Software Co., Middleton, WI) program code for calculating temperature profiles in continuous and pulsed microwave heated 2% agar gel cylinders R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 "General information" »*» Y=0.07 "sample height (m)" R=0.035 "sample Radius (m)" dT_av=32.73 K=0.6 "thermal conductivity" h=42 "heat transfer coefficient" T_air=25 " air temperature C" t=180" total heating time seconds" V=Y*RA2*pi" volume o f the sample mA3" rho=1070 "density of the sample kg/mA3" C_p=4200 "heat capacity of the sample J/kg.C" Kappa=75 "dielctric constant" Kappa_2=13 "loss factor" tangent_theta=Kappa_2/kappa lambda_0=3E+8/2.45E+9 " wavelength in free space" " is * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * " R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 "Calculation of the attenuation constant for microwave" »»*» n * tt '»** **************afe* ** * * ** *** ** ** *********** *** * ** ** * ** ** •'» alpha=2*(Pi/lambda_0)*sqrt(0.5*kappa*(sqrt( l+(tangent_theta)A2)~ 1)) " l/m" "***** ************** ******* ******* lit****** "Calculation of the total power and P_0" 11*11 < t* n » » * * * * * * * * * * * * * * * jfc* * jft * * * * * * * * * * * * * * * * * jf: j f c j f e * * * * * * * * * * * * * * * * " P_total=(rho*V*c_p*dT_av)/t P_0=(alpha*P_total)/(pi*Y*(l-exp(-2*alpha*R))) A=(pi*Y*P_0)/alpha ••* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * " "Calculation of the Radius of each nodal point R[i]" n*»r tt* ti "*********************************************************" R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 164 dR=0.001 duplicate i=0,35 R[i]=dR*i end ' I * * * * * * * * * * * * * * 4c 4c * * * 4c * * * * * * * * * * * % 4c 4c * 4c * % £ * * * * * * * * * * * * * * * * * * '> "Calculation of distance of the boundary of each subshell form the surface except the most outer shell" t t a|e t t it* " »**********************************************************11 duplicate i=0,34 X[i]=R-(R[i]+dR/2) end »***********************************************************” Duplicate i=0,35 U[i]=0 end {"Determine the power absorbed term (U[i]) for each each subshell (i=l to 34) except the center and the boundary subshell" R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 165 ''**************************************************************•* Duplicate i=l,34 U[i]=A*(exp(-2*alpha*X[i])-exp(-2*alpha*X[i-l])) end "♦Sts************************************************************" "Determine the power absorbed term for the center subshell (U[0]) and the boundary subshell (U[35])" ««*«« «*•* U[0]=A*(exp(-2*alpha*X[0])-exp(-2*alpha*R)) U[35]=A*(l-exp(-2*alpha*X[34])) "Determine the Mass of each subshell (i=l to 34) except the central and boundary subshell" "*» R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 166 »*" I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * duplicate i=l,34 M[i]=pi*((R[i]+dR/2)A2-(R[i]-dR/2)A2)*Y*rho end •>** ** ********$* ****%%****** **** ***** * *3ft************Jit**** ***#* * * » "Determine the Mass of the central (M[0]) and boundary (M[35]) subshell" »»* t i "*" "**************************************************************" M[0]=pi*(dR/2)A2*Y*rho M[35]=pi*(RA2-(R-dR/2)A2)*Y*rho "**************************************************************" Tau_d=l "time increment in second" Tau=index-1 "Total heating time" "Energy balcnce equation for each nodal point except the central and boundary point" 11*1I tt* II R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 