SIMULATION OF TEMPERATURE AND QUALITY PROFILES IN FROZEN FOODS SUBJECT TO STEP CHANGES IN STORAGE CONDITIONS ' Dissertation for the Degreeof :Ph. D. MICHIGAN STATE UNIVERSITY ELAINE PATRICIA soon ' ' 1987 “fffi'IB M" mum 1 IAYF Hwy! w "v :4wr-v-“F i I I 3 1293 00463 0038 p I R8\i ( mam STATE “WC MIC“ ”WARN 5 LIBRARY Michigan State University Mm MICHIGAN STATE UNIVERSITY LIBRAR ~ EV1531_J RETURNING MATERIALS: Piace in book drop to LIBRARIES remove this checkout from —:-—. your record. FINES will V ,3. be charged if book is THIS 390“ W‘ 0‘8“”de returned after the date FEB ‘l 1 1990 Stamped beIow. /@ .1 {+8—3- 0.1994191 SIMULATION OF TEMPERATURE AND QUALITY PROFILES IN FROZEN FOODS SUBJECT TO STEP CHANGES IN STORAGE CONDITIONS By Elaine Patricia Scott A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Agricultural Engineering Department of Agricultural Engineering 1987 ABSTRACT SIMULATION OF TEMPERATURE AND QUALITY PROFILES IN FROZEN FOODS SUBJECT TO STEP CHANGES IN STORAGE CONDITIONS by Elaine Patricia Scott Frozen food products may be exposed to fluctuating ambient condi- tions during storage. An increase in storage temperature may result in an increase in the overall quality deterioration rate and/or a substan- tial quality differential within the food product. The overall objectives of this research were to develop a mathematical multi- dimensional model to simulate transient temperature dependent quality deterioration within a frozen food product subject to step changes in storage conditions, and to estimate surface heat transfer coefficients prevailing during step changes in storage conditions. The temperature distribution history of the product, used in simulating the quality deterioration rate, was found numerically using finite differences. The transient surface heat transfer coefficients were estimated using experimentally determined temperature measurements in the sequential regularization method, developed for a class of problems called inverse heat conduction problems. A highly concentrated methyl-cellulose substance was used as an analog food substance in the experimental procedures. A systematic procedure was developed to select the optimal numerical parameters used in the finite difference and the Elaine Patricia Scott sequential regularization methods. The quality simulation model was used to determine the effects of various parameters on the quality deterioration rate. Parameters investigated included the magnitudes of the kinetic parameters and the surface heat transfer coefficient, the storage time and temperature, the magnitude of a step change in storage temperature, the food product dimensions, and the product geometry. The sequential regularization procedure was found to provide es- timates of the surface heat transfer coefficients which included the effects of the exterior packaging boundary and the accumulation and diminution of frost on the outer surface. Internal packaging boundaries ‘were found to have a significant influence on the temperature differen- tial within the food product. The magnitude of change in the quality deterioration rate was highly dependent on the magnitude of the kinetic parameters, and strongly influenced by product dimensions and the choice of a one or two dimensional heat transfer model. Approved K W FED 2i I937 Major Professor Date Approved WM/ #44377 Department Chairperson Date To Mary Scott, Thanks for your support, Elaine ACKNOWLEDGMENTS I would like to give special thanks to Dr. Dennis R. Heldman, who continued to serve as my advisor after leaving Michigan State University. I greatly appreciated his guidance, support, and patience in helping me to complete this research. I would also like to thank the members of my Committee, Dr. J. V. Beck, Dr. L. J. Segerlind, Dr. J. F. Steffe, and Dr. M. A. Ubersax. Special thanks go to Dr. Beck, for his input on the Inverse Heat Conduction Problem, to Dr. Segerlind for his thoughts on determining the numerical parameters and for his humorous remarks, and especially to Dr. Steffe who handled the majority of the paper work in the absence of Dr. Heldman. In addition, I want to thank my family, Mary, Mary Ann, and Charles, my long time friends, Cindy and Chris, and my new found Michigan friends, Sharon, Janice and Allyson, who helped me in more ways than they will ever know. Last, but not least, special thanks to Scott, ‘who gave never ending support and encouragement through my frantic hours. TABLE OF CONTENTS PAGE LIST OF TABLES ................................................... xi LIST OF FIGURES .................................................. xv LIST OF SYMBOLS .................................................. xxvi CHAPTER 1. INTRODUCTION ................................................. 1 1.1 Objectives ............................................... 4 2. LITERATURE REVIEW ............................................ 6 2.1 Quality Loss in Frozen Foods during Storage ............... 6 2.2 Simulation of Transient Heat Conduction in Frozen during Storage ........................................... 10 2.2.1 One Dimensional Analysis ........................... 10 2.2.2 Multi-Dimensional Analysis ......................... 14 2.3 Estimation of the Surface Heat Transfer Coefficient ...... l6 3. THEORETICAL CONSIDERATIONS ................................... 22 3.1 Thermal Properties ....................................... 23 3.1.1 Unfrozen Water Fraction ............................ 23 3.1.2 Density in Frozen Foods ............................ 25 3.1.3 Thermal Conductivity in Frozen Foods ............... 26 3.1.4 Apparent Specific Heat ............................. 28 3.2 Practical Evaluation of Thermal Properties ............... 29 3.3 Transient Heat Conduction during Frozen Food Storage ..... 30 vi vii 3.3.1 One Dimensional Heat Transfer Analysis ............. 32 3.3.2 Two Dimensional Analysis ........................... 35 3.4 Numerical Time-Temperature Simulation Models ............. 37 3.4.1 One Dimensional Heat Transfer Finite Difference Scheme ............................................. 38 3.4.2 Two Dimensional Heat Transfer Analysis ............. 42 3.5 Estimation of the Surface Heat Transfer Coefficient ...... 45 3.5.1 Analytical Methods ................................. 46 3.5.1.1 Forced Convection over a Flat Plate ........ 47 3.5.1.2 Free Convection ............................ 48 3.5.1.3 Combined Forced and Free Convection ........ 48 3.5.1.4 Packaging Layer ............................ 49 3.5.1.5 Overall Surface Heat Transfer Coefficient .. 49 3.5.2 Inverse Heat Conduction Methods .................... 50 3.5.2.1 The Sequential Regularization Method ....... 53 3.5.2.2 Determination of the Temperature at the Surface .................................... 58 3.5.2.3 Estimation of the Surface Heat Transfer Coefficient ................................ 59 3.6 Quality Loss Prediction if Frozen Foods during Storage ... 59 Experimental Procedures ...................................... 62 4.1 Karlsruhe Test Substance ................................. 62 4.2 Containers for the Karlsruhe Brickettes .................. 65 4.3 Temperature Measurement .................................. 68 4.4 Velocity Measurements .................................... 72 4.5 Experimental Storage Conditions .......................... 73 DETERMINATION OF NUMERICAL PARAMETERS ........................ 76 5.1 Selection of Parameters Inherent in the Finite Difference Solution ...................................... 77 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 viii Numerical Oscillations ............................. 77 Accuracy ........................................... 86 Selection of the Optimal Time Step and Spatial Step ................................... 93 Analysis of Time and Spatial Steps for Other Geometries ......................................... 98 Analysis of Time and Spatial Steps for the Two Dimensional Model .................................. 106 Summary of Observations in the Determination of Finite Difference Parameters .................... 107 5.2 Parameters Used in the Solution of the Inverse Heat Conduction Problem ....................................... 109 5.2.1 The Deterministic Bias ............................. 111 5.2.2 Mean Squared Error ................................. 113 5.2.3 Time Increment Between Temperature Measurements .... 119 5.2.4 Selection of the Optimal Parameters used in the Inverse Heat Conduction Problem of Estimating the Surface Heat Transfer Coefficient .................. 123 6. RESULTS AND DISCUSSION ....................................... 130 5.1 Estimation of the Surface Heat Transfer Coefficient ...... 130 6.1.1 Analytical Estimation of the Surface Heat Transfer 6.1.2 Coefficient ........................................ 131 6.1.1.1 Forced Convection .......................... 134 6.1.1.2 Free Convection ............................ 135 6.1.1.3 Combined Free and Forced Convection ........ 135 6.1.1.4 Packaging Layer ............................ 138 6.1.1.5 Overall Surface Heat Transfer Coefficient .. 141 Estimation of Surface Heat Transfer Coefficient using Inverse Transfer Estimation Techniques ....... 141 ix 6.1.3 Comparison of Results using Analytical and Inverse Heat Conduction Methods ............................ 149 6.2 Simulation of One Dimensional Heat Conduction Through a Food Product ....................................... 153 6.2.1 Comparison with Analytical Solutions ............... 153 6.2.2 Comparison with Experimental Results ............... 157 6.23 Simulation of Two Dimensional Heat Conduction Through a Food Product ............................................. 168 6.3.1 Verification of the Two Dimensional Model .......... 170 6.3.2 Comparison with Experimental Results ............... 171 6.4 Effects of the Magnitude of the Activation Energy Constant on the Mass Average Quality History ............. 180 6.5 Effects of Boundary Conditions on Temperature and Quality Histories of Frozen foods During Storage ................. 185 6.5.1 Influence of the Surface Heat Transfer Coefficient on Temperature and Quality Distribution Histories .. 186 6.5.2 Effects of Step Changes in Storage Temperature on Temperature and Quality Distribution Histories .. 193 6.5.3 Effects of Ambient Temperature During Step Changes in Storage Conditions on Temperature and Quality Distribution Histories ............................. 200 6.6 Effects of Size, Two Dimensional Geometry, and Geometric Shape on Temperature and Quality Histories of Frozen Foods During Storage ..................................... 216 6.6.1 Influence of Product Thickness ..................... 217 6.6.2 Effects of Two Dimensional Heat Transfer on Temperature and Quality Histories .................. 227 6.6.3 Effects of Geometrical Shape ....................... 241 7- SUMMARY AND CONCLUSIONS ...................................... 248 APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E EQUATION FOR BOUNDARY NODES IN TWO DIMENSIONAL FINITE DIFFERENCE SOLUTION .................................. 252 ONE DIMENSIONAL TRANSIENT HEAT CONDUCTION AND QUALITY RETENTION PROGRAM .................................... 262 TWO DIMENSIONAL TRANSIENT HEAT CONDUCTION AND QUALITY RETENTION PROGRAM .................................... 305 SURFACE HEAT TRANSFER COEFFICIENT ESTIMATION PROGRAM . 347 RESULTS FROM SECOND AND THIRD EXPERIMENTAL TEST REPETITIONS .......................................... 371 BIBLIOGRAPHY ..................................................... 3 8 2 Table Table Table Table Table Tdfle Tdfle Table Tflfle Table Table Table 5 5 .2a .2b .4a .4b .7a .7b LIST OF TABLES PAGE Thermal Properties of the Karlsruhe Test Substance above Freezing ........................................... 66 Storage Times and Measurement Intervals for the Three Configurations ........................................... 74 Values for Ax, p, k, Cp, Used in Evaluating Amax (hxz - 7.85 W/m'C) ....................................... 82 Eigenvalues and Resulting Critical No Oscillations Time Step (sec) for Properties Evaluated at -33°C ............. 83 Eigenvalues and Resulting Critical No Oscillations Time Step (sec) for Properties Evaluated at -12°C ............. 84 Comparison of A and Zolézi (-102) ..................... 87 Summation Terms in Series Solution and Resulting Time for Terms to Vanish (Properties Evaluated at -33°C) ...... 90 Summation Terms in Series Solution and Resulting Time for Terms to Vanish (Properties Evaluated at -12°C) ...... 91 Limiting Time Step Based on Accuracy ..................... 94 Limiting Time Steps for Cylindrical and Spherical Geometries ............................................... 99 Comparison of t: Values for Cylindrical and Spherical Geometries (Properties Evaluated at —33°C) ............... 102 Comparison of t: Values for Cylindrical and Spherical xi "L . l Table 55.8 Table 25.9a Table Table Table Table Table Table Table Table Table Table xii Geometries (Properties Evaluated at -12°C) ............... 103 Comparison of SEE; and Amax for a Two Dimensional Grid (Ax - Ay, Properties Evaluated at -33°C) ................. 108 Deterministic Bias for a - 10'9 (At - 600 seconds: Thermal Properties Evaluated at -33°C) ................... 114 .9b Deterministic Bias for a - 10'8 (At - 600 seconds: Thermal Properties Evaluated at -33°C) ................... 114 .9c Deterministic Bias for a - 10'7 (At - 600 seconds: Thermal Properties Evaluated at -33°C) ................... 115 .9d Deterministic Bias for a - 10'6 (At - 600 seconds: Thermal Properties Evaluated at ~33°C) ................... 115 .9e Deterministic Bias for a - 10'5 (At - 600 seconds: Thermal Properties Evaluated at -33°C) ................... 116 .9f Deterministic Bias for o - 10‘4 (At - 600 seconds: Thermal Properties Evaluated at -33°C) ................... 116 .9g Deterministic Bias for a - 10'3 (At - 600 seconds: Thermal Properties Evaluated at -33°C) ................... 117 .10a Average Mean Squared Error (S) and Standard Deviation of S (as) for a - 10'6 (Atm - 600 seconds; Thermal Properties Evaluated at -33°C) ........................... 120 .10b Average Mean Squared Error (S) and Standard Deviation of S (as) for a - 10'5 (Atm - 600 seconds; Thermal Properties Evaluated at -33°C) ........................... 121 .10c Average Mean Squared Error (S) and Standard Deviation of S (as) for a - 10-4 (Atm - 600 seconds; Thermal Properties Evaluated at -33°C) ........................... 122 “* .11 Average Mean Squared Error per Time Step (S ) and * Standard Deviation (as) for Atm - 600 and 1200 seconds (a - 10- 4 , r - 10) ....................................... 124 u ..- .T xiii Table 6.1 Average Nussult Numbers Resulting from Forced Convection over a Flat Plate ........................................ 136 Table 6.2 Average Nussult Numbers Resulting from Free Convection... 137 Table 6.3a Combined Free and Forced Nussult Numbers, NE, and Convective Heat Transfer Coefficients, hxcv, Assuming a Constant Temperature Boundary Condition .................. 139 Table 6.3b Combined Free and Forced Nussult Numbers, NE, and Convective Heat Transfer Coefficients, hxcv, Assuming a Constant Heat Flux Boundary Condition ................... 140 Table 6.4 Surface Heat Transfer Coefficients used in the Numerical Solution in the Comparison with Experimental Results of the Single Layer Slab with One Exposed Surface (Tests la-c) ....... 160 Table 6.5 Initial and Ambient Temperatures used in the Numerical Solution in the Comparison with Experimental Results of the Double Layer Slab with One Exposed Surface (Tests 2a-c) ....... 162 Table 6.6 Thermal Properties of Unfrozen Strawberries .............. 181 Table 6.7 Kinetic Properties of Frozen Strawberries ................ 184 Table 6.8 Definition of Boundary Condition Cases, with Step Changes in storage Temperatures over Given Storage Interval ...... 187 Table 8.1 Description of One Dimensional Transient Heat Conduction and Quality Retention Program ............................ 263 Table 8.2 Computer Code for One Dimensional Transient Heat Conduction and Quality Retention Program ................. 265 'kmle C.1 Description of Two Dimensional Transient Heat Conduction and Quality Retention Program ............................ 263 Tana 0.2 Computer Code for Two Dimensional Transient Heat Conduction and Quality Retention Program ................. 265 Taflelxl Description of Surface Heat Transfer Estimation Program.. 348 xiv Table D.2 Computer Code for Surface Heat Transfer Estimation Program .................................................. 349 LIST OF FIGURES PAGE Figure 3.1 Comparison of Constant Thermal Conductivity Values with Thermal Conductivity Values as a Function of Temperature.. 31 Figure 3.2 Direction of Assumed Heat Transfer for Different One Dimensional Geometries ......................................... 34 Figure 3.3 Direction of Assumed Heat Transfer for Different Two Dimensional Geometries ......................................... 36 Figure 3.4 Evaluation of Thermal Properties in One Dimensional Numerical Solution ..... . ....................................... 40 Figure 3.5 Evaluation of Thermal Properties in Two Dimensional Numerical Solution ............................................. 44 Figure 3.6 Inverse Heat Conduction Problem ........................ 52 Figure 4.1 Single Layer, One Dimensional Container Configuration.. 67 Figure 4.2 Double Layer, One Dimensional Container Configuration.. 69 Figure 4.3 Triple Layer, Two Dimensional Container Configuration.. 70 Figure 4.4 Thermocouple and Hypodermic Needle Assembly ............ 71 Figure 5.1 Numerical Eigenvalues as Percentages of Analytical Exponential Terms .............................................. 92 Flynn 5.2 Crank-Nicolson Approximation for the Time Derivative... 96 Figne 5.3 Numerical Eigenvalues as Percentages of Analytical mqmnential Terms for Cylindrical and Spherical Geometries ..... 105 Figure 5.4 Estimated Heat Transfer Coefficients with and without XV xvi Random Errors in Input Data with Standard Deviation, a, of 0.73°C ......................................................... 126 Figure 5.5 Ambient and Internal Temperature Input Data with and without Random Errors .......................................... 127 Figure 5.6 Estimated Heat Transfer Coefficients with and without Random Errors in Input Data with Standard Deviation, a, of 0.36°C ......................................................... 129 Figure 6.1 Ambient and Average Internal Temperature of Karlsruhe Test Substance Measurements using Single Layer Slab with One Exposed Surface (Test 1a) ...................................... 132 Figure 6.2a Analytical Determination of the Overall Heat Transfer Coefficient using Constant Temperature Boundary Condition ...... 142 Figure 6.2b Analytical Determination of the Overall Heat Transfer Coefficient using Constant Heat Flux Boundary Condition ........ 143 Figure 6.3 Estimated Heat Transfer Coefficients, hx, and Surface Heat Flux, 3, using Experimental Results with Karlsruhe Test substance from Single Layer Slab with One Exposed Surface (Test 1a) ...................................................... 144 Figure 6.4a Estimated Heat Transfer Coefficients, hx, and Surface Heat Flux, q, using Experimental Results with Karlsruhe Test substance from Single Layer Slab with One Exposed Surface (Test 1a), and Average Ambient Temperatures for Each Storage Interval ....................................................... 146 Figure 6.4b Estimated Heat Transfer Coefficients, hx, and Surface Heat Flux, 3, using Experimental Results with Karlsruhe Test Substance from Single Layer Slab with One Exposed Surface (Test 1b), and Average Ambient Temperatures for Each Storage Interval ....................................................... 147 A Figure 6.4c Estimated Heat Transfer Coefficients, hx, and Surface xvii A Heat Flux, q, using Experimental Results with Karlsruhe Test substance from Single Layer Slab with One Exposed Surface (Test 1c), and Average Ambient Temperatures for Each Storage Interval ........................................................ 148 Figure 6.5 One Dimensional Numerical Solution Compared with Analytical Solution with Constant Thermal Properties of Karlsruhe Test Substance at x - Lx/Z ............................ 156 Figure 6.6 One Dimensional Numerical Solution Compared with Analytical Solution with Constant Thermal Properties of Karlsruhe Test Substance at x - Lx/4 and x - 3-Lx/4 ............. 158 Figure 6.7 One Dimensional Numerical Solution Compared to Experimental Results from Single Layer Slab with One Exposed Surface (Test 1a) ............................................... 161 Figure 6.8 One Dimensional Numerical Solution Compared to Experimental Results from Double Layer Slab with One Exposed Surface (Test 2a) ............................................... 163 Figure 6.9 One Dimensional Numerical Solution, using Initial Freezing Point, Tif’ Given by Specht et. a1. (1981), Compared to Experimental Results from Double Layer Slab with One Exposed Surface (Test 2a) ............................................... 165 Figure 6.10 One Dimensional Numerical Solution, with and without Imposed Insulated Boundary Condition at Packaging Interface (x - Lx/2), Compared to Experimental results from Double Layer Slab with One Exposed Surface (Test 2a) ......................... 167 Figure 6.11 One Dimensional Numerical Solution, with hxo - 1 W/m2°C at x - 0, Compared to Experimental results from Double Layer Slab with One Exposed Surface (Test 2a) ......................... 169 Fignfi 6.12s Verification of Two Dimensional Numerical Solution widihx - 8.51 W/m2°C, hy - 0.0, with One Dimensional Numerical . l 0 e1..- xviii Solution with hx - 8.51 w/n2°c .................................. 172 Figure 6.12b Verification of Two Dimensional Numerical Solution with hx - 0.0, hy - 8.51 W/mz’C, with One Dimensional Numerical Solution with hy - 8.51 W/m2°C .................................. 173 Figure 6.13 Two Dimensional Numerical Solution Compared to Experimental Results from Triple Layer Slab with Two Exposed Surfaces (Test 3a) .............................................. 175 Figure 6.14 Two Dimensional Numerical Solution, with and without Imposed Insulated Boundary Condition at Packaging Interface Compared to Experimental results from Triple Layer Slab with Two Exposed Surfaces (Test 3a) ...................................... 177 Figure 6.15 Two Dimensional Numerical Solution, with hxo - hy0 - l W/m2°C, Compared to Experimental results from Triple Layer Slab with Two Exposed Surfaces (Test 3a) ....................... 178 Figure 6.16 Effect of Magnitude of Activation Energy Constant, Ea (kJ/mole), on Food Quality Deterioration Rate, for a Product Initially at -30"C, Exposed to 100 days in Storage at -5°C ...... 183 Figure 6.17 Effect of Surface Heat Transfer Coefficient, hx, on Product Temperature History at Geometric Center and Exposed Surface ......................................................... 189 Figul'e 6.18 Effect of Surface Heat Transfer Coefficient, hx, on Product Quality Deterioration Rate at Geometric Center and EXposed Surface (Ea - 182 kJ/mole, Shelf-life at -18°C - 630 days) ........................................................... 190 Figure 6.19 Effect of Surface Heat Transfer Coefficient, hx, on PI‘Oduct Quality Deterioration Rate at Geometric Center and Exposed Surface (Ea - 49 kJ/mole, Shelf-life at -18°C - 540 days) ........................................................... 191 Figure 6.20a Effect of Duration of Step Changes in Storage xix Temperatures on Product Temperature History at Geometric Center, for a Product with Initial Temperature of -30°C (Cases 1, 2 and 3) .......................................................... 194 Figure 6.20b Effect of Duration of Step Changes in Storage Temperatures on Product Temperature History at Exposed Surface, for a Product with Initial Temperature of -30°C (Cases 1, 2 and 3) .......................................................... 195 Figure 6.21 Effect of Duration of Step Changes in Storage Temperatures on Product Quality Deterioration Rate, for a Product with Initial Temperature of -30°C (Cases 1, 2 and 3; Ea - 182 kJ/mole, Shelf-life at -18°C - 630 days) ............... 197 Figure 6.22 Effect of Duration of Step Changes in Storage Temperatures on Product Quality Deterioration Rate, for a Product with Initial Temperature of -30°C (Cases 1, 2 and 3; Ea - 49 kJ/mole, Shelf-life at -18°C - 540 days) ................ 198 Figure 6.23a Effect of Magnitude of Step Change in Storage Temperatures on Product Temperature History at Geometric Center, for a Product with Initial Temperature of -30°C (Cases 1, 4 and 7) .......................................................... 201 Figure 6.23b Effect of Magnitude of Step Change in Storage Temperatures on Product Temperature History at Exposed Surface, for a Product with Initial Temperature of -30°C (Cases 1, 4 and 7) .......................................................... 202 Figure 6.24 Effect of Magnitude of Step Change in Storage Temperatures on Product Quality Deterioration Rate, for a Iiz'll'Oduct with Initial Temperature of -30°C (Cases 1, 4 and 7; En - 182 kJ/mole, Shelf-life at -18°C - 630 days) ............... 203 Figure 5.25 Effect of Magnitude of Step Change in Storage Temperatures on Product Quality Deterioration Rate, for a r-1 eo-J 1 Nu- ’- I '~-. {‘- .ix ».I '1 Product with Initial Temperature of -30°C (Cases 1, 4 and 7; Ba - 49 kJ/mole, Shelf-life at -18°C - 540 days) ................ 204 Figure 6.26a Effect of Magnitude of Step Change in Storage Temperatures on Product Temperature History at Geometric Center, for a Product with Initial Temperature of ~30°C (Cases 2, 5 and 8) .......................................................... 206 Figure 6.26b Effect of Magnitude of Step Change in Storage Temperatures on Product Temperature History at Exposed Surface, for a Product with Initial Temperature of -30°C (Cases 2, 5 and 8) .......................................................... 207 Figure 6.27 Effect of Magnitude of Step Change in Storage Temperatures on Product Quality Deterioration Rate, for a Product with Initial Temperature of -30°C (Cases 2, 5 and 8; Ea - 182 kJ/mole, Shelf-life at -18°C - 630 days) ............... 209 Figure 6.28 Effect of Magnitude of Step Change in Storage Temperatures on Product Quality Deterioration Rate, for a Product with Initial Temperature of -30°C (Cases 2, 5 and 8; Ba — 49 kJ/mole, Shelf-life at -18°C - 540 days) ................ 210 Figure 6.29s Effect of Magnitude of Step Change in Storage Temperatures on Product Temperature History at Geometric Center, for a Product with Initial Temperature of -30°C (Cases 3, 6 and 9) .......................................................... 211 Figure 6.29b Effect of Magnitude of Step Change in Storage Temperatures on Product Temperature History at Exposed Surface, for a Product with Initial Temperature of -30°C (Cases 3, 6 and 9) .......................................................... 212 Figure 6. 30 Effect of Magnitude of Step Change in Storage Telllperatures on Product Quality Deterioration Rate, for a Product with Initial Temperature of -30°C (Cases 3, 6 and 9; 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Ian. 5 y on or“ 0'0. . xxi Ea - 182 kJ/mole, Shelf-life at —18°C - 630 days) ............... 214 Figure 6.31 Effect of Magnitude of Step Change in Storage Temperatures on Product Quality Deterioration Rate, for a Product with Initial Temperature of ~30°C (Cases 3, 6 and 9; Ea - 49 kJ/mole, Shelf-life at ~18’C - 540 days) ................ 215 Figure 6.32s Effect of Large Product Thickness on Product Temperature History at the Geometric Center for Different Step Change Intervals, for a Product with Initial Temperature of -30°C (Cases 4, 6 and 8) ........................................ 218 Figure 6.32b Effect of Large Product Thickness on Product Temperature History at the Exposed Surface for Different Step Change Intervals, for a Product with Initial Temperature of -30°C (Cases 4, 6 and 8) ........................................ 219 Figure 6.33 Effect of Large Product Thickness on Product Quality Deterioration Rate at the Exposed Surface for Different Step Change Intervals, for a Product with Initial Temperature of -30°C (Cases 4, 6 and 8; Ea - 49 kJ/mole, Shelf-life at —18°C - 540 days) ..................................................... 221 Figure 6.34a Effect of Small Product Thickness on Product Temperature History at the Geometric Center for Different Step Change Intervals, for a Product with Initial Temperature of '30’C (Cases 4, 6 and 8) ........................................ 222 Figure 6.34b Effect of Small Product Thickness on Product Temperature History at the Exposed Surface for Different Step Change Intervals, for a Product with Initial Temperature of ‘30°C (Cases 4, 6 and 8) ........................................ 223 Figure 6.35 Effect of Small Product Thickness, Lx - 1.0 m, on Product Quality Deterioration Rate at the Exposed Surface for Di~fferent Step Change Intervals, for a Product with Initial xxii Temperature of -30°C (Cases 4, 6 and 8; Ea - 182 kJ/mole, Shelf-life at -18°C - 630 days) ................................. 225 Figure 6.36 Effect of Small Product Thickness, Lx - 0.2 m, on Product Quality Deterioration Rate at the Exposed Surface for Different Step Change Intervals, for a Product with Initial Temperature of ~30°C (Cases 4, 6 and 8; Ea - 182 kJ/mole, Shelf-life at -18°C - 630 days) ................................. 226 Figure 6.37 Locations of Solutions for Two Dimensional Geometry.... 229 Figure 6.38 Two Dimensional Numerical Solution for Square Rod (Lx - Ly - 2.0 m) Compared to One Dimensional Solution for a Slab with Equal Dimension, Lx, for Product with Initial Temperature of -30°C (Case 6) ................................... 230 Figure 6.39 Quality Deterioration Rate Resulting from Two Dimensional Solution for Square Rod (Lx - Ly - 2.0 m) Compared to One Dimensional Solution for a Slab with Equal Dimension, Lx, for Product with Initial Temperature of -30°C (Case 6; Ea - 182 kJ/mole, Shelf-life at -18°C - 630 days) ............... 231 Figure 6.40 Two Dimensional Numerical Solution for Rectangular Rod (Lx - 2.0 m, Ly - 1.0 m) Compared to One Dimensional Solution for a Slab with Dimension, Lx - 1.0 m, for Product with Initial Temperature of -30°C (Case 6) ................................... 233 Flynn 6.41 Quality Deterioration Rate Resulting from Two Dimen- sional Solution for Rectangular Rod (Lx - 2.0 m, Ly - 1.0 m) Compared to One Dimensional Solution for Slab with Dimension, Ix - 1.0 m, for Product with Initial Temperature of -30°C (Case 6; Ea - 182 kJ/mole, Shelf-life at -18°C - 630 days) ...... 234 Figure 6.42 Two Dimensional Numerical Solution for Rectangular Rod (1x - 2.0 m, Ly - 0.2 m) Compared to One Dimensional Solution for a Slab with Dimension, Lx - 0.2 m, for Product with Initial xxiii Temperature of -30°C (Case 6) .................................. 235 Figure 6.43 Quality Deterioration Rate Resulting from Two Dimen- sional Solution for Rectangular Rod (Lx - 2.0 m, Ly - 0.2 m) Compared to One Dimensional Solution for Slab with Dimension, Lx - 0.2 m, for Product with Initial Temperature of -30°C (Case 6; Ea - 182 kJ/mole, Shelf-life at -18°C - 630 days) ..... 236 Figure 6.44 Quality Deterioration Rate Resulting from Two Dimen- sional Solution for Square Rod (Lx - Ly - 2.0 m) Compared to One Dimensional Solution for a Slab with Equal Dimension, Lx, for Product with Initial Temperature of -30°C (Case 6; Ea - 49 kJ/mole, Shelf-life at -18°C - 540 days) ..................... 238 Figure 6.45 Quality Deterioration Rate Resulting from Two Dimen— sional Solution for Rectangular Rod (Lx - 2.0 m, Ly - 1.0 m) Compared to One Dimensional Solution for Slab with Dimension, Lx - 1.0 m, for Product with Initial Temperature of ~30°C (Case 6; Ea - 49 kJ/mole, Shelf-life at -18’C - 540 days) ...... 239 Figure 6.46 Quality Deterioration Rate Resulting from Two Dimen- sional Solution for Rectangular Rod (Lx - 2.0 m, Ly - 0.2 m) Compared to One Dimensional Solution for Slab with Dimension, Lx - 0.2 m, for Product with Initial Temperature of -30°C (Case 6; Ea - 49 kJ/mole, Shelf-life at ~18°C - 540 days) ....... 240 Flynn 6.47 Comparison of Solutions for Two Dimensional Heat Transfer Through a Square Rod and One Dimensional Heat Transfer Through a Cylinder of Equal Surface Area (Case 6) ............... 242 Fignm 6.48 Quality Deterioration Rate Resulting from Two Dimen- sional Heat Transfer Through a Square Rod and One Dimensional Heat Transfer Through a Cylinder of Equal Surface Area (Case 6). 243 Figure 6.49 Comparison of Solutions for Two Dimensional Heat Transfer Through a Square Rod and One Dimensional Heat Transfer gill-U“. I (MW. 4 xxiv Through a Cylinder of Equal Volume (Case 6) ..................... 245 Figure 6.50 Quality Deterioration Rate Resulting from Two Dimen- sional Heat Transfer Through a Square Rod and One Dimensional Heat Transfer Through a Cylinder of Equal Volume (Case 6) ....... 246 Figure E.1a Ambient and Average Internal Temperature of Karlsruhe Test Substance Measurements using Single Layer Slab with One Exposed Surface (Test 1b) ....................................... 372 Figure E.1b Ambient and Average Internal Temperature of Karlsruhe Test Substance Measurements using Single Layer Slab with One Exposed Surface (Test 1c) ....................................... 373 Figure E.2a Estimated Heat Transfer Coefficients, hx, and Surface Heat Flux, q, using Experimental Results with Karlsruhe Test substance from Single Layer Slab with One Exposed Surface (Test 1b) ....................................................... 374 Figure E.2b Estimated Heat Transfer Coefficients, hx, and Surface Heat Flux, q, using Experimental Results from Single Layer Slab with One Exposed Surface (Test 1c) .............................. 375 Figure E.3a One Dimensional Numerical Solution Compared to Experi- mental Results from Single Layer Slab with One Exposed Surface (Test 1b) ....................................................... 376 Figure E.3b One Dimensional Numerical Solution Compared to Experi- mental Results from Single Layer Slab with One Exposed Surface (Test 1c) ....................................................... 377 Fiyue E.4a One Dimensional Numerical Solution Compared to Experi- mental Results from Double Layer Slab with One Exposed Surface (Test 2b) ....................................................... 378 Figure E.4b One Dimensional Numerical Solution Compared to Experi- mental Results from Double Layer Slab with One Exposed Surface (Test 2c) ....................................................... 379 g {ETIENNE Figure E.5a Two Dimensional Numerical Solution Compared to Experi- mental Results from Triple Layer Slab with Two Exposed Surfaces (Test 3b) ....................................................... 380 Figure E.5b Two Dimensional Numerical Solution Compared to Experi- mental Results from Triple Layer Slab with Two Exposed Surfaces (Test 3c) ....................................................... 381 m | (e) CI - CP(T) - hx - LIST OF SYMBOLS Constant Surface Area (m2) Coefficient matrix for temperatures at time n+1 Constant Coefficient matrix for temperatures at time n Concentration of quality index Heat Capacity Matrix Elemental Heat Capacity Matrix Confidence Interval Specific heat (kJ/kg'C) Specific heat of frozen food as function of temperature(kJ/kg°C) Vector containing known boundary conditions Deterministic Error Activation energy constant (kJ/mole) Acceleration of gravity (m/s2) Grashof number Enthalpy (kJ/kg) Regularization difference matrix Surface heat transfer coefficient on boundaries along x-axis (W/m2°C) hx - CV hy Heat transfer coefficient from forced and free convection (W/m2 ° C) - Surface heat transfer coefficients on boundaries along y-axis xxvi xxvii (W/m2°C) th 1 order Bessel function Thermal conductivity (W/m'C) Matrix containing thermal conductivity terms - Elemental matrix containing thermal conductivity terms Thermal conductivity of frozen food as a function of temperature (W/m°C) Rate constant (sec'1 or day'l) Total number of nodes along x axis Position along x axis of first boundary (m) Characteristic Length (m) Position along x axis of second boundary (m) Position along y axis of second boundary (m) Total number of nodes along y axis Molecular weight of substance j (kg/mole) Molecular mass of substance 3 Water content (t) of methyl-cellulose Order of reaction Total number of time steps Number of thermocouples Nussult number Prandlt number Surface heat flux (W/mz) Surface heat flux vector (W/mz) Food quality (time or %) Assumed heat flux vector (W/mz) Initial food quality at reference temperature Tr(time) Universal gas constant Regularization terms (j - 0, 1, 2) £2 L.» :4 "N xxviii Raleigh number Reynolds number Least squares function Least squares function for 1th set of random numbers Mean squared error per time step Time (8) Temperature (°C or 'K) Temperature Vector ('C or °K) Time for 1th terms in summation be insignificant Time for solution to reach steady state conditions Initial freezing temperature (°K) Reference temperature ('K) Surface temperature (°C) Temperature matrix from assumed heat flux (°C) Initial product temperature (°C) Air free stream velocity (m/s) Variance Weighting factors for regularization terms (j - 0,1,2) position along x axis (m) Sensitivity coefficient (°C/W/m2) Sensitivity coefficient matrix (°C/W/m2) Mole fraction location of temperature sensor (m) postion along y axis (m) Temperature measurements at xo (°C) Temperature measurements at xo vector (°C) Regularization parameter Weighting factor at time n Weighting factor at time n in x-direction (2-D solution) fly - Weighting factor at time n in y-direction (2-D solution) fie - Expansion coefficient of air (°K'1) F - Latent heat of solvent At - Time step Atac - Maximum time step for accuracy Atm - Time increment for temperature measurements Atosc - Limiting time step for no oscillations Ax - Spatial step x direction Ay - Spatial step y direction {I - 3th root of transcendental equation n - Weighting factor at time n+1 "x - Weighting factor at time n+1 in x-direction (2-D solution) "y - Weighting factor at time n+1 in y-direction (2-D solution) v - Kinematic viscosity (m2/s) n - Thermal diffusivity (mz/s) max - Maximum numerical eigenvalue (“0-1) A;:; - Maximum elemental numerical eigenvalue (°C-1) A! - 1th numerical eigenvalue (°C-1) A; - Ith analytical eigenvalue (°C'1) 1r -Pi P - Density (kg/m3) P(T) - Density of frozen food as a function of temperature (kg/m3) 0 - Standard deviation of temperature measurements (°C) OS - Standard deviation of least squares function (°C) a; - Standard deviation of least squares function per time step (°C/sec) Q - Arbitrary temperature function XXX Sgbscripts a - air c - carbohydrate D - without input errors F - forced convection i - ice 1p - lipid Lx - second boundary in x-direction on - first boundary in x-direction Ly - second boundary in y-direction m - mineral N - free convection p - unfrozen food product pb - paper board pf - plastic film pk - packaging material pr - protein s - solids V - with input errors w - water W1 - water-ice component Wis - water-ice-solids component "0 - unfrozen water 0 - first boundary in y-direction 1 - first storage period 2 - second storage period 1” - ambient conditions for first storage period 2m ambient conditions for second storage period — e - at a location between m - at a location between m SC I: - Mean - Estimated xxxi h and (m-l)th node th and (n+1)th node CHAPTER 1 INTRODUCTION Freezing is one of the most important methods of food preservation used in the United States. The freezing process cannot improve the quality of a food product; the reduction of temperature in a food product only results in the retardation of the processes which are detrimental to product quality, such as enzyme activity, microbial growth and chemical reactions. The overall quality of the food product “lay be affected during pre-treatment (post harvest handling and Preparation), freezing, and post-freezing handling (transportation, Storage and distribution). However, with proper pre-treatment and freezing, the majority of the quality reduction occurs during the post- freezing handling phase of the overall freezing process. The rate of quality loss during this phase is primarily temperature dependent; changes in temperature during the post-freezing phase may result in a reduction in storage or shelf-life for the product (Singh and Wang, 1977). Zaritzky (1982) cited two types of temperature changes a frozen food product may be exposed to during the post-freezing phase: (1) fluctuations in the temperature of the storage chamber, and (2) Smdden increases in temperature during loading and unloading of the Product during transportation and distribution. In both of these cases, a rise in temperature may lead to an undesirable loss of product quality. Additionally, since frozen food products are commonly stored in large pallet loads, a sudden change in temperature may result in a higher rate of quality loss at the surface of the pallet load than at the center . Since product quality is largely a result of the temperature history of the product after freezing, accurate methods of predicting tempera- ture distribution histories within the product as a result of fluctuating storage temperatures are important in estimating final product quality. Substantial research has been devoted to developing analytical and numerical models for the simulation of freezing in foods and the estima- tion of freezing times (Plank, 1913, Hayakawa and Bakal, 1972, Charm et. 81. , 1972, and Cleland and Earle, 1977a). Only a limited amount of work has focused on extending these studies, particularly the numerical mOdels, to simulating the food product during the post-freezing phase. Although many similarities exist in the numerical analysis of the two problems, simulation of the post-freezing phase differs from the freez- ing process in several ways: first, the heat transfer coefficients prevalent during the freezing phase are generally much higher than those found in the post-freezing phase. Second, the size of the body con- Sidered also differs. During freezing, the individually packaged Product is considered, while during the post-freezing phase the product is often palletized, and consequently, the pallet load is the object of Consideration. In addition, the duration of the freezing process is generally much shorter than the duration of the post-freezing phase. These differences in the rate of heat transfer, size of body, and dura- tion time, suggest that care must be taken in adapting numerical methods deVeloped for the freezing process to the post-freezing process. The heat transfer rate depends on the temperature differential between the surrounding environment and the product, and the heat trans- fer coefficient. Several researchers, including Bonacina and Comini —L—____4 . e. .. ...-w o... .- (1972) and Chavarria and Heldman (1983) have investigated the measure- ment of the heat transfer coefficient during the freezing stage, but little data can be found on its value during the post-freezing stage. Values were measured by Dagerskog (1974), but only for steady-state To gain a full understanding of the post-freezing situa- conditions . tion, it is desirable to estimate the heat transfer coefficient for both transient and steady-state heat transfer. Estimation of internal frozen food temperatures is essential in estimating final product quality. If a first order quality deteriora- tion mechanism is assumed, the product quality loss my be predicted from internal temperatures and known kinetic parameters. In this study, the problem of determining temperature and quality distributions in food products subject to step changes in ambient tem- peratures in the post-freezing phase was considered. A numerical method was developed to simulate one or two dimensional heat transfer, and accommodate regular geometric shapes; a rectangle, cylinder or sphere. Additionally, a procedure is presented to estimate surface heat transfer coefficients from internal product temperature measurements throughout transient and steady state heat transfer. The predicted temperature distribution histories are used to estimate product quality loss, assum- ing a Single quality deterioration process exists which limits the Overall quality degradation rate. The uniqueness of this study lies in the coupling of temperature history with quality loss estimation, in the development of a numerical m°del to accommodate the conditions inherent to the post-freezing r p ocess s and in the estimation of the transient and steady-state heat trans fer coefficients . 1.1 Objectives The principal objectives of this study were: To develop a generalized one dimensional mathematical model to simulate transient temperature and quality distributions within frozen food products subject to step changes in storage conditions, using the implicit Crank-Nicolson finite difference scheme, and assuming a single limiting quality reaction exists, which governs the overall quality degradation rate. 2. To extend the one dimensional analysis to two dimensions using an alternating direction implicit (ADI) finite difference scheme. 3. To estimate the surface heat transfer coefficient of a frozen food product during step changes in storage conditions as a function of time, from discrete ambient and internal product temperature measurements . 4. To analyze the influence of the product boundary conditions on quality loss during storage. Parameters affecting the product boundary conditions include ambient temperature, the heat transfer coefficient, and, for step changes in storage conditions, the time interval at each storage temperature, and the magnitude of the step change in storage temperature. 5- To analyze the effects of product dimensions and kinetic parameters on the quality distribution. Specifically, it was desired to determine under what conditions the temperature and quality distributions are uniform throughout the body, and a lumped (uniform temperature) model can be used. 6' To determine the influence that geometry and the choice of a one or two dimensional model has on temperature and quality distributions. This includes determining under what conditions a one dimensional model can be used to approximate two dimensional heat flow, that is, to determine when heat flow in an infinite cylinder can be used to approximate two dimensional heat flow through an infinite rectangular rod. CHAPTER 2 LITERATURE REVIEW 2.1 Quality Loss in Frozen Foods during Storage Freezing is an important method of preserving food products for later consumption. This process reduces the rate of quality loss by restricting enzyme action, reducing chemical reaction rates and inhibit- ing microbial growth. The process does not completely preserve food quality; detrimental changes continue at a reduced rate dependent upon storage temperature and type of product. Physical and chemical changes are the primary factors affecting the overall quality of frozen foods. Important detrimental physical changes cited by Singh and Wang (1977) are ice crystallization with volume expansion, and desiccation at the surface of the frozen food product. Fluctuations in storage temperature increase the rate of desiccation at the surface of the product, especially in improperly packaged foods, resulting in dry, brown spots, particularly in poultry products, com- monly referred to as "freezer burn”. Chemical changes occurring during frozen storage, as described by Fennema et. a1. (1973), are lipid oxidation, enzymatic browning, flavor deterioration, protein insolubilization and degradation of chlorophyll and vitamins. Most of the existing data on quality changes in frozen foods repre- sent the allowable or tolerable time and temperature conditions for a specified quality retention. These tests, initiated by Van Arsdel (1957), are commonly known as the time-temperature-tolerance (TTT) experiments. The Western Regional Research Center, Berkeley, California continued these tests into the early sixties on a great many fruit, vegetable, and poultry products (Jul (1984)). Several mathematical models have been developed to predict the quality change of frozen foods during storage based on this data. Schwimmer et. a1. (1955) developed series relationships for quality losses resulting from periodic storage temperature fluctuations. Van Arsdel and Guadagni (1959) presented a procedure to predict quality changes resulting from known irregular temperature fluctuations through graphical integration of temperature history curves. In the case where a limiting reaction affecting quality exits, kinetic theory may be used to describe the change in quality at a constant temperature. The loss of quality with storage time, at a given temperature may be found from: -QQ- .9 dt kr C where C - concentration of quality index t - time kr - rate constant n - order of the reaction It was suggested by Charm (1971) that if a limiting quality factor exists, the empirical equation developed by Arrhenius (1889) can be used to describe the effect of temperature on the rate constant. Singh (1976) also utilized the Arrhenius equation to describe the first order reduction of a single component in a product. The kinetics of quality change in frozen foods were analyzed by Lai and Heldman (1982) in an effort to apply kinetic models to TTT data found in the literature. In a related study, Heldman and Lai (1983) developed a model based on the Arrhenius equation where the reaction order need not be considered. Statistical methods for the computation of kinetic parameters used in the Arrhenius equation from existing TTT data were developed by Chu (1983), Haralampu et. a1. (1985), and Cohen and Saguy (1985). Ross et. al. (1985) developed shelf-life prediction models based on nonlinear regression and contingency-table methods. Singh and Heldman (1976) modeled the diffusion of oxygen accompanied by a second order chemical reaction with ascorbic acid to simulate food quality loss in liquids during storage. Bhattacharya and Hanna (1986) estimated rate constants, the order of reaction and the activation energy constant for texture degradation of frozen beef during storage. Jul (1984) warned against the use of mathematical models based on a single quality factor in particular situations where the rate of quality deterioration may be a result of several factors and no single limiting reaction exists. The effects of temperature fluctuations during storage on product quality has been the subject of research for a number of years. Hustrulid and Winter (1943) reported fluctuating storage temperatures below -15°C had no great influence on the appearance and/or palatability of the products studied. Gortner et. a1. (1948) compared quality loss in frozen food products (pork, strawberries, snap beans, and peas) subject to three different storage conditions. One storage compartment was maintained at -17.8°C, the second was held at -12.2°C, and the third fluctuated between -17.8°C and -6.7°C in a six day cycle. Quality losses in the products held at -12.2°C and in the fluctuating compart- ment were comparable, based on palatability, thamine, and perioxide content in pork, and ascorbic acid loss in fruits and vegetables. A number of the TTT studies, completed during the late fifties and early sixties, investigated the effects of temperature fluctuations during storage. Dietrich et. a1. (1960) found, in a study of frozen snap beans, that when storage temperatures were varied in patterns, deterioration was found to be a summation of constant temperature incre- ments. In an investigation of ready-to-cook cut-up chicken, Klose et. a1. (1959) conducted constant storage tests at -6.7°C, -12.2°C, ~17.8°C, -27.8'C and -34.4°C, and periodic storage tests between -17.8°C and -6.7’C, and between -27.8'C and -12.2°C, and found only slightly greater deteriorative effects on quality in the fluctuating storage tests than found at the equivalent arithmetic mean temperature. Comparable results, using similar test conditions, were obtained by Boggs et. a1. (1960) in a TTT study of frozen peas, and by Dietrich et. a1. (1962) in an investigation of quality changes in cauliflower. Fennema and Powrie (1964) discussed the lack of evidence to extend the conclusions found in the TTT studies to the texture of fruits, and called for more investiga- tions on the effects of fluctuating temperatures on fruit texture. Ashby et. a1. (1979) studied energy savings resulting from periodic fluctuating storage temperatures ranging from -23 to -15°C. For storage periods greater than six months, it was determined that the product temperature should not rise above —18°C, and should not fluctuate more than 3°C. Moleeratanond et. a1. (1981) conducted similar studies on the energy consumption of a fluctuating temperature storage regime and its effects on quality changes in frozen boxed beef. Results indicated that product quality was not seriously affected in peripheral pallet loca- tions, provided the temperature was maintained at less than -18°C and the maximum fluctuation did not exceed 3°C. 10 Sastry and Kilara (1983) reflected the need for analysis of heat transfer in frozen foods exposed to periodic storage conditions to determine quality variations within a given pallet load. 2.2 Simulation of Transient Heat Conduction in Frozen Foods during Storage Considerable attention has been given to simulation of freezing in food products within the last two decades. Only recently have research- ers begun to investigate the thermal behavior of frozen foods during distribution and storage. In many instances, the methods utilized in freezing studies may also be utilized in storage studies. Freezing simulation models may be categorized with regards to the results gener- ated in two groups: (1) those producing freezing time estimations, and (2) those producing temperature distribution histories within the product. Only the latter group is of interest in storage simulation studies and is included in this review. 2.2.1 One Dimensional Analysis Most of the models developed to simulate freezing or frozen food storage are based on one dimensional heat transfer analysis. Analytical and numerical techniques have been proposed to estimate temperature distribution histories in both the freezing and post-freezing stages. An analytical solution to the freezing phenomenon involving a pure liquid was presented by Carslaw and Jaeger (1959). Komori and Hirai (1970) provided an analytical solution of the freezing problem in cylindrical coordinates, with the single, unique temperature and only at the solid-liquid interface. Tien and Geiger (1967) developed an ll analytical solution to the solidification of a binary eutectic system, assuming three distinct regions: a solid, liquid, and a liquid-solid region in which the solid fraction is linear with position. Grange et. a1. (1976) obtained an approximate analytical solution for freezing of salt solutions using an integral method, and assuming latent heat is released at a constant temperature. A food product is a solution or mixture, however, and freezing does not occur at a single distinct temperature. Instead, the initial freez— ing temperature of the mixture is depressed compared to that of the pure substance, such as water (Heldman, 1982, and Chen, 1986). As the mix- ture freezes, the liquid portion becomes more concentrated with solute, and the freezing point is depressed further. As a result, latent heat is produced over a range of temperatures, and thermal properties, which vary according to the solid-liquid composition, are temperature depend- ent. Several researchers have used analytical methods in developing solution techniques with modification to allow for temperature dependent thermal properties. Sastry and Kilara (1983) approximated constant thermal properties over small temperature ranges using an "apparent” thermal diffusivity which includes latent heat terms. An analytical solution of the linear one dimensional heat conduction problem with designated sinusoidal temperatures at the boundaries was then obtained to simulate the temperature response of frozen peas in fluctuating temperature storage conditions. Zaritzky (1982) developed both analyti- cal and numerical models to simulate the thermal behavior of frozen meat during its storage and distribution. In the analytical model, average values for the thermal properties, including the effects of latent heat, ‘were again used, but in this case, a boundary condition of the third ‘kind with sinusiodal ambient temperatures was imposed. One dimensional 12 analytical solutions were multiplied together to generate two and three dimensional models, and results were compared with experimental data. In a related study using similar boundary conditions, Zuritz, et. al. (1986) simulated temperature fluctuations within frozen foods stored in cylindrical containers. Zuritz and Sastry (1986) determined the effects of packaging materials on temperature fluctuations in frozen foods using an analyti- cal model to calculate the temperature distribution histories, resulting from an imposed sinusiodal ambient temperature at the surface, and assuming constant thermal properties. Many different approaches have been used in developing mathematical models of the freezing and post-freezing phases. De Michelis and Calvelo (1982) used a simplified model which uses three distinct precooling, freezing and tampering phases. An analytical solution with constant coefficients is obtained for the precooling and tempering phases, and the freezing phase is simulated assuming steady state heat transfer and constant coefficients. Chen et. al. (1984) used a method of lumping to incorporate diffusivity and latent heat terms into a temperature dependent 'effective' diffusivity. The resulting equation was solved using finite differences. Sanz, et. a1. (1986) applied the z-transfer function method to predict temperature - time history of food stuffs during chilling and cold storage. In this procedure, the z-transfer coefficients are obtained be means of an experimental method. A number of researchers used methods which assumed latent heat is released at a fixed freezing point. Charm et. a1. (1972) assumed latent heat was released at a constant temperature over a specified region. Grange et. al. (1976) also assumed latent heat at a distinct tempera- ture. An implicit finite difference solution was obtained assuming 13 Variable thermal properties and compared with an approximate analytical solution. Dix and Cizek (1971) solved the heat conduction problem replacing the usual dependent variable, temperature as a function of position and time, with the isotherm.position as a function of temperature and time. This tedhnique is termed the 'isotherm migration method' (IMM). The solution was obtained explicitly using finite differences and variable thermal properties. Chernous'ko (1970) also developed a similar methodology using isotherms for the solution of the nonlinear heat conduction problem with phase change. Talmon and Davis (1981) utilized the previously developed IMM methods in developing a new technique called the 'modified isotherm migration method' (MIMM). Unlike previous IMM methods, the MIMM uses a moving front boundary condition in the governing differential equation. Mastanaiah (1976) also incorporated a moving front boundary condition, this time by use of the transformation of coordinates. Temperature was maintained as the dependent variable, and the solution was obtained using the Crank-Nicolson finite difference method (Ozisik, 1980). A significant number of researchers have incorporated the latent heat into an 'apparent' specific heat. Freezing is assumed to take place over a range of temperatures, consequently, all thermal properties are assumed to be temperature dependent. Lescano (1973) used the Crank-Nicolson finite difference technique to simulate freezing in codfish. Heldman (1974a) also used the Crank-Nicolson finite difference method to simulate the freezing process in spherical food products. The Kopelman (1966) equation describing the relationship of thermal conduc- tivity with the temperature dependent product composition was implemented in this simulation model. Bonacina and Comini (1973a) used a second order accurate three level time scheme originally proposed by 14 Lees (1966) for the solution of the transient heat conduction equation Vita: temperature dependent parameters. Bonacina et. a1. (1973) extended this work to account for phase-change by including latent heat terms into the specific heat. Cleland and Earle (1984) noted the advantages of Lee's scheme, in that the thermal properties are evaluated at the add-point time level instead of at the beginning time level as done in other methods, such as the Crank-Nicolson method. Tarnawksi (1976) developed finite difference equations for simultaneous heat and mass transfer in frozen food products. Zaritzky (1982) used the Douglas- Jones method of finite differences (Von Rosenberg, 1969) to simulate frozen meat in storage, and compared results with an analytical method discussed previously. An alternate approach was presented by Joshi and Tao (1974). In this procedure, the finite difference equations were written in term of the enthalpy, and these were solved implicitly by assuming an exponen- tial relationship between enthalpy and temperature. 2.2.2 Multi-Dimensional Analysis Various methods have been proposed for the numerical solution of a nonlinear two dimensional heat conduction problem. Most of the solution methods may be categorized as finite element or finite difference solu- tions. The finite element method was used to solve transient, nonlinear heat transfer problems by De Baerdemaeker et. a1. (1977) in axi- symmetric products, and by Zuritz and Singh (1985) in modeling temperature fluctuations in stored frozen foods. Comini et. a1. (1974) utilized finite elements in a three dimensional analysis of a brick shaped body, including convective and radiative boundary conditions. Two dimensional finite element techniques were also utilized by De 15 ““610 et. a1. (1985) in the analysis of ice cream brickettes and by Rebellato et. a1. (1978) in the freezing of meat carcasses. Lewis et. 81. (1984) applied an alternating-direction finite element scheme to the freezing problem with a substantial savings in computation time and comparable accuracy to standard schemes. Finite difference methods were used as an alternate approach. Most of these methods may be classified as explicit, implicit or alternating- direction techniques. Dagerskog (1974) used an explicit finite difference method in three dimensions to simulate temperature distribu- tions in foods during handling and storage. This method was severly limited in its usefulness by the stability condition on the time step. Implicit solutions to the two dimensional transient nonlinear heat conduction problem require the inversion of large matrices at each time step, requiring a substantial amount of computation time (Anderson et. a1. (1984)). To overcome the difficulties of solving the two dimen- sional problem using explicit or implicit techniques, an alternating- direction implicit (ADI) scheme with second order accuracy was developed by Peaceman and Rachford (1955). The ADI method involves a two step scheme, where the temperature field is determined in different directions for each time step. This results in the inversion of two tridiagonal matrices at each time step, for which efficient algorithms exist. Douglas and Gunn (1964) developed a general ADI method for two and three dimensions utilizing a Crank- lfiicolson scheme which is of second order accuracy and unconditionally stable. Allada and Quon (1966) developed a stable explicit multidimen- sional alternation direction solution for nonhomogeneous media. Fleming (1973) utilized the Peaceman-Rachford method in simulating the freezing process with temperature dependent thermal properties. Bonacina and Comini (1973b) applied Lee's tri-level scheme in alternating directions 16 to simulate food freezing with two dimensional heat transfer. Evans and Gene (1978) solved the transient heat conduction problem for an annular ring, using the Peaceman-Rachford ADI method. Alternate approaches to the finite difference solution to the transient heat conduction problem include the splitting or fractional-step methods discussed by Yanenko (1971) and developed by Soviet mathematicians about the same time ADI methods were developed in the United States. The modified box method for the heat equation and the hopscotch methods are two additional methods discussed by Anderson et. al. (1984). The modified box method is second order accurate even with variable grid spacing. The hopscotch method is a first order accurate two step alternating explicit-implicit scheme. 2.3 Estimation of the Surface Heat Transfer Coefficient The prediction of temperature profiles within a frozen food sub- stance during storage requires knowledge of the resistance to heat transfer between the product and the cooling medium. This resistance is characterized by a surface heat transfer coefficient (h), which may be dependent on time and/or position. The importance of the surface heat transfer coefficient in estimat- ing freezing times was discussed by Heldman (1974a), Hsieh et. a1. (1977) and Tarnawski (1976), but little effort has been directed toward investigating the effects of the heat transfer coefficients during storage conditions. Some simple steady state solutions for the heat transfer coeffi- cient resulting from forced convection have been developed for regular geometries. Rays and Crawford (1980) presented solutions for constant free stream velocity flow over a constant-temperature or arbitrarily l7 specified temperature semi-infinite flat plate, and for flow over a semi-infinite plate with an arbitrarily specified surface heat flux. Since the Reynolds numbers encountered in storage conditions are typically low due to low air velocities, heat transfer due to natural or free convection may also be a significant factor. Solutions for the heat transfer coefficient resulting from natural convection of a hot or cold horizontal surface facing up were given by McAdams (1954), and modified by Goldstein, et. a1. (1973). In addition, solutions for the heat transfer coefficient resulting from induced flow parallel to a vertical wall were presented by Rays and Crawford, (1980). In the situations where both free and forced convection effects are comparable, correlating equations to include both forced and free con- vection effects have been developed by Churchill (1977, 1983). The free convection factor will tend to either enhance or decrease the forced convection effect, depending on whether or not the bouyancy force opposes or aids the forced convective motion (Rays and Crawford, 1980, and Incropera and DeWitt, 1985). Estimation of surface heat transfer coefficients using analytical methods is very difficult, especially for other cases, such as, tran- sient heat transfer, odd shaped geometries, and irregular flow patterns (Lightfoot et. al., 1965). Consequently, researchers have resorted to using experimental techniques in the estimation of surface heat transfer coefficients, and many different methodologies have evolved. Several researchers have used metal transducers to estimate heat transfer coefficients during freezing. Due to the high thermal conduc- tivity of the metal, the transducer is assumed to have negligible immernal resistance to heat transfer. Therefore, the surface heat transfer coefficient may be obtained from a logarithmic plot of dimen- sionless time against dimensionless temperature. Lescano (1973) 18 utilized geometric and kinematic similarity in using aluminum transducers to simulate heat transfer through codfish fillets during freezing. Creed and James (1985) used copper transducers in predicting heat transfer coefficients associated with plate freezers. In their study, estimations of the influence of packaging materials were made by placing a layer of the material between the transducer and the cooling medium. Bonacina and Comini (1972) used nonlinear regression between calcu- lated and measured temperatures to predict surface heat transfer coefficients. In this method the surface heat transfer coefficient was assumed to be constant, that is, not a function of time and/or position. Comini (1972) extended this study to investigate the design of optimum transient experiments for the determination of the surface heat transfer coefficient. These studies were based on work by Beck (1967, 1969). Beck utilized sensitivity coefficients in estimating thermal contact conductance, and in determining optimum, transient experiments for estimating conductance coefficients. Chavarria and Heldman (1983) also used nonlinear regression in estimating a convective heat transfer coefficient for ground beef during freezing. The coefficient was as- sumed to be constant during the freezing process, and heat transfer was assumed to be one dimensional. Succar and Hayakawa, (1986) used a surface response method for the estimation of convective and radiative heat transfer coefficients as a function of time during freezing and thawing of frozen foods. In this method, experimental and predicted temperatures were minimized using the method of least squares. Cleland and Earle (1976) presented a new method of estimating heat transfer coefficients from surface temperature measurements of a transducer with a thermal conductivity closely resembling a food product. Some of the restrictions of this method were that the l9 pro-cooling part of the freezing curve be sufficiently long, and that the center temperature of the body be unaltered for several minutes after the onset of cooling. Different numbers of cardboard sheets between Tylose samples and the cooling medium were used by Cleland and Earle (1977b) to estimate the relationship between the heat transfer coefficient and the number of sheets (thickness of packaging material). The previous methods assume the heat transfer coefficient is con- stant in the solution (with the exception of Succar and Hayakawa, 1986, who investigated transient heat transfer coefficients during freezing and thawing). Beck et. a1. (1985) presented method of estimating heat transfer coefficients as a function of time using ambient temperatures and temperature measurements from a sensor located inside the body. In this solution, the problem is treated as part of a class of problems called inverse heat conduction problems (IHCP). In the solution of the IHCP, the boundary conditions are determined instead of the internal temperature distribution which is found in the direct solution. Various methods have been proposed to solve the inverse heat conduction problem of determining a boundary condition at the surface of a body from dis- crete temperature measurements. Exact analytical solutions were developed by Burgraff (1964) and Langford (1976). These methods require continuously differentiable data. Stolz (1960) provided one of the earliest solutions to the IHCP, which was found to be unstable with small time steps. A similar method involving the numerical inversion of aiconvolution integral and utilizing future time steps was developed by Beck (1968). This method provides a solution at each time step, and is called the sequential function specification method. Osman and Beck (1987) used the sequential function specification method in estimating Imattxansfer coefficients as a function of position, using a spherical coordinate system. Other integral methods using Laplace transforms have 20 been demonstrated in one dimensional form by Imber and Khan (1972), and in two dimensional form by Imber (1974). Weber (1981) replaced the traditional heat conduction equation by a hyperbolic one to obtain a well-posed problem with established solution techniques. Regularization methods were proposed by Miller (1970), and Tikhonov and Arsenin (1977). These methods provide stability by the addition of smoothing factors and reduce the influence of measurement errors in the data. The influence of the regularization component is determined by the magnitude of a regularization parameter. Different criteria are found in the literature for the selection of this parameter. Tikhonov and Arsenin (1977) and Reinsch (1967) base their criteria on the errors in the measurements, while Murio (1985) considers in addition a bound based on the square of the L2 norm of the heat flux vector. Hills and Mulholland (1979) applied the method of Backus and Gilbert (1970) to a transient heat conduction problem. This method, adapted from geophysics, also utilizes smoothing function to stabilize the solution. Beck and Murio (1986) presented a new method which combines the sequential function specification procedure with the regularization method. This method differs from the global regularizaton methods in that the solution is found sequentially, greatly improving computational efficiency. This method was shown to be very competitive with the global regularization methods in terms of the heat flux estimates. Difference methods have been used to solve the nonlinear IHCP, whiCh cannot be solved using integral methods. Methods utilizing finite (Hfferences were demonstrated by Blackwell (1981), Beck (1970), Beck et. a1” (1982), and Williams and Curry (1984). In Beck's methods the same concepts are used to develop the algorithms as were used for the con- vohnflon based methods. A stabilizing matrix was utilized by Hensel and IHllsIJBB4) in developing a space marching finite difference algorithm. 21 Finite elements were incorporated in the solution by Krutz, et a1. (1978), and Bass (1980). It is important to note that the solution of the linear IHCP with the function specification and regularization methods are independent of the method of solution of the heat conduction equation because whether numerical convolution, finite differences or finite elements are used, nearly identical solutions are obtained (provided accurate approximations are used in each case). CHAPTER 3 THEORETICAL CONSIDERATIONS The three major problems analyzed in this study are: (1) the deter- mination of temperature profiles of food products in storage, (2) the estimation of the surface heat transfer coefficients encountered during storage conditions, and (3) the prediction of quality profiles within food products during storage. The determination of temperature profiles from known boundary conditions is called a direct problem, and it is considered to be mathematically well-posed. The surface heat transfer coefficient was estimated from internal product temperature measure- ments. This is called an indirect problem, and it is ill-posed. The three problems are interrelated in that the surface heat transfer coef- ficients are required as input for the direct problem; the temperature profiles resulting from the solution of the direct problem are required for the prediction of the quality profiles; and, the numerical solution of the direct problem is inherent in the solution of the indirect problem. In both the one and two dimensional solutions for the tempera- ture profile, and the estimation of the surface heat transfer coefficient, determination of thermal properties as a function of tem- perature is required. Also, both the two dimensional direct problem and fluzone dimensional indirect problem use the solution of the one dimen- shnml direct problem as a fundamental building block in the numerical analysis. 22 23 In the following sections, the evaluation procedure for the thermal properties is presented first, followed by an analysis of the one dimen- sional direct problem. The results in both of these sections are important in the solution of the two dimensional direct problem and the one dimensional indirect problem. The analyses and numerical procedures for these two problems are presented in the succeeding sections. Finally, the methods used to evaluate quality deterioration from calcu- lated temperature distribution histories within the food product are presented. 3.1 Thermal Properties Food products are primarily composed of water which contains various solutes. Due to the presence of solutes, the initial freezing point is depressed, compared with that of pure water. Consequently freezing occurs over a range of temperatures, and unbound liquid water can be present at temperatures associated with storage and distribution. The changing water fraction over a range of temperature results in tempera- ture dependent thermal properties in frozen food products. Accurate prediction of thermal properties is very important in estimating tem- perature distributions within the product. 3.1.1 Unfrozen Water Fraction The relationship between the unfrozen water fraction and temperature hsbased on the equality of the chemical potentials in different phases ifithin a system (Heldman, 1974b). The underlying assumptions of this 24 derivation are (l) the solution is dilute, and (2) the conditions ap- proach that of an ideal binary system. The derivation of this relationship results in the following equation (Moore, 1962) _ I 1 - 1 1n xW R [T T] (3.1) where, Xw is the mole fraction at absolute temperature (T), which is found from an experimentally determined initial freezing point (Tif)’ the latent heat of the solvent, (F), and the universal gas constant (R). substituting this value into the definition of mole fraction, shown below in Eq. (3.2), the effective molecular weight of the product solute may be found (Heldman, 1974b) Mwo/mw xw - M /m + M /m (3'2) wo w s s where mW and Mwo refer to the molecular weight and mass of the unfrozen water, respectively. In a food product, the product solute and solids are assumed to be indistinguishable; therefore, the mass and effective molecular weight, (Ms and ms), are of the combined solute and solids, and are hereby referred to as the mass and molecular weight of the solids. Furthermore, due to the binary solution assumption, the in- dividual effects of the carbohydrate, lipid, protein and mineral components of the food product are lumped together in the effective molecular weight of the solids (ms). By equating Eqs. (3.1) and (3.2) and substituting different values for temperature, T, the mole fraction of unfrozen water may be found for temperatures below the initial freezing temperature. Since thermal [upperties aré dependent on the relative amount of each component in the 25 food product, knowledge of the frozen and unfrozen water fractions as a function of temperature allows for the estimation of thermal properties during freezing and storage. 3.1.2 Density in Frozen Foods The temperature dependence of density in frozen foods can be predicted from the relative amounts of solids, liquid water, ice, and in some cases, air present in the product. The following relationship, including the air fraction contribution, is based on the density model without air utilized by Heldman and Corby (1975a), Hsieh et. a1. (1977), and Perez (1984) 1 gin/pp + (M1(T)/pi) + (MS/pg) + (Ma/pa) _ ——————————————- (3.3) MD MP where the subscripts w, i, s and a refer to the water, ice, solids and air components, and Mp is the total mass of the food product. Given the solids, unfrozen water and air mass fractions, and the product density above freezing, the solids density may be found from Eq. (3.3) as l. _ (MD/pp) + (MW/9w) + (Ma/pa) T > T (3 4) ps M8 if ' The relationship shown in Eq. (3.3) does not distinguish between the Various components of the solid fraction. The influence of the car- bohydrate, lipid, protein and mineral can be included as 26 1 (Mwumw) + /Pi> + (Me/PC) + (MID/p11) MD MP (“pr/Pvt) * ("Midlife/£2 M (3.5) P + Eq. (3.5) is difficult to utilize practically since the densities and mass fractions of the solid components must be known. Furthermore, Hsieh et. al. (1977) showed little variation in density as a function of temperature, consequently, any variability due to the individual solids components in Eq. (3.5) would generally be insignificant. 3.1.3 Thermal Conductivity in Frozen Foods Due to the large difference between the thermal conductivity of ice and water, the thermal conductivity of the unfrozen food product in- creases suddenly during freezing. Consequently, thermal conductivity is difficult to predict (Heldman, 1982). Kopelman (1966) developed relationships for thermal conductivity in two-component-homogeneous dispersed, fibrous and layered systems. These models assume that two phases are present; a continuous phase and a discontinuous phase. More than two phases are present in frozen food products (water, ice, solids and, in some cases, air); therefore, nmdifications of this model are required to include the additional phases. Heldman and Gorby (1975a) modified the Kopelman model to simulate a dues phase (water, ice and solids) frozen food product. In this model, Ufl>steps are required for the estimation of thermal conductivity. Additional modification has resulted in a three step model to include 27 air, if present. In the first step, the water fraction is considered to be the continuous phase and the ice fraction to be the discontinuous phase. For the second step, the combination of water and ice is assumed to the continuous phase, and the solids fraction is assumed to be dis- continuous. In the final step, the water-ice-solids combination is considered continuous, and the air fraction is discontinuous. The three step process is shown mathematically below. Step 1. Continuous phase: water Discontinuous phase: ice M,(T)/pi 1 ' Mw('r>/pw(r) + Him/pin) 2/3 Q1 - vi (1 - ki/kw) kwi - kw (3.6a) Step 2. Continuous phase: water-ice Discontinuous phase: solids _ 145/703 8 Mw(T)/pw(T) + M1(T)/pi(T) + AIS/pS 2/3 Q2 - VS (1 " kS/le) k k 1 - Q2 (3 6b) wis wi 1 _ Q2(1 - Vi/3) 28 Step 3. Continuous phase: air/water/ice Discontinuous phase: solids Ma/pa a ' MW - 9%? (3.7) 29 The enthalpy can be expressed in terms of the sensible heat removed from the solid, unfrozen water, ice and, in some cases, air fractions, and from the latent heat (F) as follows (Heldman and Singh, 1981) T H - I [ (Mw(T)'CPW) t + (MS‘CPS) + (Ma-ope) de(T) + Po dT dT (3.8) Note that the mass fractions of the solids and air fractions are not assumed to be functions of temperature. Substituting Eq. (3.8) into Eq. (3.7), results in the following expression for the apparent specific heat Cp + (M1~Cpi) + (MS-ops) + (Ms°CPa> de(T) dT (3.9) + r- The specific heat of solids can be found from an experimental value for the specific heat of the product above freezing Cp _ (MD/CPD) + (Mwo/pr) + (Mg/Cpél 3 Ms T > Tif (3.10) 2L2 Practical Evaluation of Thermal Properties The temperature dependence of thermal properties must be considered (fining the simulation of frozen food storage. Thermal properties may be (mmermined explicitly using Eqs. (3.3), (3.6a-c), and (3.9) for density, thermal conductivity, and specific heat, respectively. However, use of 30 explicit functions in a numerical solution is computationally ineffi- cient. Alternately, a scheme was developed to determine constant thermal property values over specified temperature intervals which were selected to minimize errors in the solution. The temperature intervals, over which thermal property values were considered constant, were selected by limiting (1) the change in the magnitude (that is, the first derivative) and (2) the change in the slope (that is, the second derivative) of the thermal property function over a given temperature interval. The derivative values were estimated numerically from Eqs. (3.3), (3.6a-c), and (3.9). The constant property values over each temperature interval were determined numerically using a five point Gauss quadrature integration method (Hornbeck, 1975). Constant thermal conductivity values and associated temperature ranges are compared in Figure 3.1 with thermal conductivity values found using Eq. (3.6a-c). (Note, in this case, kp - 0.94 W/m C, Mwo - 77%, Tif - 0.7°C.) These values were determined by limiting the increase in both magnitude and slope of the density curve over each constant property interval to 5% and 25%, respectively. Similar constant property tem- perature intervals were determined for thermal conductivity and specific heat. 3.3 Transient Heat Conduction during Frozen Food Storage To predict quality loss in frozen food storage, the temperature dhnxibution within the product as a function of time must be estimated amnnately. This involves the solution of the transient heat conduction Pnflflem. The complexity of the solution arises from the temperature dependence of the thermal properties. 