'I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * .. Duplicate i=l,34 (C_p*M[i]*(T_n[i]-T[i]))/Tau_d=2*pi*Y*K*0.5*((T[i-l]-T[i]+T_n[i-l]T_n[i])*(R[i]-dR/2)+(T[i+l]-T[i]+T_n[i+l]T_n[i])*(R[i]+dR/2))/dR+U[i]+h*pi*(T_air-T[i])*((R[i]+dR/2)A2-(R[i]-dR/2)A2) end •I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * " "Energy balance equation for the central and boundary point" 11*11 (C_p*M[0]*(T_n[0]-T[0]))/Tau_d=K*pi*Y*0.5*(T[l]-T[0]+T_n[l]T_n[0])+U[0]+h*pi*(T_air-T[0])*(dR/2)A2 (C_p*M[35]*(T_n[35]-T[35]))nau_d=2*K*pi*Y*0.5*((T[34]-T[35]+T_n[34]T_n[35])*(R[35]-dR/2))/dR+U[35]+h*(T_airT[35])*((2*pi*Y*R[35])+pi*((R[35])A2-(R[35]-dR/2)A2)) •I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * " "EES's standard programming for unsteady state energy balance equation involving time step" R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 168 " I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * '' Duplicate i=0,35 T[i]=TableValue(index-l, #T_n[i]) end R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 169 APPENDIX B.1 MATLAB (MathWorks Inc., Natick, MA) program code for the calculating of temperature profile within 4*cm radius agar gel cylinders according to the Maxwell’s equations R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. t=inputCEnter a microwave processing time:'); PR=inputCEnter a pulsing ratios (PR):'); Tml=zeros(l,41); Rm=zeros(l,41); MPm=zeros(l,41); am=zeros(l,40); bm=zeros(l,41); cm=zeros(l,40); dm=zeros(l,41); Xm=zeros(l,41); C_lm=zeros(l,41); C_2m=zeros(l,41); C_3m=zeros(l,41); KKl=zeros(41,41); KK2=zeros(41,l); TXl=zeros(41,l); R=0.04; c=3.00e+8; f=2.45e+9; omega=2*pi*f; epsilon=le-9/(36*pi); mu=4*pi*le-7; R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. kappa_const=75; kappa_loss=13; KKKl=(RA2*omegaA2*mu*epsilon*kappa_const); KKK2=(RA2*omegaA2*mu*epsilon*kappaJoss); alpha_0=omega/c; xx=R*alpha_0; xx 1=(besse Ij(0,xx) )A2+(bessel y(0,xx))A2; kkk3=R*alpha_0*(besselj( 1,xx)*besselj(0,xx)+bessely( 1,xx)*bessely(0,xx))/xx 1; kkk4=2/(pi*xxl); kkk5=(-4/pi)*bessely(0,xx)/xx 1; kkk6=(-4/pi)*besselj(0,xx)/xx 1; bb=sqrt(KKKl+i*KKK2); deno=-bb*besselj(l,bb)+(kkk3-i*kkk4)*besselj(0,bb); KK=(kkk5+i *kkk6)/deno; dR=0.001; Y=0.07; k=0.6; h=42; T_air=22.5; tau_d=l; C_p=4200; R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 172 rho=1070; V=Y*RA2*pi; dT_av=27.2; t_p=180; P_total=rho*V*C_p*dT_av/t_p; P_00=P_total/(2*pi *R* Y); E_0=sqrt(P_00*pi*alpha_0*R/(c*epsilon)); for n=l:41 Tml(n)=4; Rm(n)=0.001*(n-1); Xm(n)=R-(Rm(n)+dR/2); if n = l C_lm(n)=(pi*(dR/2)A2*Y*rho*C_p)/tau_d; C_2m(n)=0.5*pi*Y*k; cm(n)=-C_2m(n); dm(n)=C_lm(n)+C_2m(n); RR(n)=(n-l)*dR; RB(n)=RR(n)+dR/2; R V(n)=RB (n) A2*pi *Y; elseif n<41 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. C_lm(n)=(pi*((Rm(n)+dR/2)A2-(Rm(n)-dR/2)A2)*Y*rho*C_p)/tau_d; C_2m(n)=pi*Y*k/dR; am(n-1)=(-C_2m(n))*(Rm(n)-dR/2); cm(n)=(-C_2m(n))*(Rm(n)+dR/2); dm(n)=C_lm(n)+C_2m(n)*2*Rm(n); RR(n)=(n-l)*dR; RB(n)=RR(n)+dR/2; RV(n)=(RB(n)A2-RB(n-1)A2)*pi *Y; else C_ I m(n)=(pi *(RA2-(R-dR/2)A2)* Y*rho*C_p)/tau_d; C_2m(n)=pi*Y*k/dR; am(n-1)=(-C_2m(n)):,t(Rm(n)-dR/2); dm(n)=C_ 1m(n)+C_2m(n):(!