31 .CHSDNHCQECH mo CoHuocsm m we mozam> hua>wuospcoo HmEMCLH £uw3 PO mosam> xufi>muospcoo Hmanose ucmumcoo mo comwuodeoo H.m madman Gov ouzyouanoa. ml O—l m—I ONI ......NI onl mnl owl D IP H b b b b b I I— - II—I b h D .023... 35025 .63 x “c3300 ..I 0.0 ES. o.coto> .. . Ind I 0.3.. .. osod screen as... use I “con—coo 2330: IO._. use; :3 .... 33328 .855. nabs. 32 .. assoc sentenced «cavern. senate: In; (GOLD/M) Ail/\Qonpuoo [DwJGLLL 32 The assumptions used in this study were: 1. Heat transfer between the surroundings and the food product occurs by convection only, and heat transfer within the product proceeds by conduction. 2. The frozen food product is isentropic and homogeneous. 3. The surface heat transfer coefficient associated with convective heat transfer is a constant or a function of time, but not a function of position. 4. Moisture loss from the product is negligible, and total product mass remains constant. 5. Internal packaging boundaries within a large mass (such as a pallet load) of product have negligible affect on the heat transfer rate. 3.3.1 One Dimensional Heat Transfer Analysis In this analysis, it is assumed that heat transfer occurs in one dimension only. The Fourier one dimensional transient heat conduction equation describes one dimensional conductive heat transfer through an isentropic medium (Carslaw and Jaeger, 1959). The governing partial differential equation for regular geometries is given by . J. 5’1 - . fl xj a [x 1<('1‘)ax p(T) Cp(T)at (3.11) j - 0: infinite slab j - l: infinite cylinder j - 2: sphere An infinite slab is finite in the direction of heat transfer, and it is infinite in the other two dimensions. An infinite cylinder is finite is 33 the radial direction (direction of heat transfer) and has infinite length. Heat transfer is assumed to occur only in the radial direction in the sphere. The various geometries and indicated directions of heat transfer are shown in Figure 3.2. Since the product properties are functions of temperature, the problem is nonlinear. An initial condition is required for the solution; the product is assumed to be at a uniform temperature, or at a temperature distribution that is a known function of position. T-To IaXoSXSIax or T - To(x) t - o (3.12) Note, on - 0 for a slab, solid infinite cylinder and sphere, and on # O for a hollow infinite cylinder. Two boundary conditions at x - on and at x - Lx are required be- cause of the second order differential with position in the governing partial differential equation (Eq. (3.11)). Convective heat transfer is assumed to be occurring at the surface of the product. For the case of symmetrical boundary conditions on both sides of an infinite slab, or a solid infinite cylinder, or a sphere, the boundary conditions are OI - o x - o (3.13s) 6x x-O t > 0 km£21 - hx (t)-[T - T (t)] x - Lx (3 13b) ax x-Lx Lx m,Lx t > 0 . Iftim.boundary conditions are asymmetrical, or if the geometric shape is a hollow cylinder, the convective boundary condition at x - 0 is 34 .mownuoeooo HmconcCEHQ 8 mac utouommwa mom Hommcmuh use: poEdmm< mo coHuooHHQ N.m ouswfim .x\ _ xx/ / Lr \ - \ ///:t:.lir:i\I\\\\ x .iIIIII poem ounceucH _ \\\.IIITIII/ \\ // 8 / _ .J \r ouonem \ //I \\\ _ 8 Decrease ouudHMdH 35 Q1 k(T)ax x-Lx - thxo(t)o[T - Tm’on(t)] x - on (3.13c) 0 3.3.2 Two Dimensional Analysis Heat transfer in two dimensions is considered for rectangular and cylindrical geometries. The governing partial differential equation is 11.3; [ xj-k(T)%§ ] + g; [ k(T)%§ ] - p(T)°CP(T)%% (3 14) x j - O: infinite rectangle j - l: finite cylinder An infinite rectangle is finite in two dimensions (directions of heat transfer) and infinite in the other dimension; in this case, a finite cylinder has a finite radius and length (directions of heat transfer), with no angular heat flux. The geometries and assumed directions of heat transfer are shown in Figure 3.3. The initial condition is assumed to be constant, or a known function of position IAO ‘< IA IA x I" IA s4 or T - To(x) t - O (3.15) idmre on - 0 for an infinite rectangle or solid cylinder and on # 0 for a hollow cylinder. 36 nopcmamo opach .mowuuoaooo HmCOHmCCEwa 038 uConommHa How Howmcmwe use: poEdmm< mo Gawuoouwn m.m ouswwm oawcmuoom ouwaamcH 37 Four boundary conditions are needed in this case because of the second order differential in both dimensions. The boundary conditions for symmetrical heat transfer along the x-axis of an infinite rectangle, or along the radial direction of a solid cylinder are given by Eqs. (3.13a,b) for O s y s Ly. Equations (3.13a,c) describe the boundary conditions for unsymmetrical heat transfer along the x-axis, or along the radial axis of a hollow cylinder for O s y S Ly. The boundary conditions along the y-axis are k§§| - hYo(t)'LT - Tm’o(t)l y - 0 (3.16a) Y'O on _ x _<_ Lx t > o k(T)fl| h (t)-['r - T (t)] - L (3 16b) 63' _L yLy «nLy y y ‘ y y on S x S Lx t > o 3.4 Numerical Time-Temperature Simulation Models The one dimensional simulation model for frozen foods during storage was based on the predictive models developed by Lescano (1973) and Heldman and Gorby (1975b) for estimating freezing times. The Crank-Nicolson implicit finite difference scheme was used to numerically solve the governing partial differential equation given in Eq. (3.11). The two dimensional prediction model was based on the Douglas and Gunn (1964) Alternating Direction Implicit (ADI) scheme. The scheme was modified to allow temperature dependent thermal properties. In both Inadels, thermal properties were assumed to be constant over specified temperature ranges, using the procedure described in Section 3.2, while 38 allowing for variation over the total temperature range in considera- tion. In addition, both models permit a number of different storage periods with different storage temperatures and convective heat transfer coefficients during each period. Input parameters required for the solution of both the one and two dimensional problems are: 1. Initial freezing temperature. 2. Unfrozen water and air fractions. 3. Thermal properties of the unfrozen food product. 4. Temperature of the frozen product prior to storage. 5. Product thickness or radius, or length (for the two dimensional model). and for each storage period: 6. Length of storage period. 7. Ambient storage temperature. 8. Heat transfer coefficients (on all sides of product). In addition, for the two dimensional model: 9. Product length. 3.4.1 One Dimensional Heat Transfer Finite Difference Scheme The one dimensional heat transfer given by Eq. (3.11) was solved nmmerically for an infinite slab, infinite cylinder and sphere using the Crank-Nicolson finite difference scheme, and temperature dependent thermal properties. The resulting implicit finite difference equation for'an interior node is 39 k-A- Tn+1 _ (k-A- + k+A+) + (( C ) + ( C ) )AX°A Tn+1 + k+A+ Tn+1 " Ax 2-1 " Ax P P - P P + 2-At 2 " Ax 2+1 k A (k A + k A ) Ax-A k A - - n - - - + + n + + n ' ’ Ax Tz-l + I P Ax ' ((pCp)-+ (pCP)+)2-At I T2 ' 5 Ax I2+1 (3.17) where B - n - 0.5 for the Crank-Nicolson method. The thermal properties ( p, k, and Cp ) are evaluated at the nth time step, and at the loca- tions indicated in Figure 3.4. Equation (3.17) may be used for various geometries by using the appropriate cross sectional areas as shown below. A - A - A - 1.0 } for infinite slabs A - 2t-[on + Axo(£-3/2)] A - 2x-[on + Axo(£-1)] for infinite cylinders A - 2x-[on + Ax-(2-1/2)] + A - a«-[on Ax-(2-3/2)]2 ‘ Ax-(£-1)]2 > I 4“. [LXO + for spheres + A - a«-[on Ax-(2-1/2)]2 ‘where I - 1 at the location on. The finite difference representations of the boundary conditions given in Eqs. (3.13a,b,c) are 40 - A I + . I 221 I" + 23.1 I I (AX/4*] A ”b .......__ O __._.. 'U v I To. A b 0.. 'U v + Figure 3.4 Evaluation of Thermal Properties in One Dimensional Numerical Solution. 41 At x - 0, (2 - 1) I '7 Ax P P + 2-At 1 ’7 Ax 2 k A Ax-A k A + 1+ 1 n + 1+ n - In:— - “MW—mt I'Tx + A? T2 n n n+ n+1 .. (fio'rm,onethxo + ".TQ,LXo.th-XO)A1 (3.18a) At x - Lx, (£ - L) " Ax L-l Ax 2-At L k k Ax-A k -:£L: rn+1 - [ n——5L; + <_>- L J-T“+1 - - s—;§L= T“ k Ax- + I p-géL; - <_)-§:;%L I-TE - (3'T2,Lx.hx2x + ".TZTLx'th;1)AL X (3.18b) Note: in Eq. (3.18a), hxl is equal to zero for the insulated slab, solid cylinder, and solid sphere, and in Eq. (3.18b), L is the total number of nodes. The set of finite difference equations may be expressed in matrix form as A-Tn+1 - nor“ + D (3.19) ‘The coefficient matrix, A, is tridiagonal. The set of equations is solved by using the Thomas Algorithm (Thomas, 1949), which makes use of the large numbers of zeros in the coefficient matrix to solve the equa- tions efficiently. 42 A flow chart for the one dimensional numerical temperature solution, including the prediction of the quality profile, and the code for the computer program, written in Fortran 77 on a Vax 11/750 computer, are presented in Appendix B. 3.4.2 Two Dimensional Heat Transfer Analysis The two dimensional Alternating Direction Implicit (ADI) method proposed by Douglas and Gunn (1964) was modified to include temperature dependent thermal properties. In this method, the Crank-Nicolson scheme is utilized, and a two step procedure was employed. The modified dif- ference equation for an interior node is shown below. Step 1. Use Crank-Nicolson approximation in the x-direction k A n+6 k A + k 71%; T1-1,m - [ "X -x «Ax +x +x + ((pCp)_x+ (pCp)+x+ (.o_ x+ (pep)+x+ for solid cylinder A+x - 2«o[on + Axo(£-l/2)]Ay J A - Ay - A+y - n-Axo[on + Axo(£-1)] where 2 - 1 at the location on. The Douglas-Gunn ADI method differs from the methods proposed by Peaceman and Rachford (1955) and Douglas (1955), because, in the first sweep, a full time step is used to provide an estimate of the tempera- ture values at n+1; these values are used in the second equation and are denoted as n+5. The traditional methods evaluate the first sweep at n+1/2, the one-half time step. Eight different expressions for the boundary nodes (four surfaces and four edges) were derived in a similar manner as for the one dimen- sional case. These difference equations are found in Appendix A. A flow chart, describing the numerical solution of the two dimen- sional heat conduction problem, and the corresponding computer code (Fortran 77) are given in Appendix C. The two subroutines, 'PROPER', and ‘CONSP', are identical to the same subroutines in the one dimen- sional model, and are omitted in the code listing. 3.5 Estimation of the Surface Heat Transfer Coefficient Analytical, numerical and experimental methods have been proposed to determine the surface heat transfer coefficient. Traditional analytical methods of determining the surface heat transfer coefficient are presented first. This is followed by a investigation of a numerical method which uses experimentally determined internal temperatures to estimate the surface heat transfer coefficient. In this case, the 46 problem of estimating the surface heat transfer coefficient is treated as part of a group of problems called inverse heat conduction problems. 3.5.1 Analytical Methods The surface heat transfer coefficient is a function of the air stream velocity, the temperature difference between the surface being heated or cooled and the surroundings, and the packaging layer between the free air stream and the product. The Reynolds number (ReLx) is used to characterize the free stream velocity, and the Grashof number (Ger) is used to characterize the temperature gradient as ReLx - —°5—-—° (3.21a) g-fi,-°3 Ger - 2 (3.21b) V where Udo is the air free stream velocity, u is the kinematic viscosity of the air, g is the acceleration due to gravity, Be is the expansion coefficient of air, T8 is the temperature of the surface, and Lxc is the characteristic length. The characteristic length is defined as the ratio of the cross sectional area and the perimeter of the surface (Goldstein, et. al., 1973). Forced convection, resulting from forced air flow, and free convec- tion, resulting from temperature induced density gradients, are both important when the square of the Reynolds number and the Grashof number are approximately equal. Forced convection effects dominate when (Relx)2 is much greater than Ger, and free convection effects dominate when.the opposite is true. These relationships are summarized below. 47 Gr -—%3—— >> 1 Free Convection only (3.22a) ReLx GrI 2 z 1 Both Free and Forced (3.22b) ReLx Convection Gr -—%5- << 1 Forced Convection only (3.22c) ReLx 3.5.1.1 Forced Convection over a Flat Plate The steady state one dimensional for the determination of the mean Nussult number (ESE) resulting from a constant free stream velocity flow along a constant temperature semi-infinite plate, and along a semi- infinite plate with an arbitrarily specified surface heat flux are (Rays and Crawford, 1980) Nu - 0.664-Pr1/3-Re11‘;2 Constant Temperature (3.23a) Laminar Flow, Pr 2 0.6 EEF - 0.906-Pr1/3-ReL/c2 Constant Heat Flux (3.23b) Laminar Flow where h—oLx fig _ _EE__. F ka E - 2-hxF(x) k - thermal conductivity of fluid (air) 48 3.5.1.2 Free Convection Free or natural convection is a result of temperature induced den- sity gradients in the fluid. These temperature gradients cause free convection currents as the denser fluid (cooler fluid) falls, and the less dense fluid (warmer fluid) rises. The Nussult numbers resulting from free convection (NuN) for the upper surface of a horizontal heated or cooled plate, are given below (McAdams, 1954). 1. Upper surface of a heated plate (Ts > T ). Q - 1/4 4 7 NuN - 0.54-RaLx 10 s RaL S 10 (3.24a) —- 1/3 7 11 NuN - 0.15-RaLx 10 S RaL s 10 (3.24b) 2. Upper surface of a cooled plate (T0° > T8). -— _ . 1/4 5 10 NuN 0.27 RaLx 10 s RaL s 10 (3.25) where the Rayleigh number, Ra is defined as Lx 3 3% -(T - T )-(Lx) Ra - Gr oPr - e g’ m C Lx Lx V-n 3.5.1.3 Combined Forced and Free Convection For the case where (Ger/Reix) z 1, a correlating equation has been recommended as (Churchill, 1983) -—n -n -—n - + Nu NuF _ NuN (3.26) 49 where n - 3 or 7/2 for parallel or perpendicular flows respectively. The natural convection component may increase or reduce the influence of forced convection depending on whether or not the free convection in- duces motions in the same or opposite direction as the forced air flow. The heat transfer coefficient resulting from free and forced convec- tion is hx - -——43 (3.27) 3.5.1.4 Packaging Layer The thin packaging layer between the product surface and the sur- rounding air may result in additional resistance to the flow of heat to the product. This resistance may be characterized by an effective packaging resistance, defined as k __pL hpk L.Pk (3.23) where kpk and ka are the overall effective thermal conductivity and length of the packaging layer, including the packaging material, air interfaces, and frost build-up, if any. 3.5.1.5 Overall Surface Heat Transfer Coefficient The overall surface heat transfer coefficient is a result of the com- ‘binsd thermal resistance from forced and free conduction, and the 50 packaging layer. The overall surface heat transfer coefficient is defined as hx h (3.29) 3.5.2 Inverse Heat Conduction Methods. The estimation of the surface heat transfer coefficient as a func- tion of time is based on the solution of the inverse heat conduction problem (IHCP). In contrast to the direct problem of determining inter- nal temperature distributions from known boundary conditions, the IHCP involves the determination of a boundary condition from internal tem- perature measurements. In this study, one dimensional heat transfer in considered through a regular geometrically shaped food product with variable thermal properties. A temperature sensor is located at x - x0, where measure- ments are taken at a discrete time interval, Atm. The boundary at x - O is insulated, and an unknown heat flux is imposed at the opposite bound- th ary. The temperature distribution is assumed to be known up to the n time step, tn, (tn - n-Atm). The problem is mathematically expressed as 13.3; [xj-unfi] - p-Cp t (3.31) Note this problem can be divided into two parts, the direct problem from x - 0 to x - x0, and the inverse problem from x - x0 to x - L, as shown in Figure 3.6. The surface heat transfer coefficient was determined by first estimating the surface heat flux at x - Lx. In this analysis, both the heat flux and the surface heat transfer coefficient, were assumed to be a function of time, but not position. The heat flux was first estimated as aIfunction of time using the regularization method and finite dif- ferences in the solution. The temperature at the boundary, T(Lx,t), was calculated using the finite difference solution described for the direct 52 Y(t) Inverse Figure 3.6 Inverse Heat Conduction Problem. \\\--_______,,.—””’/’ \ \ \ \ \ \ \ 7:. \ x0 \ \——-—-- X \ \ \ \ \ Direct Problem Problem q(t) unknown 53 problem in Section 3.4.1, and substituting the calculated surface heat flux values for the convective boundary condition in Eq. (3.18b). Finally, the surface heat transfer coefficient was calculated from Eq. (3.30e) given the predicted surface heat flux and surface temperature, and the known ambient temperature. Two variations of the regularization method used to estimate the surface heat flux are found: (1) the whole domain technique, and (2) the sequential technique, proposed by Beck and Murio (1986), in which the heat flux components are estimated sequentially. The sequential procedure is more computationally efficient than the whole domain method because, this procedure estimates only a few heat flux components at a time instead of simultaneously estimating all of the components, as done in the whole domain procedure. Due to its computational efficiency, the sequential regularization method was used in this study. 3.5.2.1 The Sequential Regularization Method The modified least squared function of the regularization method in matrix form is 2 s - (Y-T)-(Y-T)+a-}Rj (3.32) j-O where'Y and T are the measured and estimated temperature vectors, and the terms in the summation (R0,R1,R2) represent the zeroth, first and second order regularization components. The scalar term, a, is the regularization parameter which is adjusted to determine the degree of the influence the regularization terms have on the least squares func- tion. 54 Tikhonov and Arsenin (1977) suggested the following expression for the various regularization terms t J - . £14 _ Rj W3 I [ drj dr j o, 1, or 2 (3.33) Each regularization term acts to minimize its corresponding derivative of the estimated heat flux when the least squared function, 8, is mini- miz ed. Therefore, the zeroth order regularization term tends to bias the heat flux towards zero, the first order pushes the heat flux towards a constant value, and the second order term forces the heat flux towards a constant slope, Beck et. a1. (1985). Forward differences were used to approximate the regularization terms in Eq. (3.33). The expressions for j - O, l, 2 are shown below in EqS - (3.34a,b,c), respectively. _ 2 R0 ' wo'} (qn+i 1) (3.34a) i-l r _ 2 R1 - 14,- E (qn+1 - qn+1 1) r 2 1 (3.341)) i-l r ° _ 2 R2 _ W2. E (qn+i+l - qn+1 + qn+1 1) i-l r 2 2 (3.34C) wh Qre the W1 values are weighting factors. In the sequential procedure, tlb) e summation is carried out over r-future time steps, and qn+i+l, qn+1 h a - hd an' 1 are evaluated at the (n+i+1)t , (n+i)th and (n+1-l)th time 55 steps, respectively. The heat flux at qn'1 is assumed to be known. Eqs - (3.34a,b,c) can be written in matrix form as: T R -W-H H -0,1,r2 3.35 where the estimated heat flux components are contained in the vector q, the forward difference approximations for dq/dt are contained in the H J matrices, and the weighting factors, WJ, are scalar. Beck et. a1. (19 85) proposed the following expressions for the H3 matrices Ho - I (Identity matrix) (3.36a) ' -1 1 o o I O -1 l 0 0 H1 - o . (3.36b) O O -l 1 I o o o _ P l -2 1 0 . . . O I O 1 -2 l 0 0 H2 - ' ' (3.36b) 0 0 l -2 1 O 0 O _ O O O _ The H matrices are r X r with j + 1 non-zero diagonals, and with all J z Q“Toe. in the last j row(s). The regularization orders were analyzed individually by setting the .th 3 weighting factor (Wj) equal to one, and the remaining weighting 56 factors (W1, 1 I‘ j), equal to zero, and substituting Eq. (3.35) into Eq - (3.32) to obtain 3 - (Y-T)2(Y-T) + a-W -(qu)T(qu) j - o, 1, or 2 (3. .1 The unknown heat flux q is estimated by minimizing the least squares 37) function in Eq. (3.37). The expression is differentiated with respect to q and then set equal to zero. T A T A vqs - 2 [Vq(Y-T) ](Y-'1‘) + 2a-Vq(qu) (qu) -o j-0,l,or2 (3. The temperature matrix '1' may be expanded in a Taylor's series as '1‘ - 13* + Vq'r(q - q*) (3. * Whe re '1' is the resulting temperature vector from an assumed imposed * heat flux q . A sensitivity coefficient matrix, X, is defined as x - v1 (3 q F 1301:: Eq. (3.40), Vq(Y-T)T - -xT (3 Vq(qu)T - a? j - o, 1, or 2 (3. J 38) 39) .40) .41) 42) 57 Beck et. a1. (1985). Substituting Eqs. (3.39), (3.40), (3.41) and (3 - 42) into Eq. (3.38) produces XTX(q - q*) + a-HJTHJq - XT(Y-T*) - 0 j - 0, 1, or 2 (3.43) No t ing that con Tn q* - o (3.44) J J allows Eq. (3.43) to be rearranged and solved for q: q — q* + (xTx + a-H T11 SloxTa-r") j - o, 1, or 2 (3.45) J J The temperature vector T? was found by solving the direct problem “‘3ii:r1g the Crank-Nicolson finite difference method as discussed in SeCdzion 3.4.2, and substituting an assumed imposed heat flux (1* at x “' 'Lx, in place of the convective boundary condition shown in Eq. (3 - 18b). The sensitivity coefficient matrix X, is found by differen- t5i~£11:1ng Eqs. (3.30a-d) with respect to q(t), the unknown heat flux at x :I—fit. The resulting equations are shown below. -1-.-3— [xjok(T)g—)'}:] - p(T)-Cp(T)g')t‘: (3.46a) j - 0: infinite slab 1: infinite cylinder (.1. I j - 2: sphere 8x x-O t a tn (3.46b) 58 kmfi m-l x-m x- t 2 tn (3.466) X-O OSXSLx t (3.52) Substituting Eq. (3.50) into Eq. (3.52) and performing the indicated d1 fferentiations yields V(Q(x.t)) - 1°V(Qor) + {E I exP{'%§[ T(x t) ' %_I I ’ r . {% ITTI'CB' - t] }0At]}2oV(Ea) (3.53) In summary, this procedure allows the estimation of quality loss and L ts associated variance in frozen foods as a function of time and posi- tion, based on predicted internal temperature measurements and estimated h: ihetic parameters and associated variances. CHAPTER 4. EXPERIMENTAL PROCEDURES Frozen foods in fluctuating temperature storage conditions were simulated experimentally using the Karlsruhe Test substance (Gutschmidt, 1960), as a substitute food product. The test substance was alternately placed in two adjacent storage chambers (A and B) with average ambient temperatures of -6 and -30‘C, respectively. Temperature distributions within the test substance were determined from thermocouple readings recorded by a data aquisition computer. These measurements were used in confirming the one and two dimensional finite difference models, dis- cussed in Section 3.4, and in estimating the surface heat transfer coefficients, discussed in Section 3.5.3. 4.1 Karlsruhe Test Substance The Karlsruhe Test Substance, developed at the Federal Research Institute, West Germany, is a highly concentrated methyl-cellulose mixture, which has similar thermal properties and freezing characteris- tics as common food products. Methyl-cellulose mixtures have been used in studying the freezing process in foods by many researchers, such as, Bonacina, et. a1. (1973), Cleland and Earle (1977b), and Succar and Hayakawa (1986). 62 63 One half gallon paperboard ice cream containers, measuring 0.170 m long, 0.125 m across, and 0.90 m high, were used as the product con- tainers. The test substance was shaped into brickettes to fit into the boxes, wrapped twice with plastic film to prevent moisture loss, and the placed in the paperboard containers. The thickness of then paperboard boxes and the two layers of plastic film were measured with a micrometer. The paperboard box thickness averaged 1.7 mm, and the plastic film averaged 0.3 mm. The volume of the containers was 0.0019 m3. Assuming a density of 1040 kg/m3 for the Karlsruhe Test Substance (Specht et. al., 1981), 1.98 kg of test substance were required for each container. The following procedure was followed to prepare the test substance (Gutschmidt, 1960): 1. The water content of the supplied methyl-cellulose (MC) was found from the manufacturers (Dow Chemical, U.S.A), and the additional water required for a test substance with 77% moisture content was determined. This amount of water was increased by approximately 4% of the weight of the methyl-cellulose, since, as Gutschmidt, (1960) has shown, this amount of water vaporizes in the warming of the water, and in the stirring, kneading and forming of the methyl—cellulose. 2. Salt (NaCl) was added to depress the freezing point of the methyl-cellulose from -0.6°C to -1.0°C. The amount of salt added was determined from Adballa and Singh (1984). SALT (kg) - 0.0240(1 - 0.01-MC) 3. In addition, 1 gm parachorometacresol was added as a Preservative for every 100 grams added water. 6h 4. The water was warmed in a large beaker to approximately 60-70°C, and then the heating source was turned off. 5. The salt and parachorometacresol were added while stirring continuously. 6. The methyl-cellulose powder, Methocel A4m Premium, supplied by Dow Chemical, U.S.A., was slowly poured into the salt water mixture, and was stirred continuously until the solution became homogeneous. 7. The test substance was allowed to cool to 35-40’C. 8. The substance was kneaded until it formed a bread-like dough. 9. Brickettes were formed by flattening the sides on a smooth planar surface. They were then wrapped in plastic film, and placed in the paperboard containers. The following quantities were required for every 1 kg Karlsruhe Test Substance: 0.772 kg water 0.237 kg methyl-cellulose (4500 cps, 2% moisture content) 5.57 gm NaCl 0.76 gm parachlorometacresol For better mixing, the test substance was prepared in 0.99 kg batches, and two batches were combined just prior to kneading to form each brickette. There is some discrepancy between the various thermal properties for the Karlsruhe Test Substance given in the literature. The initial freezing point given by Gutschmidt (1960) has not been used by other researchers (Specht, et. al., 1981). Values for Tif’ pp, kp, and Cpp 65 are shown in Table 4.1. The properties given by Specht, et. a1. (1981) were used in succeeding calculations for consistency, unless otherwise noted. 4.2 Containers for the Karlsruhe Brickettes The Karlsruhe Test Substance was used in the following three ways: (1) in the determination of the surface heat transfer coefficients; (2) for comparison with the one dimensional numerical model; and (3) for comparison with the two dimensional model. This resulted in three configurations for the containers to hold the Karlsruhe samples. The first configuration was used in estimating the surface heat transfer coefficient. A container was constructed to hold three ad- jacent pairs of brickettes with an exposed top surface, and insulation around the remaining three sides. An open topped box or trough, con- sisting of 0.0125 m plywood, formed structural support for the container. Foam insulation, 0.077 m in thickness, was glued to the interior sides of the box. This insulation thickness was based on that used by other researchers using the Karlsruhe Test Substance (Cleland and Earle, 1977b, and Succar and Hayakawa, 1986). Values for the ther- mal conductivity of the foam insulation found in the literature varied around 0.28-0.31 W/m°C (Baumeister, et. al., 1978) and 0.35 W/m°C (Cleland and Earle, 1977b). The box was constructed in two parts, and clamped shut after fill- ing, for better packing of the Karlsruhe Test Substance brickettes. The container with brickettes in place is shown in Figure 4.1. For the second configuration, a similar container was constructed, but without a bottom, to hold six brickettes in a single layer. This layer was secured to the top of the first configuration to form a double 66 Table 4.1 Thermal Properties of the Karlsruhe Test substance Moisture Initial Densigy Thermal Specific Reference Content Freezing (kg/m ) Conductivity Heat (%) Temperature (W/m'C) (kJ/kg’C) (°C) 77% -l.0 Gutschmidt (1960) 77% -0.7 1040 0.944 3.8 Specht (1981) 67 12.7 cm Hypodermic Boxes with Needles (13 83) Karlsruhe Test Substance (6) L1“ [.47 “,1 -/ [.17 .MI/ Insfiii'é'ion T;' /\\IN/V///é 1.2 :1: agiywood m 2 | = 2:. 229,57 A? L .;//.\\;\\4;/ 3: Thermocouple /‘ TOP VIEW Bead¥$:c;?p33 3a) I; 81 cm / :I 6% cm I 21.1% I -’ I-- h - I - :r// I 13in: l/,///////\\\\\v \2 Figure 4.1 Single Layer, One Dimensional Container Configuration. 68 layer slab with only the top side exposed. The configuration is shown is Figure 4.2. In the final case, two adjacent sides were exposed to allow two dimensional heat conduction. An insulated plywood box was constructed to hold twelve brickettes, in three layers. The container holding the brickettes is shown in Figure 4.3. 4.3 Temperature Measurement Internal product temperature measurements were obtained using 30 gage, Type T, thermocouples (Omega, 1985). The thermocouples were placed at the geometric center of each brickette through 0.127 m long 18 ga hypodermic needles. Prior to placement in the brickettes, the thermocouple wire was threaded through the needles, and soldered at the point of the needle (Figure 4.4). The end cavity of the needle was filled with epoxy to keep the thermocouple in place. The hypodermic needle was placed through a hole drilled through the plywood and the insulation at the desired thermocouple location. This served as a guide for the hypoder- mic needle to minimize the variability of the thermocouple location within the brickette. The ends of the thermocouples were insulated with foam to limit conduction down the thermocouple wire and hypodermic needle. The placement of the thermocouples in the three configurations are also shown in Figures 4.1, 4.2, and 4.3. The ambient temperatures were measured using a thermocouple placed above the Karlsruhe container, midway between the container and the ceiling of the storage chamber. All thermocouples measurements were recorded by a Hewlett Packard 3497A Data Aquisition/Control unit, coupled to a Hewlett Packard 85 desk 69 12.7 cm Hypodermic Needles (18 ga) Thermocouple Bead (Type T, 30 ga) 9 cm 1.2 cm Plywood ‘j ll:— 17 cm‘—>l I t:- 17 cm 76 I /////I Foam I ////// Insulation I V . 35.4 CHEEEEEi ' . I ;:iffj1;1;. 1 I\ \4 \ ' —I ///2I///.\\L\\\//ji- / 9 cm I Ii II il \ Thermocouple.z/’ TOP VIEW Boxes with Connector Karlsruhe Test Substance (12) 81 cm Ifi . I //f///\\\\\ Thermocouple VIEW Boxes Joined Locations SIDE Together Figure 4.2 Double Layer, One Dimensional Container Configuration. TO Foam TOP VIEW 1.2 cm Plywood Insulation Frame / "‘VWQ I 6.3 cm 21.2 cm . /..-- “-2. 12.5 cm‘ ijj; éjjjg 6.3 ch'é-7 “'t: "-12.5 cm Thermocoupljl I._17 cm _,I age: with Bead (Type T. 30 88) Karlsruhe Test Substance (12) I 2 //7 9c;rr_:/-—-— ---- I 9cm I ****** '""'" I I / 9cm V l7““‘“ “"“" .\ ....» /////////// 2:322:25“ 12.7 cm Hypodermic Needles (18 ga) I‘—-l7 cm 4.. SIDE VIEW 9 cm ”I Figure 4.3 Triple Layer, Two Dimensional Container Configuration. Tl .haaaomm< oaoooz owauooonhm use oaasoooauocfi ¢.q ouswwm any useooz fl. :owuocsh oaaaoooEHoLH pas: coauamflsv< some aouu owed oamzoooapoce ou uouooscoo /. OHH3 oaasoooaHoSH am On \ caemoz oaauoeoesm a kua.o an am we 72 top computer. The capacity of the data aquisition device was limited to a maximum of 16 simultaneous thermocouple measurements over a total of 216 time steps. 4.4 Velocity Measurements The air free stream velocity was required for the analytical deter- mination of the surface heat transfer coefficient. The two storage chambers used in the study measured 2.20 m across by 6.90 m long by 2.53 m high for chamber A, and 1.81 m across by 6.90 m long by 2.53 m high for chamber B. The forced air flow in both chambers resulted from two 0.37 kW fans., placed in the upper end corners of both chambers. The fans were at the same end in chamber A, and they were at opposite ends in chamber B. Velocity measurements were taken at 0.045 m incre- ments in a grid pattern about the region where the Karlsruhe containers were placed, using a hot wire anemometer (Model 2440, Weathertronics, Inc.). The measurements were taken in the middle of a defrost cycle. The horizontal velocities in chamber A ranged from 0.1 - 0.6 m/s and averaged 0.25 m/s, and ranged from 0.8 to 1.25 in chamber B, and averaged 1.0 m/s. The airflow in chamber A was lower due to a large obstruction in the storage room near the fans. Vertical velocity measurements in chamber B varied from i 0.5 to i 1.0 m/s, indicating mixed air flow conditions. Streamers were used as a visual conformation of the flow conditions. The streamers, made of magnetic tape, were hung from string placed at 0.45 m intervals, in a grid pattern, across each chamber. Consistent with the velocity measurements, the streamers fluttered up and down randomly in chamber B, and showed comparatively streamline characteristics in chamber A. 73 4.5 Experimental Storage Conditions The Karlsruhe brickettes were prepared just prior to placement in the plywood containers, and then immediately placed in chamber B until equilibrium conditions were obtained (approximately four to five days). One complete step change cycle in storage temperature was used for all configurations. The Karlsruhe Test Substance, initially at equi- librium in chamber B, was placed in chamber A for a given storage time period, and then placed back in chamber B for an equivalent time period, to complete the cycle. The storage period was limited by the capacity of the data aquisition unit. Since the temperature measurements ob- tained using the first configuration, shown in Figure 4.1, were required in the sequential regularization solution for the surface heat transfer coefficients, a smaller time interval for the temperature measurements was used, than in the other two cases. The time steps and the storage time periods for the three cases are shown in Table 4.2. Each test was repeated three times, using the same Karlsruhe samples for each configuration. The samples were allowed to equilibrate in chamber B after the conclusion of each test. In subsequent sections, the three repetitions using the single layer, double layer, and two dimensional triple layer configurations will be referred to as Tests la-c, Tests 2a-c, and Tests 3a-c, respectively. The defrost cycle period was approximately 1.5 hours in storage chamber A, and four hours in chamber B. Some frost accumulated on the surface of each container after storage in chamber B. The average storage temperature in chamber A was -6°C for all test cases. The average storage temperature varied from ~33 to ~34°C for all three tests using the first configuration (single layer), but due to a failing Table 4.2 Storage Times and Measurement Intervals 74 for the Three Configurations Storage Period Measurement Test in each Chamber Interval (hours) (mins) 1 18 10 (single layer) 2 24 30 (double layer) 3 48 30 (two dimensional) 75 compressor, increased to -26°C by the end of the final test using the third configuration (two dimensional case). At the conclusion of the tests, the hypodermic needles were removed from the Karlsruhe brickettes, and the brickettes were removed from the containers. Due to the starchy nature of the Karlsruhe Test Substance, the hole left by the hypodermic needle was left in tack. To determine the exact location of the thermocouples, red food dye was placed in the hole using a glass pipette, and each brickette was cut open to revel the end point of the hole. This location was then measured and recorded. Finally, the distance between the Karlsruhe brickette and the outer surface of the paperboard carton was measured. The thickness of the paperboard carton and the plastic film were subtracted from this value to obtain the air interspace thickness between the carton and the Karlsruhe test substance. The air interface was found to vary between one and ten millimeters. CHAPTER 5. DETERMINATION OF NUMERICAL PARAMETERS The numerical analysis in this investigation of frozen foods during storage focuses on three major problems: (1) the development of a numerical procedure to estimate heat transfer coefficients typical of storage conditions (indirect problem); (2) the development of a multi- dimensional model to simulate temperature changes within frozen foods during storage (direct problem); and (3) The estimation of temperature dependent quality deterioration. These models were described in detail in Chapter 3. Application of these models require input of parameters inherent to the problem being studied, that is, product properties and geometry, and user specified parameters inherent to the numerical proce- dure. Input parameters, determined by the user, intrinsic in the finite difference solutions of both the direct and indirect problems, include the number of nodes and the time step. In addition, the user must select the magnitude of the regularization parameter, the order of regularization, the time between temperature measurements, and the number of future temperature measurements used in the solution for the indirect problem. An investigation of the influence of the user definable parameters on the numerical models described above was completed to provide a systematic procedure for optimal parameter selection. Since the analytical solution of the nonlinear problem with temperature dependent PIOduct properties is not readily obtainable, this investigation of user 76 77 definable parameters focused on the constant property solution, for which exact solutions exist. The resulting observations and conclusions from the study of the linear problem was used as a basis for the selec- tion of the parameters in the nonlinear case. 5.1 Selection of Parameters Inherent in the Finite Difference Solution The one dimensional direct problem is an intrinsic part of the solution of the inverse problem, and it provides a basis for the solu- tion of the two dimensional problem. Because of its importance in both problems, the influences of the user adjustable parameters; node spacing and time steps, on the accuracy and numerical oscillatory tendencies of the one dimensional solution, were studied in detail. Results from this analysis were also used in the indirect problem, and expanded upon in the analysis of the two dimensional problem. The number of nodes and the time step, along with the product properties and boundary conditions, are important factors determining the numerical oscillatory tendencies and the accuracy of the numerical solution. In both cases, the eigenvalues resulting from the set of finite difference equations, Eq. (3.19), play an important role in the analysis. 5.1.1 Numerical Oscillations In evaluating the numerical oscillation criteria, the matrix equa- tion shown in Eq. (3.19) is considered. Multiplying Eq. (3.19) by Afl results in: '1'“+1 - {113-'1'" + A'ln (5.1) 78 Numerical oscillations are related to the eigenvalues of the matrix A713; oscillations will occur in a stable solution if some of the eigen- values are negative, but greater than -1, (Meyers, 1971). Therefore, to avoid oscillations, AEIB must be positive definite. Segerlind (1984) showed that the study of the eigenvalues of A713 may be reduced to a study of A.and B where: L A - "Atc+x (5.2a) _l_ B - fiAtc- K (5.2b) The C and K.matrices contain the heat capacity (p-Cp) terms and the thermal conductivity (k) terms, respectively. For the simplified case of a one dimensional finite slab with constant product properties, and insulated at the first boundary (x - O), and with a convective boundary condition at the second boundary (x - Lx), the C and K.matrices are given below. P egpr 0 . - - 0 q 2 O pCpr C - pCpr E (5.3a) O O O O O 22%; T9 ' i :3 . . . 1 Ax Ax 0 0 ,1; .23 ,3 I Ax Ax Ax . .3 2k .3 K " O -Ax Ax -Ax o (5'3b) .15 _k _ 0 O . . -Ax Ax + thxJ For Ale to be positive definite, A.and B must both be positive definite. The condition for A.is satisfied since from Eqs. (5.3a,b), C is positive definite, and K is positive definite in this case because of the thx term (Segerlind, 1984). The second matrix in question, B is positive definite if (Fried, 1979) det[ K - A16 ] - 0 2 - l, 2,..., L (total No. of nodes) where l *2 ' MN (5.4) and fl - the weighting coefficient used in the numerical method Therefore, considering the worst case where A2 - Amax’ the maximum eigenvalue, numerical oscillations can be avoided if (fl - 0.5) 80 At < osc (5.5) ’l“ max The following hypothetical problem, similar to the conditions used during the experimental procedures for the single product layer described in Section 4.5, with the exception of the use of constant thermal properties, was considered in the determination of the oscil- latory criteria 2 LE .. Ill-22.3% Osstx (5.6a) 8x O (5.12) max max for the finite difference grid. The maximum eigenvalues calculated from P the elemental matrices and the global matrices are compared in Table “1k 5.3. 5.1.2 Accuracy t“ The accuracy of the numerical solution of the non-linear problem i described by Eqs. (3.11), (3.12), and (3.13a,b) is difficult to deter- mine, because the analytical solution is impractical to obtain. Insight ‘ into the accuracy of the nonlinear problem was achieved, however, by «if! considering the linear problem with constant properties. The analytical solution to the simplified one dimensional problem described by Eqs. (5.6a-d) is given by Carslaw and Jaeger (1959) as 8 T(x,t) - To + (Tco - To)- 1 - R e (5.13a) where 87 Table 5.3 Comparison of A ~102 and A(e)-102. max max Temperature (No. of Nodes) -1 -33°c -12°c (sec ) (5) (9) (13) (17) (5) (9) (13) (17) Amax 0.356 1.415 3.180 5.651 0.212 0.842 1.891 3.359 A;:; 0.353 1.412 3.177 5.648 0.210 0.839 1.889 3.357 s 99.2 99.8 99.9 99.9 99.0 99.7 99.9 99.9 88 2-hkocos(§£x) R _ k: [(hk2+ (3)-Lx + hk]-cos(§£Lx) - (k/pCp> «,2 hk - thx/k Rah h and (2 is the it root of the transcendental equation g,-tan(g£-Lx) - Lx-hk (5.13b) The eigenvalues found in the numerical solution, A1, are approxima- M tions to the exponential term, Ac, of the analytical solution. Since there are a finite number of eigenvalues in the numerical solution, it 1 is important to determine how many of the terms in the summation shown in Eq. (5.13a) are significant for any given time. Considering terms < 0.01'C to be insignificant, the time for any given term in the summation to be insignificant can be calculated for a specific location of x. Values for R£,x were calculated, with thermal properties evaluated at -33 and -12°C, for the first nine values of {l (Abramowitz and Stegun, 1965), at the insulated boundary (x - 0), the mid-section (x - Lx/2), and at the convective boundary (x - Lx). Due to the insulated boundary condition and resulting damping effects, the values for R1, at x - 0, (R£,0)’ were greater in magnitude than those at the other locations. Consequently, the values of R£,0 were the most crucial values in deter- mining the times (t?) for each term in the summation to be insignificant. The values for t” were calculated for each term such 2 that -(A20t3) R2 O-e < 0.01 (5.14) 89 e m Values for A2, R1,0’ R£,Lx/2’ R£,Lx and t! first nine terms in the summation are found in Tables 5.4a,b for thermal evaluated using R£ 0 for the properties evaluated at -33 and -12°C, respectively. The values for t: monotonically decreased to less than one after only four eigenvalues; therefore, the the damping effects of the exponential factors were more influential on the behavior of t: than the sinusiodal effects of the R2 terms. Since only the first three eigenvalues were significant after 100 seconds, these terms were considered the most the important when comparing the eigenvalues of the numerical solution (A2) to the exponen- tial factors of the analytical solution (A2). In addition, due to the differences in thermal properties, the to; values evaluated at -12°C were 8“ greater than those values evaluated at -33°C (Eq. (5.7)), and conse- I. ’ quently, there were more significant terms using the thermal properties A evaluated at -12°C than at -33°C. 1‘ Since the first three exponential terms were the most significant in '4' ‘27 '"1 '— " “I the analytical solution, it was decided that the first three eigenvalues of the numerical solution be within 5% of the corresponding analytical I _.. . . .« ..-. terms. The eigenvalues were calculated as percentages of the associated A; terms for the eight conditions shown in Tables 5.2a,b with 2 s 9 in Figure 5.1. From these results, three eigenvalues were found within 5% of A; using nine or more nodes, for the thermal properties evaluated at both -33 and -12°C. Therefore, Ax - 0.01625 meters (Table 5.1) was the maximum spatial increment considered, given the 5% accuracy criteria. Segerlind (1986) has proposed that the limiting time step with regards to accuracy be based on the time for the solution to reach steady state (tss). The first eigenvalue controls the time required to reach steady state. Considering the exponential term as the dominant factor at tss’ he has suggested the following criteria for determining the maximum time step for accuracy (Atac) 90 Table 5.4a Summation Terms in Series Solution and Resulting Time for Terms to Vanish (Properties Evaluated at ~33°C). *; R2,0 R2,Lx/2 R2,Lx t: 2 (sec'l) (°C) (°C) (°C) (sec) 1 2.173-10’5 1.065 1.01 0.86 214,818 2 5.932-10'4 -8.05.10'2 5.55-10'2 7.98-10'2 3,517 3 2.227-10'3 2.22-10'2 -2.22-10'2 2.21.10‘2 358 4 4.949-10‘3 .1.01o10'2 -2.74-10'“ 1.01.10‘2 2 5 8.760-10'3 5.71-10'3 -5.71-1o‘3 5.71-10‘3 <1 6 1.369-10‘2 -3.66o10'3 6.00.10'“ 3.67-10'3 <1 7 1.965-10'2 2.55-10‘3 .2.55-10'3 2.55-10'3 <1 8 2.672-10’2 -1.88-10'3 -1.94-10‘5 1.88-10'3 <1 =====__. 9 3.492-10'2 1.44.10’3 1.44.10‘3 1.44-10‘3 <1 91 Table 5.4b Summation Terms in Series Solution and Resulting Time for Terms to Vanish (Properties Evaluated at -12°C). *2 R2,0 R2,Lx/2 R£,Lx t: 2 (sec'1> (°C) (°C) (°C) (sec) 1 1.435-10'S 1.07 1.01 0.85 325,782 2 3.563.10'“ -8.97-10'2 6.95-10‘3 8.86-10'3 6,158 3 1.328-10'3 2.51-10'2 -2.51-10‘2 2.50-10'2 698 4 2.946-10'3 -1.14-10'2 -3.10-10'4 1.14.10'2 45 5 5.210-10'3 6.47-10'3 6.47.10“3 6.47-10'3 <1 6 8.124-10‘3 .4.16.10'3 6.80-10'5 4.16-10'3 <1 7 1.168102 2.89-10'3 .2.89-10'3 2.89-10'3 <1 8 1.589-10’2 -2.13-10‘3 -2.5o.10'5 2.13-10'3 <1 9 2.074.10‘2 1.63-10'3 1.61.10'3 1.61-10'3 <1 92 .mEumB angucwcodxm Hwofiuhamc< mo mmwmucmoumm mm mmSHm>Come Hmowuofisz H.m muswfim mEo>cm9m to tonEnz H — _ m m + m m ... m 4 m mopoz Sou ole mopoz 092:. Elm mouoz of. «(.4 6662 25 9.1.. mmuoz So... 0 .o mOUOZ 002:. BIG oonnl uo mottoaotm mouoz of «In 302 0:0 9 .0 ooNFl “o moztoaotm A ‘r r f,/// / 6050: 8:0th otcE // ,// E pom: mmpoz co tmnEzz // J/ // ll. l/l/ ./ .///;/¢r/ // /// /l/- /../. / .rlolz’ / / / ’lr,’ I/‘v/ [/l/ / ION .04 low uuei lonueuodxg loan/(louv 3,; 93 At - —§§ (5.15) where, The values for the time step based on accuracy are shown in Table 5.5. From comparison of these values (tss) with the times steps resulting from the oscillation criteria (Tables 5.2a,b), the oscillation criteria was far more limiting than the accuracy criteria, in this particular problem. 5.1.3 Selection of the Optimal Time Step and Spatial Step To ensure an efficient and accurate solution without oscillations, maximum spatial and time steps were chosen to satisfy both the oscillac tion and accuracy criteria discussed in the two previous sections. From the analysis of accuracy, the maximum spatial step was found using a minimum of nine nodes, and the limiting time step, based on the oscilla- tion criteria for nine nodes, was 141 seconds, using thermal properties evaluated at -33°C, and 238 seconds, using thermal properties evaluated at -12°C. A time step of 120 seconds, which satisfies the oscillation criteria for both sets of properties was used. Segerlind (1986) has suggested a procedure for analyzing the long time solution using the values of Ax and At chosen above. In this analysis, the first exponential term of the analytical solution after N (tmax/At) time steps with the numerical approximation. In this case the maximum time is 18 hours or 64,800 seconds, which is greater than t: using thermal properties evaluated at both -33 and -12°C; therefore, 94 Table 5.5 Limiting Time Step Based on Accuracy. At ac (sec) Temperature (No. of Nodes) (5) -33°C -12°C (9) (13) (17) (5) (9) (13) (17) 2288 2300 2301 2249 3471 3475 3535 3476 95 only one eigenvalue of the numerical solution is significant. The numerical approximation to exp[ -A:-t ] was found using the Padre expan- sion of the Crank Nicolson finite difference method with N time steps. The Padre expansion for the first exponential term was found by considering both the analytical and numerical solutions to the time dependent differential equation is the separation of variables solution of the linear heat conduction problem, Ozisik (1980) $199.. at + A1 0 0 (5.16) The analytical solution is (5.17) where, a '- cons tant From Figure 5.2, the Crank-Nicolson approximations to the time deriva- tive, and 0 are determined as 1 0 ¢ 0 - 6 g; - -—ZE——— (5.18) 1 O 8 - 9;_§_2_ (5.19) Substituting Eqs. (5.18), and (5.19) into Eq. (5.16) and replacing A: by the Crank-Nicolson approximation (A1), and solving for 01 yields 1 (1 ' AlAt/Z) o ( 1 + AlAt/Z )‘8 (5.20) 96 O A ¢(t) Crank-Nicolson {’0 __ __ _ __ Approximation M of at I l i . l l l I At 2At Figure 5.2 Crank-Nicolson Approximation for the Time Derivative. 97 After n time steps, the Crank-Nicolson approximation is n [ ( 1 - AlAt/2 ) ]“ 0 ¢ _ .6 (5.21) ( 1 + A,At/2 ) 0 Evaluating Eq. (5.17), at t - 0, gives 0 - a, and therefore, ¢ - a, and the Crank-Nicolson approximation to the exponential term is e (5.22) -A§t ( 1 - AlAt/2 ) n z ( 1 + A,At/2 ) where t - n-At. The exponential term of the analytical solution after t1 - 64,800 seconds, and the approximate numerical term after n - 540 time steps (At - 120 seconds) for properties evaluated at -33 and -12°C are 1. Thermal properties at -33°C: -A:°tl A e - 0.2446 1 - AI-At/Z > 99.9% Accuracy - 0.2444 1 + AI-At/Z J where: Af - 2.173-10‘S A, - 2174.105 2. Thermal properties at -12°C: 98 -A§-tl ‘ e - 0.3946 1 A At/2 > 99.7% Accuracy . 1. - 0.3936 1 + A,oAt/2 , 5 where: A: - 2.173-10‘ A, - 2.174-10‘S 5.1.4 Analysis of Time and Spatial Steps for Other Geometries Cylindrical and spherical geometries were considered using nine nodes and compared with results from the previous sections. The eigen- values from the numerical solution were found by modifying the conductivity and heat capacity matrices, Eqs. (5.3a,b), to account for the different geometries, and solving Eq. (5.4) as described in Section 5.1.1. The limiting time steps for the numerical oscillation criteria and for the accuracy criteria proposed by Segerlind (1986) were found using Eq. (5.5) and Eq. (5.15), respectively, for the thermal properties evaluated at -33 and -12°C. Results are shown in Table 5.6 for both cylindrical and spherical geometries using nine nodes. Again, the oscillatory criteria was more restrictive on the time step than the accuracy criteria of Eq. (5.15). Comparing Table 5.6 with Table 5.2 a,b revealed that Atosc for the sphere was significantly less than that for the cylinder, and that the Atosc for the infinite slab was least restrictive of all. For the thermal properties evaluated at both -30 and -12°C, the critical time steps for no oscillations using spherical and cylindrical geometries were 63 and 83%, respectively, of those for an infinite slab. 99 Table 5.6 Limiting Time Steps for Cylindrical and Spherical Geo- metries. Cylindrical Spherical Geometry Geometry -33°C -12°C -33°C ~12°C H. At (sec) 117 197 89 150 osc Atac (sec) 1072 1674 706 1089 1 100 The eigenvalues of the numerical solution were compared with the exponential terms (A2) of the analytical solution; the analytical solu- tions for an infinite solid cylinder and a solid sphere with a convective boundary condition at r - 0.13 m are given below (Carslaw and Jaeger, 1959). 1. For an infinite solid cylinder T(r,t) - I, + (Tco Lx- To)-[ 1 - E CI-exp(-A2-t¢)] (5.23a) where 2(hk)-Jo(§£r) 02 ' 2 2 Lx-(g + hk >-Jo<§,Lx> A‘ - (k/p0p>-:2 2 2 hk - thx/k th root of the transcendental equation and, (1 is the 2 c,°Jo(:,-Lx) - (hk)-Jo 2. For a solid sphere T(r,t) - To + (Tdo Lx- To)-[ 1 - E Sfioexp(-A£ote)] (5.24a) where lOl 2-Lx-hkogi + (Lx-hk-1)2 s - )2 2 ~sin(Lx-§£)-sin(ro§£) + Lx-hk-(Lx-hk-l)] r°§i [(Lx'§£ e A, - (mom-cf, hk - thx/k and, {1 is the 2th root of the transcendental equation Lx-(lcot(§£-Lx) - l - Lx-hk (5.24b) The time for each term in the summation to be insignificant (t?) was also determined for the cylinder and the sphere, using the criteria presented in Eq. (5.14), giving a. Cylinder: S 0.01 (5.25a) b. Sphere: IA 0.01 (5.25b) The times for each summation term to become insignificant were calculated and compared for the solid sphere and the infinite solid cylinder. Results for the first six eigenvalues are shown in Table 5.7a,b for thermal properties evaluated at -33 and -12°C, respectively. 102 Geometries (Properties Evaluated at ~33°C). Table 5.7a Comparison of t: Values for Cylindrical and Spherical Cylindrical Spherical Geometry Geometry ‘;_1 C2,0 t: *;_1 82,0 ‘3 (sec ) (°C) (sec) (sec ) (°C) (sec) 4.48-10'5 1.106 105,066 .80-10'5 1.131 69,550 8.58-10’4 0.144 3,111 .16-10'3 0.202 2,582 2.76-10'3 6.04.10'2 651 .34-10'3 0.118 738 5.76.10’3 3.49-10'2 217 .61o10'3 8.35-10'2 321 9.84-10‘3 2.34-10‘2 86 .10-10'2 6.48-10'2 170 1.50-10'2 1.70-10'2 35 .64.10'2 5.29-10'2 102 103 (D 2 Geometries (PrOperties Evaluated at -12°C). Table 5.7b Comparison of t Values for Cylindrical and Spherical Cylindrical Spherical Geometry Geometry *;_1 C2,0 t: *;_1 52,0 ‘3 (sec ) ('C) (sec) (sec ) (°C) (sec) 2.98-10'5 1.120 158,543 4.61-10'5 1.149 102,945 EFNL 5.15-10’“ -0.162 5,410 7.00-10'4 -0.228 44,670 , 1.65-10'3 6.83-10'2 1,167 2.00-10'3 0.133 1,294 ’ 3.43-10'3 3.96.10'2 401 395103 946402 568 5.85-10'3 2.65-10'2 167 6.57-10'3 7.34-10'2 304 8.93-10'3 1.93-10'2 74 9.81-10‘3 5.99-10‘2 183 104 The first two terms in the summation approach zero faster for the sphere compared with the cylinder, and for the cylinder compared with the infinite slab (Table 5.4a,b), as would be expected due to the increased surface area to volume ratio of the sphere and cylinder, compared with the slab. However, the time for the remaining terms in the summation to become insignificant is longer for the sphere compared with the cylinder, and longer for the cylinder compared with the slab, indicating that more eigenvalues are significant when evaluating the accuracy criteria for the sphere and cylinder. From Table 5.7a, with thermal Properties evaluated at -33°C, there are four and six significant eigen- values after 100 seconds for the cylinder and the sphere, respectively, °°mpared with three for the infinite slab (Table 5.4a). For properties eva'-‘-‘--1a.ted at ~12°C, the number of significant terms increases to 5 terms for the cylinder, and at least six terms for the sphere (Table 5.7b), compared with three for the infinite slab (Table 5.4b). The numerical eigenvalues, expressed as percentages of the analyti- cal BXponential factors (A2) for the infinite solid cylinder and the solid sphere are shown in Figure 5.3, using nine nodes and thermal properties evaluated at -33 and -12°C. Only the first two eigenvalues for bOth the cylindrical and spherical cases satisfy the 5% accuracy criteria set for the infinite slab, indicating that a smaller position Step (Ar) is required for the cylinder and the sphere, than for the infinite slab to satisfy the given accuracy criteria. In summary, using observations from the infinite solid cylinder and the Solid sphere: (l) the limiting time step for the oscillation criteria (Atosc) decreased as the surface area to volume of the geometry increased; (2) the accuracy of the eigenvalues expressed as Ai/AZ-lOOQs decreased with increased surface area to volume ratio; and (3) the time be quired for each exponential term to be insignificant decreased with 105 .mowuumaomu Hmoaumzam paw HmowupCHHmo How mEuoH HmHquCOme Hmoauhamc< mo mowmucmouom mm modam>Come Hmowumezz m.m owswwm 828:me 40 53:52 m w m N F _ _ _ _ O 59:30 ale oonnl Lo mmfiemaoem A forum 9.0 empcio me . OoN—I.eo mmzemaocm A “Begum Elm ..ON 66:82 886:5 BE: E pom: mopoz 052 row Limei [onueuodxgl loonflgouv g; 106 increased surface area to volume ratio for the first two terms, but increased for the remaining terms. 5.1.5 Analysis of Time and Spatial Steps for the Two Dimensional Model The analysis of the oscillation criteria and accuracy of the one dimensional problem required the determination of the numerical eigen- values as shown in Eq. (5.4) for an L by M matrix (L, M - number of nodes in x and y directions). The matrices associated with the two dimensional problem are considerably larger: for equal number of nodes in two dimensions, the coefficient matrices will be L2 by M2, for which it is generally impractical to solve for the eigenvalues, except for small values of L and M. Therefore, a simplified analysis was sought which would give insight into oscillation criteria and the accuracy of the solution. In the previous discussion of the oscillation criteria, it was found that the maximum eigenvalue of the elemental matrices (AéiL) provided an excellent estimate of Amax of the global matrix. For the two dimen- sional element, the elemental matrices are shown below (Belytschko and Hughes, 1983) a + b -b 0 -a (e) -a a + b -b 0 k _ (5.26a) 0 -a a + b —b 107 where a - BA! and b - EA; c - g-gp-(Ax-Ayzol (5.26b) where I is the identity matrix. Assuming Ax - Ay, and substituting these matrices in to Eq. (5.10), resulted in the following expression for A(e). max ‘13:; - 464L2- (517) p°Cp(AX) This expression for 88:; is four times the value of 88:; for the one dimensional element, Eq. (5.11); therefore, the no oscillation time step is four times as restrictive for the two dimensional case, as for the one dimensional case with equal position increments. The values for Amax calculated from the global matrix are compared with Aéi; for both a 3x3 and a 4x4 two dimensional grid with Ax - Ay, using thermal properties evaluated at -33°C. Again, results indicate that for a finite difference grid, A(e) provides a good estimation of max A x within 5%, as shown in Table 5.8. ma 5.1.6 Summary of Observations in the Determination of Finite Difference Parameters Two criteria were used in the selection of the time step and posi- tion increment used in the finite difference solution. The position increment (Ax) was selected to satisfy the accuracy criteria, and the time step was determined according to the oscillation criteria. The oscillation and accuracy analysis is summarized below. 1. The no oscillation time step is determined using the largest 108 Table 5.8 Comparison of A(e) and A for a Two Dimensional Grid max max (Ax - Ay, Properties Evaluated at -33°C). Grid Size (position increment) 3x3 4x4 (Ax - 0.065m) (Ax - 0.0433m) A 1.822-10'3 3.873-10'3 max A(e) 1.765.10‘3 3.972-10'3 max (%) 96.9% 102.5% 109 eigenvalue (Amax) of the set of finite difference equations. 2. The elemental eigenvalue (A(e)) is approximately the value of A max max within 99% accuracy for the one dimensional case and 95% accuracy for the two dimensional case. 3. The no oscillation time step decreases as the surface area to volume ratio of the geometric shape increases: Atosc(slab) > Atosc(cylinder) > AtO c(sphere). s 4. The no oscillation time step of a two dimensional grid with equal position increments in the x and y directions is half the value for the one dimensional grid, using the same position increment. 5. The oscillation time step at one temperature, given Atosc at a sec- “‘ ond temperature, is proportional to the inverse ratio of their res- pective thermal diffusivities. *j 6. The position step was determined according to the accuracy of the .3 significant eigenvalues in the series solution. 7. The time for each term in the series solution to be insignificant increased with decreasing thermal diffusivity. 8. The number of significant eigenvalues increased with increasing sur- face to volume ratio. 9. The limiting time step for accuracy, proposed as 1% of the total time to steady state conditions, was not as restrictive as the time step for the no oscillation criteria. 5.2 Parameters Used in the Solution of the Inverse Heat Conduction Problem The sequential regularization method, using finite differences, was used to estimate surface heat transfer coefficients. This method re- quires input values for: (l) the regularization parameter, a; (2) the llO order of regularization, W0, W,, or W2; (3) the number of future time steps, r, used the sequential procedure; (4) and the time increment between temperature measurements, Atm, in addition to the user adjus- table parameters inherent in the finite difference technique discussed in Section 5.1. The values selected for the spatial and times steps (Ax and At) in the one dimensional direct problem were also used in the one dimensional indirect problem. The time increment between temperature measurements, Atm, was limited by the data aquisition unit used in the experimental procedures, to a minimum value of 600 seconds. Two criteria, the deterministic bias and the variance, were used in determining optimal values for the parameters inherent to the inverse f3, problem. The deterministic bias is defined as a measure of the bias or ly!‘ error in the estimator when input temperature measurement errors are equal to zero. The variance is a measure of the estimators sensitivity to random measurement errors (Beck et. al., 1985). The deterministic bias and variance of the estimated heat flux for the nth time step, D? and Vn, are defined by Scott and Beck (1985) as D? - E(q3) - q“ (5.28a) Vn - £1133 - 8033)?) (5.281») where qn is the true heat flux, 8(83) is the expected value of the estimated heat flux with no errors in the temperature input values, and E(q$) is the expected value of an with random temperature measurement errors. It was desired to find parameter values which minimized both types of errors. The mean square error, S, was defined as the sum of the variance and the square of the deterministic bias (SN)2 - vN + (01:)2 (5.29) where N is the total number of time steps. The problem used to evaluate the spatial and time steps (Equations 5.6a-d), assuming constant thermal properties and a surface heat trans- fer coefficient of 7.85 W/m°C, was also used in the determining of the sequential regularization parameters, with the exception that, in this case, two 18 hour storage periods were used. The first storage period was -30°C, and the ambient temperature for the second storage period was -5°C. The initial temperature was set at -30°C. The analytical solu- tion of this problem was solved to obtain temperature values at x - Lx/2, corresponding to the location of the thermocouple in the experimental procedures for the single layer slab described in Section I . i f I 3 4.3. These temperature values were used as input data for the sequen- tial regularization solution of the IHCP. The surface heat flux was first estimated using exact temperatures from the analytical solution as input, and the deterministic error was calculated for various values of a and r, and for the zeroth, first and second regularization order. The process was repeated using imposed random measurement errors in the temperature input values to estimate the mean squared error. 5.2.1 The Deterministic Bias The deterministic bias was estimated by Scott and Beck (1985) N .5 A 0 n 2 De - [ E (thx - thx) ] (5.30) n-l 112 where hxzx is the estimated heat flux at the nth time step, and thx is the constant heat flux value used in generating the input temperature values for the IHCP. The regularization parameter (a) regulates the degree the regular- ization terms, shown in Equation (3.34a-c), influence the solution. The number of future time steps, r, influences the stability of the solution and computation time. As r becomes large, the solution approaches the whole domain solution, and computation time increases significantly (Beck, et. al., 1985). To determine the critical values for both of these parameters on the solution, the surface heat flux values were estimated using various values of a and r, for the zeroth, first and second order regularization orders, with the time step for temperature measurements, Atm, equal to 600 seconds. Thermal properties were evaluated at ~33°C, since from Tables 5.2a,b, the time step for no oscillations was most restrictive at that temperature. The range of values for r were conservatively chosen from two to fifteen, based on observations by Scott and Beck (1985), who noted that the sequential regularization solution of the IHCP is independent of r for r z 8. In determining the range of values considered for the regularization parameter, a, it was noted that the regularization term is added to the sensitivity coefficient matrix product, XIX, in the regularization method. Therefore, the magnitude of the coefficients in the X?X.matrix, with a equal zero and r equal fifteen, were used to determine the range of values for a. The magnitudes of the XIX.matrix 9 3 product ranged from a 10' to a 10' T 1 “1' are of order one; therefore, values of a ranging from 10' The regularization parameter, a, is multiplied by H i - 0, l, or 2, which, from Equations (3.36a-c) 9 to 10.3 were used to determine its influence on the solution. :- 113 The deterministic error was calculated and compared for discrete values of a and r. Results are shown in Tables 5.9 a-g, for r equal 2, 4, 6, 8, 10, 12, and 15; for the zeroth, first and second regularization orders; and for a - 10'9, 10'8, 10'7, '6, '5, 10'“, and 10‘3, 10 10 respectively. The time step between temperature measurements, Atm, was equal to 600 seconds and thermal properties were evaluated at -33°C in all cases. The results for exact temperature input values support the observa- tions by Scott and Beck, (1985), in that the deterministic bias shows little dependence on r, for r greater than eight. The critical values for the regularization parameter, a, ranged from 2 10-4 to z 10.7 for I“ all regularization orders. For a 2 10-5, the deterministic bias, De, 6’ D increased with increasing order of regularization, and for a s 10- e decreased with increasing a. 5.2.2 Mean Squared Error The hypothetical problem used to determine the deterministic bias was also used to estimate the mean squared error. Input ambient and internal temperatures were modified by the addition of normally dis- tributed random numbers. The standard deviation used for the random numbers was determined from the error limits of the thermocouples used in the experimental procedures. The error limits of the T-type thermocouples were given as the greatest value between (Omega, 1985) i 1 °C or i 1.5 % of maximum |°CI 114 9 Table 5.9a. Deterministic Bias for a - 10- (Atm - 600 seconds; thermal properties evaluated at -33°C). 2 4 6 8 10 12 15 W0 ------ 75.44 78.26 78.24 78.24 78.24 78.24 W, ------ 54.72 69.09 68.88 68.91 68.91 68.91 W, ------ 102.44 28.16 28.78 28.99 27.62 28.14 Table 5.9b. Deterministic Bias for a - 10-8 (Atm - 600 seconds; #1? 1' thermal properties evaluated at -33°C). 2 4 6 8 10 12 15 W0 22.02 55.73 54.86 54.88 54.89 54.89 54.89 W, ------ 45.13 37.48 37.52 37.55 37.55 37.55 W2 ------ 22.21 23.78 26.22 26.66 26.70 26.71 115 Table 5.9c. Deterministic Bias for a - 10.7 (Atm - 600 seconds; thermal properties evaluated at -33°C). 2 4 8 10 12 15 w, 25.02 24.98 24.96 24.94 24.93 24.93 24.93 w, 29.40 13.57 18.06 19.16 19.29 19.29 19.29 15‘ w, ------ 18.30 15.57 15.25 15.49 15.64 15.67 A?» Table 5.9d. Deterministic Bias for a - 10'6 (Atm - thermal properties evaluated at ~33°C). 600 seconds; 2 4 6 8 10 12 15 W0 12.33 11.77 14.78 14.81 14.84 14.84 14.84 W, 24.12 9.92 12.36 12.63 13.13 13.13 13.14 W, ------ 7.90 9.41 11.82 11.77 11.90 11.96 116 Table 5.9e. Deterministic Bias for a - 10.S (Atm - 600 seconds; thermal properties evaluated at -33°C). 2 4 6 8 10 12 15 W0 16.88 12.37 13.17 14.06 14.09 14.42 14.43 W, 81.51 10.39 13.42 13.49 13.88 13.90 13.95 ._ W2 ------ 31.64 14.53 15.66 15.62 15.66 15.71 3%.» Table 5.9f. Deterministic Bias for a - 10'4 (Atm - thermal properties evaluated at -33°C). 600 seconds; 2 4 6 8 10 12 15 W0 64.51 22.52 19.05 20.05 20.05 19.93 20.02 W, 129.51 35.43 21.46 24.05 23.01 22.67 22.83 W2 ------ 102.44 28.16 28.78 28.99 27.62 28.14 117 Table 5.9g. Deterministic Bias for a - 10'3 (AtIn - 600 seconds; thermal properties evaluated at -33°C). r 2 4 6 8 10 12 15 ,, . 4 w, 125.83 113.61 51.72 39.38 36.78 37.07 37.64 w, 137.32 107.44 56.38 42.82 46.31 47.29 44.81 j w, ------ 133.23 144.48 51.26 55.81 58.32 53.87 3 118 The experimental data ranged from a —33°C to -5°C, therefore, the first criteria (i 1°C) was the greatest. The standard deviation was calcu- lated assuming that i 1 “C represented the 99.5% confidence interval of the thermocouple temperature measurements. The standard deviation was calculated from the ta v probability distribution, with a - 99.5%, and /2. the number of degrees of freedom, u, equal to five, corresponding to the six thermocouples used in the experimental procedures. Therefore, the standard deviation was estimated from the confidence interval (CI), where (Walpole and Myers, 1978) t 00 01 - 4; 1°c - : £&”—5- (5.31) .14.. “‘9 where the number of thermocouples, Nt’ equaled six, and ta/Z u - 3.365. ,1 From Equation (5.31), a was estimated to be 0.73°C. This was considered it I to be a conservative estimate, since the largest variation between ihw’ thermocouples located within the Karlsruhe test substance from the experimental results was only 0.7°C. Beck, et. a1. (1985) recommended using a single temperature measure- ment error as an estimation of the variance. An alternate approach, used by Scott and Beck, (1985) is sometimes referred to as the Monte Carlo Method. In this case, the heat transfer coefficient is estimated from input values with added random temperature measurement values. The mean square error was estimated from the estimated heat transfer coeffi- cient using random errors in the input temperature values as follows A N . 2 0.5 n - [g [9... - thx] ] (5.32. 119 A Since the value of S1 depends on the set of random numbers added to the input temperatures, different sets of random numbers (i) were added to A the input temperature values to calculate different values for 81' The mean squared error, S, was determined from the average of the Si values from twelve different sets of random numbers N r S - E (Si/i) (5.33) n-l where twelve different sets of random numbers (Nr) were used. From Tables 5.9a-g, the range of a for which the deterministic error was less 25, was selected as the range over which the mean squared error 7 was determined. The critical values of a ranged from a 10- to - 10-4. Values of S were calculated for a - 10-4, 10-5, and 10-6, r - 4, 6, 8, A 10, and 12, and for W0, W,, and W2. Results for the average mean squared error S, and the standard deviation of S are shown in Tables 4 -5 5.10a-c, for a - 10' , 10 , and 10-6, respectively. Since the results for S using a - 10.6 were much greater the those using a - 10-5, the mean squared error using a - 10'7 was not considered. The results indicate that S calculated using a - 10'4 provided the lowest mean squared error 8, and that the solution is independent of r, for r 2 10. The zeroth order regularizer provided a slightly better estimator than the first or second order estimators. 5.2.3 Time Increment between Temperature Measurements Since the time increment between temperature measurements was limited to 600 seconds by the data aquisition device used in the ex- perimental procedures, the values of Atm were limited to 600 seconds or 120 A Table 5.10a. Average Mean Squared Error (S) and Standard Deviation of s (as) for a - 10'6 (Arm - 600 seconds; thermal properties evaluated at ~33°C). 4 6 8 10 12 W0 S 231.08 205.57 200.42 197.60 194.18 as 38.06 45.12 43.45 43.77 44.18 W, S 167.32 158.89 131.57 125.26 123.85 as 17.53 18.54 15.44 14.94 14.35 W2 5 254.36 100.32 108.39 106.21 99.74 as 23.72 6.52 7.94 8.38 8.67 121 A Table 5.10b. Average Mean Squared Error (S) and Standard Deviation of s (as) for o - 10'5 (Arm - 600 seconds; thermal properties evaluated at -33°C). 4 6 8 10 12 W0 8 92.52 72.87 72.32 64.70 62.92 a 7.51 5.96 6.94 8.85 8.70 W, S 100.91 49.51 55.08 48.02 48.10 a 7.61 4.66 4.19 4.09 4.47 W2 S 44.20 55.86 44.94 42.15 41.41 a 2.36 5.17 3.97 4.36 4.15 122 A Table 5.10c. Average Mean Squared Error (S) and Standard Deviation of s (as) for o - 10'“ (Atm - 600 seconds; thermal properties evaluated at -33°C). 4 6 8 10 12 W0 S 36.37 30.89 27.25 26.99 26.90 aS 1.77 1.88 1.56 1.58 1.60 W, S 40.34 32.84 28.84 29.19 28.34 aS 1.09 1.70 1.22 1.33 1.31 W2 S 102.50 34.72 31.72 31.46 30.22 aS 0.32 0.93 0.58 0.52 0.48 _ a . __ _'—:- _-u-; '9-— 123 a multiple of 600 seconds. To determine the influence of Atm on the mean squared error, the solution was determined for the zeroth, first and second regularization orders using a Atm of 1200 seconds, with ten future time steps and a - 10-4. Random errors were again added to the input temperature values, and the average mean squared error was calcu- lated from Equation (5.33) using twelve sets of random numbers (Nr - 12). Since the mean squared error is dependent on the total number of time steps, N, as shown in Equation (5.32), to compare these results with those found using AtIn - 600 seconds, the resulting average mean squared values were divided by the total number of time steps used in each case to obtain an average mean squared error per time step, 8 These values for 8* are shown in Table 5.11, along with the respective standard deviations, a3, (aS divided by the total number of time steps), for Atm - 600 seconds and Atm - 1200 seconds, with a - 10-4 and ten future time steps. The results show that the average mean squared error per time step is significantly higher using Atm - 1200 seconds than that found using Atm - 600 seconds, for all regularization orders. 5.2.4 Selection of Optimal Parameters used in the Inverse Heat Conduc- tion Problem of Estimating the Surface Transfer Coefficient The values for the regularization parameter, a, the number of future time steps, r, the order of regularization, W,, i - 0, l, or 2, and the time increment between temperature increments, Atm, were selected to minimize the average mean squared error (8) in the solution. From the results shown in Tables 5.9a-g, 5.10a-c, and 5.11, the following values were selected for the parameters inherent in the sequential regulariza- tion IHCP solution: a - 10-4, 124 A * Table 5.11. Average Mean Squared Error per Time Step (S ) and * Standard Deviation (as) for Atm - 600 and 1200 seconds 4 (0 - 10- , r - 10). w0 w1 w2 Atm- 600 s* 0.129 0.139 0.150 a; 0.008 0.006 0.002 Atm- 1200 3* 0.188 0.203 0.238 a: 0.012 0.014 0.007 125 r - 10, W0 - 1, (W, - W2 - 0), and Atm - 600 seconds. A Surface heat transfer coefficients (hx estimated using both Lx>. exact input temperatures and input temperatures with added random er- rors, and the above values for the input parameters, are shown as a function of time in Figure 5.4. The mean squared error for this par- ticular set of random numbers (81) was 23.7 W/m2°C. (From Table 5.10c, S - 26.99 W/m2°C). Several observations may be noted from the results shown in Figure 5.4. (1) At time, t - 0, the simulated product was exposed suddenly to a change in ambient temperature from -30°C to -5°C. During this time, the damping effect of the sequential regularization procedure on thx A was evident in the gradual increase of the predicted value of thx’ using exact data, to a constant value at t z 3 hours. (2) At t - 18 hours, when the simulated ambient temperature changed suddenly from -5°C to -30°C, thx suddenly decreased, and then increased gradually to a A constant value as before with t - 0. (3) The sinusiodal nature of thK using input temperatures with added random errors was a result of the estimators tendency to smooth out variations in the input temperature values. (4) The high variability of the estimated heat flux was a result of the large standard deviation used in the random errors added to the input temperature values. The input temperatures (ambient temperatures and internal temperatures) using exact data and using added random errors are shown in Figure 5.5. Since the maximum variation in the input temperatures with random errors was higher than that observed from the actual tem- perature measurements, it is expected that the resulting variability in the estimated surface heat transfer coefficient using temperature 126 .o.ms.o mo .6 .aoaooe>oo ownocoom so“: some one: H :H muouum Eopcmm usonufi3 use £uw3 mufimaowmmoou wommccuh use: poumEHumm q.m ouswwm A9305 oEP mcouoEocom Sac. Eotm Scene: 0: 5? 300 Sec. I... 8. n56 ..u by 23.0 Eopcom 55 Ban Sec. 0 on on ¢N m. N_ m o t E 4 1r L h 17 b F L L ml 085? was 1 eooo .96.: too: E 2.0 fl mmocxofk no_w r 3:0: onlmw .ooonl ..u Eco; 210 .90. u .che .95. ooonl fl vasectanoh BEE no I In 1 CD P ......— Dwnsa UJ/M) WGPEHGOO (0°: JGISUDJl loeH pa), 127 .muouum Eopcmm unonuHB paw Sufia mama usacH ousumuoafiow HmcwoucH paw uCoHQE< m.m owswfim flmc30cv oath (3°) eJnloJedLuei on on tum 9 NP 0 o b L b IF h k P h 1.»! [F b I‘DIJlmnll 4 on! . T cube..- . .. TON... .:o.hs .x 4 .... o . r.. r .2; .. ION: ... . ...... . .. um.. T 4 , . .. . 4. ;..n . Tmpl E Ed 1 865.2,: com .. . 0.8: 1 deep 62:. _ ....e. T .. 0onE\3 no.5 n $000 .25.; com: _. o— 0. 0.0 n 8 ll 3 oSHEoQEoH 0o nnd acoBE< o. 0.0 .1 SBanEo... o. mud BEBE 128 measurements as input will be less than that shown in Figure 5.4. The surface heat transfer coefficient was estimated using a standard devia- tion one half of that used previously (a(new) - 0.5-a(old) - 0.500.73 - 0.365°C). Results are shown in Figure 5.6; the maximum variation in thx for this case was z 2 W/m2°C, corresponding to a maximum variation in input temperatures of z 0.5°C. 129 mm Fl. .oowm.o mo .6 .coHumH>ma pumpcwum £uw3 mama uzmcH CH muouwm Soccmm uzozuwa paw £ua3 muCofionmooo Hommcmue use: poumEHumm w.m ouswwh A3305 oEP on em 9 E o o [F I? L L [F In: F L L Li maul use? was ...... eooo .25: too: E nto n «8562: cam . o 950... mnlm— .ooonl ..u too; Bio 6%.. 1 deep .95 3 r. S Pom... n 23286.: .635 o men mtoemEotom 39: e w ”....o 1.1+. quad mo. WH )m Mm1+ /// w m an. row was J 4 mtotm Eoocom o: 53 Soc «39.: .II 8. mend n 3 Eotm source 53 Soc Sec. o mp CHAPTER 6. RESULTS AND DISCUSSION Both analytical and inverse heat conduction methods were used in estimating the surface heat transfer coefficient; results are discussed in Section 6.1. The one and two dimensional direct numerical solutions were verified by comparison with analytical solutions, assuming constant product properties, and experimental results, assuming variable thermal properties. These comparisons are described in Sections 6.2 and 6.3. Some of the parameters affecting the temperature and quality dis- tribution histories of a simulated food product were also investigated. This study concentrated on two areas: (1) the effects of boundary condi- tions (Section 6.4), and (2) the effects of size and geometry (Section 6.5). The primary objectives in this analysis were to determine how the parameters associated with these areas affected the overall rate of quality deterioration, and the variation of quality deterioration within the product. 6.1 Estimation of the Surface Heat Transfer Coefficient The surface heat transfer coefficient was estimated using both analytical and inverse heat conduction methods. Both methods required ambient and product temperatures, and in addition, the analytical solu- tion required knowledge of the velocity profile over the surface of the 130 131 product. The ambient and average product temperature measurements, obtained in the first repetition for the single layer slab with one exposed surface (Test la, Section 4.5), are shown in Figure 6.1. (Measurements obtained in the second two repetitions are shown in Figures E.la,b.) These values were used in both analytical and inverse solutions, and the velocity measurements for the analytical case were determined experimentally, as discussed in Section 4.4.. 6.1.1 Analytical Estimation of the Surface Heat Transfer Coefficient The surface heat transfer coefficient was estimated using the analytical methods described in Section 3.5.1. For both forced and free convection to be significant, Eq. (3.22b) must be satisfied. The Reynolds number in Eq. (3.22b) was determined using average velocity measurements. The average air velocities over the product during the first and second storage intervals, Ulao and U from 0 to 18 hours and 2G’ from 18 to 36 hours, respectively, were estimated from the velocity measurements, found using a hot wire anemometer, in each storage chamber as described in Section 4.4. The kinematic viscosity was calculated using the average ambient temperature values shown in Figure 6.1 (Incropera and Dewitt, 1985). For the first storage interval, the average ambient temperature, T1co equaled -6°C, and for the second storage interval, the average ambient temperature, T2co equaled -33°C. (Similar values were found for the second two test cases, shown in Figures E.la,b.) The Reynolds number was calculated for both 18 hour storage intervals from Eq. (3.21a). Results are shown below. 132 .Ama ummev mommwsm pomomxm oco fiuwa scam uozmg oawcfim wCHm: mquEousmmoz oocmumnSm umoh ofiswmaumx mo oudumuomeoe HmcuoucH owmwo>< paw DCoHnE< H.e ouswfim Amcaocv witp mm, on cam m? NP w 0 p h e _ L _ . _ . P . 0.?! ocauotanoh €05.54. 0 N\fi an .anh Bounced 658:. .m>< :1 .. ooonwo o 00.0. 1 l 1.6.... ..M. ......w ... ....ws...... ciao... 8 00 00 00 000 $50000 0 10805600080 0 1 .0 00 o o O O O 0 10mm] 0 I Tmml T ION! Infill 1.0—II .w. 3. ayfisffisewfifissficfge - . .. E 2.0 fl .xu .mmmcxflf n05 Donn... fl otfiotodEop BEE :1 (3°) aJnloJedLue 133 U oLx Re - ~23————9 - 2.74.103 0 2 c > 18 hours Lx,l v1 fi oLx , Re - -—&3————9 - 1.10.104 18 2 c a 36 hours Lx,2 v2 where: Lxc - 0.118 m film - 0.25 m/s 62m - 1.0 m/s V1 - 1.296-10'5 m2/s v2 - 1.067-10'5 mz/s The Grashof number was calculated from Eq. (3.21b) by approximating average internal product temperature measurements for the surface tem- perature measurements. The expansion coefficient, fie, was determined assuming an ideal gas, and using the average ambient storage temperature values. Using the extreme average product temperature values at the beginning and ending of both storage intervals, (§ at 0, 18', 18+, and 36 hours) the maximum and minimum Grashof numbers were found at these times, as First storage interval: 3°Be,1-(?(0) - T1m>oLx2 Ger - Vi - 9.35-10 t - 0 hrs s-fi --Lx3 e,l 1w c - 2 - 2.16-106 t - 18‘ hrs ”1 Second storage interval: 13h swam-(Yam - $2,,»sz Gr - 2 - 1.24.107 t - 18+ hrs Lx y2 g-fi -(?(36) - T )-Lx3 e,2 2m c - 2 - 2.95.106 t - 36 hrs ”2 The criterion for both forced and free convection was calculated from Eq. (3.22b) at O, 18',18+, and 36 hours. Results are shown below. Ger 2 - 1.25 t - 0 hrs ‘ Re 0 Lx * Tlm - -6 C - 0.29 t - 18'hrs J Ger + 2 - 0.10 t — 18 hrs ‘ ReLx T - -33°c r 200 - 0.02 t - 36 hrs J The upper and lower limits of the criterion shown in Eqs. (3.22a-c) were arbitrarily set from 0.1 to 10.0. Both forced and free convection were found to be significant in all but one case; therefore, both forced and free convection were considered in the analysis. 