(Rm(n)-dR/2); RR(n)=(n-1)*dR-dR/4; RB(n)=R; R V(n)=(RB(n)A2-RB(n-1)A2)*pi *Y; end RRD(n)=RB(n)/R; Zr=KK*besselj(0,bb*RRD(n)); MPm(n)=RV(n)*(omega*epsiIon*kappa_loss*E_0A2)*abs(Zr).A2; end R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 for n=l:41 KKl(n,n)=dm(n); if n<41 KK1(n+1,n)=am(n); KKl(n,n+l)=cm(n); end KK1; for tau=l:t*PR tau; index l=rem(tau,60); index2=60/PR; if P R = 1 Pm=MPm; elseif index2~=0 & (indexl>index2 | in d ex l= 0 ) Pm=zeros(l,41); else Pm=MPm; R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 end for n=l:41 if n = l bm(n)=C_ 1m(n)*Tm 1(n)+C_2m(n)*(Tm 1(n+ 1)-Tm 1(n))+Pm(n); elseif n<41 bm(n)=C_lm(n)*T m 1(n)+C_2m(n)*((Tm 1(n-1)-Tm 1(n))*(Rm(n)-dR/2)+(Tm 1(n+1)Tml(n))*(Rm(n)+dR/2))+Pm(n); else bm(n)=C_ 1m(n)*Tm 1(n)+C_2m(n)*(Rm(n)-dR/2)*(Tm 1(n-1)Tml(n))+Pm(n)+2*pi*R*Y*(T_air-Tml(n)); end end bm; KK2=bm.’; TX=KK1\KK2; TX=TX.’; Tml=TX; end Tml plot(Rm,Tml) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 APPENDIX B.2 PREDICTED AND MEASURED TEMPERATURE DISTRIBUTION OF 3.5- AND 4-cm RADIUS 2% AGAR GEL CYLINDERS AT 250-W OVEN SETTING • PR2=2, after 1, 2 and 3 min o f pulsed microwave heating. • PR= 1, after 2 min of continuous microwave heating. • PR=3, after 3 min of pulsed microwave heating. 2 Pulsing ratio R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 60 MP. 1 min M, 1 min LP. 2 min 'MP. 3 min M, 3 min 50 Mff. 2 min M, 2 min LP, 3 min £ 40 bo> 5 30 d> a | 20 10 0 0 1 2 3 4 Radi al distance from center (cm) Figure B.l Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 3.5* cm radius 2 % agar gel cylinders after 1,2 and 3 min of microwave heating with PR=2. R eproducedw ith perm ission of the copyright owner. Further reproduction prohibited without permission. 178 60 Temperature (°C) 50 MP, 1min LP, 1 min M, 1 min MP, 2 min - - - LP, 2 min M, 2 min 'MP, 3 min LP, 3 min M, 3 min 40 30 20 10 0 0 1 2 3 4 5 Radial distance from center (cm) Figure B.2 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 4-cm radius 2 % agar gel cylinders after 1,2 and 3 min of microwave heating with PR=2. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 60 50 Temperature (°C) KP 40 20 0 1 2 3 4 Radial distance from center (cm) Figure B.3 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 3.5cm radius 2 % agar gel cylinders after 2 min of microwave heating with PR=1. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 60 50 Temperature (°C) MP 40 0 2 3 4 5 Radial distan ce from center (cm) Figure B.4 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 4-cm radius 2 % agar gel cylinders after 2 min of microwave heating with PR=1. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 60 50 MP LP 10 0 0 2 3 Radial distance from center (cm) 1 4 Figure B.5 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 3.5cm radius 2 % agar gel cylinders after 2 min of microwave heating with PR=3. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 182 60 Temperature (°C) 50 MP -LP 40 30 20 10 0 0 1 2 4 3 Radial distance from center (cm) 5 Figure B.6 Predicted temperature profiles based on Maxwell’s (MP) and Lambert’s (LP) models compared to the measured (M) temperature profile in 4-cm radius 2 % agar gel cylinders after 2 min of microwave heating with PR=3. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 183 APPENDIX C.1 General linear model with time and covariate interaction terms using MiniTab (Minitab Inc., State Park, PA) analysis. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission General Linear Model: Temperature versus Time, Pulse and Radius Factor Time Pulse Radius Type Levels Values fixed 3 1 2 3 fixed 2 1 2 fixed 2 3 .5 4.0 Analysis of Variance for Temperature, using Adjusted SS for Tests Source Radial D RDA2 Time Pulse Radius Time*Pulse Time*Radius Pulse*Radius Time*Pulse*Radius Time*Radial D Pulse*Radial D Radius*Radial D Time*RDA2 Pulse*RDA2 Radius*RDA2 Error Total DF 1 1 2 1 1 2 2 1 2 2 1 1 2 1 1 140 161 Seq SS 184.86 568.36 7380.09 1041.20 865.64 267.53 277.67 75.26 26.12 167.89 89.79 100.21 16.61 113.10 96.15 410.78 11681.25 Adj SS 802.47 606.03 2931.40 824.78 732.27 281.73 213.13 77.11 26.12 61.84 170.15 154.45 16.61 113.10 96.15 410.78 Adj MS 802.47 606.03 1465.70 824.78 732 .27 140.86 106.57 77.11 13.06 30.92 170.15 154.45 8.31 113.10 96.15 2.93 F 273.50 206.55 499.54 281.10 249.57 48.01 36.32 26.28 4.45 10.54 57.99 52.64 2.83 38.55 32.77 P 0.000 0. 000 0.000 0.000 0.000 0.000 0.000 0.000 0.013 0.000 0.000 0.000 0.062 0.000 0.000 185 4-1 <u 0 CJ o o o o o o o o o • • • o o o o o O o C o o • • o o o o o • o i n Ch Nl on C" r - 00 I—1 ID ID r•^i ID ID r~ rji m m o o CN ro r-l • w O • • • O o o t> o o o o • o Hm « in o *• • o o <7l id 10 C— 00 m m m • o o o • o r-l • -d' r-l r-l r~ 00 CN CN t-H r-l • o o o • o • • o O O O o o 24 44 i n i n 00 <U CM i n ■<d< 0 m t—i a ID "d< ID • • • ID 1 at e -H * £ J-) p p jj i—i (0 (0 CO -rl CN •r-l G T3 < 'O (0 <UO co q a at 2 ID ID VO in ** x—1 • • CN O 1 in • CN 1 00 m rH 00 r-l G <a id CT\ e' e n aV at a) CO i—1 G *—1 r~ CN ■d1 in CN tH oo r o 00 Ol CN O • CN 1 o • CN in in ID • o o 1 o CO G •rl T3 m (0 e •H i—I (0 •H TJ r-l CO -r-l Tt (0 at (0 at (0 • rH CN f t H a t cn * ■K <u P P 0) CO rH G E-C r-l CN f t H G -H 'O in (0 * ■it CN < CN < CN at 2 q 2 P q • at m * < R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 186 APPENDIX C.2 General linear model without time and covariate interaction terms involved using MiniTab (Minitab Inc., State Park, PA) analysis (including the Probability Plot of the Residuals and Residuals versus the Fitted Values) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. °Q. C o 1 CD Q. with permission of the copyright owner. Further reproduction General Linear Model: Temperature versus Time, Pulse and Radius Factor Type Time Pulse Radius Levels fixed fixed fixed Values 31 2 3 212 2 3.5 4.0 Analysis of Variance for Temperature, using Adjusted SS for Tests prohibited without p erm ission. Source Radial D RDA2 Time Pulse Radius Time*Pulse Time*Radius Pulse*Radius Time*Pulse*Radius Time*Radial D Pulse*Radial D Radius*Radial D Pulse*RDA2 Radius*RDA2 Error Total DF 1 1 2 1 1 2 2 1 2 2 1 1 1 1 142 161 Seq SS 184.86 568.36 7380.09 1041.20 865.64 267.53 277.67 75.26 26.12 167.89 89.79 100.21 113.10 96.15 427.39 11681.25 Adj SS 802.47 606.03 3812.53 824.78 732.27 281.73 198.51 77.11 26.12 167.89 170.15 154.45 113.10 96.15 427 .39 Adj MS 802.47 606.03 1906.26 824.78 732.27 140.86 99.26 77.11 13.06 83.94 170.15 154.45 113.10 96.15 3.01 F 266.62 201.35 633.35 274.03 243 .30 46.80 32.98 25.62 4.34 27.89 56.53 51.31 37.58 31.95 P 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.015 0.000 0. 000 0.000 0.000 0.000 188 o o o o o o o o O H o o o o o o o o o o o o o o o o o o o o co cn o o ..4 fej 4H O cn CTl co ro cn cn rH rH • • • o o o rH rH • . o o CO cn CN cn co O o 4-1 in in 00 0 CN in 0 co rH U vo• • vo• VO i—i i rH in co oo rH cn rH • rH O 1 vo VO VO in• CN 1 0 C" CN in 0 r - cn u CN co 24 w CO 0 3 0 n • in • 00 00 in rH rH • o rH O CN in CN in in vo. o o in •'tf rH 00 • CN 1 o o 0 3 •rH rH T3 in 0 • E-* *H CN Pi rH Pi co 0 * * * 0 E •H •U p c 0 rH ■U (0 Vi -H CN c T3 < 0 0 P u Pi rH 3 0 3 •rl p P P iH rH 0 ■H T3 i—I rH Pi rH Pi CO ■K ■n CN CN 0 Pi 0 -H 73 0 Pi 0 •H 73 0 Pi 3 73 in 0 • < P < 3 q 3 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (Response is temperature) Fig. C. 1 Residuals versus the