6.1.1.1 Forced Convection Steady state analytical solutions are available for simple boundary conditions such as a constant temperature or a constant heat flux at the surface of a specified geometry. The actual conditions prevailing r ‘ Mn» 1" )7: h.-- 135 during the experimental procedures were not bound to steady state condi- tions or either of the two boundary criteria; however, the solutions to these simplified conditions were used to provide order of magnitude estimates of the surface heat transfer coefficients. The Nussult numbers resulting from forced convection were calculated from Eqs. (3.23a,b), for both the constant temperature and constant heat flux boundary conditions, and for both 18 hour storage interval. Prandlt numbers were found from the properties of air at T1”, and T2co (Incropera and Dewitt, 1985). Results are shown in Table 6.1. 6.1.1.2 Free Convection Nussult numbers resulting from free convection were calculated using Eq. (3.25) for the first storage interval (Tlco > T), and Eq. (3.24a or b) for the second storage interval (sz < T), depending on the magnitude of the Rayleigh number. Since the maximum Rayleigh number for the second storage interval was found to be < 107, Eq. (3.24a) was used. Nussult numbers were calculated using average ambient and internal product temperature measurements from the results shown in Figure 6.1. Both Rayleigh and Nussult numbers are shown in Table 6.2. 6.1.1.3 Combined Free and Forced Convection. Equation (3.26) with n equal 7/2 was used to calculate Nussult numbers from combined free and forced convection for both aiding and abating free convection. Heat transfer coefficients were determined from the Nussult numbers using Eq. (3.27). Nussult numbers and result- ing heat transfer coefficients for both aiding and abating flows are 136 Table 6.1 Average Nussult Numbers Resulting from Forced Convection over a Flat Plate. Boundary Average Reynolds Prandlt Average Conditions Ambient Number Number Nussult Temperature Tdo ReLx Pr NuF Constant Temperature 31.09 -6°C 2.74-103 0.716 Constant Heat Flux 42.42 Constant Temperature 62.52 -33°c 1.10-104 0.723 Constant Heat Flux 85.30 137 Table 6.2 Average Nussult Numbers Resulting from Free Convection. Time Rayleigh Average (hours) Number Nussult Number RaLx(-106) KEN 0 6.67 13.72 Storage 3 5.13 12.85 Interval 6 3.85 11.96 No. l 9 2.89 11.13 (-6°C) 12 2.44 10.67 15 2.05 10.22 18' 1.80 9.88 18+ 8.34 29.02 Storage 21 7.51 28.27 Interval 24 6.26 27.01 No. 2 27 4.59 24.99 (-33°C) 30 3.75 23.77 33 2.50 21.48 36 2.09 20.52 E.___ .___1:n‘-" 138 shown in Table 6.3a for the constant temperature assumption, and in Table 6.3b for the constant heat flux assumption. 6.1.1.4 Packaging Layer. The effective packaging resistance was found using Eq. (3.28), from the thickness and thermal properties of the packaging material. Since the packaging layer was actually composed of three substances; the paperboard box, the plastic film wrapping, and air trapped between the two materials, (Section 4.5) Eq. (3.28) was modified as follows to account for all three substances k k k h - -29 + —§ + —2i (6 1) k L L L ' p pb a pf where Lpb’ La and Lpf are the thicknesses of the paperboard, air inter- face and plastic film, respectively. The thickness of the paperboard, Lpb' was found to be 2 1.7 mm, and the plastic film was z 0.3 mm. The air interface varied from 1 to 10 mm. Thermal conductivities were found to be 0.18 W/m°C for the paperboard, and 0.2256 W/m°C for air at the average temperature between -6 and -33°C (Incropera and Dewitt, 1985). The thermal conductivity of the plastic film was 0.20 W/m°C (Modern Plastics Encyclopedia 1984-85). This resulted in an effective packaging coefficient, hpk’ ranging from 2.20 W/m°C to 18.10 W/m°C, and averaging 4.75 W/m°C. 139 Table 6.3a Combined Free and Forced Nussult Numbers, ES, and Convective Heat Transfer Coefficients, hxcv, assuming a Constant Temperature Boundary Condition. Time Aiding Flow1 Abating Flow2 (hours) EC hxcv RE hxcv 0 31.59 6.33 30.57 6.13 3 31.49 6.31 30.68 6.15 6 31.40 6.30 30.77 6.17 9 31.33 6.28 30.84 6.18 12 31.30 6.28 30.88 6.19 15 31.27 6.27 30.91 6.20 18' 31.25 6.27 30.93 6.20 18+ 63.71 11.61 61.27 11.16 21 63.61 11.59 61.38 11.18 24 63.45 11.56 61.55 11.22 27 63.23 11.52 61.79 11.26 30 63.12 11.50 61.91 11.28 33 62.94 11.47 62.09 11.31 36 62.88 11.46 62.16 11.32 1. Free convection aiding forced convection. 2. Free convection opposing forced convection. 140 Table 6.3b Combined Free and Forced Nussult Numbers, EC, and Convective Heat Transfer Coefficients, hxcv, assuming a Constant Heat Flux Boundary Condition. Time Aiding Flow1 Abating Flow2 (hours) EB hxcv EH hxcv 0 42.65 8.55 42.19 8.46 3 42.60 8.54 42.23 8.47 6 42.56 8.53 42.28 8.48 9 42.53 8.53 42.31 8.48 12 42.52 8.52 42.32 8.49 15 42.50 8.52 42.34 8.49 18‘ 42.49 8.52 42.35 8.49 18+ 85.86 15 64 84.74 15.44 21 85.81 15.63 84.79 15.45 24 85.73 15 62 84.86 15.46 27 85.63 15.60 84.97 15.48 30 85.58 15.59 85.02 15.49 33 85.49 15.58 85.10 15.51 36 85.47 15.57 85.13 15.51 1. Free convection aiding forced convection. 2. Free convection opposing forced convection. 141 6.1.1.5 Overall Surface Heat transfer Coefficient. Overall surface heat transfer coefficients, including the effects of free and forced convection and the packaging layer, were found from Eq. (3.29), using the results shown in Tables 6.3a,b, and in Section 6.1.1.4. Results for both aiding and abating flow conditions are shown in Figures 6.2a,b, for constant temperature and constant heat flux boundary conditions, respectively. There were insignificant differences in both the constant temperature and constant heat flux solutions for aiding and abating flow conditions, indicating that the free convection term had very little influence on the solution. 6.1.2 Estimation of Surface Heat Transfer Coefficients using Inverse Heat Transfer Estimation Techniques. Surface heat transfer coefficients were estimated using the inverse heat conduction (IHCP) techniques described in Section 3.5.2 from the three test results using the single layer slab of Karlsruhe Test Substance with one exposed surface (Tests 1a-c), described in Section 4.5. The experimental ambient and internal temperature values, shown in Figure 6.1, for the first repetition of Test 1, and in Figures E.la,b for the second two repetitions of Test 1, and the optimum parameters determined in Section 5.2.4 were used as input to the solution algorithm outlined in Appendix D. Results for the surface heat flux and surface heat transfer coefficients are shown in Figure 6.3, for the first repetition, and in Figures E.2a,b for the second two repetitions. Estimation of the surface heat flux, q, and the surface heat trans- fer coefficient, thx’ produced similar results in all three A repetitions. In all cases, there is a sudden increase in both q and 142 .coHuHUCoo %umccsom musumqu50H ucmumsoo mafia: quHowmmoou nommcwufi use: Hamum>o 0:» mo :oHuMCHEuouon Hmowuhflmc< mm.e ounwwm A339: oEF mm on ¢N w— NF 0 o p — L _ h — b b b 1? 1p! L 0.0 lfi. 1% fl.,ltliui}li¢lalizh -Iilix .Iilii! 1: nfim E.lH1711..ml.l.l&meiim1liJInwuwlln.u T--lxm.ll..:1l1ufii.fl-lyfi 10+ 4.311011141111105... Hal 488””; . Tod ”Unfllnwllllihllnnmutnlu lllwulhllllmnllllg 1| 0 fi cozom>coo 30.6.... EE 0.? n 1. «In @5330 . EE mg. n 0.. mlm rod 53002.50 02... r EE 0; n om ole cozoo>coo poocoh. EE 0.0? n o... a. .4 mcfioxood E 056?. . EE mm. 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These observations corresponded to the sudden increase in the estimated surface heat transfer coefficient at t - 0 hours, and sharp drop followed by an increase in hx at t - 18 hours, Lx in the solution using exact data from a constant heat transfer coeffi- cient as input (Figure 5.4). Therefore, the sudden changes in the estimated surface heat flux, as shown in Figure 6.3, are assumed to be a direct consequence of the solution method, and not characteristic of the actual boundary conditions. The estimation procedure provided relatively smooth curves for the surface heat flux compared with those for the estimated surface heat transfer coefficient. The algorithm to estimate a was designed to dampen the effects of irregularities in the solution, by including regularization terms and by incorporating several future time steps in the procedure. The surface heat transfer coefficient, however, is a direct function of a, which is a smoothed function, and the ambient temperature, which fluctuates with the defrost cycle, as shown in figure 6.1. To dampen the irregularities in the solution resulting from the defrost cycles, the ambient temperatures for each storage interval were A averaged in the estimation of hx Results for the three repetitions, Lx’ using averaged ambient temperatures from Figures 6.1 and E.la,b, are shown in Figures 6.4a-c. The estimation of the surface heat flux was unchanged; however, the irregularities in the estimation of hx were Lx damped using the average ambient temperatures. Comparing hx for the two storage intervals indicated that dif- Lx ferent boundary conditions prevailed in each storage room. The two fans in the first room were at the same end, and operated continuously during each defrost cycle. This resulted in estimated surface heat transfer coefficients which decreased smoothly, indicating that free convection 1h6 .Hm>noucH mmeOum comm you mmusumumneoe ucownfi< owmuo>< mam .AmH ummHV oommu3m pomoaxm oso Sofia anm Hoxmq oawcwm Eouw mocmumasm umoe oSSHmHumx £ua3 muafimom Housmawuomxm mafia: .v .xSHm umo: communm pad .x: .mucofiowmmoou Monacouh poo: poumEHumm< No.0 ouswwm < $505 oEfi mm 0% RN m_— N; w 0 c Booszmm -..- x; nososzmm ..n m: r 00—11 . i... ..o 3 OW]. L ~ .— WM.” mm; r __ . mm C m 1 \ ”MU.- rrn/I/ O i \ TW 0 a UW if. \I \\ $.01 S( x U I . or: _ a X . 4H 1% W __ z \a x x \ )MW .mm- :1. .5 . K a M... a s ’\In\ I. / mm 87 , (K o. w/wi .l D Lanny. r “cu 71H. OO— 0.67 i ( . I: 5 com, 3:2 on]? 6.93.. n meson m—lo ”Wood; H .anH .nE< .m>< mCOEUCOU a i 0.2”: u . ES 8:333 $3 65E. , 0mm E n—d H .xu £35.25 305 EwcmECQme ON 1h? .Hm>umu:H owmu0um 30mm pom moHSuMHoQEoH ucoH£E< owmuo>< paw .AQH umoHv mOQMHSm pomoaxm oco LuHB nmam Hohmq mawcfim Eoum mocmumnzm umoe onzumaumx £uw3 muadmom HmuGoEwuomxm wawm: .v .xSHm boom oommu5m paw .x: .mucoaowmwooo ummmcmua pom: wouoEHumm< no.0 oudwfim < $305 oEfi mm. on ¢N mp N. m b r n L b — b L b _ L w pouoEzmw II x: oouoEzmm .. I L < 007... ; OWIJ s). mm) . _ ‘ m m 01 a .. W i a , J \ .\ I II I I. S/W\ t. X 1 z I x d X oml — \Jl. \ l/ / mm L _ > \ / OF .4) \ ’ \z \ // p s — x / .\\ / 5 m LL OO — 1 . x . \ x. I I a ....UoCu ,\.l\ ’\l\ /\\./\ s . i EH of... 1 930: wmlmp ”oomfinl H com 28; mTo 6.0.9. n .95: 9E .m>< 952950 0.27 n .ane 8:333 $2 .025 i E 2.0 u .x._ .82on 92m BEoEtoaxm T f to O T CD (OozUJ/M) welomeoo Jelsqul inc-3H peiowgsg OnN 1A8 .Hm>umucH ammuoum 50mm How moudumuoaaok ucoaQE< owmuo>< paw .AoH umohv oommu:m pomomxm oco £ua3 swam noqu onCwm Eoum mosmquSm umoe onsumfiumx nuHB muadmom Hmucoswuoaxm magma .c .xSHm umo: commuSm paw .x: .mucoqofimwoou nommcmua woo: poumEHumM< oq.o ouswwm A9505 9:? mm on . em mp m. o o P .— + F n h b if r b P w 866sz I. x; 8655mm - u Estimated Surfo Heat Flux ( 1 2:2 819 6.92.. n com 28: 3:0 6%.? u as»: .95 .o>< 983680 i 0.0:“: u .QES @2325 68 HE... E n70 .|.. .x;_ .mmmcxof... 92m _OuC®EC®me 5 1483 2: w/M) 1U913U4900 (3oz Jal'SUDJi lDeH paiotu lh9 was a significant factor, and that it abated the influence of forced convection. The two fans in the second storage room were at opposite ends, providing a mixed flow, and they cycled on and off in synchroniza- A tion with the defrost cycle. This resulted in a cyclic curve for thx which followed the defrost cycle. Free convection appeared to have less influence during this storage interval as shown by the small overall change in magnitude in thx with time. This was assume to be a result of the mixed flow conditions prevailing in the storage room. 6.1.3 Comparison of Results using Analytical and Inverse Heat Conduc- tion Methods In comparing the analytical and inverse heat conduction solutions for the surface heat transfer coefficient, the assumptions used in generating the solutions were first examined. The assumptions made in generating the analytical solution are given below. 1. Steady state conditions, 2. Constant temperature or constant heat flux boundary conditions, 3. Laminar, unidirectional air flow over surface of product, 4. Known temperature at surface of product, 5. Negligible changes at packaging interface, such as the build up of frost, 6. One dimensional heat transfer, 7. Negligible moisture loss, and 8. Constant ambient temperature.. The assumptions used in the inverse heat conduction solution are as follows. 1. One dimensional heat transfer, 150 2. The surface heat transfer coefficient is a constant, or a function of time only, 3. Homogeneous, isentropic thermal properties, 4. Temperature measurements are made at a known position within product, and 5. Negligible moisture loss. The assumptions used in the IHCP solution are consistent with the conditions prevailing during the experimental procedures; however, many of the assumptions used in the analytical solution conflict with these conditions. Examples of conditions which conflict the assumptions used in the analytical case are transient heat transfer, mixed flow condi- tions, and known internal product temperatures (not at surface). The assumptions used in the analytical solution were very restrictive, compared to those used in the IHCP solution, which was designed to accommodate a wide variety of boundary conditions and variable thermal properties. Both methods yielded surface heat transfer coefficients ranging from 2 to 18 W/m°C, as shown in Figures 6.2a,b and 6.4a-c. In the first storage interval, the IHCP solution indicated that either free convec- tion or the diminution of frost was the limiting factor influencing the magnitude of the surface heat transfer coefficient, while in the analytical solution, forced convection dominated. The four primary factors influencing the magnitude and influence of free convection in the analytical solution were: (1) the magnitude of the coefficients in Eqs. (3.24a,b) and (3.25); (2) the magnitude of the difference between the ambient and product temperatures, used in determining the Rayleigh number; (3) the value of n used in Eq. (3.26); and, (4) the magnitude of the air free stream velocity used in calculating the Reynolds number for forced convection. The magnitudes of the coefficients in Eqs. (3.24a,b) 151 and (3.25) were given for very specified flow conditions, and the op- timum coefficients for the flow conditions used in this study may have actually been greater or less than those presented in these equations. Futhermore, since internal product temperatures were used instead of A surface temperatures, these estimated values of thx are assumed to higher than those that might have been obtained using surface tempera- tures. The value of n in Eq. (3.26) was also very influential: as n increases, the influence of the higher Nussult number (free or forced) increases exponentially; therefore, if n - 3 had been used, free convec- tion would have been more influential. Finally, the variability of the velocity measurements was high, and the velocity was assumed to be constant over each storage interval, disregarding the defrost cycles. In summary, small changes in the analytical determination of the surface heat transfer coefficients may have resulted in more or less influence from free convection. Both solutions yielded little overall change in the surface heat transfer coefficient with time during the second storage interval; however, the IHCP solution responded to the defrost cycle, with decreases and increases in the estimated surface heat transfer coeffi- cient as the fans turned off and on. To detect similar variations in air velocities using the analytical method, velocity measurements would have been required throughout the storage interval. Since the conductivity of air is very low, especially at low tem- peratures, the air interface thickness, La’ provided the highest resistance to heat transfer, and it was most influential in the deter- mination of the overall surface coefficient. Therefore, using different values for La changed the solution significantly, as seen in Figures 6.2a,b. The air interface thickness was very difficult to measure accurately, and the measurements varied significantly, within a given 152 package. The inverse heat conduction solution does not require knowledge of the surface conditions in the estimation procedure; there- fore it is not subject to the high variation in the analytical solution resulting from the interface measurements. Furthermore, the analytical solution does not account for the accumulation and diminution of frost as the product is cooled and heated, both in the package interface and on the surface of the product. The heat transfer coefficients of snow and ice are 10 to 100 times greater than that of air; therefore, as the frost layer accumulates or diminishes, the resistance to heat transfer decreases or increases accordingly. This results in an increasing or decreasing heat transfer coefficient. The analytical solution provided a continuous, smooth estimation of the surface heat transfer coefficients during each storage interval, with a step change in the estimation of hx between storage intervals. A Lx The IHCP method predicted, however, a sharp increase in thx at the beginning of the first storage interval, a sudden dip in thx between storage intervals, and a sharp drop in thx at the end of the second A storage interval. As discussed previously, these sudden changes in thx are all assumed to be characteristic of the estimation method, and not the actual boundary conditions. This indicates that this method does not respond well to step changes in surface conditions. In summary, although use of the analytical and IHCP methods yielded surface heat transfer coefficients within similar ranges, (from 2 to 8 W/m°C for the analytical solution, and from z 5 to 17 W/m°C for the inverse solution), several differences in the solution methods must be noted. Unlike the analytical method, the IHCP method provided a solu- tion valid for time dependent boundary conditions, without any restrictions on the airflow pattern, or changes at the product surface, such as frost accumulation. In addition, the analytical solution was 153 highly dependent on experimental measurements of air velocity and air interface thickness, which were both extremely variable and difficult to measure accurately. In comparing the results for the three test cases, A shown in Figures 6.4a—c, almost identical estimations for thx were obtained, indicating that the inverse method produces repeatable results. The primary drawback of the inverse heat conduction method was its poor response to step changes in surface conditions. Poor response to step changes at the beginning and end of the overall experimental test time can be avoided, however, by taking temperature measurements before, and continuing measurements after the designated testing time (Beck et. al., 1985). 6.2 Simulation of One Dimensional Heat Conduction Through a Food Product The one dimensional heat conduction program (Appendix B) was verified by comparison with analytical and experimental results. Analytical solutions obtained using constant thermal properties, and experimental data (Section 4.5) from the single layer slab with one exposed surface (Tests la-c), and the double layer slab with one exposed surface (Tests 2a-c) were used in the evaluations. 6.2.1 Comparison with Analytical Solutions The analytical solution, assuming constant thermal properties, was determined for a step change in ambient temperature. The temperature distribution within a body, initially at a uniform temperature, To, then eXposed to constant ambient conditions, T160 and thx, at x — Lx, and then subject to a step change in ambient conditions (T100 and hx at Lx) 15h time t is given below (Carslaw and Jaeger, 1959). (The boundary at x 1! - O is insulated throughout.) -A t 2 T(x,t) - To + (Tlm- To)-[ 1 - Rg-e ] 0 < t 5 t1 (6 2a) , -A£t T(x,t) - T2co - RI-To-e t1 < t s (tl+t2) (6.2b) where A - (k/ c )-c2 2 p P 2 F!. 2-hk-cos(§£x) _§1L R3 - ' [((hk)2 + (3)-Lx + hk]-cos(§£Lx) , I -A£tl q To - T10 " To + ( T2” ..- T1”).e 'i hk - thx/k The times t1 and t2 were both set at 18 hours, and the thickness, Lx, , was set equal to 0.13 meters; all of these values were consistent with ‘f the values used in the experimental procedures for the single layer slab with one exposed surface (Tests la-c). The ambient temperatures, T100 and T2”, and the heat transfer coefficient, hx were based on the 1x, overall average values found from Tests la-c. The initial temperature was set at -33°C, and the ambient temperatures, Tloo and T were set at 2w’ -6°C and -33°C, respectively. The average overall heat transfer coeffi- cient, was found from Figures 6.4a-c to be 8.5 W/m2°C, and was used for thx in the analytical solution. The thermal properties were calculated 155 using the average values between -33 and -12°C. This range was deter- mined from the minimum and maximum product temperature values shown in Figures 6.1 and E.la,b. The numerical one dimensional solution was found using identical initial and boundary conditions, and using both constant and temperature dependent thermal properties. A time step of 120 seconds and a position increment of 0.016 m were used in the numerical solution. These are the same values proposed for use in Section 5.1.3. The temperature histories at the mid-section of the slab (x - 0.065 m), from the analytical solution, using constant thermal properties, and from the numerical solution, using both constant and temperature dependent thermal properties, are shown in Figure 6.5. The numerical solution with constant thermal properties provided an excel- lent approximation to the analytical solution (using constant properties). The effect of the temperature dependent thermal properties was found to be very significant: at the end of the first storage inter- val, the solution using variable thermal properties was 27% higher than both analytical and numerical solutions using constant thermal properties. This indicates the importance of accurate estimation of thermal properties of foods during the freezing and post-freezing processes. Similar solutions were obtained using the same conditions, described in the experimental procedures, for the double layer slab, with a thick- ness (Lx) of 0.25 m, and only one exposed surface (Tests 2a-c). In these tests, the methyl-cellulose boxes were layered two high, and both storage intervals were increased to 24 hours. The same initial and boundary conditions, average thermal property values, time step, and Spatial increment used for the comparison with the single layer slab CFigure 6.5), were also used in this comparison. Solutions for the 156 .N\xq I x on oocmumnSm ummH assumauwx mo mowuuoaoum Hmauosh ucmumcou zua3 coHusaom Hmowuzawc< nuHB poumaaoo :oHuSHom Hmowumesz HmconCoEHQ oco m.o ouswmm A3305 mEfi Amozcoaotn. 233.53 cozaom 38.6832 .. I Amfltoaotd acoumcoov co_...3_om BotcEaz II Amozcoaotd “coumcoov co_u:_om 60:30.2 0 on on em 9 S o o b F . F . b . _ F L . 0.0.?! o u x so 036? 0.0 u x._ u x so 63$? 3 u .38 .96: Sm: f 2:2 3:2 .92: was; 2:0 .90: u .92: .92 roan: 0.8: u .ane .055 E . H .x 68: o. o 2 o ._ x E a .m. lode: (3°) eJnimedLuei 157 temperature histories at Lx/h (x - 0.0625 m), and 3-Lx/4 (x - 0.1875 m) are shown in Figure 6.6. Again, the analytical and numerical solutions using constant thermal properties were almost identical, while the numerical solution using variables thermal properties yielded tempera- ture values significantly higher. 6.2.2 Comparison with Experimental Results. Results obtained using the numerical one dimensional heat conduction program were compared to the experimental results obtained for the single layer slab, with one exposed surface, (Tests la-c), and for the double layer slab, with one exposed surface, (Tests 2a-c). In simulat- ing the conditions in the experimental procedures, the predicted heat transfer coefficients, shown in Figures 6.4a-c, were averaged over specified time increments for each storage interval. These values were used as input to the numerical model and compared with the experimental results. In using the numerical model to compare with the experimental results, the surface heat transfer coefficient was determined in two ways. First, the surface heat transfer coefficients were averaged over the total storage interval, and second, the surface heat transfer coef- ficients were averaged over two to four hourly increments over the total storage interval. In each case, variable thermal properties, and iden- tical product thicknesses (Lx - 0.13 m) were used. The thermal properties were calculated using the initial freezing temperature, Tif’ given by Gutschmidt (1960), and the thermal properties of the unfrozen methyl-cellulose, p, k, and Cp, given by Specht et. a1. (1981), shown in Table 4.1. The initial and ambient temperatures were obtained from average initial product temperature and overall ambient temperature 158 .¢\xq.m I x paw «\x4 I x on mocmumnsm umoe mnaumaumx mo mofiuumaoum assuonh ucmumcoo Lodz cowusaom Hmofiumamc< £ua3 poquEoo :oHuSHom Hmofiuoasz HononCoEHa oco w.o owsmfim Amazocv oEfi E 2.0 n .5 585.2,: new :53 n xv cozéom _oo:>_oc< o I I / \ \ \\ \ .\ I I. V ..l \ .\ \\ x \ \ \ A¢\x.__..n n x $330 .633 cozaom _oo_..oE:z .. I moztoaem o_no_._o> A¢\x._ n x “0330 326.; coraom EctoEaz I I A¢\x._..n u x 6230 conga cozaom BotcEaz II ?\x.. n x 6230 330.5 cozaom .ootoEaz II. mozcoaen. «coamcoo $531... u xv co::_om 623.92 0 me 9. an ...m m: m o _ u . u . I; r _ . .L I nflowuu o n x to use}, 0.0 .I. x4 ...... x «o .oan\3 0.0 u .300 .mco._._. you: . 950: melem .0..an ”Ego; ...NIo .oooI u .38 .nE< row? oonnI u as...» 625 (3°) SJDIDJedLUSi 159 measurements over each storage interval. Again, an insulated boundary condition was imposed at x - 0. The initial and boundary conditions used in the numerical solution are shown in Table 6.4. Results are shown in Figure 6.7 for the first repetition using the single layer slab with one exposed surface, (Tests la), and in Figure E.3a,b for the second two repetitions (Tests lb,c). In all cases the solution obtained using incrementally averaged surface heat transfer coefficients yielded very similar values as the experimental results. This was expected, since the one dimensional solution was used directly in estimating the surface heat transfer coefficients.' The temperature solutions obtained using overall averaged surface heat transfer coeffi- cients yielded slightly lower values than the experimental data. To compare the numerical solution with the results of the two layer slab with one exposed surface (Tests 2a-c), the surface heat transfer coefficients shown in Table 6.4, calculated using the experimental results for the single layer slab, were used. The time increments for the surface heat transfer coefficients were adjusted to account for the longer storage intervals (24 hours) used in the experimental procedures for the double layer slab, compared with the intervals (18 hours) used for the single layer slab. The initial and ambient temperatures used in the numerical solution were based on the average initial and ambient temperatures for the experimental results of the double layer slab. These values are shown in Table 6.5. Numerical solutions were found at Lx/4 (x - 0.63 m) and at 3-Lx/4 (x - 0.188 m), and compared with temperatures measurement at ap- proximately the same locations in the double layer slab configuration, described in Section 4.2. These comparisons are shown in Figure 6.8, for the first repetition of the experimental test using the double layer slab, and in Figures E.ha,b, for the second two repetitions of the test. 160 Table 6.4. Surface Heat Transfer Coefficients used in the Numerical Solution in the Comparison with Experimental Results of the Single Layer Slab with One Exposed Surface (Tests 1a-c). Surface Heat Transfer Coefficient (W/m2°C) Time Test 1a Test 1b Test 1c Average Storage Interval 1: 0 - 2 hr 12.31 11.90 12.00 12.07 2 - 4 hr 12.16 11.81 11.99 11.99 4 - 6 hr 11.18 11.09 10.78 11.01 6 - 8 hr 9.11 9.48 9.10 9.23 8 -10 hr 7.81 7.85 .55 7.74 10 -12 hr 6.44 6.63 6.47 6.51 12 -14 hr 5.65 5.93 5.88 5.82 14 -16 hr 5.56 5.66 5.71 5.65 16 -18 hr 5.49 5.45 5.58 5.51 Storage Interval 2: 18 -20 hr 5.37 6.62 6.65 6.21 10 -12 hr 7.34 9.20 9.46 7.44 12 -14 hr 9.24 10.86 11.10 10.40 14 -16 hr 9.56 10.86 10.91 10.38 Average for Storage Interval l: 8.41 8.42 8.34 8.39 Average for Storage Interval 2: 8.37 9.73 9.93 9.34 Initial and Storage Temperatures (°C) To -33.0 -31.2 ~31.5 -31.9 T -6.0 -6.0 -5.9 -6.0 «3,1 T ~33.6 -33.5 -33.0 -33.4 00,2 161 on .Ama ummHv oommu3m pmmomxm oco fiuHB anm “0%64 meCHm Eonm muasmom HmucoEHuomxm ou poummeoo :oHuSHom Hmowuossz HmconCoEHa oco n.o ouzwwm A3305 mEc. on ¢N @— L h L [— IF N_. a *-(D L L E @006 u x yo .Eom _oo_._oE:z x... u x yo AoonE\>>v .aooo coamcofi How: u x; E .26 n .x.. 52m *0 mmocxflf. PE onle .90an ”mt: male .906! n .anF .nE< 0.0.3.. I .ane .235 PE owl? .o.ml¢.m ”mt: mFIo .nhlnd— n 5.. ...o> we: mmlm— find uPE mFlo .36 H x; .umcoo E nmod u x be 3.33.. .390 .o>< o .0235 omocoam 36 x: .330 I I .0235 003on ..o>o x: o_no_._o> II. (3°) eJmoJeduJai 162 Table 6.5. Initial and Ambient Temperatures used in Numerical Solution for Comparison with Experimental Results using the Double Layer Slab with One Exposed Surface (Test 2a-c). Initial and Storage Temperatures (°C) Test 2a Test 2b Test 2c I, -25.9 -27.7 .24.0 T - 6.0 - 6.1 - 6.1 «0,1 T -29.2 -26.9 -26.0 «“2 163 .Amm umoev momwudm comomxm oco £ua3 nmam HommA cannon Eoum muasmom kucoawuomxm ou woummeoo coausaom HooHHoESZ HmconCoEwa oco w.o ouswwm A9505 oEfi m... 9. mm em 3 m o IF IF L F L F I? i? L 0.0¢| 2; meIem .13de HE vao .m.ml.m. .I. 5. ...o> x4 I. x «o AoonE\3v .eooo coamcot. you: n x; L L E mad ..u .x._ 52m *0 mmocxofh ro.mnl 9E 9.le .ooN.mNI uPE hwmlo .ooo.ml fl dEo._. .nE< f 05.2: .... as»: .055 ..oan AE mate a xv cozaom _oo_._oEaz II 1 AS 806 u xv cozaom _oo_._oE:z II rofil AE 02.0 n xv 333m .396 .m>< .76 AE nwod fl xv 3.331 .coaxm .m>< I (3°) SJDIDJedwel 164 In all cases, the predicted values from the numerical solution using variable thermal properties were significantly lower than those obtained experimentally. In addition, the predicted values nearer the surface at x - 0.188 were less accurate than those obtained near the insulated surface at x - 0.063 m. (After 24 hours, results yielded a 37% error at x - 0.188 m, while the error at x - 0.063 m was 9%.) The following explanations for these results were proposed: (1) the initial freezing point of the methyl-cellulose was actually -0.7°C (Specht et. al., 1981), and not -l.0°C, as given by the original source, Gutschmidt (1960); (2) the assumption of negligible resistance to heat transfer due to the packaging interface within the total product mass was invalid; and, (3) the assumption of perfect insulation along the first boundary (at x - 0), used in the experimental procedures, was invalid. To test these hypothesis, the numerical simulation was first repeated using Tif - -O.7°C, instead of Tif - -1.0°C, as given by Gutschmidt (1960), to determine the temperature dependent thermal properties. All other input values for the numerical model remained unchanged from those used for the solution shown in Figure 6.8. The I numerical solutions at x - Lx/4 and x - 3-Lx/4 are compared with the experimental results in Figure 6.9. Comparing the numerical solutions in Figure 6.8, using Tif - -l.0°C, and in Figure 6.9, using Tif - -0.7°C, there was very little difference in the numerical solutions, indicating that the difference in the numerical and experimental values was not primarily a result of the differences in the magnitude of Tif found in the literature. The second explanation, proposed to explain the differences between the numerical and experimental results, was that the assumption of 165 .Amm umoev oocmusm pomoaxm oso Lua3 anm Mazda cannon Eoum muasmom Houcoaauonxm ou coucafioo .Aammav .Hm .uo uzooam an co>wu .m H .ucHom weauooum HwHuHcH magma .coauzaom amoduoasz HmconCoEaa mco m.o ouswfim A0505 oEc. In W b ww 0* NM wrN Q— m L. P mt; m¢l¢N .wdwlud uPE. .vNIo .mfil—NF n x; ...o> 03 n x we AconE\>5 .aooo 0295; you: n x... E 00.0 .... .x._ .020 to 8052.: 2; SILK 0.0.0? “...; 0T0 .0.0.0I ..I. .053 .95 0.0.0NI I. .90.: .002. 0.5.0.. I E .1060 0532... .0001. AE mm—d n so cozzom BoroEaz I I AE nmod u Xv cozgom _oo_._oE:z II. AE mm—d n xv 3.3001 .toaxm .m>< Olo AE mood .u xv 3.303. .0090 .m>< I -007 1.0.0? f 10.nl 0.0 (3°) SJnlDJedLuei 166 negligible resistance to heat transfer due the internal packaging be- tween the containers was invalid. To test this hypothesis, the extreme case of infinite resistance to heat transfer at the packaging boundary was considered. The numerical simulation shown in Figure 6.8 was repeated using an insulated boundary condition at the location of the internal packaging boundary (Lx/2). This problem was similar to the case of the single layer box with one exposed surface (Test la-c). The numerical solution at 3-Lx/4 (x - 0.188 m) for this hypothesis was compared to the experimental and previous numerical results, given in Figure 6.8 at the same location. Results are shown in Figure 6.10. The experimental results fall between the two extreme cases of no resistance to heat transfer and infinite resistance to heat transfer at the packag- ing interface, supporting the hypothesis that the assumption of negligible resistance to heat transfer at the packaging interface was invalid. The third hypothesis was that the assumption used in the experimen- tal procedures, of a perfectly insulated boundary condition at the inner container surface (x - 0), was invalid. The container which held the methyl-cellulose paperboard boxes was designed to limit the heat flow at the unexposed surfaces to less than 1% of the expected heat transfer rate at the exposed surface, assuming equal surface heat transfer coef- ficients on all sides of the container. However, the bottom of the container rested on a stainless steel cart (high conductivity), so that in the extreme case of perfect conductance between the container and the cart, the bottom of the container may have been at the ambient tempera- ture. Based on the thicknesses of the insulation board and plywood in the container (Section 4.2), a conservative estimate of the effective heat transfer coefficient, defined as (insulation thickness)/(thermal conductivity), through the insulation materials was 1 W/m2°C. The 167 .Amm 0mmHv moMmuSm pomoaxm oco SuHB nmam Momma mandoa Eowm muedmou AmuCoEHuomxm 00 poumaeoo .AN\KA I xv oommuoucH wcwmmxomm um cowuwpcoo humpcsom pouQHSmCH momanH uzonufi3 0:0 £0w3 .cowusaom Hmowuoesz HmconCoEHo 0:0 0H.c ouswwm A0505 oEfi m... 00 mm 0m ma m o . _ F _ . L . _ . a r 0.0¢| mE mvlwm ..vdplqm u0E wmlo .mfiIFNP H x; 00> 3 .II. x 00 AoonE\>>V .0000 030:0: 000: n x... E mmd n .03 .no_m .0 0005.25 rohnl mE mwlvm .ooN.mNI u0...; .vNIo .ooo.ml H .anh .nE< .. 0.0.0.? I .05.: .055 -007 (30) einqoiedwai E 0.0 u x “0 bopczom 03030:. 53 .Eom .Eaz I I Iohl x 00 bopcaom 03030:. 53, .c_om .Eaz II 33001 .toaxm .02.. I E 09.0 n x 00 .500 E 02.0 168 simulation shown in Figure 6.8 was repeated using a heat transfer coef- ficient (thx ) of 1.0 W/m2°C at x - 0. Results are shown in Figure 0 6.11. The temperatures at Lx/4 from O to 24 hours were over-estimated, while the temperature values from 24 to 48 hours were under-estimated, supporting the third hypothesis. Comparing Figure 6.10 and Figure 6.11, suggests that both assump- tions of negligible internal packaging resistance to heat transfer, and a perfectly insulated boundary at x - 0 were invalid. These results indicate that there is a need for further study in estimating the resistance to heat transfer due to the packaging inter- face. The packaging interface can be thought of as a contact resistance, with a contact resistance coefficient associated with it. This coefficient may be estimated using the same methods used for es- timating the surface heat transfer coefficients. In this case, there are two unknowns, the surface heat transfer coefficient and the contact resistance, requiring additional internal temperature measurements. At least one thermocouple would be required on each side of the packaging interface. The existing computer program developed to estimate the surface heat transfer coefficient (Appendix D) could be used with minor modification to also estimate the contact resistance coefficient. 6.3 Simulation of Two Dimensional Heat Conduction Through a Food Product. The two dimensional heat conduction program (Appendix C) was verified by comparison with the one dimensional model and experimental results. The one dimensional model, discussed in Section 6.2, was used to demonstrate that the two dimensional solution can be reduced to the one dimensional solution. Consistant with the comparisons described in 169 .Amw ummav oommuzm pmmomxm mco nuw3 anm momma mansoo Eouw muaamou Housmafiuoaxm ou oouoafiou .o I x um oo~E\3 H I ox; Sofia .aofiusaom Hmoauoasz HmCmeCmEHQ oco HH.© ouswam A9505 mEfi AE mad I xv cosgom _oo_._oE:2 n AE nmod n Xv cozaom _oo_._oE:z II AE mmfio fl xv 3:601 .390 .m>< olo AE nmod fl Xv 3.33m .toaxm .m>< I m¢ o¢ mm ¢N m. m o p _ r L p _ b L F _ . O.O.V|. 9:. 31% 6.213 as sale .mhifi u E ..o> . x4 H x yo Aoan\>>v .wooo toumcot. “on: H x; E nmd n .3 52m to mmmcxef roan: 9; Elem .033: at; ...mlo .906: u .95: .2: . 90.2: n .95: 62:. rodnu 2:2 $10 .0382, o; u 92 o H x go .300 5.6.5.: “cm: ...I. 0x: s ‘ (3°) eJnioJedLuei 170 Section 6.2.1, the two dimensional solution was compared with the one dimensional solution using constant thermal properties. The experimen- tal data (Section 4.5, Tests 3a-c) was compared with two dimensional numerical solution using variable thermal properties. 6.3.1 Verification of Two Dimensional Model. The one dimenisonal numerical model, used in verifying the two dimensional numerical model, was shown to approximate the analytical solution excellently, using constant thermal properties, in Section 6.2.1. The two dimensional model was compared with the one dimensional numerical model for two reasons: (1) to verify the accuracy of the two dimensional model; and (2) to show that the two dimensional model may be reduced to the one dimensional model. The geometry of the problem considered here was based on the geometry used in the experimental procedures for Tests 3a-c. A rectan- gular rod, measuring 0.25 m by 0.27 m, with perfectly insulated surfaces along the third dimension was considered in the two dimensional model. A surface heat transfer coefficient of 8.5 W/m2°C was alternately im- posed along two of the remaining parallel surfaces, and an insulated boundary condition was imposed along the other two surfaces. The initial temperature was -33°C, and the simulated ambient conditions were taken to be ~6°C from O - 48 hours, and -33°C from 48 to 96 hours. These were the same storage time intervals used in Tests 3a-c. First, the surface heat transfer coefficient was imposed at x - 0.25 m, and the surface at y - 0.27 m was taken to be insulated. This solution was compared with the one dimensional solution using a slab, 0.25 m in 2 thickness, with a surface heat transfer coefficient of 8.5 W/m °C at x - 0.25 m, and an insulated condition at x - 0.0 m. Results are shown in 171 Figure 6.12a. A similar comparison was made using a surface heat trans- fer coefficient of 8.51 W/m2°C along the surface at y - 0.27 m, and an insulated condition on all other surfaces, as the boundary conditions for the two dimensional model, and similar boundary conditions (hy - 0, at y - 0; hy - 8.5 W/m2°C, at y - 0.27 m), with y - 0.27 m, for the one dimensional model, as shown in Figure 6.12b. in both instances, the solutions proved to be identical, which demonstrates that the two dimen- sional numerical solution may by reduced to the one dimensional solution, and that, since the one dimensional solution accuracy was previously verified (Section 6.2.1), the two dimensional numerical solution is accurate. 6.3.2 Comparison with Experimental Results. The two dimensional numerical heat conduction program was compared with the experimental results using the triple layer, two dimensional configuration (Tests 3a-c), shown in Figure 4.3. Numerical solutions were found using the same geometry for the configuration used in Tests 3a-c (Lx - 0.25 m, Ly - 0.22 m), with insulated boundary conditions imposed along the surfaces at x - 0, and y - 0. The heat transfer coefficients shown in Table 6.4 were used as input for hx and hyLy in Lx the numerical model, (hx and hyLy were considered to be equal). Since Lx the storage interval used for the experimental tests was longer than either of the one dimensional tests (Test la-c and Test 2a-c), the time increment for the last heat transfer coefficient of each storage inter- val, shown in Table 6.4, was adjusted to account for the longer storage time of the two dimension tests. This was done on the assumption that the surface heat transfer coefficients, shown in Figures 6.4a-c, ap- proach a constant value at the end of each storage period. (The sudden 172 mm .o.~a\3 Hm.m u x; cuss coausaom Hmoaumssz HmcoamcoeHo «so so“; .o.o n a; .o. s\3 Hm.m u x; LuHB cowuaaom Hmoauoasz HmconCmEaa 038 no :ofiuwommHno> mNH.o muswwm A9505 9:; ma 9. em 0 L b L F L — b L O.O¢I 23; 81m... .92.. 352 91.0 .99. n .QES .nE< 0.0.3: I SBESES 625 ohm: i 3 u XV 8226 “03E? Ed I x; .... a E BN6 fl 3 ".EE F r 8 n .0 toucoo “oonEbs Ed I x: l Tohl E 3.0 n 3 03 n 5 82.8 66 a E .o....E\; E.» n x; o r E 3.0 ... j H.Ea mi. 8 n xv 555 66 n .3 .anbs Ed I x; o (3°) SJnlDJedLuel 173 .o. s\3 Hm.” I a; cuaa cowussom Hmouumssz Hmcoamcmaao 20 gas: .0. a\z Hm.w n a; .o.o : x; Lufi3 GoduSHom HmoHuoEdz amcowmcoafia o3e~mo COwuonmHuo> an.o muswwm A9305 oEfi mm Q. 9. ¢m o .r t . L, L \L L t nfiO¢II ego; calm... .92”... "232 male .9? ... dEfl EE< 0.0.3... ... 232033 BEE can: .fi 9.. u a 825m ”9%; Ed .. E .. s E “Nd fl 3 ”.55 F r 3 n b 3.50 "oonEbs Ed I A: .I. rohl E Rd ...... .3 .3 n a 82.8 6.0 n x; .an? Ed I a. o f E mud n 3 ".EE m i. 8 ... 3 .350 ad a 5. .o....E\3 :3 n .3 . (3°) SJnlDJedLuei 174 changes in the surface heat transfer coefficients at 18 and 36 hours, as in Figures 6.4a-c, discussed previously as being characteristic of the solution method, were not included in the determination of the incremen- tally averaged values shown in Table 6.4.) Numerical solutions were determined at the locations x - 0.063 m, y - 0.035 m, and x - 0.225 m, y - 0.19 m, which were the average ther- mocouple locations of the experimental measurements in Tests 3a-c. A time step of 60 seconds was used, with a spatial increment of 0.016 m, for Ax, and 0.017 m for Ay, based on the analysis given in Section 5.1.5. Thermal properties, given by Gutschmidt (1960) and Specht et. a1. (1981) were again used to determine the thermal properties below the initial freezing point. Results are shown in Figure 6.13 for the first repetition using the triple layer configuration with two exposed surfaces (Test 3a), and in Figures E.5a,b, for the second two repetitions of the test. Similar results were found in all repetitions. These results have the same characteristics as the results found for the double layer slab con- figuration with one exposed surface (Test 2a-c), shown in Figures 6.8, and E.4a,b. The temperatures at the location nearest the surface (x - 0.19 m, y - 0.225 m) were under-estimated during the first storage interval, and over-estimated during the second storage interval. At the end of 48 simulated storage hours, the numerical solution gave tempera- tures 1.5 C° lower than the experimental results at the same time and location. The numerical solution varied less than 1 C° from the ex- perimental values at the location x - 0.063 m, y - 0.035 m. As was found in the comparisons shown in Figures 6.8 and E.4a,b, the numerical solution predicted a smaller variation in temperature between the loca- tions shown, than the experimental results. The latter two hypothesis 175 mm .Awm umoev mooomusm oomomxm oae Suds Adam poms; manage Eouw muaommm Hmucoaawoaxm ou woumafioo CoHuSHom Hooau0332 HmGonamEgo 039 ma.o muswfim Amtnocv mEP Nx. we. VN o f L h L p L » 0.0.VII AE 0006 ...u x .E nmod fl xv 3.3QO .390 .m>< I AE mud n > .E 26 fl Xv 33QO .596 .m>< ole . E 986 ... s .E Bod ... xv 3:28 _8_EE32 I roan: AE «Nd I x .E 9.0 n xv cozaom 628832 -I AE BN6 ..u 3v m_xol> 9.0.0 c0583 H x f rodnl - AE mad I x._v m_xolx mco_o cozooom H x , ,. - - _. ,, .. .- , - .31; .x._nx go S: .5. .AoonE\>>v .300 .93.; “on: .x 1‘ E; 4.07.: a; male fining u E u x; ...o> PE calm... .903: a: mile .93.. u .QES .nE< 9ST ... 232033 62:. some: _oo_._oE:z ..ov mtoumEotom “sac. (3°) SJnlDJedLUSi 176 presented in Section 6.2.2 were proposed as explanations for the results. In the first case, the assumption of negligible influence due to the packaging interface was hypothesized to be invalid. This hypothesis was tested in the same manner as was done in Section 6.2.2. An insulated boundary condition was imposed at the packaging interface around the corner methyl-cellulose box (x - 0.13, y - 0.18 m), and the numerical simulation was repeated using all other input values as before. Results are shown in Figure 6.14. As was found in the one dimensional com- parison (Figure 6.10), the experimental results fell between the two numerical solutions with and without the insulated condition at the packaging interface. This indicated that the resistance due to the packaging interface was greater than zero, but less than infinity, again supporting the hypothesis that the assumption of no resistance to heat flow by the packaging material was invalid. The third hypothesis, presented in Section 6.2.2, that the assump- tion of perfect insulation at the interior boundaries (x - 0, y - 0) was invalid, was tested by imposing a surface heat transfer coefficient of l W/m2°C along the these boundaries. Again, similar results were obtained as was found in Section 6.2.2. The estimated temperatures are compared with the experimental results in Figure 6.15. The solution nearest the interior boundary (x - 0.63 m, y - 0.35 m) over-estimated the experimen- tal results during the first storage period and over estimated the results during the second storage period. Comparing Figures 6.13 and 6.15, the experimental values at x - 0.63 m, y - 0.35 m, were bounded by the numerical solutions using an insulated condition and a surface heat transfer coefficient of l W/m2°C at the boundary at x - 0, y - 0. This again supports the hypothesis that the heat transfer coefficient at the interior boundaries was somewhat greater than zero. 177 mm .AMm umoav moommusm oomoaxm 039 spas Adam uo%m4 madame scum muasmou Hmucmefiuomxm ou pmumanu ooomuoucH wcuwoxomm um :oHuHocoo humocsom oouaezmcH oomoaem usonuws was Sofia .coHuSHom Hooauoasz HmconCoEHQ oBH «H.o oustm Amtzozv mEP mm. WV .va O 3:501 .396 .m>< ale 0 owl E 2.0".» .E nfoflx «o c0323.... 53 .c_om .832 II . Emfofl» .E n—dflx yo cozofimc. 3053 Sam .Eaz .. I rohml AE nmd M iv m_xol> mco_o cozooo... n A W AE .36 n xqv m_xolx 9.5.0 c0583 u x 0.0m... ....E 4.9:: a: $10 .2..-an .... .3 u x; ...o> . .jnx .x._ux to $2 .xc .Aoo«E\>>v .Looo .mcot. boo: (3°) eanoJeduJei mt: calms» .03.le ”mt: mwlo .016! 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BEE . ”Boo: _oo_..oE:z tot mthEEod 39.: (3°) SJnlDJedLlJei 179 Therefore, the results shown in Figures 6.14 and 6.15 suggest that both assumptions of negligible internal packaging resistance to heat transfer, and perfectly insulated interior boundaries were invalid, supporting the results found in Section 6.2.2. This again indicates that there is an need for further study in estimating the resistance to heat transfer due to the packaging interface. _. -_‘_ .4____—_——-r:=" 180 6.4 Effects of the Magnitude of the Activation Energy Constant on the Mass Average Quality History The degree of temperature dependence of quality degradation is dependent on the magnitude of the activation energy constant (Ea); typical values for food products range from 40 to 200 kJ/mole (Bonner, et. al., 1984). Two quality criteria are commonly used to determine the activation energy constant: (1) High Quality Life (HQL), and (2) Practical Storage Life (PSL). High Quality Life is defined as the length of product storage time until a detectable change in frozen food quality exists when compared to the same product stored at conditions where quality change is limited. Practical Storage Life is defined as the length of product storage time until the frozen food has unaccep- table quality (Scott, et. al., 1984). Activation energy constants reported for HQL are generally higher than those reported for PSL (Bonner, et. al.,l984). The effects of the magnitude of the activation energy constant (Ea) on the mass average rate of quality deterioration was demonstrated by comparing the predicted mass average quality histories for a range of values of Ea. One dimensional heat transfer was considered through a rectangularly shaped product mass, measuring two meters in the direction of heat transfer, and initially at a uniform temperature of -30°C. A surface heat transfer coefficient of 8.5 W/m2°C was imposed on the boundaries perpendicular to the direction of heat flow, and all other boundaries were considered to be perfectly insulated. The hypothetical food product was subject to storage conditions of 100 days at -5°C. The thermal properties of strawberries were used in generating the tempera- ture distribution histories. These values are shown in Table 6.6. Note the thermal properties given for strawberries are very similar to those 181 Table 6.6 Thermal Properties of Unfrozen Strawberries. Density Thermal Specific 3 Conductivity Heat (kg/m ) (W/m°C) (kJ/kg°C) 1040 0.54 3.93 (Heldman, 1982) 182 given for the Karlsruhe Test Substance. Quality distribution histories were determined using a hypothetical reference shelf-life of 500 days, and activation energy constants ranging from 20 to 200 kJ/mole. The resulting quality was expressed as a percent of the initial shelf-life at the reference temperature, in this example, 50% quality would mean the product would have 50% of 500 days, or 250 days remaining storage life at -18°C (reference temperature). Results are shown in Figure 6.16 for activation energy constants of O, 20, 40, 60, 100, 120, 140, 160, 180, and 200 kJ/mole. An activation energy constant of zero indicates no temperature dependence for the rate constant, and was used as the base line for comparison. From Figure 6.16, products with ac- tivation energy constants of sixty or less have little temperature dependence, products with activation energy constants from 60-100 kJ/mole have moderate temperature dependence, and products with Ea greater than 100 kJ/mole have high temperature dependence. Strawberries were chosen as the food product considered in all of the subsequent quality analysis, since the activation energy constants found in the literature for the HQL and PSL criteria cover both ex- tremities of the range for typical food products. Guadagni (1969) reported an activation energy constant (Ea) of 182.37 kJ/mole, with a reference shelf life of 630 days at —18°C for HQL of bulk frozen straw- berries. Tressler, et. a1. (1957), reported an activation energy constant of 49.13 kJ/mole, and a reference shelf life of 540 days at ~18°C for the PSL of sliced frozen strawberries. The kinetic properties for strawberries are shown in Table 6.7. 183 AHOON nfiom— aflomp nfio¢p .Houp nfioo— nfiom nfiom aflow “HON OAv fl IIEIIIIIIII om om om om om om om om om om om oop .Uom- um ommuoum a“ ammo OOH ou vomoaxm .U.Om- um haamwuHGH uosooum o How .oumm soaumuowuouoo xuaamod coon so .AoHoE\wxv mm .ucwumcoo hwuosm cowum>wuo< mo monugcwmz mo uoommm ma.o ouswfim 9«E\3 Om .... E O.N " Oomwl «0 m>OU Amxonv oEfi x: x4 com I £31.25 Ex O¢ ON Tom r9. Tom (z) mono 184 Table 6.7 Kinetic Properties of Frozen Strawberries. Activation Reference Reference Shelf Energy Constant Shelf Life Life Temperature (kJ/mole) (days) (°C) 1 HQL Criterion 182.37 630.0 ~18 2 PSL Criterion 49.13 540.0 -18 l. Guadagui, 1969. 2. Tressler, 1957. 185 6.5 Effects of Boundary Conditions on Temperature and Quality Histories of Frozen Foods during Storage. Variations in boundary conditions, such as step changes in storage temperatures, or a change in the surface heat transfer coefficient, can change the rate of heat transfer, which consequently, change the tem- perature history of the food product. Since quality degradation was assumed to be temperature dependent (Eq. 3.49), variations in boundary conditions will also affect the quality profile within the product. It was desired to determine under what conditions the quality distribution history was affected the most. Several different parameters affecting the boundary conditions were investigated: (1) the magnitude of the surface heat transfer coefficient; (2) the storage time interval for step changes in the ambient storage temperature; and, (3) the magnitude of the storage temperature, and the amplitude of step changes in the storage temperature. One dimensional heat transfer through a rectangularly shaped food product of thickness, Lx, equal to 2.0 m, was considered. The initial temperature was uniform and equal to -30°C in all cases. This might correspond to a pallet load of frozen foods with two non-adjacent ex- posed outer surfaces, and perfectly insulated on all other sides. The total simulated storage time was set at 100 days. From Tables 5.4a,b, only one eigenvalue was found to be significant after approximately two hours, and the numerical approximation for the first eigenvalue was very accurate, even for large Ax (Figure 5.1). Therefore, only 25 nodes (Ax - 0.042 m) were used for the long storage time solutions. A time step of 3600 seconds was used; this value is near the upper end of the accuracy criterion, shown in Table (5.5), for thermal properties evaluated at -12°C, and it is much higher than the 186 time step criterion requirement for no oscillations (Tables 5.2a,b). Use of this time step and spatial increment was justified by comparison with results using 32 nodes and a time step of 600 seconds. No dif- ference in the temperature distribution history was found after 50 hours of simulated storage time at -5°C, with the surface heat transfer coef- ficient, hx, equal to 8.5 W/m2°C. The influence of the magnitude of the surface heat transfer coeffi- cient was demonstrated using a single storage interval at -5°C for 100 days. The effects of step changes in storage temperatures were inves- tigated using nine different combinations of storage temperatures and storage time intervals. These nine cases are defined in Table 6.8, and will be referred to by their respective case number, for example, Case 1 refers to fluctuating storage conditions between one day storage periods at -5°C, and ten day storage periods at -30°C, for a total of 14 storage periods over 100 days. 6.5.1 Influence of the Surface Heat Transfer Coefficient on Temperature and Quality Distribution Histories. The effects of the surface heat transfer coefficient (hx) on the temperature and quality distributions within a frozen food product were demonstrated by simulating storage conditions by using the extreme range of values for hx found in the literature (Dagerskog, 1974, and Zaritzky, 1982). For the upper limit of the range, hx - 20 W/m2°C was selected, and for the lower limit, hx - l W/m2°C was used. One dimensional heat transfer was simulated in a product, initially at -30°C, and subject to storage for 100 days at -5°C. Results obtained using hx - 20 W/m2°C and hx = 1 W/m2°C were com- pared with the results found with hx a 8.5 W/m2°C. (This is the overall 187 Table 6.8 Definition of Boundary Condition Cases, with Step Changes in Storage Temperatures over Given Storage Interval. High Storage Low Storage Temperature Period Temperature Interval (°C) (daYS) (°C) (daYS) Case 1 ~5 1 ~30 10 Case 2 -5 5 ~30 10 Case 3 ~5 10 ~30 10 Case 4 ~5 1 ~18 10 Case 5 ~5 5 ~18 10 Case 6 ~5 10 ~18 10 Case 7 ~13 1 ~18 10 Case 8 ~13 5 ~18 10 Case 9 ~13 10 ~18 10 188 average surface heat transfer coefficient determined from the experimen- tal results, described in Section 6.1.2.) The effects of the surface heat transfer coefficient on the temperature history at the exposed surface and at the geometric center are shown in Figure 6.17. The temperature differential between the two surfaces was very small for hx - 1.0 W/m2°C, compared with the results shown for hx - 8.5 W/m2°C, and hx - 20.0 W/m2°C, during the first 20 days of the simu- lated storage period. However, as the magnitude of the surface heat transfer coefficients increased, the solution approached lumped steady state conditions (constant temperature with time and position) faster. In addition, the shape of the curves changed dramatically by in- creasing the surface heat transfer coefficient from hx - 1.0 W/m2°C, to hx - 8.5 W/m2°C, compared with the change in the shape of the curves shown by increasing hx from 8.5 to 20.0 W/m2°C. This indicates that a change in magnitude of the surface heat transfer coefficient has a greater influence on the solution at lower values of hx than at higher values of hx. The effects of the magnitude of the surface heat transfer coeffi~ cient on the quality distribution histories, determined for strawberries from Eq. (3.49), are shown in Figure 6.18 for the criterion with Ea - 182 kJ/mole, and reference shelf-life of 630 days at ~18°C, and in Figure 6.19, for the criterion with Ea - 49 kJ/mole, and reference shelf-life of 540 days at ~18°C. The high activation energy constant resulted in a high temperature dependence of the quality deterioration rate. All of the simulations using the high activation energy constant resulted in zero shelf life by the end of the 100 day storage period. The shelf life was completely diminished by the end of 79 days storage time for the case where the surface heat transfer coefficient equaled RE .moomusm oomomxm mam noucoo owuuoaoou um known“: ousumuoasm9 uosooum so .xn .uCoHonmooo Hommcwu9 use: ooomuam mo uoommm 9H.o ouswwm Agovv 6E9 oo— om om o¢ ON 0 I IF .. + T .I t . t . mnl O.ON u x... I on ... x; I r mootam G..«E\>>v .tooo teamed; “om: .... x; . 0.9 n x... Elm. ...? md n x; I v .2ch \ . O.—. n X: ..I... L \ ; Imml . row! |I|I|n‘u'w n u u u IIIVII E as I .5 .82on 86 com! I SEEKS: acoBE< . 0.09.... n 3330qu SEE (3°) eJnioJedwei 190 .Amhmo One I coma- um mmfiH-MHo£m .mHoE\wx «ma I mmv communm oomoaxm was noucoo cauuoaoou no mum“ coHumuoHumuoo muwamso uQSUon so .x: .usowoammooo wommcmu9 use: oommuam mo uoommm ma.o ouowfim ’ E ad I «353;... 2.5 /, _ 9? n .ant EE< , Tom 98.. ... aEE. BEE / / AoanE\3v .300 .25.... you: I x; ,., . . 0.8 a x: I. E ,., . 06 n x: ole . @0325 r , , 3...: Ta 11 0.8 n E I 0.” Nu X...— ..I. v Lmucmo o; n x; T- A303 6E9 09 mm 0% we Jmm Ls o L 9 I.-. .I .- AlllIl-PIIIIIIIIIII _.I AA ... J / a ,. ... .1, v .. , 10m ., I v / I . , ,, ,, / -3 oo— (24) mono 191 .Amzmo can I mama- um owHH-MHo£m .oHoE\wx me I mmv oomMHSm pomoaxm was umuSmo owuumsooo um comm Godumuowuouwn muHHmSO poopoum :0 .x: .ucoaowmmooo nommsmu9 use: ooomu3m mo uoommm mH.o ouawfim Amxouv 6E: 00, cm ow ow ON 0 I IF . L _ p L... b o E 9N I 305.029 no_m o; n. x... Iii obI I .ane .nE< mootsm . ad I x; I o.onI I dES. BEE odu u x; I . - o; u E min .3350 . m6 I x... ole WON rodm n. 5.. «La AoonEbsv .300 .95.: goo: n x... r -9. Wu /,, DI .inhw. .I. //,../,...////d/ v. GA... ,/ Uoflf -8 Z /. i/ I. / ( ..om 00F 192 1.0 W/m2°C, while for the cases where the surface heat transfer coeffi- cient equaled 8.5 W/m2°C and 20.0 W/m2°C, shelf-life was diminished by the end of 35 and 21 days, respectively. The difference in quality of the product at the outer surface com~ pared to the quality at the geometric center increased as the surface heat transfer coefficient was increased. When the high activation energy constant was considered, the differences in quality between the two surfaces were substantial: ninety percent of the quality at the center of the product mass was retained at the time when the quality at the surface was diminished, using hx - 20.0 W/m2°C, while 77% of the initial quality was retained using hx - 8.5 W/m2°C, and 33% of the quality remained at the center using hx - 8.5 W/m2°C. Due to the lower temperature dependence on the quality deterioration rate for the criterion with the low activation energy constant, there was very little difference between each of the solutions using hx - l, 8.5, and 20 W/m°C, and even smaller variations in the product quality between the center and the exposed outer surfaces, as shown in Figure 6.19. In none of the cases was the product shelf life exceeded, and the difference in quality between the center and the outer boundary at the end of 100 days storage was less than 5% of the initial quality, for all cases. In summary, changes in the surface heat transfer coefficient at low values (5 8.5 W/m2°C) had more influence on the temperature distribution than equivalent changes at higher magnitudes (2 8.5 W/m2°C). High values for the heat transfer coefficient yielded solutions which had a greater temperature differential between the two boundaries initially, but approached the steady state solution rapidly. The effects of the magnitude of the surface heat transfer coefficient were strongly depend- ent on the magnitude of the activation energy constant. For low values 193 of Ea (s 50 kJ/mole), increasing the surface heat transfer coefficient increased the quality degradation rate moderately, but with only small variations in quality within the product (Lx s 2.0 m). 0n the other hand, for high activation energy constants (2 180 kJ/mole), increasing the surface heat transfer coefficient decreased product shelf life substantially, and resulted in high quality variations within the product. 6.5.2 Effects of the Frequency of Step Changes in Storage Temperatures on Temperature and Quality Distribution Histories. The effects of the fluctuation frequencies of step changes in the storage conditions were investigated using three different intervals for the step changes: (1) one day at ~5°C, and ten days at ~30°C (Case 1); (2) five days at ~5°C, and ten days at ~30°C (Case 2); and, (3) ten days at ~5°C, and ten days at ~30°C (Case 3). Each of these cycles were repeated for a total storage time of 100 days. Again, one dimensional heat transfer was considered through a 2.0 m product, with an initial temperature of ~30°C, and with a constant heat transfer coefficient of 8.5 W/m2°C on the exposed boundaries. The resulting temperature histories are shown in Figure 6.20a, at the geometric center, and in Figure 6.20b, at the outer surface. For the boundary conditions described in Case 1, the temperature at the geometric center changed only slightly, while the temperature at the outer surface changed significantly (14°C), but only for a very short period of time, compared to the overall product shelf life. Using the boundary conditions described in Case 2 resulted in temperature fluctua- tions at the center of 6°C, and fluctuations of 19°C at the outer 194 .Am pcm N .H mommuv Guam- mo ousumuano9 HmfiuwcH Luaa uospoum w you .uoucoo owuuoeoou um known“: muoumuanm9 uosooum so mousumuano9 owmuoum CH mowcmso doom mo coHuwuso mo uoommm mem.m onswwm Amzonv 6E: OS on om ow ON 0 I L . L . _ . _ . t mnI. UonE\ CS 0. m" uC0_0_tQOU LOvaOI—L. “OUT— GOOwLJW E 3.. n nsm .6 8056:: a. I > x . (Ax ”/SX/K XAXK JQ/ \ , \\ .. . \\ . _ I .. n T (K /< r/\ r( r. lle i I an 38v 9%.. so 9% o. .99. so 28 S .I.. 1.... 252280 382m Q 088 90...... so 98 2 .99. so 9% m. ... C $ch 92.... so 9% o. .99. so sou _ u.- (30) eJnioJedwei l95 .Am 6cm N .H mummov ooom- mo ouzumumaaoe HmwuwcH Sada nonvoum a you .momuu5m cwmoaxm um known“: mudumumaEwH uosvoum co mmudumummama mmmuoum CH mmwcmno amum mo sawumuza mo uoomwm aoN.o whamwm A933 mEfi oo— om 00 0+» ON 0 F p L mml oonE\>> m.m H ucfloEmoo 9:95: How: mootam E ow n 32m 3 32:2,: / a V ,_ i A... 9.68 99... «o £8 E .99. 6 28 S «La -m- 22:38 383m 8 088 92... yo 28 2 .9m: yo 98 m I C 88v 9%. 6 £8 E .9? 6 3n _ arm h F \P P r — F (QC) eJruDJedwei 196 surface, for a slightly longer period of time. When the storage inter- vals were equal for each storage temperature (Case 3), fluctuations of 10°C were found at the center, and fluctuations of 20°C were found at the exposed surface. The mass average product temperature increased from 29°C for Case 1, and 23°C for Case 2, to 18°C for Case 3. The effects of these step changes in storage conditions on the quality histories at the geometric center and at the outer surface are shown for the criterion with an activation energy constant of 182°C in Figure 6.21, and for the criterion with an activation energy constant of 49°C in Figure 6.22. In the cases where the storage intervals were less than or equal to five days at -5°C (Cases 1 and 2) the step changes in the storage conditions had very little effect on the quality history at the center for both high and low activation energy constants. The effect of the short high temperature interval (Case 1) on the quality at the exposed surface was also very small, with very little variation in quality within the product mass. The steeper slope of the curves resulting from storage conditions of one day at -S°C (Case 1) in Figure 6.22, compared to the corresponding curves in Figure 6.21, were a result of the shorter reference shelf life for the PSL criterion (540 days), compared with the HQL criterion (630 days). Increasing the storage time interval at the higher storage tempera- ture to five days (Case 2) resulted in a substantial loss of quality at the outer surface (50%), with very little change in quality at the geometric center, using the quality criterion with the higher activation energy constant. For the quality criterion with an activation energy constant of 49 kJ/mole, there was a slight drop (8%) in overall quality, compared to the results using one day at ~5°C (Case 1), and only a 4% change in quality between the geometric center and the outer surface. 197 .Ammmt 0mm I come. um QMHH-mHo£m .oaos\nx NwH I mm ”m can N .H mommov ooom- mo oHSumHmQEOH HawuHGH suds uozooum o How .oumm GodumuoHuouoo huwawzo uofip -oum co moHZuoHoaaoH owououm ca mowcmco moum wo cowumusa mo uoomwm HN.© ouzwwm A963 oEfi 00F om om 9. ON 0 p p F I r . k . b P O n 300 «In E ON H x._ c0383 mootam N 030 olo . . . _. omOO Elm. 9%? m m ... x; n 38 I cozoog ..Bcoo N 38 I ..ON — 300 I... [9‘ - - .. .00 n u u n (I? . 10m rummimnmulnluflflmflflfluflf|-h fl -.“I-u.-.r up I I - - v 009 (z) mono 198 .Ammwc can u o.ma- um moaa-oam;m .mHoa\nx as . mm mm was N .H mammov o.om- mo eunumuoaaoa HmHuHaH saga uosooum w you .ouom coaumuoHuouoa huwamso uoav -oum co mouSumuanoH mmmuoum CH momcmso moum mo cowumusa mo uommmm NN.e ouswwm Am>oov with oo_ pm pm ow pm 0 in 88V 9%.. so 28 9 .97 so £8 E I o . a 808 99.... to 38 S .99. to 3% m. To co_ooo 000;: . .s ._ c m u C 238 9%. so 28 E .99. so >8 . mum An 808 9%.. to 96v 2 .99. so 28 o. 4-.. 8583 5:53 a 88v 92.... so :8 E 9...: so 38 n I row . C 308 98.. so £8 E .99. to son P a... r E 9N r. x._ 9%.: man u x; -9. ..om 4 . .ium s .mmuau..- .I.-. . oo. (z) mono 199 In Case 3, where the storage interval at -5°C was equal to the interval at -30°C, there was significant quality loss for the higher activation energy constant at both the geometric center and the outer surface, and a substantial difference in quality between the two loca- tions. The product quality was diminished at the surface after 67 days of storage, while the product quality at the center was only reduced to 84% of its initial value. When the lower activation energy constant was used (Figure 6.22), there was also a decrease in overall product quality and an increase in the quality differential within the product, but the degree of change was much smaller in both cases. Product quality at the surface dropped to 75% of its initial value, compared with 93% remaining quality after 100 days in storage, using step change intervals of only one day at -5°C (Case 1). Again, the quality differential within the product was less than 5% of its initial quality. To summarize, increases in ambient temperature for short time inter- vals, compared with the overall storage time at the lower temperature (5 10% overall storage time), had negligible effects on overall product quality. Step changes in storage conditions with equivalent storage time intervals at high and low storage temperatures had much greater influence on quality reduction, especially when using high activation energy constants. Step changes in the storage temperatures with the storage interval at -5°C equal to one half the interval at -30°C, resulted in substantial reductions in quality at the surface, with high quality variations within the product, when using high activation energy constants.' However, when using low activation energy constants under the same conditions, only small changes in both the overall quality and the internal quality variation were observed. 200 6.5.3 Effects of Ambient Temperature during Step Changes in Storage Conditions on Temperature and Quality Distribution Histories., The effect of the magnitude of the ambient storage temperature in fluctuating storage conditions was demonstrated using the same three repeating step change intervals described by Cases 1, 2 and 3 (Table 6.8), but with different magnitudes for the ambient temperatures. Three different sets of high and low ambient temperature were considered for these three step change cycles (for a total of nine cases): (1) low storage temperature - -30°C, high storage temperature - -S°C, (same temperatures used in Cases 1, 2, and 3); (2) low storage temperature - - 18°C, high storage temperature - ~5°C, (Cases 4, 5, and 6); and, (3) low storage temperature - -18°C, high storage temperature - -l3°C, (Cases 7, 8, and 9). These cases are also defined in Table 6.8. The same product geometry, initial temperature, and surface heat transfer coefficients described in Section 6.4.2 were also used in this analysis. The temperature histories at the geometric center and at the outer surface for the step change intervals of one day at -S°C and ten days at ~30°C (Case 1), one day at -5°C and ten days at -18°C (Case 4), and one day at -13°C and ten days at -18°C (Case 7), are shown in Figures 6.23a,b. The mean temperatures were higher for both Cases 4 and 7 due to the increase in the minimum storage temperature. The temperature at the geometric center fluctuated slightly (1°C) using Case 4 boundary conditions, and negligibly using Case 7. The outer surface temperature fluctuated only 2°C. The effects of the changes in ambient temperature on the quality deterioration rate for the HQL and PSL criteria are shown in Figures 6.24 and 6.25, respectively. Note that the scale for % Quality ranges form 60 to 100%, instead of 0 to 100%. The solid boxes represent the 201 .Am tam o .H mommov Daem- mo chaumquEmH Hmwuacm Lows uoavoum a How .uouCoo cauuoaooo um huoumaz unduouanoH uosooum co mousumuoaaofi omnuoum CH owcmno amum mo unauacwmz mo uoommm MmN.o ouswwm Agog 9t: 00— om om 04. ON 0 p — s — n — p h b — mm} 0onE\>> m.m ..u ucflofooo tocmcofi goo: mootsm E QN n 85 to 805.2,: IKE/fl 183.2 Infill C 088 92- so 3% 2 .92.. so .6“. . 4L. -n: 203680 002on a. $08 99.. so 28 E .99. so .26 _ I C 988 90?. so 28 8 .9n- so >8 _ I-.. (30) emimadwei 202 :o moHSUQHoQEoH OO_ .AN paw o .H mommov O.Om- mo eunumuoaaoh HmHuHGH LuNB uodfioum a How .oUQMMSm bwmonxm um xuoumwz annumuoasoe weapoum ommuoum CH owcmso moum mo ovsuwcwmz mo uowmmm QMN.O ouswwm mcozficoo mootoym A933 9:? OO OO OV ON O _ _ L _ . b . 5 mm] oonE\>> md H 1.665000 toamcofi OooI mootzm E S u saw to mmmcxefi njuu "J nu uu.nunwnu.uunfll-lm .u... ONI ...\..\... : Q. 088 92- so 22. E .92- 8 son F «I... -nl O. 808 9?- so 22. 9 .9m- 6 >8 _ one C 088 92... so 2% 9 .9m- so sou _ me (QC) amimedwei 203 .Amsme One a o.mH- um mmaH-mHonm .oaoe\sx «as I am ”a can a .H manque o.om- mo mucumuoaaoa HoHuHCH sows posooum a Mom .oumm coguwuoauouoo Nowamso uosv -oum :o mousuouoasoa owmuoum Ga owcoso noum mo monogamoz mo uoommm dN.O ouswwm OO A983 oEfi Ow OO O¢ ON O Ir P b L r L! b 0* as 268 92.. so £8 9 .92.. so sou . T... 5:83 825m . O. 268 99.. so 9A8 2 .90.. so son _ o1... .. C 238 98.. so 38 S .9? so >3 . mum 9 238 93.. to £8 9 .92.. so son . I a i c. 308 99.. 3 98o 2 9...- so sou _ .8 cosoog toucoo r C 268 99...... so axon OF dam! “o zoo F l..- ..om E 0N n .3 03E? 9m u x; . ..Om OO— (z) mono 204 .Fmsma can - o.aF- um omFF-oFm;m .mFoE\ns as - mm ”F can a .F mammov 0.0m- mo chauoquEoH HmauHcH £ua3 auspoum a you .oumm cONUmuoHuouoa muHHmso uozp -oum co mounumquEoH owououm CH owcmfio noum mo onsuacwmz mo uoomwm mN.o ouzwwm Am>oov acts OOF OO OO Ow ON O P F F IF p L L L 1.4 0.? As 088 9mF- so 28 9 69.7 6 >8 F 41... 8:83 825m . FF. 088 9mF- so 28 2 .9m- ,8 .8 F o-é .. FF .88 9cm- 8 28 OF .9...- 6 >8 F ...-m f 808 9mF- 8 :8 OF .92- 8 .8 F I F co. 80 8 cm . O. 808 92- 8 98 3 .9m- so .8 F I .F a F o . FF 888 9cm- 8 28 3 .9c- 8 .8 F .T- -OO E 3.. n .3 9.E...3 9m .- x; .. . TOO .1! ulu'u- : .. - .I..»n/ul..- l l. null-Ill- ...n - ....l- l- - .. .. ...-H/u”... -..n--- - - ,-- u--.- -..--uuwuwmw..mw.- OOF (z) (3.1-0m 205 results of using step changes in the boundary conditions of one day at «5°C and ten days at -30°C (Case 1) in both figures, and are identical to the corresponding curves shown in Figures 6.21 and 6.22. Both condi- tions of step changes of one day at —5°C and ten days at ~18°C (Case 4), and one day at -l3°C and ten days at -l8°C (Case 7) resulted in a lower quality retention at the end of 100 days, compared to the results using Case 1 (one day at -S°C and ten days at -30°C) results, because of the overall higher mean storage temperature. Step changes in storage temperatures between -5°C and -l8°C (Case 4) resulted in a 10% quality differential across the two surfaces using the activation energy constant of 182 kJ/mole, and 1.5% using the lower activation energy constant. The results from using temperatures fluc- tuating between -18°C and ~13°C (Case 7) indicated very little influence on the quality distribution history in all cases. The quality differen- tial between the center and the surface was 2% using the high activation energy constant, and 1% using the low activation energy constant. The same sets of ambient temperatures (-S°C and -30°C, -5°C and -18°C, and -l3°C and -18°C) were used in Cases 2, 5, and 8, but with storage periods alternating between five days at the higher storage temperature, and ten days at the lower storage temperature. The solu- tions for the temperature histories at the geometric center and at the outer surface are shown in Figures 6.26a,b. The solutions for Case 2 (five days at -5°C and ten days at -30°C) in both figures are the same as the corresponding curves in Figures 6.20a,b. The temperatures at the geometric center fluctuated approximately 3°C for the conditions where the high storage temperature equaled -5°C, and the low storage temperature equaled ~18°C, (Case 5) and only 1°C using -13°C for the high storage temperature, and -l8°C for the low storage temperature. The temperature fluctuation at the outer surface was substantial in all 206 .Am cam m .N mommov ooOm- mo ousumuoaaob HmHuaCH £uw3 uodfioum w you .Hmucoo oHHuoEomo um huoumwz ouzumuanoB uosooum co mouduouoaaoa omnuoum a“ omcono amum mo opsuwcwoz mo uoommm «ON.O ouswwm A983 oEP ooF om om 0... ON L p p — L — p _ b O 0onEx..>> Md N 3205000 toamcofi Foo: mootam E O.N fl n2m F0 3930:: FR» < :R/N/ 3.9.: k .11-$11} , 23 /. 6 088 9F:- 8 98 OF .9nF- .0 28 m 1.... 82:88 883m 3 268 9mF- Fe 28 9 .9m- 8 28 m a... Q 088 9%- 8 98 OF .9...- 8 98 m 9.- _ mm! I LO ‘7‘ TOF- (Qo) eJnlDJedLLJSi 207 OOF .AO tam m .N mommov Doom- mo ounumummeoh HmHuwcH Sofia uosooum a now .oommu3m oomoaxm um huoumH: annumuodaoe uosooum co moHSumquEoH owmuoum CH owcmno doom mo opsuacwmz mo uoommm AON.O ouswwm A983 mEfi . b L — L _ . h L mmul oonE\.>> m.m N 2205000 Lochofi ”Foo: mootam E ON u 86 F0 8920:: .mm- / [ml—.I- 8 808 93- .0 m>8 oF .92- F0 28 n ...-a -m- mcosficoo 898m 3 088 93- 8 28 oF .99- 8 mb8 m I . . a 088 92..- 8 98 OF .9...- 8 38 m m-m (3°) SJfllDJedLuai 208 cases, as shown in Figure 6.26b. The surface temperature fluctuated 10°C for Case 5, and 4°C for Case 8, compared to 19°C for Case 2. The quality distribution histories for the step changes in boundary condition alternating five days at the higher storage temperature ten days at the lower storage temperature, with storage temperatures of ~S°C and -30°C (Case 2), -5°C and -l8°C (Case 5), and -l3°C and ~18°C (Case 8), are shown in Figure 6.27, using the high activation energy constant, and in Figure 6.28, using the low activation energy constant. The higher minimum storage temperature used in Case 5, compared with that used in Case 2, had a very significant effect on the quality deterioration rate using the high activation energy constant. The shelf life at the outer surface was diminished after 78 storage days, while at the same time, 60% of the initial quality was retained at the geometric center. Moderate differences (6%) in quality were determined between the center and the surface using step changes in the storage temperature between -13°C and ~18°C (Case 8), and the high activation energy con— stant. For the quality criterion with the lower activation energy constant, little variation in quality within the product mass was found for both step changes between -5°C and -18°C (Case 5), and between -l3°C and -18°C (Case 8). In Cases 3, 6, and 9, the step change intervals were ten days for both high and low storage temperatures, and again, the magnitudes of storage temperatures were -30 and ~5°C, -18 and -5°C, and ~18 and -l3°C, for Cases 3, 6, and 9, respectively. The solutions for the temperature histories at the geometric center and the outer surface are shown in Figures 6.29a,b. The longer storage period at the high storage tempera- ture resulted in a greater fluctuation in temperatures at both locations. Fluctuations of 9°C, 5°C, and 3°C were observed at the 209 .Ammmp One I coma- um omda-mao£m .oHOE\Ox NOH I um ”O was m .N mommov 0.0m- mo ouzuouoaaoh HowuficH Sada uoscoum a you .ouom coauwuoauouon %ua~mfio uozp -oum co mouaumuoaaofi omnuoum :« owcwco moum mo oOSuucwoz mo uoomwm NN.O ouswwm OO— nm>oov oEfi om om 0% ON 0 . b p _ b p p O m omoo «it cozooom mootam . m $60 I N 300 Elm. . ..m 300 I . cozooom toucoo . m 800 o... WON / fiN 300 I... 10¢ r? . ...-If! . CLAHN H XO 4), 9%sz 9m ... .2 Tom 4. I ‘1”. n H J r u n 0 1.0m -'.'..’ r mlullu- I r lflluilflrlflllil!1!1!aIIIATAPIYI-I - w.1£d. . q - -.. OOF (z) mono 210 .Amhmp oqm I coma- um awaH-mHo£m .oHoE\Hx me I on ”a can n .N mommov ooOm- mo ousumuoaaoa HwFuHCH cows uosooum o How .oumm Godumu0auouoa huaaoso uoso -oum co mouauouoasoa owmuoum :« owcmgo moum mo oczuacmmx mo uoommm ON.O ouswwm A963 mEfi 09 mm 0% 0.... om 3 268 9.3- so 28 oF .9nF- so 98 m. - .F . - F 8:83 8233 F... 088 92 so m 8 0F .9... so ... 8 m . .. a 808 92..- 8 :8 9 9n- 8 28 m. 8 088 9mF- 8 98 oF .9nF- 8 28 m cozooom .5550. an $68 0.2.. Fo goo OF .oonl so 98o m r 3 238 98- 8 28 3 .9m- 8 28 n .E 3 - oonEls 9m .... I Q-IO Elm 4L4 o-.. a .- ..ON .0... ..OO -om OO— (23) mono 211 .AO paw O .m mommov 00Om- mo ousumquEoH HmHuHCH Luw3 uosooum o How .uoucoo oHuuoaooU um kuoumwz ousomuodEoH uospoum co moufiumuomawa omnuoum :F owcmfio amum mo oOSuacwmz mo uoommm MON.O ouswfim OO— om A983 oEfi OO . OF. ON O L b L . L _ LP mcoEpcoo mootoum Q onE\>> Ob u Efiomtooo 0395....r Foo: mootam E O.N u nEm Fo mmochFC a 088 9mF- .0 0.8 oF .92.- F0 28 0F 0 I 8 0008 90F... .0 0.8 0F .9...- 00 0.8 oF 0 6 F... 0008 9cm- 8 98 OF .90- 8 98 S It TON..- (Oo) ambled-149i 213 geometric center, for storage temperatures alternating between -5°C and ~30°C (Case 3), -5°C and -18°C (Case 6), and -l3°C and -l8°C (Case 9), respectively. At the outer surface, temperatures varied 15°C for Case 3, 10°C for Case 6, and 3.5°C for Case 9. In all cases, the retention time near the higher storage temperature was longer than that found for the previous step change cycles. This resulted in a substantial increase in the rate of quality deterioration using the high activation energy constant, as shown in Figure 6.30. The shelf-life at the surface, using storage temperatures between -5°C and ~30°C, was predicted to be exhausted after 67 storage days. However, using storage temperatures between -18 and -5°C resulted in the total lost of quality throughout the entire product after 88 days. The outer surface exceeded it recommended shelf life in 45 days, while the product at the geometric center still retained 64% of its initial shelf life. When using the low activation energy constant, as shown in Figure 6.31, the slope of the curves was slightly greater than those shown in Figure 6.28 for the five day step change interval time at the high storage temperature. Again, the step changes in storage conditions had very little effect on the quality distribution history compared to that shown in Figure 6.30 for the activation energy constant of 182 kJ/mole. In all cases, the variation in quality within the product was less than 5%. In comparing the results from Figures 6.24, 6.25, 6.27, 6.28, 6.30, and 6.31, several observations were noted. Step changes with short storage intervals (one day or less) at high storage temperatures had very little effect on the quality distribution for most cases using both high and low activation energy constants. The exception to this obser- vation occurred when temperatures were allowed to fluctuate between -18 211- .Ammmp One I coma- um omHH-wHo£m .oHOE\Hx NOH I mm no men O .m mommov 00Om- mo opsumumaaoh Howuch nuwz uosooum w you .ouom cowumuowuouoo huwamzo wont -oum co mousumuoaaoe owmuoum CH omcoco noum mo opauucwoz mo uoommm Om.o ouswam A883 mEfi OOF OW CFO - Ohv . O_N . 0 i9 4 F3 m omoo «in O I]! r .... cosooom mootam . m 0000 .010 {/0 r . -n 0000 arm 4 ,,( ,5, F c0580.. ..Bcoo m 0000 T4 ., dl-m-Im-IE .. . g m omoO 9... fiON ,Fo / (lo/0r n 0060 l.- / F/ IA, f f y/ l, 1.19 FF .E ON I x... TO¢ J! ,MT-mTl9lfi F! r .. F. ...... Tom (3.4;) mono 215 .F0F00 000 - 0.0F- 00 000F-0F000 .0F00\0x 00 u 0m .0 000 0 .0 000000 000m... 00 00:00H0QE0H HmfiuwcH nuwz uosooum 0 now .000“ cowumuofiu0u0a zugamso 0030 .00m co m0usumu0na0h 0w0uoum CH 0mcmno m0um mo 00:uacwmz mo uo0mmm Hm.o 0u3wwm A933 0Ec. 00F pm. 3% ow am I 0 0008 907 00010.8 OF .907 00 0.80 0F .. s . .. s 000000.. 88.8 A... 0008 90F .0 0 8 S .90 .0 0 8 S . F0 0008 900- 00 0.8 OF 90- 00 000 S m 0008 907 00 0.60 0F .9nF- .0 0.80 3 005000... 03:00 6 0008 Dow—.- ”.0 0x00 OF .000! “0 0.30 OF 3 0008 900- 00 0.8 S .90- 00 0.8 E c: O.N n. 000E}. 0.0 .... IIIEII rON TO». #00 OOF 5000 ) .0- 0 1-0 ( 216 and -5°C, and the high activation energy constant was used; in this case, a 10% variation in quality within the product was found. There was very little change in the quality deterioration rate using the low activation energy constant, in all cases. In contrast, use of the high activation energy constant resulted in substantial quality changes, expecially for the situations where the storage interval at the higher temperature was greater than or equal to five days, and the storage temperature fluctuation was greater than or equal to 13°C. Step changes in storage temperatures between ~18°C and -l3°C (Cases 4, 5, and 6) had very little affect on the overall quality deterioration rate of the product. This observation supports the work by Moleerantanond, et. al. (1982), who found very little change in the quality of frozen beef resulting from 3°C fluctuations in storage temperatures, and who proposed use of cyclic storage temperatures as a means of energy conser- vation. 