fitted values 189 •V*. •4 •t m R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Fig. C.2 Normal probability plot of residuals 190 83 2u o. <u s3 TJ a C/3 ® a: ocn C o a. ucn OS CO CM o CM d j o o s leuiiOM R eproduced with perm ission of the copyright o w n e r Further reproduction prohibited without permission 191 APPENDIX D Determination of the experimental error due to 30-s lag during temperature measurement R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. The experimental errors due to the 30-s delay during the sample temperature measurement were determined. The 3 min of continuous microwave power at 250-W oven setting were applied to 2% agar gel cylinders (3.5-and 4-cm radius). Temperatures at center and every centimeter from the center were measured in triplicates and recorded at 5, 10, 15, 20, 25 and 30 s after the microwave heating (instrument described in Section 3.3). The collected temperatures are shown in Figures D -l and D-2. The effect of 30-s delay (time effect) on temperature measurement is not significant (P > 0.01, Table D). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 193 40 35 O "jr 3 0 - 3 © E 25 © I— 20 - P, 0 c m M, 0 cm M, 1 c m A M, 2 cm M, 3 cm M,4cm — 0 5 10 15 20 25 30 35 Time (s) Figure D-l Temperature variation at 0 ,1 ,2 ,3 and 4 cm from center in a 4-cm radius 2 % agar gel cylinder (P is predicted data and M is measured data), during 30 s after 3-min microwave heating. R eproduced with perm ission of the copyright o w n e r Further reproduction prohibited without perm ission 194 Temperature (°C) 45 40 35 30 25 a P , 0 cm M, 0 cm M, 2 cm M, 3 cm 0 Time (s) Figure D-2 Temperature variation at 0,1,2 and 3 cm from center in a 3.5-cm radius 2 % agar gel cylinder (P is predicted data and M is measured data), during 30 s after 3-min microwave heating. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Table D Effect of experimental variables on 30-s delay of temperature measurements- statistical analysis Source of variation Radial distance (cm) Sample radius (cm) Time (s) Replication Error Total Degrees of Freedom 4 1 5 2 149 161 F ratio 112.83 271.43 0.75 1.81 P value <0.01 <0.01 0.584 0.167 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. APPENDIX E The evaporation cooling effect during microwave heating R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 197 The agar gel cylinders were heated as described in Chapters 4 and 5. The sample mass before and after 3 min of total microwave power input was measured (XT-3000 DR, Fisher Scientific, Pittsburgh, PA) in triplicates. The decrease in mass (Aw) during the process indicated the loss of moisture content in the sample. The latent heat (h/g) of water is 2,438 J/g at 27 °C and at atmospheric pressure. The power correction term (Pc) due to evaporative cooling was calculated and applied to the five outermost incremental shells near the sample radial surface. Aw * h . ________ SJj_ p c TMA* 5 The Aw and Pc for different microwave applications were listed in Table E below. Table E. Loss of moisture (Aw) and the corresponding power correction (Pc) for 2 % agar gel cylinders during microwave heating PR 1 2 3 1 2 3 Sample radius (cm) 3.5 3.5 3.5 4.0 4.0 4.0 Aw (g) 0.75 1.03 1.44 0.86 1.27 1.54 Energy loss (J) 1833 2511 3506 2097 3088 3755 Pc(W) 1.45 1.00 0.94 1.45 1.05 0.85 R eproduced with perm ission of th e copyright owner. Further reproduction prohibited without permission.

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