6.6 Effects of Size, Two Dimensional Geometry, and Geometric Shape on Temperature and Quality Histories of Frozen Foods During Storage Size and geometry are important factors in determining the rate of heat transfer though a mass of food product. Three aspects were con- sidered here: (1) the product size or thickness; (2) the ratio of thickness versus length; and, (3) the geometric shape. In the first case one dimensional heat transfer was assumed. The objectives for this case were to determine under what conditions a product with a low ac- tivation energy constant will have a significant quality variation between the two sides perpendicular to heat flow, and to determine when a high activation energy product will have insignificant quality varia- tion between these two boundaries. In the second case, two dimensional 217 heat transfer was assumed, and the influence of the length versus thick- ness ratio was investigated. Finally, the influence of geometric shape was studied. In this case, it was desired to determine if and under what circumstances a one dimensional geometry could be used to approximate a two dimensional shape. For example, one dimensional heat transfer through a cylindrical mass, with insulated ends, might be used to simulate two dimensional heat transfer through cube shaped mass, insulated on two opposite sides. 6.6.1 Influence of Product Thickness The influence of product thickness was first considered using the criterion with the low activation energy constant (49 kJ/mole). In Sections 6.5.2, and 6.5.3, a one dimensional slab of thickness 2.0 m was considered. The quality differential between the center and outer surfaces never exceeded 6% in all of the nine cases (Table 6.8) con- sidered in these sections. A very large product mass was simulated to determine if the quality distribution within the product increased significantly. One dimensional heat transfer through a product 6.0 m in thickness, using step changes in boundary conditions of one day at -5°C and ten days at ~18°C (Case 4), five days at -5°C and ten days at -18°C (Case 6), and ten days at -5°C and ten days at -18°C (Case 8), was considered. The resulting temperature histories at the geometric center and at the outer surface are shown in Figures 6.32a,b. The temperature at the inner surface gradually increased with negligible fluctuations (Figure 6.32a), while the temperature at the outer surface approached the storage temperature in less than three days, and fluctuated continuously in synchronization with the step changes in storage temperatures. .F0 0:0 0 .0 000000 0.00- 00 00000000000 F0F000H 0003 uosnoum 0 you .mH0>u0ucH 0wcm£u Q0um uC0u0mmHO Mom H0uC0O oHHu0EO0O 050 um zuoumwz 0usumu0QEOH uos0oum :0 000con£H poacoum 0wu04 mo 000mmm 0Nm.e 0uswwm 218 OOF 0|- A983 0EP 00 00 0... ON 0 _ L l'lr------I.FIl--!--LI- :1-..- L. .- -. L .. CL -.. -.-. .1- mm]- OonE\>> mm H 0.0065000 000006.; 0001 83:5 4 E 0.0 .... 00.0 .0 0005.8: F0 0008 90F- 00 008 0F .90- 00 008 OF «L. -m- mcoéucoo 000.55 3 00008 0.07- Fo .030 OF door so 0.60 n. 0.0 O. 0008 99.. .0 0.8 OF .90-. 00 .8 F ....-L V 10100-140- (3°) em. .F0 000 0 .0 000000 0.00- 00 00000000000 F00000F 0003 uosnoum 0 Mom .0H0>u0ucH 0wcmno 000m uC0u0mmHO you 0ommuzm 00000xm 050 um xuoumwx 0u500u0aa09 uosuoum co 000con£H uos0oum 0wu04 no 000000 nNm.o 0uswfim 219 O0 F 003 0E; 00 00 o... om o b . 0 F 000E\>> 0.0 H 00065000 000:0; 000: 0ootam E O0 H 020 .6 0000x030 L0 ‘7‘ Infill (30) emimadwai F0 0008 907 00 0.80 0F .90- .0 £8 2 0L. .....- 8 0008 90F- .0 0.80 0F .90- .0 0.80 0 9.0 O. 0008 907 .0 0.80 0F .90- .0 F8 F m-m 0:020:00 0000000 220 During the initial phase of the total storage time, the temperature differential within the product was as much as 28°C; at the end of the total storage time, the temperature differential reduced to about 8°C. The effects of the temperature differential on the quality histories at the two locations are shown in Figure 6.33. Even with high tempera- ture variations, the quality variations within the product were very moderate, especially when compared to those shown for the 2.0 m product mass, using the high activation energy constant (Figure 6.27, and 6.30). In the situation where the step change interval for the storage tempera- ture at -5°C was one day, ten days for the storage temperature at -18°C (Case 4), a 9% variation in quality was found after 100 storage days. Increasing the storage period at -5°C to five days increased the quality differential to 14%, and for equal storage periods at -5 and ~18°C, the quality differential increased to almost 18%. Therefore, for low ac- tivation energy constants, significant (< 10%) quality variations within the product mass were only found using very large product thicknesses (> 6.0 m). Next, the product using the high activation energy constant was considered, with the objective of determining if and under what condit- ions, the quality variability within the product was negligible. Again, Cases 4, 6, and 8 (step changes of: one day at -5°C and ten days at ~18°C, five days at -5°C and ten days at ~18°C, and ten days at -5°C and ten days at -l8°C), were used for the simulated boundary conditions. The quality variation between the geometric center and the outer surface was significant (> 10%) using these cases, and a product mass of 2.0 m, as shown in Figures 6.24, 6.27, and 6.30; therefore, the product thick- ness was reduced to 1.0 m, and the simulation processes were repeated. The temperature histories at the geometric center and the outer surface are shown in Figures 6.34a,b. With the exception of the curve 221 on: om Om CV 1% L .Amann new u n.n~- an anaa-naonm .¢_c5\nx on u an an new n .n mnmnnv n.0m- no unsununaana smashes no“; uoan -oum w you .mHm>hou:H omcmzo doom ucouommfia pom oommuam vomoaxm msu um mama cofiumuofiuouoa huwawad uofipoum co mmoconnh woncoum swung mo uoommm mm.o shaman A963 oEfi c0380.. mootam cosooom toucoo m .I litvILla An omoov o.na..sn axon o. .o.m:.un mson o_ 8 088 0.2.. to gen 3 .9? so can m an «88 0.9: so inn o. 6%.. no son a .An omoov o.n_..nn.m>on o_ .o.m...n «son op An omnov o.n_..sn msnn o. .o.m..on mxnn n An omnov o.n_l.so mson oi .o.mx.sn son _ E on use. a on ON - L:. ; 1L! iirlfi. 4116 I film. I .I. .1- x1. x: 0 ..on I C) co a I C) a: On.— (24) mono 222 .Aw van 0 .n mommov ooOm- mo ousumuoaeoe HmHuHCH LUHB oozpoum a Mom .mam>houcH owcmno doom ucouommHo Mom Houcmo oauumeooo on» on xpoumfiz ouzumuanoH uosvoum co mmocxownh uospoum HHmEm mo uoowmm mnm.w ousmwm on: om A303 mEfi 00 0+. ON 0 — F ~ » r L [— mMII mcoEUcoo 000..on oonE\>> md n “0205000 coamcot. “on: oootam E o; .I. 02m 0.0 $05.02.: W LO N i .1. (3°) 9,2.QDJ9dLU8' / , J! in If m ..m—. \ ,, u an omoov Dom—l yo gov 0' deal 00 axon or HoucH owcmno doom ucouommaa you oommusm pomoaxm 050 um xuoumam ouduouoaaoh uospoum co mmocxofink nospoum HHmEm mo uooMMM nom.w ousmwm Amsonv 9:: OS on on o... ow o b — r r p i _ oooE\.>> Wm N 0565000 todmcofi 000: mootzm E o; .... 02m 00 $05.02.; .3 0008 Oompl “0 0.30 OF don... 00 930 o— «in An omoov 0.3.. no «son or .0.m.. 0o 28 n no mcoEUco 00900. 0 m An omoov oom_|.uo m>00 o— .oom|.0o x00 _ .uzn 22b for resulting from one day at -5°C and ten days at -l8°C (Case 4), the temperature fluctuations increased significantly, compared to those found using a thickness of 2.0 m. Comparing Figures 6.34a and 6.34b, the resonance time near the high storage temperature at the outer sur- face was about twice that at the geometric center. The resulting quality histories, using the criterion with an activa- tion energy constant of 182 kJ/mole, at the geometric center and at the outer surface, are shown in Figure 6.35 for the three cases considered. The quality variation within the product was small (8%) using Case 4 boundary conditions (one day at -S°C, ten days at -18°C), while the quality variations using the other two cases were high. For step changes of five days at -5°C and ten days at -18°C in the storage condi- tions (Case 6), the shelf-life at the geometric center was exhausted after 65 days, at which time 35% of the initial quality was retained at the outer surface. When the boundary conditions described by Case 8 (10 days at both -5°C and ~18°C) were used, the quality at the surface was diminished after 29 days, while 32% of the initial quality remained in the product at the geometric center. Although the quality variation between the two locations was less than that found using Lx - 2.0 m, there was still a high variation in quality within the product using Lx - 1.0 m using the boundary condit- ions defined by Cases 6 and 8. The same simulations were repeated using Lx - 0.2 m. The quality variation between the center and the outer surfaces was about 6.5% of the initial quality, for the Case 4 boundary condition; this value changed only slightly, compared to the results using Lx - 2.0 m, due to the short storage interval at -5°C. The tem- perature variations between the surfaces, using Cases 6 and 8, were found to be very small ( -0000H 0&0050 @000 0:0000000 you 0000uzm 00modxm 0:0 00 000m 6000000000000 zuwamao auscoum so .5 o.H I 34 .000GonSH poacoum Hamam no 000000 mm.0 opswwm A0>00V 0::0 00F 00 00 Ow ON 0 . — . - F p I L .I _ . O J m 0000 «In .. mootnmm m 0000 olo 0 ¢ 0000 mlm .. . .0 0000 1i£ - .. .. 000009 0 0000 I row .n 0000 Ill 036;) 0.0 u E. E 04 H x4 ..on If” . ..00 an n u nan u n a row 'u'fl. I u.;,. on. (34) mono 226 .Anxon 000 u 0.00- on 0000-00030 .oHoa\nx ~00 I o0 ”w 0:0 0 .0 000000 0.0m- mo 00500000508 ~000HCH £003 0030000 0 000 .0H0> -000:H ;wC0£0 Q00m 0:000000o 000 000003m 00modxm 0:0 00 000m 0000000000000 >0m~0zo 00:000m to .E N.o I x0 .000cx0059 00:000m Hamem mo 000000 0m.0 003000 A0>00v 0Ct0 000 on 00 L 038;, 0.0 x; a E Nd H 03 00 L ir A. 0000 0.1L 0000 0.0 . c03000.“ 0025.5 0000 EL 0000 In 0000 I v 0000 IL 003000.. 03:00 V‘EDIDVI'COQ (2;) mono 227 To summarize, for low activation energy products, quality variations within the product were found to be significant only for very large product masses (2 6.0 m), considering one dimensional heat transfer, and a surface heat transfer coefficient of 8.5 W/m2°C on two parallel sur- faces, with insulated conditions on all other boundaries. For high activation energy products, the quality variation was insignificant only for very small product masses (5 0.2 m), using the step changes in boundary conditions of at least five days at -5°C, and ten days at -18°C. In addition, the overall rate of quality deterioration increased substantially as the product size considered decreased. 6.6.2 Effects of Two Dimensional Heat Transfer on Temperature and Quality Histories. Two dimensional heat transfer was simulated in a rectangularly shaped product mass, using step changes in the boundary conditions of five days at -5°C and ten days at ~18°C (Case 6). A constant heat transfer coefficient of 8.5 W/m2°C was considered on the surfaces per- pendicular to the directions of heat transfer, and the surfaces parallel to the heat flow were considered to be insulated. Various width versus height ratios were considered for the exposed boundaries. Temperature histories, assuming two dimensional heat transfer in the x and y directions, were first determined using equal lengths of 2.0 m for both the width, Lx, and height, Ly. A constant heat transfer coef- ficient of 8.5 W/m2°C was imposed along all boundaries perpendicular to the direction of heat flow. The two dimensional temperature history was determined at the geometric center (x - O, y - 0), the midpoint of the exposed sides (x - O, y - Ly/2; and x - Lx/2, y - 0), and at the exposed 228 corner, (x - Lx/2, y - Ly/2) for all cases considered in this section. These locations are shown in Figure 6.37. The resulting temperature histories were compared to the one dimen- sional solution of a 2.0 m product mass subject to the same initial and boundary conditions, shown in Figure 6.26a,b. The temperature histories are shown in Figure 6.38, for the two dimensional simulation at the locations shown in Figure 6.37, and for the one dimensional simulation at the geometric center and exposed surface. Comparing the one and two dimensional solutions, the temperature at the geometric center using the one dimensional solution was slightly higher than that found with the two dimensional solution. The temperature at the surface, using the one dimensional solution was bounded by the temperatures at the midpoint of the sides and the corners. Note that the temperatures around the perimeter of the two dimensional case were bounded by the temperatures at the corners and the midpoint of the sides, and due to symmetry, the temperature histories at (O,Ly/2) and (Lx/2,0) are identical. The quality histories, using the activation energy constant of 182 kJ/mole, at these locations are shown in Figure 6.39. In all cases, the rate of quality deterioration increased, compared with the one dimensional simulation. The one dimensional solution predicted quality 8% higher than the two dimensional solution after 100 storage days. The predicted quality at the exposed surface, using the one dimensional solution, was similar to that predicted at the side midpoints in the two dimensional solution. The quality at the most extreme point (Lx/2,Ly/2) deteriorated after only 47 days. Although the difference in temperature around the perimeter of the product mass was small (two dimensional case), the difference in predicted quality around the outer surface was high, due to the high temperature dependence of the rate constant. 229 (LX.L)’) Ly/2 (Lx,0) Figure 6.37 Locations of Solutions for Two Dimensional Geometry. 230 .A0 00000 ooom- mo 00000000509 H0H0HCH £003 00:000m pom .xq .000000500 H0000 £003 £0Hm 0 000 00009Hom H0cofimc0500 000 00 00000500 A5 o.m I %A I x40 000 000000 000 £0003Hom H00000esz #0000000500 039 mm.0 003mwm A0>00V 0Cte 000 00 00 . 0o 00 0 .. _ . _ . _ . _ . DWI. in» x._ux .010 00500 >4") .OHIIX I on» .3". on... w 80 E 0.0“} .E 00...: Ho 0 oH> .onx III 00:00 x..uxolo oEm _t . u x ” oux9$ Eco: 00 0h: 001 dynamufls . -00: ... r r [ml 0..E\>> 0.0 ... E u .3 A0 omoov 0.0_..0o oaon 0_ .0.0..oo maon 0 (3°) eJfllDJedLuai 231 .0m0av one I 0.00- an «000-00mnm .o0o5\nx ~00 I am no ammov 0.o«- 00 00000000509 0000000 £003 0000000 000 .00 .000000500 000cm £003 n00m 0 000 0000000m 00000000500 000 00 00000500 A5 o.~ I 00 I 00v 00m 0000Um 000 0000000m 00000000500 039 5000 w00000m0m 000M 0000000000000 0000000 mm.o 000w00 A9003 00:0 00— cm \r 0? ON 9 {lb L o 010 1 I-.. 02m 000.00 00500 02m 000000 0 ”EB ”E5 N L! 0.2.. “a 3% o. .o.n| Ho 960 m 0:5} 0.0 u E n x; c.02N n 04nu xgucgo m use? no u x; 0:.20 u x0 ago _ ION row row oo— (23) mono 232 The two dimensional simulations were repeated using Lx - 2.0 m and Ly - 1.0 m; results are shown in Figure 6.40. The temperature varia- tions at the geometric center, using the two dimensional solution, were higher than that shown for the previous case. Due to lack of symmetry, the temperature histories at (O,Ly/2) and (Lx/2,0) were not the same. The two dimensional solution was compared to the one dimensional solu- tion for an infinite slab 1.0 m in thickness. Again, the one dimensional temperature solution was bounded by the two dimensional solution. The predicted quality histories resulting from these tempera- ture histories are shown in Figure 6.41, using the activation energy constant of 182 kJ/mole. The quality deterioration at the geometric center of the two dimensional case was greater compared to that found for the 2.0 m by 2.0 m rod; however, little change was found in the quality profile at the exposed corner, compared to the 2.0 m by 2.0 m solution. The one dimensional solution for a 1.0 m slab over-estimated the quality at the geometric center, and closely approximated the quality along the sides, while greatly over—estimating the quality at the corners. Finally a 2 m by 0.2 m rod was considered. Due to the small height versus width ratio, the temperature distribution was very similar at all points in the rod, with high temperature fluctuations in all cases, as shown in Figure 6.42. The resulting one dimensional solution (not shown here) for an infinite slab, 0.2 m thick, was bounded by the solution for the two dimensional case. The resulting quality histories were very similar for all locations (Figure 6.43). Due to the high temperature fluctuations, the quality deterioration rate was very rapid, and little variation in quality within the product (< 10%) was found. The one dimensional temperature solution produced very similar quality results, 233 .00 00000 0.0m- 00 00000000509 0000000 £003 0000000 000 .5 0.0 I x0 .0000 -00500 £003 £000 0 000 00000000 00000000500 000 00 00000500 05 0.0 I %0 5 0.0 I 000 000 00000000000 000 00000000 000000502 00000000500 039 00.0 000000 Am>00V 00:0 000 Ow Do 00 0m 0 v: . i‘I,l L- -I.! .IlllI—Ilfvlutlb . .ll—v-. -vqlliulfL! 2|!!Lll‘l1lu IIIFII - .,-.I\ .Ll‘ 1 lb. 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T r00 5 n. x .1. 02m; . . ouxOIo 000:00 EOFHfib— 1 . x om X 010 .x._lx “0 02m E 0.0”? .E O.Nnx._ HQ N on» .oflx III 00:00 000 ) mono / (o 235 .00 00000 0.00- 00 00000000509 0000000 0003 0000000 000 .5 «.0 I x0 .0000 -00500 0003 0000 0 000 00000000 00000000500 000 00 00000500 05 «.0 I 00 .5 o.~ I 00v 000 00000000000 000 00000000 000000502 00000000500 039 «0.0 000000 0033 00:0 000 00 00 0.0 ON 0 L [p r L b P F L L L mm! 3n» .x..nx .010 00500 >4fl> .oflx I . ona .xJIx I w 020 E Nd“? E odflvfl " 0 N or...) .0000 Ill 000000 mm! .0 1.... 0.05.3 0.0 u E n x; 00 808 0.07. E goo S .90.. Ho 908 m (3°) aJmoJedLuei 236 .Amxmn One I coma- um GMHH-MHm£m .mHOE\fix mma I am no mmmov ooOm- mo undumuoasme HmwuHcH zuw3 uosvoum How .5 N.c I xA .:o«mcmEHo £uw3 anm you :oHuSHom Hmcofimcmaao $60 ea vmuwmeoo AE N.o I %A .E o.~ I xqv com unaswcmuomm Mom cofiusaom HQGOMchEHa O39 Eoum wcHuHSmwm ouwm GodumuoHumumo >ufiamso md.o muswfim Amxovv m::% 8. on oo 3 o P P r h P IF 0 0.2: “a :8 S .99. “a 96v m UonE\3 m.m H >5 H x: f flow fluov floo T XI— “ X I Qumm . . o H x I Loucoow E NO H x; .op row flux .vflnx film 5500 \jflx .oflx I E . ”A .E . "x u on) .OHX I LQHCOO 00. (z) Mllono 237 indicating that for the limiting case where Lx >> Ly, the one dimen- sional solution for an infinite slab may be used to approximate two dimensional heat transfer through a rectangular rod. The same three cases (Lx - Ly - 2.0 m; Lx - 2.0 m, Ly - 1.0 m; and, Lx - 2.0 m, Ly - 0.2 m) were repeated using the activation energy con- stant of 49 kJ/mole. Results are shown in Figures 6.44-6.46. In all cases, the slope of the curves were very similar, and very little varia- tion in quality was found within the product. Since the temperature solution for the one dimensional case was bounded by the solution for the two dimensional case, the resulting one dimensional quality profiles would be bounded by the two dimensional solutions shown in Figures 6.44- 6.46. The implications from these results are that for products with low activation energy constants (> 60 kJ/mole), the one dimensional solution may be used to approximate quality deterioration rates result- ing from two dimensional heat transfer with little error. In conclusion, quality variations within the product, using the high activation energy constant, were significant for width versus height ratios greater than or equal to 50%, and insignificant for ratios less than or equal to 10%. However, for low activation energy products, the quality distribution was relatively independent of the width vs height ratio, indicating that a one dimensional model, or a model using the mass average temperature (lumped capacitance model) might provide an excellent estimation of the product quality. When comparing the two dimensional solution, using equal width and height, to the one dimen- sional solution of equivalent geometry, the quality at the geometric center and the corners of the two dimensional case were over estimated by the one dimensional solution. The one dimensional solution closely approximated at the outer surface closely approximated the two dimen- sional solution at the midpoint of the sides. 238 .Amhwp oem I coma- um owHH-wHo£m .oaos\wx as I am no ommov Unem- mo ouSumuoaaoH HowuHaH £uw3 uUSboum How .xA .sofimsmswo Honda saw: nmam w you coHuSHom Hmcofimcoswa oso ou pmumdaoo As o.~ I %A I XAV pom mumscm you coHuSHom HmcofimcoEHQ O39 Eouu wGHUHSmoM oumm sofiumuofiuouoo huaamso qq.o ouswwm 363 9:: » L i F l? b F P if 0 33A .xuux film .5500 on» .jnx T» 2% E 3.13 .5 cans no N on» .oux Iz- toycoo T ION :0... 03E}; mam n .3 u x; G 308 0.9.: Lo 36.0 OF don... Lo axon n loo (as) MEIDDO 2h1 6.6.3 Effects of Geometrical Shape Different geometrical shapes were compared to determine if a two dimensional model might be better approximated by a one dimensional model of a different geometric shape, than that found for the one dimen- sional model using the same geometry in Section 6.6.2. Heat transfer through a two dimensional square rod and a one dimensional cylinder were used in the comparison. Two criteria were used to determine the radius (R) of the cylinder: (1) shapes of equal surface area; and, (2) shapes of equal volume. In both cases, no heat transfer was assumed along the axis of the rod and the cylinder, and the surface area and volume were calculated using unit length along this axis. Two dimensional heat transfer through a square rod with dimensions 2 m by 2 m (Figure 6.37). Uniform boundary conditions, of five days at -5°C, and ten days at ~18°C (Case 6), were imposed on all surfaces perpendicular to the direction of heat transfer, and an insulated boundary was imposed in the axial direc- tion. This was the same problem solved for first in Section 6.5.2, and shown in Figure 6.38. The radius of a cylinder with equal surface area was found to be 1.273 m. The solution for one dimensional heat transfer through the cylinder is compared to the solution of the two dimensional rod in Figure 6.47. The solutions at the geometric centers for both geometries were very similar; and the solution at the outer surface of the cylinder was approximately the average of the solutions at the midpoint of the sides and the corners for the square rod. The quality histories were determined using the activation energy constant of 182 kJ/mole, and compared at the same locations. Results are shown in Figure 6.48. The difference between the quality histories predicted for the rod and the cylinder at the geometric center, was less than 2% at the end of 100 days in simulated storage. The solution at 2h2 .Ao ommov mou< mUQMHSm Hmscm mo Honcaaho m szoune Hommcmue woo: amConEoEfio one new box oumavm m stoufih pommcmue use: HmCOMmcoEHa 039 you mcoauaaom mo comwumdeoo no.0 ousmwm A983 0E: 00* om om ow ON 0 h L P F p _ p — h _ mNI in» .x._ux min .6500 >4") .O"x E oux .xmux 279* 02m E ON"? 6: o.muxm ”com o N oflx .Oflx III 53:00 1 n .. 9o oEmw E mm; H m Coptic o — m o u ._ O... .3350 .. s 2,. f TWP! r 0.2.... n .38 625 fa.-- 038% 2 n P. u E 8 $08 0.2.. so as S .90... 6 98 m (3°) emeedtuei 2&3 .Ao omwov wou< commuSm Hozcm mo Hopewaxo m cwsouce Howmcmue umo: HmsonCmeo one new mom mumscm m nwsouce powmcmue umo: Hmcowmcoawa 039 Eoum wCAUHSmom mum“ GodumHOauouoa huHHmDO m¢.w ouswfim A963 mEP OO— Wm Wm (OW EN Im In L( L( 19 -1- L ..u L ,3! 1-- IF III: 1.0 fin J. . .4. 0.91 so 28 or .0}... so 28“. m (V, y r19.4919)J UonE\¢$ 0.0 nu >£ ...In x: fl «7.17) r. .... E q... n 3 u 5 ”Em ea N (To/z. ..,. 032:; no u x; flow / _ y E R; n a ”so 55 _ u , r ,, Quon! n .anH EEC; fl. my. MI? wow .mflflflflHVAvflu iarxy mvird. f {(9} r/ ,__, fl a -8 y. {19(910 ,, fl . If m u r I 222 .30 Ed 41.1.1. (I; ..m... Tom C n t I toucmo . . . fl. /. 3n» .5" film .550“: on; .xuux pl» 02m ”pom E5 N on) . "X I ..Bcoo ooF ) mono o/ [-0 ( 21... the outer surface of the cylinder, was between the two solutions at the side midpoints and the corners of the rod. This solution provided an estimate of the average quality at the surface, by under estimating the quality along the sides, and over estimating the quality at the corners. This comparison was repeated using the equivalent volume criterion. The radius satisfying this criterion was found to be 1.128 m. The resulting temperature histories are shown in Figure 6.49. The smaller radius resulted in a poorer estimate of the temperature at the geometric center, than that found using the equal surface area criterion. On the other hand, the temperature at the surface of the cylinder was almost the same as that found using R - 1.273 m. The predicted quality profiles at the geometric center and the outer surface are compared with the solution for the rod in Figure 6.50. (The solution for the rod is the same as that shown in Figure 6.48.) Comparing Figures 6.48 and 6.49, the equal surface area criterion resulted in a better estimation of the quality history of the two dimensional rod, than either the equal. volume criterion or the one dimensional solution using the same geometry (Section 6.6.2), due to the better estimation of the temperature at the geometric center, and the outer surface for the solution using the same geometry. A 10% difference in quality at the geometric center was estimated using the volume criterion, compared to only a 2% error using the equal surface area criterion. Therefore, using one dimensional heat transfer through a cylinder provided a good model for the estimation of the quality at the geometric center and the average quality around the perimeter of a square rod, with uniform boundary conditions on four adjacent sides, and insulated on the other sides. The equivalent surface area criterion provided a better estimate of the quality at the geometric center than either the equivalent volume criterion or using the same geometry. Extending this 216 .Ao omwov mESHo> Hmzcm mo Hovctho a zwzouch ummmcmue 0mm: AcconcoEHa oco paw pom mumsum m stounh nonmcmuh woo: HmconcoEfio oze How mcowusaom mo comaumaeoo o¢.w unawam $on mEfi (— L OO — 0m 00 O? I» u LP 3a.». flx Elm. $5004 >4u>.oflx «low ....>. 5.... T» 02m .E odnj .E 0.0!... fiom 0 m DNA .oflx III coucmOI m n L $10 20%.. E 2; H m ConcEAo 0 F O u L I Lmucmo . \ a _. . _ mm! LO ‘7‘ I In I .0 a K 0.9.... .... .950 BE... 0....Ex3 0.0 n P .... x; 6 3000 0.07. to 98c 0. 0.0... so 960 m T nil (3°) emgmedwei 2116 .Ac mmoov oasao> Hmavm mo nonsuaao w awaounh Houmcouh poo: HwCOHmcoaao one can pom oucavm a gwsouse Howmcmuh use: HmGOamCGEwa 039 scum wcwuazmom ouum Godumanuouoo huwamso om.m ouswfim A963 mEE m u t 9.0 85* .30 E5 Fill/.11 o H t 0.0 ..Bcoo x Hz . H - flu/do ._ x4 x mm 5500 .I.. lo/o/ 0.- -.A .xJHx 9..» 02m ”Dom EE N IIU or -2. 0.1.x I.- ..cho - V.» .P 0.Eoun3n fibomEam 03E; 0.0 .... E -om )- E2.. .530 E5. 1 1,. 0.0m- .. .QEB .055 . I? i (... 119.1.9 10¢ / 0.07. so 9% o. 0.0- so 980 m if o..«E\Bm.mH>;ch_ .. 00F 2117 application to a low activation energy product would result in a better estimate of the surface quality, since, as shown in Figure 6.44, there was very little difference in the quality around the surface of the rod using the PSL criterion. This leads to a means of quickly estimating the quality profile of a large mass of product. Running the one dimen- sional heat transfer program for the cylindrical geometry, required approximately eight minutes of CPU time on a VAX 11/750 computer, while running the two dimensional program on the same computer took over six hours of CPU time. This concept could be extended to approximate three dimensional heat transfer through a rectangular cube by using a two dimensional finite cylinder. This method, however, should only used for uniform boundary conditions around the outer surfaces of a square rod, and may not provide accurate results if extended to other situations, such as, non-uniform boundary conditions and unequal sides. CHAPTER 7. SUMMARY AND CONCLUSIONS A finite difference model, including temperature dependent thermal properties, was developed to simulate one dimensional heat transfer through frozen foods exposed to step changes in temperature storage conditions. The resulting temperature distribution histories were used to predict quality retention at different locations within the product, based on a temperature dependent quality deterioration rate constant, and a reference shelf-life. The one dimensional finite difference model was modified to simulate two dimensional heat flow, by utilizing the Crank-Nicolson approximation in an Alternating Direction Implicit finite difference model. Two dimensional quality profiles were estimated using this model. The temperature simulation models were verified by comparison with analytical solutions using constant thermal properties, and with ex- perimental temperature measurements, obtained in controlled storage conditions, using the Karlsruhe Test Substance, a highly concentrated methyl-cellulose mixture (Gutschmidt, 1960), as an analog for the food product. The interface between two interior product packages was found to increase the resistance to heat transfer within the product mass, resulting in a higher temperature differential between interior and exterior parts of the product, than found with the solution with no interface. Therefore, the packaging interface reduces the quality 248 249 deterioration rate at the interior of the product, and increases in quality deterioration rate in the exterior portions of the product. The surface heat transfer coefficients prevailing during step changes in temperature storage conditions were estimated as a function of time using the sequential regularization method of estimating the surface heat flux from internal product temperature measurements, again using the Karlsruhe Test Substance. The one dimensional direct finite difference program was utilized in the solution. The effects of the boundary conditions, size, geometrical shape, and the activation energy constant on the temperature and quality distribu- tion histories were studied. The following conclusions were drawn from this investigation. 1. The sequential regularization method provided estimates of the transient surface heat transfer coefficients which included the effects of the exterior packaging layer, and the accumulation and diminution of frost on the outer surface. 2. The interior product packaging interface increased the resistance to heat flow within the product. This resulted in a higher temperature differential, and a potentially higher quality differential within the total product mass, than predicted for by using the assumption of negli- gible internal packaging resistance to heat transfer. 3. Variations in storage conditions affected the retention of quality within the product. Higher surface heat transfer coefficients resulted in a lower quality retention; changes at lower magnitudes of the surface heat transfer coefficient (1.0 - 8.5 W/m2°C) had greater influence on the results than changes at higher magnitudes (8.5 - 2O W/m2°C). Step 250 changes in storage temperature from -18°C to -5°C for one day or less, and temperature fluctuations between ~18°C and -13°C had very little effect on the retention of product quality. 4. The magnitude of change in quality retention was highly dependent on the magnitude of of the activation energy constant. In most situations of step changes in ambient conditions, small internal variations in quality can be expected for products with low activation energy con- stants (< 60 kJ/mole), suggesting that average product temperatures (lumped capacitance solution) could be used to estimate quality his- tories. Products with activation energies above 60 kJ/mole are considered to be sensitive to temperature variations within the product, such that the lumped capacitance solution should not be used. Products with high activation energy constants (> 120 kJ/mole) are very sensitive to variations in storage conditions, such that variations in storage conditions would result in large quality differentials within the product mass. 5. Quality differences within the product mass for low activation energy constant products (< 60 kJ/mole), assuming one dimensional heat transfer and step changes in storage conditions, were significant (> 10% variation) for very large product masses (2 6.0 m), using a surface heat transfer coefficient of 8.5 W/m2°C, and ambient temperature < ~5°C. Products with high activation energy constants had insignificant quality differences within the product only when the product thickness was very small (5 0.2 m), using identical boundary conditions. For two dimen- sional heat transfer in low activation energy constant products, the quality differences within the product have limited dependence on the width versus height ratio, assuming identical boundary conditions in 251 both directions of heat flow. For high activation energy products under the same conditions, the quality differences are insignificant only for small width versus height ratios (5 10%). 6. The quality distribution history resulting from two dimensional heat transfer through a rectangular rod can be approximated within 5% ac- curacy by the one dimensional model heat transfer and quality retention, for an infinite slab of equal width, only if the height versus width ratio for the rectangular slab is greater than or equal to 10:1. 7. The quality retention resulting from two dimensional heat transfer through a square rod can be approximated by the one dimensional heat transfer and quality retention model, for an infinite cylinder of equal surface area, with over 90% reduction in computation time. The equal surface area criterion provided a better estimate of the two dimensional model than either a cylindrical geometry with equal volume, or the equivalent one dimensional rectangular geometry. The cylindrical model provided an excellent estimation of the quality at the geometric center, and the average quality around the perimeter of the square rod, however, the model greatly over-estimated the quality at the corners of the rod, given a product with a high activation energy constant (<120 kJ/mole). APPENDIX A APPENDIX A EQUATIONS FOR BOUNDARY NODES IN TWO DIMENSIONAL FINITE DIFFERENCE SOLUTION The following equations were derived for the corresponding boundary nodes shown in Figure A.1 for an infinite rod and a solid cylinder, using the Douglas-Gunn alternating direction implicit finite difference scheme (Douglas and Gunn, 1964). Eight unique equations were required to determine all of the temperature values at the boundaries. 1. l - 1, I.- 1 Step 1. Sweep in x-direction. n k A A A - —3 n+6. .13.:x . 3 y n+5 [ 2 [ thxo Ax + Ax J + ((PCP)+x+ (PCP)+y) 8At ] T1,1 k A +x +3 n+6 + "x 2Ax T2,1 5 k A A k A _ .3 n . +3 +3 n._y +1 +1 - [ 2 [ thxo Ax + Ax J + byo 2 + 2Ay ((PCP)+x + ( C ) ).£XAJ Tn _ Efil Tn _ ELLA—fl Tn P P +y 8At 1,1 2Ay 1,2 fix 2Ax 2,1 A hynA . n . n . n+6. n+6 _x _ 0 x. n - (13x thxo T... + 17x thxo Too )2 2 Too (A.la) 252 253 Step 2. Sweep in the y-direction. - 1! h n°A + EilAix + (( C ) + ( C ) ).駧l Tn+1 2 yo y Ay P P +x P P +y 8At 1,1 + k+1A+y Tn+1 "y 2Ay 1,2 k A k A _ 25 hxn+€oA + +3 +3 Tn+£ _ 0 +3 +3 Tn+5 2 on x Ax 1,1 x 2Ax 2,1 - (( C ) + ( C ) ),232x Tn - fl EiXEix Tn ’ 3 k+ A+x Tn P P +X P P +y 8At 1’1 )7 2A)! 1,2 X ZAX 2,1 A . n . n . n+6. n+5 _x (fix thx0 To + "x on Tan )2 A n n n+1 n+1 _y - (fly-hyoo Tco + ny-hyo 0T” )2 (A-lb) 2. 1 - 1, 1 <‘I < H Step 1. Sweep in x-direction. k - [ ax [MKE'AX + 41‘5“] + <(pCp>+x+ <_y+ (pep)+y>>/2> 253 Step 2. Sweep in the y-direction. 21 h “-A + 511211 + <( c > + < c > >-5351 TP*1 ' 2 yo y Ay P P +x P P +y 8At 1,1 + k+xA+y Tn+1 "y 2Ay 1,2 _ Zr. hxn+E,A + E13213 Tn+5 _ E1321; Tn+€ 2 on x Ax 1,1 "x 2Ax 2,1 13 k A ,6 k A .3 n , .1xrix .x n, .11.:1 + [ 2 thx0 Ax + Ax + 2 hy0 Ay + Ay - (( C ) + ( C ) ),éxé¥ Tn _ fl k-I-xA-I-x n _ fl k+xA+x Tn p p +x p p +y 8At 1,1 y 2Ay 1.2 X ZAX 2.1 A e n e n Q n+5. n+6 ‘25 - (Bx thXo Tco + "x o Tco )2 n n n+1 n+1 fly - (flyohyoe T“, + "yehyo 0T” )2 (A.1b) 2. 1 - 1, l <‘l < M Step 1. Sweep in x-direction. - hx“*5-A + 51351; + << c > + <( c > + < c > >)/2) "x on x Ax p p +x p p -y p p +y k_ A- n n k+ A+ k Y Y k+vA+Y 2Ay T1,m-l + fix thxo.A + Ax + 2Ay 4. A A n k+ A+ n ' ((PCP)+X+ ((PCP)_y+ (pCP)+y))/2).4At T1,!!! ' 2Ay Tl,m+l - 19 —-’5——3k+ 1“” T“ - (19x - T: + oh “5 Tn+€)Ax x Ax 2,m thx0 "x xon (A.2a) Step 2. Sweep in the y-direction. k A k A + k A -y -y n+1 - 5y 1y +y +y "y 2Ay Tl,m-l [ "y 2Ay + ((PCP)+X+ ((PCP)-y + < C ) )/2)eéx-Al Tn+1 4. "him n+1 p p 4At 1,m "y 2Ay l,m+l _ fl hxn+€oA + k+3A+x Tn+5 _ k+xA+x Tn+§ - fl k-yA-y Tn x on x Ax 1,m "x Ax 2,m y 2Ay 1,m-1 k A k_ vA +k A n . +3 +3 -y +y,+y + [ fix [ thxo Ax+ Ax ] + fly 2Ay A A k _K_l n _i¥_il n <+x+<(pCp>_y+ (pop)+y)/2>-4At ] T1,m - fly 2Ay T1,m,1 A _ _¢x_ix n _ . n . n . n+5. n+5 fix Ax T2,m (fix thxo Tm + "x thxo To )Ax (A.2b) 255 3. l - 1, Im-.M Step 1. Sweep in x-direction. n k A A A , .2: ME, _+x_tx ,_1Lx n+6 + £13121 Tn+€ "x 2Ax 2,M - 5:112): Tn + 83 hxn ,A + 122132: + k-vA-v + hyLy°Ax 2Ay l,M-l 2 on x Ax 2Ay 2 AxAy n k+xA+3 n ' “POW”.+ (PCP)-y)'8Ac T1,M ' Bx 2Ax T2,M hy 0A _ . n . n . n+5. n+£ _ __Ly__y. n (Bx thxo T... + 11x 0 T” )1.x 2 Too (A.3a) Step 2. Sweep in the y-direction. k- A- n+1 0 k- A- n+1 '44 T J 41.1 + by Ay + ((pCp>+x+ (pCp>_y> "y 2Ay l,M-l- 2 Ay Ly. .AxAx Tn+1 8At 1,M _ 25 hxn+§ A + k+xA+x Tn+£ - n k+xA+x Tn+5 _ fl k-yA-y Tn 2 o x Ax 1,M x 2Ax 2,M y 2Ay 1 M-1 256 A _ . n . Tn n+5 n+5 3 (fix thxoT w + "x thxo Tm ) 2 A . . n+1 Tn+1 _y - (19yhyhy T: + "y hyLyT m )2 (A.3b) 4. 1 < l'< L, l,- 1 Step 1. Sweep in x-direction. k A k_ xA + k A -x -x n+6 _ x. +3 +x "X 2Ax 2_1,1 [ "x 2Ax + (((PCP)_X+ (PCP)+x)/2 A A A 3 y %+ k+ + n+5 + (PCP)+y .4At ] T21,1+ ”xM 2Ax T2+1,1 k A k_ A + k A k A -x - + +x . +1 +1 Aflx 2Ax + H[ fix + hy‘ Ay + Ay A A n k+ A+ n ' (((PCP)-X+ (PCP)+x)/2 + (PCP)+y)'Z§Ex T£,1' __IZ;¥ T£,2 Eixfii‘ T“ h “A T“ A 4 ' fix 2Ax 1+1,l - y0 y. m ( ' a) Step 2. Sweep in the y—direction. n+1 k+ A+ - "y hyo .Ay+ —L1Ay + <(_x+ (pop>+x>/2 + +y> .5351 Tn+1+ Hk+y +1 Tn+1 4At 2,1"y Ay 1, 2 _ _ k-gA-g Tn+5 + k-xA-x+ k+xA+x.Tn+§ _ k+xA+x Tn+5 ”x 2Ax 2-1,1 "x 2Ax 1,1 "x 2Ax £+1,1 k_ _ n k_xA_ X+ k+xA+x n k+ A+ fix 2Ax T[-1.1 + fix 2Ax + fly hyO.A + Ay AxAy n k+yA+y n ' (((PCP)_x+ (Pcp)+x)/2 + (PCP)+y)'4At T2’1 ' fly Ay T£,2 _ fl Eixfix Tn _ (By h Tn + h n+lTn+1)A x 2Ax 2+1,1 yo "y yo m y (A.4b) 5. 11< 1 < L, -.-.M Step 1. Sweep in x-direction. k A k A x+ k 15 1; n+5 _ ~1L-x +xA +x nx 2Ax T£_1 M [ nx 2Ax + (((pCP)_x+ (pCP)+x)/2 A A k A .41.! n+6 M n+5 + (pCP)-y) At ] T1,H + "x 2Ax T£+1,M k A k A k_ x-A x+k A k A _ _ 5 -x -x Tn _ -y -y Tn + 3 +x +x + -x -y x 2Ax £-l,M Ay 1,H-1 x 2Ax Ay n 52.5.. n + hyLyAy - <</2 + (pep)_y>-4At 'T1,M k A +x +x n - fix 2Ax T£+1,M hyLyAy Too (A.5a) Step 2. Sweep in the y-direction. E¢££=1 Tn+1 - E=15=¥ + h “+1 A J + (<( c > + < c > >/2 ny Ay £,M-1 "y Ay yLy y p p -x p p +x ).A3Ay ] Tn+1 258 _ _ k-xA-x Tn+£ + 13 k-xA-x+ k+xA+x Tn+6 _ k+ggA+35 Tn+§ "x 2Ax £-1,M x 2Ax 2,m "x 2Ax 2-1,M k A k A k A + k A _ B -x -x Tn _ 5 -y -y Tn + 6 -x ax +x +x x 2Ax 2-1,M y Ay £,M-l x 2Ax k A + .3 [ 4m + hyfiy-Ay] - <</2 +_y> Y AV 0% Tn _ fl Egg Tn _ (fl 'h n . Tn 4At £,M x 2Ax 1-1,M y yLy w + n h “+1-1“+1)A (A.Sb) y' yLy w y 6. l - L, -.- 1 Step 1. Sweep in x-direction. k_ A-“ Tn+5 _ 23 k_ A_“ + h n+E.A + (( C > + ( C ) ) "x 2Ax L-l,l 2 Ax xLx x P P -x P P +y A A . _x_z Tn+£ 8At 1,1 k_xA_ n Ex k_ A_ n hyO-A k+ A+ - 'flx 2Ax TL-l,l + 2 Ax + thx.Ax + 2 + 2Ay AKA! n k+xA+y n ' ((PCP)-x+(PCP)+y)'8Ac TL,1 ’ 2Ay T1,2 A hynA - . n . n+ . “+5. —£ - ——Q—£. n (fix thx Ico i "x hx, Tm) 2 2 Tco (A.6a) 259 Step 2. Sweep in the y-direction. n k A A A _ _y n+1. +1 +1 . x y n+1 [ 2 [hyO Ay+ A), ] + ((PCP)_X+ (PCP)+y) 8At ] TL,1 + k+xA+y Tn+l "y 2Ay L,2 _ _ fl ELEfLX Tn+€ + 23 E;xf;g + hxn+€oA Tn+5 x 2Ax L-l,1 2 Ax x L,1 k_ A_ n 53 k_ A n ‘Px 2Ax TL-1,1+ 2 Ax + thx'A & n k+xA+1 AKA! n + 2 hyo'Ay + 2Ay - ((pCP),x+ (pCp)+y)-8At TL,1 A A _ _i!_i1 n _ . n . n . n+5. n+5 _g 5y 2Ay TL,2 (5x thx Tm + "x thx Tm )2 _ (fl 1'! 11.1.11 + h n+l.Tn+1)fx (A 6b) Y Yo a, "y Yo o 2 ° 7. 1 - L, 1 < lw<.H Step 1. Sweep in x-direction. k A k A -x -3 n+5 _ -3 -x n+5. "x Ax TL-1,m [ "x [ Ax + thx Ax J + ((pCP)-x 52.51 n+5 + ((pCp)_y+ (PCP)+y))/2)'4Ac T + k_ A_ n k_ A_ n k-xA-x n - - fix Ax TL+l,m - 2Ay T + fix + bx .Ax 260 k’yA'v+ k+yA+¥ C + C + C 2 + - ((p p)_x ((p p)_y (p p)+y))/ ) 2Ay A A k A . x y n _ +1 +y n _ . n . n . n+§ n+5 4At ] TL,m 2Ay TL,m+1 (fix thx Tm + "x Lx Tm )Ax (A.7a) Step 2. Sweep in the y-direction. +Y +y + ((pCp)_x+ <0C Kp - Product thermal conductivity (W/mK) } >0C Cp - Product specific heat (kJ/kgK) } >OC Misc. Variables- Xx - Intermittent value in determining Ms or T0. Tc - Temperature for printout. Yn - Character- Y or N integer type,prpscr,prpfil double precision cl,c2,c3,c5,c6,wf0,ms,dp,kp,cp,t0,ds, &di,dw,kw,cpi,cpw,moist,dens,conduc,ki,spheat,tc,xx,t0inv, &tl,th,eta(20),w(20),tavg,tdif,x(20),avgd,avgk,avgc,ynavg character yn*1,prpf11*10,prpf12*12,tit1e*20,tt1fil*4 268 Table B.2 (cont'd). logical itmode external moist,dens,conduc,spheat common/prop/wf0,ms,dp,kp,cp,t0,/cons/c1,c2,c3,/d/ds,/densi/di,dw, &/COND/KW,/SPH/CPI,CPW,/ITM/ITMODE,/ttl/title,ttlfil,/profil/prpfil &/mod/mode1,/pavg/th,tl,avgd,avgk,avgc,ynavg S ave cl-18.015d0 c2-1.0d0/273.15d0 c3-6003.0d0/8.314d0 c6-273.15d0 IF(.NOT.ITMODE)THEN READ(10,*)WFO,TO,DP,KP,CP,TYPE IF(TYPE.NE.1)MS-T0 GO TO 20 ELSE write(5,2000) 2000 format('l',72('-'),/,'0',t27,'Product Properties',/,'0',72('-')) 5 write(5,100) 100 format(' ',/,' ','Enter initial moisture content(%): ',$) READ*,WFO if(wf0.LT.1..OR.WFO.GT.100.)THEN print*,'Try again!!' goto 5 ENDIF 10 WRITE(5,300) 300 FORMAT('O','Choose:',/,' ',' 1. Initial freezing temperature',/, &' ',' 2. Molecular weight of solids') READ(5,*)type if(type.NE.l)THEN WRITE(5,400) 400 FORMAT(' ',/,' ','Molecular weight of solids: ',$) READ(5,*)MS ELSE WRITE(5,500) 500 FORMAT(' ',/,' ','Initial freezing temperature (C): ',$) READ(5,*)T0 ENDIF write(5,600) 600 format(' ',/,' ','Enter unfrozen product property values:',/, &' ',2x,'density (kg/mh3): ',$) READ(S,*)DP WRITE(5,700) 700 FORMAT(' ',2x,'thermal conductivity (W/mK): ',$) READ(5,*)KP WRITE(5,800) 800 FORMAT(' ',2x,'specific heat (kJ/kgK): ',$) READ(5,*)CP c Enter temperature range for determination of mean property values. write(5,850) 850 format(' ',/,' ','Enter temperature range for mean property ', 269 Table B.2 (cont'd). C & 860 880 900 200 'values:'/,2x,'low temperature (C): ') read(5,*)t1 write(5,860) format(' ',2x,'high temperature (C): ') read(5,*)th t1 - t1 + 273.150d0 th - th + 273.150d0 write(5,880) format(' ',/,' ',2x,'Use average temperatures in PD solution ? '(O-no; l'y) ') read(5,*)ynavg WRITE(5,900) FORMAT(' ',/,' ','Are these values correct? (y/n) ',$) read(5,200)yn FORMAT(A) if(yn.ne.'y'.and.YN.NE.'Y')goto 5 ENDIF 20 wf0-wf0/100.0d0 if(type.NE.1)THEN Initial freezing point tO-l.0d0/(c2-log(wf0/c1/(wf0/cl+(l.0d0-wf0)/ms))*1.0d0/c3) ELSE t0-t0+273.15d0 Molecular weight of solids tOinv - 1.0d0/t0 xx-exp(c3*(c2-tOinv)) Ms-(l.0d0-wf0)*xx*c1/(wf0*(1.0d0-xx)) ENDIF Determine mean property value over specified range using Gauss Quadrature integration: 20 pt. quad. Eta values: eta(l) - -0.99312859918509 eta(2) - -0.96397192727791 eta(3) - -0.91223442825133 eta(a) - -0.83911697182222 eta(S) - -0.74633190646015 eta(6) - -0.63605368072652 eta(7) - -0.51086700195083 eta(8) - -0.37370608871542 eta(9) - -O.22778585114165 eta(lO) - -0.07652652113350 do i - 1,10 eta(21-i) - -eta(i) enddo Weighting factors: w(1) - 0.01761400713915 270 Table B.2 (cont'd). C C w(2) - 0.04060142980039 w(3) - 0.06267204833411 w(h) - 0.08327674157670 w(S) - 0.10193011981724 w(6) - 0.11819453196152 w(7) - 0.13168863844918 w(8) - 0.14209610931838 w(9) - 0.14917298647260 w(10) - 0.15275338713073 do i - 1,10 w(21-i) - w(i) enddo Transform eta onto (th - t1) interval tavg - (th+tl)/2.0d0 tdif - (th-t1)/2.0d0 avgd - 0 avgk - 0 avgc - 0 do i - 1,20 x(i) - tavg + tdif*eta(i) Sum integral approximation avgd - avgd + w(i)*dens(x(i)) avgk - avgk + w(i)*conduc(x(i)) avgc - avgc + w(i)*spheat(x(i)) enddo avgd - 0.50*avgd avgk - 0.50*avgk avgc - 0.50*avgc IF(ITMODE)THEN prpscr - 0 prpfil - 0 write(5,905) 905 format(' ',/,' ','Display product properties on screen? (y/n) ',$) read(5,200)yn if(yn.eq.'y'.OR.YN.EQ.'Y')prpscr - 1 write(5,906) 906 format(' ','Do you want product properties - f(T)', & ' saved in a file? (y/n) ',$) read(5,200)yn if(yn.eq.'y'.OR.YN.EQ.'Y')prpfil - 1 IF(PRPSCR.EQ.0.AND.PRPFIL.EQ.0)GO TO 90 ELSE READ(5,*)PRPFIL ENDIF IF(PRPFIL.EQ.1)THEN WRITE(PRPFL2,910)TTLFIL,'PRP.VAR' 910 FORMAT(' ',A,A) OPEN(UNIT-12,NAME-PRPFL2(1:12),TYPE-'NEW',CARRIAGECONTROL='LIST' 6:) write(12,1000)tit1e write(12,1050) 271 Table 8.2 (cont'd). write(12,1100)t0-273.150d0 write(12,1200)ms write(12,1300) write(12,1400) WRITE(12,1500) TC-AINT(t0+2.0dO)+.150dO 40 IF(TC.GE.t0)THEN WRITE(12,1600)tc-c6,wf0*100.,DP,kp,cp ELSE WRITE(12,1600)tc-c6,MOIST(TC)*100.,DENS(TC),conduc(tc), & SPHEAT(TC) ENDIF IF(TC.GE.t0+1.0d0)THEN TC - TC-l.0d0 GO TO 40 ENDIF IF(TC.GT.AINT(t0-l.0d0)+.15)THEN TC - TC-.250d0 GOTO 40 ENDIF IF(TC.GT.AINT(t0-4.0d0)+.15)THEN TC - TC-.50d0 GOTO 40 ENDIF IF(TC.GT.AINT(t0-10.0d0)+.15)THEN TC-TC-1.0d0 GOTO 40 ELSE TC-TC-2.0d0 IF(TC.GE.233.150dO)GOTO 40 ENDIF write(12,1700)t1-273.15,th-273.15,avgd,avgk,avgc CLOSE(UNIT-12) ENDIF C Printout on screen IF(ITMODE)THEN IF(PRPSCR.EQ.1)THEN write(5,1001)tit1e write(5,1050) write(5,1100)t0-273.150d0 write(5,1200)ms write(5,1300) write(5,1400) WRITE(5,1500) 1000 format(' ',T31,A) 1001 format('l',T31,A) 1050 format(' ',t27,'Product Properties',/,' ',t22,'as a ', & 'Function of Temperature') 1100 format(' ',/,' ','Initial freezing temperature(C)- ',f6.2) 1200 format(' ',/,' ','Equivalent molecular weight- ',f6.2) 1300 format(' ',/,' ',28x,'Product properties'//) 1400 FORMAT(' ',/,' ',3X,'Temperature',5X,'Unfrozen',7X,'Density', & 4X,'Conductivity',2x,' Ap.Sp.Heat') 1500 FORMAT(' ',7X,'(C)',10X,'Water',8x,'(kg/m“3)',7X,'(W/m*K)', 272 Table B.2 (cont'd). & 6X,'(kJ/kg*K)',/,3X,5(' ------------ ')) 1600 FORMAT(' ',T4,F7.2,T22,F6.2,T36,F6.1,T49,F6.3,T62,F8.3) 1700 format(' ',//,' ','Average property values over the interval ' & ,f6.1,' to ',f6.1' :',/,' ',5x,'Average density - ',t40,f6.1, & /,' ',5x,'Average thermal conductivity - ',t40,f5.3,/,' ',5x, & 'Average specific heat - ',t40,f5.3) TC-AINT(t0+2.0dO)+.150d0 50 IF(TC.GE.t0)THEN WRITE(5,1600)tc-c6,wf0*100.,DP,kp,cp ELSE WRITE(5,1600)tc-c6,MOIST(TC)*100.,DENS(TC),conduc(tc), & spheat(tc) ENDIF IF(TC.GE.t0+1.0dO)THEN TC - TC-1.0d0 GO TO 50 ENDIF IF(TC.GT.AINT(t0-1.0d0)+.15)THEN TC - TC-.250d0 GOTO 50 ENDIF IF(TC.GT.AINT(t0-4.0d0)+.15)THEN TC - TC-.50d0 GOTO 50 ENDIF IF(TC.GT.AINT(t0-10.0d0)+.15)THEN TC-TC-1.0d0 GOTO 50 ELSE TC-TC-2.0d0 IF(TC.GE.233.150dO)GOTO 50 ENDIF ENDIF ENDIF write(S,1700)t1-273.15,th-273.15,avgd,avgk,avgc 90 return END DOUBLE PRECISION FUNCTION MOIST(X) c This function determines the unfrozen water c fraction of a food procduct below the initial freezing temp- c perature. c the variables used in this function are: cl- 18.015 kg/kmole; molecular weight of water c2- 1./273.15 KA-l c3- 6003./8.314 K; latent heat of ice/mole/R wf0- moisture content of unfrozen product ms- molecular weight of solids 00000 273 Table 8.2 (cont'd). 000 0 00000 double precision cl,c2,c3,xx,ms,tinv,t0,wf0,dp,kp,cp common/prop/wf0,ms,dp,kp,cp,t0,/cons/c1,c2,c3 c1-18.015d0 c2-1.0d0/273.15d0 c3-6003.0d0/8.314d0 tinv - 1.0d0/x xx-exp(c3*(c2-tinv)) moist-xx*(1.0d0-wf0)*cl/(ms*(1.0dO-xx)) return end DOUBLE PRECISION FUNCTION DENS(X) this function determines the density of a food product below the initial freezing temperature,as a fuction of un- frozen water fraction. the variables used in this function are: di- density of ice (kg/mAB) dp- density of unfrozen product (kg/m‘3) ds- density of solids (kg/mAB) dw— density of water (kg/m23) wf0- moisture content of unfrozen product double precision moist,wf0,ms,dp,kp,cp,t0,di,dw,ds external moist common/prop/wf0,ms,dp,kp,cp,t0,/densi/di,dw,/d/ds S ave Solids density ds-(1.0d0-wf0)/(l.0d0/dp-wf0/dw) dens—1.0d0/(moist(x)/dw+(1.0d0-wa)/ds+(wf0-moist(x))/di) return end DOUBLE PRECISION FUNCTION KI(X) c Thermal conductivity of ice as a function of temperature (k) ki-7.3640d0-0.02850d0*x+3.525d-5*x**2 return end 274 Table B.2 (cont'd). DOUBLE PRECISION FUNCTION CONDUC(X) c This function subroutine determines the thermal conductivity c of a food product below the initial freezing temperature. c Thermal conductivity is a function of moisture content and c solids content, therefore, 'conduc' is a function of temper- c ature. c The variables used in this function subroutine are: c di- density of ice c dw— density of water c ki(x)- thermal conductivity of ice as a function of c temperature c kp- thermal conductivity of product c ks- thermal conductivity of solids c kw- thermal conductivity of water c wf0- unfrozen product moisture content c moist(x)- moisture content as a function of temperature c kl- intermittent value c k2- intermittent value c k3- thermal conductivity of water-ice phase c k4- intermittent value c k5- intermittent value c va- intermittent value c val- intermittent value c c4- 2./3. double precision moist,dens,ki,wf0,ms,dp,kp,cp,t0,di,dw,ds, &kw,c4,va,va1,ks,k1,k2,k3,k4,k5 common /prop/wf0,ms,dp,kp,cp,t0,/densi/di,dw,/d/ds,/cond/kw external moist,dens,ki c4—2.0d0/3.0d0 c Solids density ds-(l.0d0-wf0)/(1.0dO/dp-wa/dw) c Thermal conductivity of solids va-(l.0d0-wf0)/ds val-(va/(va+wf0/dw))**c4 ks-kw*(val-((kw-kp)/kw—kp*(l.0d0-va1**.5)/val)) c Phase I: ice--water k1-(wf0-moist(x))/di/(moist(x)/dw+(wf0-moist(x))/di) k2-kl**c4*(1.0d0-ki(x)/kw) k3-kw*(1.0d0-k2)/(1.0d0-k2*(1.0d0—k1**(1.0d0/3.0d0))) c Phase II: solids--water/ice kh-va/(va+(wf0-moist(x))/di+moist(x)/dw) 275 Table B.2 (cont'd). 000 0 0000 00000 k5-k4**c4*(1.0d0-ks/k3) conduc-k3*(1.0d0-k5)/(l.0d0-k5*(l.0d0-k4**(l.0d0/3.0dO))) return end DOUBLE PRECISION FUNCTION SPHEAT(X) This function determines the apparent specific heat of a frozen food product as a function of unfrozen water below the initial freezing temperature. The external function moist (unfrozen water fraction) used to determine the apparant specific heat is a function of temperature; therefore, 'spheat' is also a function of temper- ature . the variables used in this function are: cp - specific heat product >0C (kJ/kgC) cpi- specific heat of ice (kJ/kgC) cps- specific heat of solids (kJ/kgC) cpw— specific heat of water (kJ/kgC) wa - moisture content of unfrozen product double precision moist,wf0,ms,dp,kp,cp,t0,cpi,cpw,dh,cps,dcp,c5 external moist common/prop/wf0,ms,dp,kp,cp,t0,/sph/cpi,cpw c5-6003.0d0/18.015d0 dh - 0.0010d0 c solids specific heat cps-(1.0d0-wf0)/(1.0d0/cp-wf0/cpw) if(x+dh.ge.t0)then dcp - (moist(x)-moist(x-dh))*c5/dh else dcp - (moist(x+dh)-moist(x-dh))*c5/(2.0d0*dh) endif spheat-(l.0d0-wf0)*cps+moist(x)*cpw+(wa-moist(x))*cpi+dcp return end SUBROUTINE CONSPR 276 Table B.2 (cont'd). c This subroutine provides constant property approximations to the c properties as a function of temperature. parameter(maxd - 101, maxc - 51, maxs - 201) integer prpfil,nsd,nsc,nss,ynavg double precision tc,c5,aa,a,b,t0m1,apb, &erdens,ercond,erpsh, &avgdi,avgki,avgci, &bstep,erldd,er1dc,erlds,er2dd,er2dc,er2ds &ddera,dderab,rh02d,rhold,cdera,cderab,k2d,k1d, &sdera,sderab,sph2d,sphld c Declare variables in common blocks double precision wf0,ms,dp,kp,cp,t0, &c1,c2,c3, &denst(maxd),densc(maxd),condt(maxc),condc(maxc), &spht(maxs),sphc(maxs), &tl,th,avgd,avgk,avgc, &MOIST,DENS,CONDUC,SPHEAT character title*40,ttlfil*10,prpf13*12,fildat*16 EXTERNAL MOIST,DENS,CONDUC,SPHEAT common/prop/wf0,ms,dp,kp,cp,t0, &/cons/c1,c2,c3, &/CONSTP/DENST,DENSC,CONDT,CONDC,SPHT,SPHC, &/NCONSTP/NSD,NSC,NSS, &/tt1/title,ttlfil, &/profil/prpfil, &/pavg/th,t1,avgd,avgk,avgc,ynavg AA - 233.150d0 NTSD - 100 NTSC - 50 NTSS - 200 ERlDD - .050d0 .050d0 .050d0 .250d0 .250d0 .250d0 BSTEP 0.10d0 TOM1 - T0-0.050d0 ‘5"; U U IIIIII C Find values for density 5 I - 1 A - AA DENST(1) - A B - BSTEP 1P1 - 1+1 APB - A+B 277 Table B.2 (cont'd). C Determine % second derivative for density. 10 DDERA - (DENS(A+0.010d0)-DENS(A))/DENS(A) DDERAB - (DENS(APB+0.010dO)-DENS(APB))/DENS(APB) RH02D - (DDERAB-DDERA)/DDERA C Determine % first derivative for density. RHOlD - (DENS(A)-DENS(APB))/(DENS(A)) C Check if second derivative is greater than allowable error; if so, store C temperature endpoint value and determine the average property value for that C section, assuming property is linear with temperature over each segment. IF(ABS(RH02D).GT.ER2DD.OR.ABS(RH01D).GT.ER1DD)THEN DENST(IP1) - APB call integr(apb,a,avgdi,avgki,avgci,1) densc(i) - avgdi A - A+B I - IPl IPl - I+1 C Check if number of steps is greater than array dimensions; if so, C double the allowable error in the second derivative and repeat calcu- C lations. IF(I.GT.NTSD)THEN ER2DD - ERZDD*2.0d0 ERlDD - ERlDD*2.0d0 GO TO 5 ENDIF B - BSTEP ELSE B - B+BSTEP ENDIF APB - A+B IF(APB.LT.T0-bstep)GO TO 10 DENST(IP1) - T0 call integr(t0m1,a,avgdi,avgki,avgci,1) densc(i) - avgdi NSD - I C Find values for thermal conductivity 20 I - 1 A - AA CONDT(1) - A B - BSTEP 1P1 - I+1 APB - A+B C Determine % second derivative for thermal conductivity. 278 Table B.2 (cont'd). 30 CDERA - (CONDUC(A+0.010dO)-CONDUC(A))/CONDUC(A) CDERAB - (CONDUC(APB+0.010dO)-CONDUC(APB))/CONDUC(APB) K2D - (CDERAB-CDERA)/CDERA C Determine % first derivative for thermal conductivity. KID - (CONDUC(A)-CONDUC(APB))/(CONDUC(A)) C Check if second derivative is greater than allowable error; if so, store C temperature endpoint value and determine the average property value for that C section, assuming property is linear with temperature over each segment. IF(ABS(K2D).GT.ER2DC.OR.ABS(K1D).GT.ER1DC)THEN CONDT(IPl) - APB call integr(apb,a,avgdi,avgki,avgci,2) condc(i) - avgki A - A+B I - IPl 1P1 - 1+1 C Check if number of steps is greater than array dimensions; if so, C double the allowable error in the second derivative and repeat calcu- C lations. IF(I.GT.NTSC)THEN ER2DC - ER2DC*2.0d0 ERlDC - ERIDC*2.0d0 GO TO 20 ENDIF B - BSTEP ELSE B - B+BSTEP ENDIF APB - A+B IF(APB.LT.T0-bstep)GO TO 30 CONDT(IPl) - T0 call integr(t0ml,a,avgdi,anki,avgci,2) condc(i) - avgki NSC - I C Find values for specific heat 40 I - 1 A - AA SPHT(1) - A B - BSTEP 1P1 - 1+1 APB - A+B C Determine % second derivative for specific heat. 50 SDERA - (SPHEAT(A+0.010d0)-SPHEAT(A))/SPHEAT(A) SDERAB - (SPHEAT(APB+0.010d0)-SPHEAT(APB))/SPHEAT(APB) 279 Table B.2 (cont'd). SPH2D - (SDERAB-SDERA)/SDERA C Determine % first derivative for specific heat. SPHlD - (SPHEAT(A)-SPHEAT(APB))/(SPHEAT(A)) C Check if second derivative is greater than allowable error; if so, store C temperature endpoint value and determine the average property value for that C section, assuming property is linear with temperature over each segment. C IF(ABS(SPH2D).GT.ER2DS.OR.ABS(SPH1D).GT.ER1DS)THEN SPHT(IP1) - APB call integr(apb,a,avgdi,avgki,avgci,3) sphc(i) - avgci A - A+B I - IP1 IP1 - 1+1 Check if number of steps is greater than array dimensions; if so, C double the allowable error in the second derivative and repeat calcu- C lations. IF(I.GT.NTSS)THEN ER2DS - ER2DS*2.0d0 ERlDS - ERlDS*2.0d0 GO TO 40 ENDIF B - BSTEP ELSE B - B+BSTEP ENDIF APB - A+B IF(APB.LT.T0-bstep)GO TO 50 SPHT(IP1) - T0 call integr(t0m1,a,avgdi,avgki,anci,3) sphc(i) - avgci NSS - I C Print out approximated property values in file. 85 88 90 100 110 IF(PRPFIL.EQ.1)THEN WRITE(PRPFL3,85)TTLFIL,'PRP.CON' FORMAT(' ',A,A) OPEN(UNIT-12,NAME-PRPFL3(1:12),TYPE-'NEW',CARRIAGECONTROL—’LIST') WRITE(12,88)TITLE FORMAT(' ',//,' ',T31,A) WRITE(12,90) FORMAT(' ',/,' ',T25,'CONSTANT PROPERTIES') WRITE(12,100)T0-273.150d0 FORMAT(' ',/,' ','Initial freezing temperature(C)= ',F6.2) WRITE(12,110)MS FORMAT(' ',/,' ','Equivalent molecular weight- ',F6.2) 280 Table B.2 (cont'd). WRITE(12,120) 120 FORMAT(' ',/,' ',//,28X,'Product Properties'//) WRITE(12,130) 130 FORMAT(AX,'Temperature',5X,'Unfrozen',7X,'Density',4X, +'Conductivity',2X,' Ap.Sp.Heat') WRITE(12,140) 140 FORMAT(' ',7X,'(C)',10X,'Water',8X,'(kg/m“3)',7X,'(W/m*K)', +6X,'(kJ/kg*C)',/,3X,5(' ------------ ')) TC-AINT(TO+2.0dO)+.150dO 150 IF(TC.LT.T0)GOTO 170 WRITE(12,160)TC-273.150d0,WFO*100.0d0,DP,KP,CP 160 FORMAT(SX,F7.2,T22,F6.2,T36,F6.l,T49,F6.3,T62,F8.3) GOTO 240 170 DO I - 1,NSD IF(TC.GE.DENST(I).AND.TC.LT.DENST(I+1))THEN DE - DENSC(I) GOTO 180 ENDIF ENDDO 180 DO I - 1,NSC IF(TC.GE.CONDT(I).AND.TC.LT.CONDT(I+1))THEN CO - CONDC(I) GOTO 190 ENDIF ENDDO 190 DO I - 1,NSS IF(TC.GE.SPHT(I).AND.TC.LT.SPHT(I+1))THEN SP - SPHC(I) GOTO 200 ENDIF ENDDO 200 WRITE(12,160)TC-273.150d0,MOIST(TC)*100.0d0,DE,CO,SP 240 IF(TC.GE.T0+1.0dO)GOTO 270 IF(TC.LE.AINT(T0-1.0d0)+.15)GOTO 250 TC-TC-.250d0 GOTO 290 250 IF(TC.LE.AINT(T0-4.0d0)+.15)GOTO 260 TC-TC-.50d0 GOTO 290 260 IF(TC.LE.TO-10.0d0)GOTO 280 270 TC-TC-1.0d0 GOTO 290 280 TC-TC-2.0d0 290 IF(TC.GE.233.150dO)GOTO 150 WRITE(12,300)NSD,NSC,NSS 300 FORMAT(' ',/,' ','No. const. density values - ',I3,/, &X,'No. const. conductivity values - ',I3,/, &X,'No. const. specific heat values - ',13) WRITE(12,305)ER1DD,ER1DC,ER1DS,ER2DD,ER2DC,ER2DS 305 FORMAT(' ',/,' ','Error 1D dens. - ’,F6.4,' Error 1D cond. - ', &F6.4,/' ','Error 1D sp. heat - ',F6.4,' Error 2D density - ',F6.4, &/,X,'Error 2D cond. - ',F6.4,' Error 2D sp. heat - ',F6.4) WRITE(12,306)BSTEP 306 FORMAT(' ','Step increment - ',f8.5) CLOSE(UNIT-12) 281 Table B.2 (cont'd). ENDIF C Store results in data file C Results for density, thermal conductivity and specific heat C approximations are stored in 'TTLFILprp.dat' WRITE(FILDAT,310)tt1fil,'PRP.DAT' 310 FORMAT(' ',a,A) OPEN(UNIT-12,NAME=FILDAT(1:16),TYPE='NEW',CARRIAGECONTROL-'LIST') WRITE(12,*)WFO,T0,MS WRITE(12,*)DP,KP,CP WRITE(12,*)NSD,NSC,NSS write(12,*)tl,th,avgd,avgk,avgc,ynavg DO I-1,NSD WRITE(12,*)DENST(I),DENSC(I) ENDDO DO I-1,NSC WRITE(12,*)CONDT(I),CONDC(I) ENDDO DO I-1,NSS WRITE(12,*)SPHT(I),SPHC(I) ENDDO CLOSE(UNIT-l2) RETURN END SUBROUTINE INTEGR(thi,tlow,avgdp,avgkp,avgcp,ncase) integer np,ncase double precision eta(25),w(25),thi,tlow,tavg,tdiff,avgdp, &avgkp,avgcp,x(25), &dens,conduc,spheat external dens,conduc,spheat c Ncase - 0: 20pt. quad for density, conductivity, sp.heat over tHi - tLo c Ncase - 1: 5pt. quad for density c Ncase - 2: 5pt. quad for thermal conductivity c Ncase - 3: 5pt. quad for specific heat 0 Determine mean property value over specified range using Gauss Quadrature integration: 0 if(ncase.eq.0)then 0 20 pt. quad. Table B.2 (cont'd). C C C C C np - 20 Eta values: eta(l) eta(2) eta(3) eta(4) eta(S) eta(6) eta(7) eta(8) eta(9) -0 -0 -0. -0. -0. -0. -0. 99312859918509 96397192727791 91223442825133 83911697182222 74633190646015 .63605368072652 —0. -0. 51086700195083 37370608871542 22778585114165 282 eta(lO) - -0.07652652113350 do i - 1,10 eta(21-i) - ~eta(i) enddo Weighting factors: W(1) W(2) W(3) W(4) W(5) W(6) W(7) W(8) W(9) 000000000 .01761400713915 .04060142980039 .06267204833411 .08327674157670 .10193011981724 .11819453196152 .13168863844918 .14209610931838 .14917298647260 w(10) - 0.15275338713073 do i - 1,10 w(21-i) - w(i) enddo else 5 pt. quad. np - 5 Eta values: eta(l) eta(2) eta(3) eta(4) eta(S) -0.90617984593866 -0.53846931010568 0.0 -eta(2) -eta(1) Weighting factors: W(1) W(2) W(3) W(4) W(5) 0. 23692688505619 0.47862867049937 0. W(2) w(1) 56888888889 283 Table B.2 (cont'd). C C 0 0000 endif Transform eta onto (th - t1) interval tavg - (thi+tlow)/2.0d0 tdif - (thi-tlow)/2.0d0 andp - 0 avgkp - 0 avgcp - 0 do i - 1,np x(i) - tavg + tdif*eta(i) Sum integral approximation if(ncase.eq.0.or.ncase.eq.1)avgdp if(ncase.eq.0.or.ncase.eq.2)avgkp if(ncase.eq.0.or.ncase.eq.3)avgcp enddo if(ncase. if(ncase. if(ncase. return end avgdp + w(i)*dens(x(i)) avgkp + w(i)*conduc(x(i)) avgcp + w(i)*spheat(x(i)) eq.0.or.ncase.eq.1)avgdp - 0.50*avgdp eq.0.or.ncase.eq.2)avgkp - 0.50*avgkp eq.0.or.ncase.eq.3)avgcp - 0.50*avgcp BLOCK DATA CONST the following values are defined in this block data: di- dw- kw- cpi- cpw— density of ice (917. kg/m23) } block density of water (998. kg/m‘3)} /densi/ thermal conductivity of water } block (0.569 w/mk) } /cond/ specific heat of ice } (2.1 kj/kgk) } block specific heat of water } /sph/ } (4.187 kj/kgk) double precision DI,DW,KW,CPI,CPW COMMON /DENSI/DI,DW,/COND/KW,/SPH/CPI,CPW SAVE /DENSI/,/COND/,/SPH/ DATA DI,DW/9l7.0d0,998.0d0/,KW/0.5690dO/, &CPI,CPW/2.10d0,4.1870d0/ END 284 Table B.2 (cont'd). 0000 0000 SUBROUTINE INPUTl This subroutine provides the input for the boundary condi- tions on the product for the case where the ambient temper- ature and surface heat tranfer coeffient are known and assumed to be constant over a given storage period. Input varibles include, initial product temperature, sym— metry of boundary conditions, number of constant temperature storage periods, length of storage period, and surface heat transfer coefficient. parameter(maxp-ZO) integer ct(maxp),cct(maxp),per,sym,sstep,unit,shape,m, &htype(maxp) double precision ti,temp(maxp),stor(maxp),h1(maxp),h2(maxp), &tunit(maxp),h,l,dz,per1,per2,ampl,amp2 character yn*1,title*20,ttlfil*4,fildat*12,inpdat*12 logical itmode common/bound/per,ti,temp,stor,h1,h2,tunit, &/geom/shape,h,l,dz,sym,m,mp1,sstep/ttl/title,ttlfil,/mod/model, &/itm/itmode,/datfil/fildat,inpdat,kindat S ave if(itmode)go to 3 read*,per,ti,sym do i - 1,per read*,temp(i),unit,stor(i),hl(i),h2(i) if(unit.eq.l)then tunit(i)-3600.0d0 else tunit(i)-86400.0d0 endif enddo read shape if(shape.lt.3)then read*,l h - 1.0d0 else read*,l h - 0.0d0 endif go to 500 write(5,1) 1 format('l',72('-'),/,'0',27x,'Storage Conditions',/,’0',72('-')) write(6,10) 285 Table B.2 (cont'd). 10 format(' ',/,' ','Enter number of constant temp. storage ', &'periods: ',$) read*,per write(6,20) 20 format(' ',/,' ','Initial product temperature (C): ',$) read*,ti write(6,30) 30 format(' ',/,' ','Are the boundary conditions symmetrical? ', &'(0-No,l-Yes) ',/,' ','(Enter "1" for cylinder & sphere ', &'geometries) ',$) read*,sym write(6,40) 40 format(' ',/,' ','Are these values correct? (y/n) ',$) read(5,2)yn 2 format(a) if(yn.ne.'y'.and.yn.ne.'Y')goto 5 c input boundary conditions for each storage period do 120 i-1,per 45 write(6,50)i 50 format(' ',/,' ','Enter data for period ',i3,':',$) write(6,60) 60 format(' ',5x,'Storage temperature (C): ',$) read*,temp(i) 70 write(6,75) 75 format(' ',5x,'Enter units for storage temp.:',/,' ',7x,'1-= hours' &./.' '.7x.’2- day8') read*,unit if(unit.1e.0.and.unit.gt.2)then print*,'try againl' goto 70 endif if(unit.eq.l)then tunit(i)-3600.0d0 else tunit(i)-86400.0d0 endif 77 write(6,80) 80 format(' ',5x,'Length of storage period: ',$) read*,stor(i) if(stor(i).lt.0)then print*,'try again!’ goto 77 endif write(6,82) 82 format(' ',5x,'Enter curve type of heat transfer coefficient:', &/,' ',7x,'1. constant',/,' ',7x,'2. sinusiodal') read*,htype(i) if(htype(i).eq.1)then write(6,90) 90 format(' ',5x,'Enter surface heat transfer coefficient (W/m“2C):', &/,' ',7x,'side l - ',$) read*,hl(i) if(sym.ne.1)then write(6,100) 286 Table B.2 (cont'd). 100 format(' ',7x,'side 2 - ',$) read*,h2(i) else h2(i)-0.0d0 endif else write(6,102) 102 format(' ',5x,'Enter amplitude (C) and period (hrs) of sinusiodal', &' curve for h (side 1) : ',$) read*,amp1,per1 if(sym.ne.1)then write(6,104) 104 format(' ',7x,'side 2 - ',$) read*,amp2,per2 else amp2-0.0d0 per2-0.0d0 h2(i) - 0.0d0 endif endif 110 write(5,115) 115 format(' ',/,' ','Are these values correct? (y/n) ',$) read(5,2)yn if(yn.ne.'y'.and.yn.ne.'Y')goto 45 120 continue c input geometry and size 140 write(6,150) 150 format('O','Enter product geometry: ',/,' ',5x,'1 - slab',/,' ',Sx +,'2 - cylinder',/,' ',5x,'3 - sphere') read*,shape if(shape.gt.l.and.sym.eq.0)then print*,'Boundary conditions must be symmetrical for cylinder and', &' sphere; try again!!' go to 3 endif if(shape.eq.1)then write(6,160) 160 format(' ',/,' ','Enter dimensions for slab:',/,' ',Sx, &'thickness in direction of heat transfer (m) - ',$) read*,l h - 1.0d0 else if(shape.eq.2)then _ write(6,180) 180 format(' ',/,' ','Enter dimensions for cylinder',/, +' ',5x,'radius (m)- ',$) read *,1 h - 1.0d0 else write(6,200) 200 format(' ',/,' ','Enter dimensions for sphere (m)',/, +' ',5x,'radius (m)- ',$) read *,1 287 Table B.2 (cont'd). 500 590 600 700 710 800 810 900 1000 h-0.0d0 endif endif ti - ti+273.150d0 if(sym.eq.l.and.shape.eq.l)L - L*0.50d0 do i - 1,per temp(i) - temp(i)+273.150d0 stor(i)-stor(i)*tunit(i) if(sym.eq.1.or.shape.ne.1)then if(shape.eq.1)then h2(i) - 0.0d0 else h2(i) - hl(i) hl(i) - 0.0d0 endif endif enddo if(1.ge.0.80d0)then m-40 sstep-lO go to 590 endif do i-1,10 if(l.lt.i*0.080d0)then m-i*4 sstep-i go to 590 endif enddo write(inpdat,600)ttlfil,'inp.dat' format(' ',a,a) open(unit-12,name-inpdat(1:12),type-'new',carriagecontrola'list') write(12,700)per,sym,ti format(' ',12,2x,il,2x,f6.2) do i - 1,per write(12,710)htype(i) format(' ',il) if(htype(i).eq.1)then write(12,800)temp(i),stor(i),tunit(i),hl(i),h2(i) format(' ',2x,f6.2,2x,fl8.2,2x,f6.0,2(2x,f8.2)) else write(12,810)temp(i),stor(i),tunit(i),amp1,perl,amp2,per2 format(' ',2x,f6.2,2x,f18.2,2x,f6.0,4(2x,f8.2)) endif enddo write(12,900)shape,L,h format(' ',il,2(2x,f8.4)) write(12,1000)m,sstep format(' ',12,2x,12) close(unit-12) return end 288 Table B.2 (cont'd). SUBROUTINE INPUT2 Kinetic properties to determine quality loss in a food product are entered in this subroutine. A file titled 'TTLFILkin.dat' containing the kinetic properties is created. This file is re- opened in the solution, and it may be reused again in subsequent runs. 00000 integer model double precision q0,tref,ea,vea,vq0 character title*20,tt1fil*4,fildat*12,inpdat*12,kindat*12 logical itmode common /ttl/tit1e,tt1fi1,/mod/model,/itm/itmode, &/datfil/fildat,inpdat,kindat c Read batch file data (if itmode - .false.) if(itmode)go to 1 read*,q0,tref,ea if(model.eq.4)then read*,vea,vq0 endif go to 30 c Read interactive input 1 write(5,2) 2 format('l',72('-'),/,'0',t23,'Reference Shelf-life data',/,'0', &72('-')) 10 write(5,100) 100 format(' ',/,' ','Enter reference shelf-life (days) : ',$) read*,qO write(5,200) 200 format(' ',/,' ','Enter reference temperature for reference', &' shelf-life (C) : ',$) read*,tref write(5,300) 300 format(' ',/,' ','Enter activation energy constant (kJ/mole) &.$) read*,ea if(model.eq.4)then write(5,400) 400 format(' ',/,' ','Enter standard deviation of activation ', &'energy const. (kJ/mole) : ',$) read*,vea write(5,500) 500 format(' ',/,' ','Enter standard deviation of ref. shelf-life’, &' (dayS) : ',$) read*,qu endif 289 Table B.2 (cont'd). write(5,550) 550 format(' ',/,' ','Are these values correct? (y/n) ’,$) read(5,20)yn 20 format(a) if(yn.eq.'n'.or.yn.eq.'N')go to 10 30 tref - tref+273.150d0 ea - ea*1000.0d0 if(model.eq.4)then vea - (vea*1000.0d0)**2.0d0 qu - vq0*vq0 endif write(kindat,600)tt1fil,'kin.dat' 600 format(' ',a,a) open(unit-12,name-kindat(1:12),type-'new',carriagecontrol-‘list') write(12,700)q0,tref,ea 700 format(' ',2x,f8.l,2x,f7.2,2x,f13.0) if(model.eq.4)then write(12,800)vea,vq0 800 format(' ',2(2x,e11.3)) endif close(unit-12) return end SUBROUTINE SOLN parameter(maxd - 101, maxc - 51, maxs - 201) parameter(maxm-lOl,maxp-20,tol-0.10d0,r-8.3140d0) integer per,shape,m,sym,sstep,htype(maxp) double precision wf0,ms,dp,kp,cp,t0,ea,q0,tref,vea,vq0, &h,1,dz,ti,temp(maxp),stor(maxp),hl(maxp),h2(maxp),tunit(maxp), &jj,kjj,eabs,eeabs,ssum,HH1(2),HH2(2),ta(2),tavg,qavg, &qua1(maxm),dsum,vsum,eex,avdl,dqdea,ct(maxp),cct(maxp),dq(2), &vq,DENST(maxd),DENSC(maxd),CONDT(maxc),CONDC(maxc), &SPHT(maxs),SPHC(maxs),cc(maxm),dd(maxm),a(maxm),b(maxm), &c(maxm),d(maxm),t(maxm,2),dt,ds,tl,th,avgd,avgk,avgc,ynavg, &tdt,amp1,per1,amp2,per2,pi character tit1e*20,tt1fil*4,fildat*12,inpdat*l2,kindat*12 logical itmode common/bound/per,ti,temp,stor,h1,h2,tunit,/tt1/title,ttlfil &,/geom/shape,h,l,dz,sym,m,mp1,sstep,/mod/model,/itm/itmode, &/datfil/fildat,inpdat,kindat,/NCONSTP/NSD,NSC,NSS,/dff/dif, &/CONSTP/DENST,DENSC,CONDT,CONDC,SPHT,SPHC,/prop/wf0,ms,dp, 290 Table B.2 (cont'd). &kp,cp,t0,/she1f/ea,q0,vea,vq0,tref,/d/ds &/pavg/th,t1,avgd,avgk,avgc,ynavg,/toldt/tdt save c Read in boundary and initial conditions write(inpdat,600)ttlfil,'inp.dat' 600 format(' ',a,a) open(unit-12,name-inpdat(1:12),type-'old',carriagecontrol-'list') read(12,*)per,sym,ti do i - 1,per read(12,*)htype(i) if(htype(i).eq.1)then read(12,*)temp(i),stor(i),tunit(i),h1(i),h2(i) else read(12,*)temp(i),stor(i),tunit(i),amp1,per1,amp2,per2 endif enddo C Input geometry and dimensions read(12,*)shape,L,H read(12,*)m,sstep close(unit-12) c Read in constant property assumptions WRITE(FILDAT,310)TTLFIL,'PRP.DAT' 310 FORMAT(' ',A,A) OPEN(UNIT-12,NAME-FILDAT(1:12),TYPE-'OLD',CARRIAGECONTROLP'LIST') READ(12,*)WFO,T0,MS READ(12,*)DP,KP,CP READ(12,*)NSD,NSC,NSS read(l2,*)tl,th,avgd,avgk,avgc,ynavg DO I-1,NSD READ(12,*)DENST(I),DENSC(I) ENDDO DO I-1,NSC READ(12,*)CONDT(I),CONDC(I) ENDDO DO I-1,NSS READ(12,*)SPHT(I),SPHC(I) ENDDO CLOSE(UNIT-l2) c Read in kinetic data if(model.ge.3)then write(kindat,600)tt1fil,'kin.dat' open(unit-12,name-kindat(1:12),type-'old',carriagecontrol='list') read(12,*)q0,tref,ea if(model.eq.4) read(12,*)vea,vq0 close(unit-12) endif 291 Table B.2 (cont'd). pi - dacos(-l.0d0) dt - 120 dz - L/m mp1 - m+1 DO k - 1,2 TA(k)-TEMP(1) if(htype(l).eq.1)then HHl(k)-Hl(1) HH2(k)-H2(1) else if(amp1.ne.0)then hh1(k) - ampl*cos(2*pi*(k-1)*dt/(perl*3600)) else hh1(k) - 0 endif if(amp2.ne.0)then hh2(k) - amp2*cos(2*pi*(k-l)*dt/(per2*3600)) else hh2(k) - 0 endif endif DO I - 1,mp1 t(I,k)-ti enddo enddo tavg - ti time-0 count-0 JJ-O JJJ-O DO I - 1,mp1 IF(MODEL.GE.3)THEN QUAL(I)-Q0*86400.0d0 ELSE QUAL(I) - 0.0d0 ENDIF ENDDO qavg - q0*86400.0d0 if(model.eq.4)then dq(l) - 0.0d0 dq(2) - 0.0d0 qu-vq0*86400.0d0 vq-qu endif nprint - 0 IF(MODEL.LT.3)THEN HEADTQ-l else headtq-2 endif call output(nprint,headtq,t,tavg,time,jj,qual,qavg,vq,ii,eend,dt) j-l c finite difference solution Table C 292 B.2 (cont'd). do 160 ii-1,per eend—O if(ii.ne.l)then time-time+stor(ii-1) nprint - 2 call output(nprint,headtq,t,tavg,time,jj,qual,qavg,vq,ii,eend,dt) endif JJ-O JJj-O c check if time is > length of storage period 65 if(jj.ge.stor(ii))goto 155 c check if product temp. is close to ambient temp. 60 eabs-abs(t(1,2)-temp(II)) do 60 i-2,mp1 eeabs-abs(t(i,2)-temp(ii)) if(eabs.lt.eeabs)then eabs-eeabs endif continue if(eabs.lt.tol)goto 105 dtmax - stor(ii)-jj C set count-count+1 ambient temperature - storage temperature ta(2)-temp(ii) c boundary conditions If(htype(ii).eq.1)then hhl(2)-h1(ii) hh2(2)-h2(ii) else if(amp1.ne.0)then hhl(2) - ampl*cos(2*pi*(j-1)*dt/(per1*3600)) else hhl(2) - 0 endif if(amp2.ne.0)then hh2(2) - amp2*cos(2*pi*(j-1)*dt/(per2*3600)) else hh2(2) - 0 endif endif 293 Table B.2 (cont'd). c thomas algorithm c find coefficients for thomas algorithm call coeff(ii,hhl,hh2,ta,dtmax,t,a,b,c,d,dt) CC(1)-c(1)/b(1) dd(1)-d(1)/b(1) do k-2,mp1 kk-k-l cc(k)-c(k)/(b(k)-a(k)*cc(kk)) dd(k)-(d(k)-a(k)*dd(kk))/(b(k)-a(k)*CC(kk)) enddo t(mp1,2)-dd(mpl) tavg - t(mp1,2) do k-2,mp1 kk-m-k+2 t(kk,2)-dd(kk)-cc(kk)*t(kk+1,2) tavg - tavg+t(kk,2) enddo tavg - tavg/mp1 JJ - JJ+dt C c find quality distribution and adjust time step if(model.ge.3)then dsum-O vsum-O endif ssum—O qavg - 0 do 85 i-1,mp1 if(model.ge.3)then eex-1.0d0/t(i,2)-l.0d0/tref d1-exp(-(ea/r*eex)) qual(i)-qua1(i)-d1*dt qavg - qavg+qual(i) if(model.eq.4)then dqdea—d1*eex*dt/r vsum~vsum+dl*eex*dt/r endif dsum-dsum+d1 endif 85 continue c find mass average quality if(model.ge.3)then qavg - qavg/mp1 if(model.eq.4)then dqdea-vsum/(mpl) dq(2)-dq(1)+dqdea vq-vq0+dq(2)**2*vea dq<1)-dq(2> 294 Table B.2 (cont'd). endif endif count-count+l if(count.ge.60)then c printout nprint - 1 call output(nprint,headtq,t,tavg,time,jj,qual,qavg,vq,ii,eend,dt) count-0 endif C c initial t for next time step do 100 i-1,mpl 100 t(i,1)-t(i,2) ta(1)-ta(2) hhl(1)-hh1(2) hh2(l)-hh2(2) J-J+1 JJJ-JJJ+1 goto 65 c end of finite difference calculations C ********************************************************* c set product temp. - ambient temperature; determine quality 105 tavg - 0.0d0 do 110 i-1,mp1 do 110 ji-l,2 t(i,ji)-temp(ii) 110 continue tavg - temp(ii) cct(ii)-jjj+1 nx-4 dt-(stor(ii)-jj)/nx do 130 ij-1,nx JJ-JJ+dt if(model.ge.3)then eex-1.0d0/temp(ii)-1.0d0/tref d1-exp(-(ea/r*eex)) qavg - 0.0d0 do i-l,mp1 qua1(i)-qua1(i)-d1*dt qavg - qavg+qual(i) enddo qavg-qavg/mpl if(model.eq.4)then dqdea-d1*eex*dt/r dq(2)-dq(l)+dqdea vq-vq0+dq(2)**2*vea 295 Table B.2 (cont'd). dQ(1)-dQ(2) endif endif c printout 130 140 150 155 160 nprint - 1 call output(nprint,headtq,t,tavg,time,jj,qual,qavg,vq,ii,eend,dt) J-J +1 JJJ'JJJ+1 count-0 continue ct(ii)-jjj+l do 150 i-1,mpl t(i,1)-t(i,2) ta(l)-ta(2) hhl(l)-hh1(2) hh2(l)-hh2(2) if(count.ne.0)then nprint - 1 call output(nprint,headtq,t,tavg,time,jj,qual,qavg,vq,ii,eend,dt) endif count-0 if(ii.eq.per)then eend-l nprint - 2 call output(nprint,headtq,t,tavg,time,jj,qual,qavg,vq,ii,eend,dt) endif continue return end SUBROUTINE COEFF(ii,hh1,hh2,ta,dtmax,t,a,b,c,d,dt) parameter(maxm-lOl,maxp-20,maxd-101,maxc-51,maxs—201) integer shape,m,mpl,ii double precision beta,nu,omega,gama,hhl(2),hh2(2), &aar,ar(maxm),ar1(maxm),area,avgl,avg2, &da,db,dc,ddd,ta(2),DENST(maxd),DENSC(maxd), &CONDT(maxc),CONDC(maxc),SPHT(maxs),SPHC(maxs),ck(maxm), &csd(maxm,2),a(maxm),b(maxm),c(maxm),d(maxm),t(maxm,2), &dtmax,dt,pi,dzz,dtt,wf0,ms,dp,kp,cp,t0,h,l,dz,ds, &th,t1,avgd,avgk,avgc,ynavg,tdt common/geom/shape,h,1,dz,sym,m,mpl,sstep,/prop/wf0,ms,dp,kp, &cp,t0,/CONSTP/DENST,DENSC,CONDT,CONDC,SPHT,SPHC,/dff/dif, &/NCONSTP/NSD,NSC,NSS,/d/ds,/pavg/th,tl,avgd,avgk,avgc,ynavg, 296 Table B.2 (cont'd). &/toldt/tdt pi - dacos(-1.0d0) c weighting functions for finite difference method c modified crank-nicolson method c weight. coeff. for d2t/d22 c for time t: beta-0.50d0 c for time t+1: nu-0.50d0 c weight. coeff. for dt/dt c for time t: omega-~1.0d0 c for time t+l: gama-1.0d0 q1-0.0d0 q2-0.0d0 dzz-l.0d0/dz if(shape.eq.2)then aar-2.0d0*pi*h else if(shape.eq.3)then aar-4.0d0*pi endif endif do 10 i-1,mp1 c slab if(shape.eq.1)then ar(i)-h ar1(i)-h else c cylinder if(shape.eq.2)then ar(i)-aar*(i-1)*dz ar1(i)-ar(i)+aar*dz/2.0d0 else 0 sphere ar(i)-aar*((i-1)*dz)**2.0d0 arl(i)-aar*((i-l)*dz+dz/2.0d0)**2.0d0 endif endif 297 Table B.2 (cont'd). 10 continue CALL PFIND(T,M,CK,CSD,DT,DZ,dtmax) C **********:k*********************************************** c lst boundary point AVGl - (AR(1)+AR1(1))*0.50d0 a(1)-0.0d0 c(1)-nu*dzz*CK(1)*ar1(1) dc--beta*dzz*CK(1)*ar1(1) b(1)--gama*CSD(1,1)*avg1-nu*hh1(2)*ar(1)-c(1) db-omega*CSD(1,1)*avgl+beta*hh1(l)*ar(l)-dc ddd--beta*(ta(1)*hh1(l)*ar(1)+ql)-nu*(ta(2)*hh1(2)*ar(l)+q1) d(1)-db*t(l,1)+dc*t(2,1)+ddd c ********************************************************** c c interior points do 20 i-2,m AVGl - (AR(I)+AR1(I))*0.50d0 AVG2 - (AR(I)+AR1(I-1))*0.50d0 a(i)-nu*dzz*CK(I-1)*ar1(i-1) da--beta*dzz*CK(I-1)*ar1(i-l) c(i)-nu*dzz*CK(I)*ar1(i) dc--beta*dzz*CK(I)*ar1(i) b(i)--gama*(CSD(I,1)*avg2+CSD(I,2)*avgl)-a(i)-c(i) db-omega*(CSD(I,1)*avg2+CSD(I,2)*avg1)-da-dc d(i)-da*t(i-1,1)+db*t(i,1)+dc*t(i+l,1) 20 continue c *********************************************************** c 2nd boundary point AVG2 - (AR(mp1)+AR1(M))*O.50d0 c(mp1)-0.0d0 a(mp1)-nu*dzz*CK(M)*ar1(m) da--beta*dzz*CK(M)*arl(m) b(mp1)--gama*CSD(mp1,2)*avg2-nu*hh2(2)*ar(mp1)-a(mpl) db-omega*CSD(mpl,2)*avg2+beta*hh2(l)*ar(mpl)-da ddd--beta*(hh2(1)*ta(1)*ar(mp1)+q2)-nu*(ta(2)*hh2(2)*ar(mpl) &+q2) 298 Table B.2 (cont'd). d(mpl)-da*t(m,1)+db*t(mp1,l)+ddd re turn end SUBROUTINE PFIND(T,M,CK,CSPD,DT,DZ,dtmax) PARAMETER (MAXm-lOl, MAXC-Sl, MAXD=101, MAXS-201) INTEGER NC(8),NSC,NSD,NSS double precision TAVGK,TAVGSD(2),CK(MAXM),CSPD(MAXM,2), &kc,DC(2),SPC(2),t(maxm,2),dt,dtmax,dz double precision CONDT(MAXC),CONDC(MAXC),DENST(MAXD), &DENSC(MAXD),SPHT(MAXS),SPHC(MAXS), &wf0,ms,dp,kp,cp,t0,ds, &th,tl,avgd,avgk,avgc,ynavg, &dens,conduc,spheat COMMON/CONSTP/DENST,DENSC,CONDT,CONDC,SPHT,SPHC, &/NCONSTP/NSD,NSC,NSS,/prop/wf0,ms,dp,kp,cp,t0,/d/ds, &/pavg/th,t1,avgd,avgk,avgc,ynavg/toldt/tdt external dens,conduc,spheat MP1 - M+1 eigen(1) - 0. emax - 1.0e10 cc - 1000.0d0*dz/2.0d0 cl - 1.0d0/(cc*dz) DO 100 I - 1,mp1 if(ynavg.eq.l.0)then ckl - 3.0d0*avgk ck(i) - avgk do iii - 1,2 spc(iii) - avgc dc(iii) - avgd enddo go to 90 endif DO KK - 1,5 NC(KK) - 0 ENDDO IF(I.LE.M)THEN TAVGK - (T(I,l)+T(I+1,1))*0.50d0 TAVGSD(1) - 0.750d0*T(I,1)+0.250d0*T(I+l,l) ENDIF IF(I.GT.1)THEN TAVGSD(2) - 0.750d0*T(I,1)+0.250d0*T(I-1,1) 299 'Table B.2 (cont'd). ENDIF DO 10 J - 2,NSC+1 IF(NC(1).EQ.1)go to 10 IF(I.Eq.Mp1)go to 10 if(tavgk.ge.t0)then ck(i) - kp nc(1) - 1 else if(tavgk.ge.t0-4.0d0)then ck(i) - conduc(tavgk) nc(1) - 1 else IF(TAVGK.LE.CONDT(J))THEN ck(i) - CONDC(J-1) NC(l) - 1 endif endif endif 10 continue if(i.gt.1)then ckl - ck(i-1)+ck(i)+(ck(i-1)*ck(i))**0.5 endif DO 40 J - 2,NSD+1 DO 30 RR - 1,2 IF(NC(KK+2).EQ.1)go to 30 IF(I.EQ.mp1.AND.KK.eq.1)GO TO 30 IF(I.EQ.1.AND.KK.eq.2)GO TO 30 if(tavgsd(kk).ge.t0)then dc(kk) - dp nc(kk+2) - 1 else if(tavgsd(kk).ge.t0-4.0d0)then dc(kk) - dens(tavgsd(kk)) nc(kk+2) - 1 else IF(TAVGSD(KK).LE.DENST(J))THEN DC(KK) - DENSC(J-l) NC(KK+2) - 1 ENDIF - endif ENDIF 30 CONTINUE 40 CONTINUE DO 60 J - 2,NSS+1 DO 50 KR - 1,2 IF(NC(KK+5).EQ.1)go to 50 IF(I.EQ.mp1.AND.KK.eq.1)GO TO 50 IF(I.EQ.1.AND.KK.eq.2)GO TO 50 if(tavgsd(kk).ge.t0)then spc(kk) - cp nc(kk+5) - 1 else if(tavgsd(kk).ge.t0-4.0d0)then Table 50 60 90 100 300 B.2 (cont'd). spc(kk) - spheat(tavgsd(kk)) nc(kk+5) - 1 else IF(TAVGSD(KK).LE.SPHT(J))THEN SPC(KK) - SPHC(J-l) NC(KK+5) - 1 ENDIF endif ENDIF CONTINUE CONTINUE CONTINUE CONTINUE do i - 1,mp1 DO KK - 1,2 CSPD(I,KK) - SPC(KK)*DC(KK)*cc/dt ENDDO enddo RETURN END SUBROUTINE OUTPUT(nprint,headtq,t,tavg,time,jj,qual,qavg,vq, &ii,eend,dt) parameter(maxp-20,maxm-lOl) integer per,shape,model,sym,sstep,m,eend,day,dead double precision wf0,ms,dp,kp,cp,t0,ea,q0,tref,vea,vq0, &h,l,dz,ti,temp(maxp),stor(maxp),h1(maxp),h2(maxp),tunit(maxp), &abc(5),hr,c7,time,jj,c8,t(maxm,2),qual(maxm),abcd,qavg,vq,tavg &,tavg1,dt character tit1e*20,ttlfil*4,outfi1*12,hh11*29,hh22*21 common/ttl/title,ttlfil/mod/model/shelf/ea,qO,vea,vq0,tref &/prop/wf0,ms,dp,kp,cp,t0/bound/per,ti,temp,stor,h1,h2,tunit &/geom/shape,h,l,dz,sym,m,mpl,sstep COCO NPRINT - 0 if printing input parameters and headings NPRINT - 1 if printing temperature distribution and/or quality distributions NPRINT - 2 if printing period no. and end line IF(NPRINT.EQ.0)THEN GO TO 1100 ELSE if(nprint.eq.l)then 301 Table B.2 (cont'd). go to 1200 else go to 1300 endif endif 1100 write(outfil,1000)tt1fil,'out.dat' 1000 format(' ',a,a) open(unit-12,name-outfil(1:12),type-‘new',carriagecontrol-'list') write(12,l)title l format(' ',///,3x,'Title: ',a20,/3x,' ----- ',//,14x,'Input Para', +'meters',/,14x,l6('-')//) if(model.ge.3)then write(12,3) 3 format(' ','Kinetic Parameters') write(12,4)q0 4 format(' ',/,' ',2x,'Reference shelf-life (days) ............ ', &f7.l) abcd-tref-273.150d0 write(12,5)abcd 5 format(' ',2x,'Reference temperature (C) .............. ',f6.2) abcd-ea/1000.0d0 write(12,6)abcd 6 format(' ',2x,'Activation energy constant (kJ/mole)...',f8.2) abcd-vq0**0.50d0 if(model.eq.4)then write(12,8)abcd 8 format(' ',2x,'St. dev. of ref. shelf-life (days) ..... ',f6.2) abcd-vea**0.50d0/1000.0d0 write(12,9)abcd 9 format(' ',2x,'St. dev. of ea (kj/mole) ............... ',f6.2) endif endif write(12,10) 10 format(' ',/,' ','Unfrozen Product Properties',/) abcd-wf0*100.0d0 write(12,11)abcd 11 format(' ',2x,'Moisture content (%) ................... ',f6.2) abcd-t0-273.150d0 write(12,12)abcd 12 format(' ',2x,'Initial freezing temperature (C) ....... ',f6.2) write(12,13)ms 13 format(' ',2x,'Molecular weight of solids (kg/mole)...',f8 2) write(12,14)dp 14 format(' ',2x,'Unfrozen product density (kg/m‘3) ...... ',f8.2) write(12,15)kp 15 format(' ',2x,'Thermal conductivity (W/mK) ............ ',f6.3) write(12,16)cp 16 format(' ',2x,'Specific heat (kJ/kgK) ................. ',f7.3) abcd-ti-273.150d0 write(12,17)abcd 17 format(' ',/,' ','Initial Condition:',/,' ',2x,'Product temp.’ +,' (C) at time-0 ........... ',f6.2) c product geometry 302 Table B.2 (cont'd). if(shape.eq.1)then if(sym.eq.1)l - l*2.0d0 write(12,18)l 18 format( ' ',/,' ','Slab Geometry:',/,' ',2x,'thickness (m)', +26('.'),f10.6) else if(shape.eq.2)then write(12,20)1 20 format(' ',/,' ','Cyclindrical Geometry:',/,' ',2x,'radius (m)', +29('.'),f10.6) else write(12,22)l 22 format(' ',/,' ','Spherical Geometry:',/,' ',2x,'radius (m)', +29('.'),f10.6) endif endif c boundary conditions do 40 i-l,per write(12,24)i 24 format(’ ',/,' ','boundary conditions for period ',i2,':',/) abcd-stor(i)/tunit(i) if(tunit(i).eq.3600.0d0)then write(12,25)abcd 25 format(' ',4x,'storage time(hours) .......... ',f7.2) else write(12,26)abcd 26 format(' ',4x,'storage time (days) .......... ',f7.2) endif abcd-temp(i)-273.150d0 write(12,27)abcd 27 format(' ',4x,'storage temperature (C) ...... ',f6.l) write(12,28) 28 format(' ',4x,'convective heat transfer coeff. (W/m“2K):') if(shape.eq.1)then write(12,29)h1(i) 29 format(' ',6x,'side 1-',f7.2) if(sym.ne.1)then write(12,30)h2(i) 30 format(' ',6x,'side 2-',f7 2) endif else write(12,35)h2(i) 35 format(' ',6x,'at surface— ',f7.2) endif 40 continue write(12,45)dt 45 format(4x,’Time step - ',f6.2) write(12,100)tit1e 100 format(' ',/////,' ','Title- ',a20,/) if(sym.eq.1)then write(12,110) 303 Table B.2 (cont'd). 110 format(' ','Note: Distribution is symmetrical;'/,6x,'results', +' are shown for half-thickness only.'/) endif hh22-'DISTRIBUTION HISTORY' if(headtq.eq.l)then hh11-' TEMPERATURE (C) ' else hhll-'TEMPERATURE (C) & QUALITY (%)' endif write(12,120)hh11,hh22 120 format(' ',/,' ',19x,a,/,23x,a,/,19x,27('-'),/) if(model.lt.3)then write(12,130) 130 format(' ',28x,'position (m)',/' ',5x,'time',5x,':',42x, &'Avg Temp') else write(12,135) 135 format(' ',28x,'position (m)',/' ',5x,'time',5x,':',42x, &'Avg Temp Qual.') endif do i - 1,5 abc(i)-(i-1)*sstep*dz enddo if(model.lt.3)then write(12,137)abc(1),abc(2),abc(3),abc(4),abc(5) 137 format(' ',4x,'hours :',5(f8.4)) else if(model.eq.3)then write(12,140)abc(1),abc(2),abc(3),abc(4),abc(5) 140 format(' ',4x,'days + hr :',5(f8.4),2x,'or Qual') else write(12,145)abc(l),abc(2),abc(3),abc(4),abc(5) 145 format(' ',4x,'days + hr :',5(f8.4),2x,'or Qual StD(%)') endif endif if(model.ne.4)then write(12,150) 150 format(' ',65('-')) else write(12,155) 155 format(72('-')) endif write(12,160) 160 format(' ','Period 1 :') c Printout time heading 1200 c7-86400.0d0 tavgl - tavg-273.150d0 do i - 1,5 abc(i)-t((i-1)*sstep+1,2)-273.150d0 enddo write(12,190)(time+jj)/3600,abc(1),abc(2),abc(3),abc(4),abc(5), 304 Table B.2 (cont'd). &tavg1 190 format(' ',f8.2,' hour:',6(f7.2,1x),'C') if(headtq.eq.2)then C Printout quality distribution 210 c8-100.0d0/(86400.0d0*q0) do i - 1,mp1 if(qual(i).lt.0)then dead-1 endif enddo do i - 1,5 abc(i)-qua1((i-1)*sstep+1)*c8 enddo if(model.eq.3)then write(12,210)abc(1),abc(2),abc(3),abc(4),abc(5),qavg*c8 format(' ',14x,':',6(f7.2,1x),'%') else write(12,215)abc(1),abc(2),abc(3),abc(4),abc(5),qavg*c8, &(vq)**0.50d0*c8 215 220 format(' ',l4x,':',6(f7.2,1x),1x,e7.1,'%') endif if(dead.eq.1)then write(12,220) format(' ',18x,'she1f-life has been exceeded') endif endif return c Printout end line 1300 300 305 310 if(model.ne.4)then write(12,300) format(' ',65('-')) else write(12,305) format(' ',72(’-')) endif if(eend.eq.0)then write(12,310)ii format(' ','period',13) else close(unit=12) endif return end APPENDIX C APPENDIX C TWO DIMENSIONAL TRANSIENT HEAT CONDUCTION AND QUALITY RETENTION PROGRAM The two dimensional transient heat conduction program, including estimation of quality retention, discussed in Chapter 3, is presented here. An outline of the program is given in Table C.1, and the listing for the program, written in Fortran 77 for a Vax 11/750 is given in Table C.2. 305 306 Table C.1 Description of Two Dimensional Transient Heat Conduction and Quality Retention Program. Subroutine Title PROGRAM FREEZE SUBROUTINE PROPER DOUBLE PRECISION FUNCTION MOIST(X) DOUBLE PRECISION FUNCTION DENS(X) DOUBLE PRECISION FUNCTION KI(X) DOUBLE PRECISION FUNCTION CONDUC(X) DOUBLE PRECISION FUNCTION SPHEAT(X) SUBROUTINE CONSPR SUBROUTINE INTEGR BLOCK DATA CONST SUBROUTINE INPUTl SUBROUTINE INPUT2 SUBROUTINE SOLN SUBROUTINE COEFFl SUBROUTINE COEFF2 SUBROUTINE THOMAL Description Main program; contains program menu. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. Allows interactive input of ambient conditions and product geometry. Writes output to data file. Allows interactive input of kinetic properties. Writes output to data file. Computes temperature distribution and quality retention as a function of temperature. Calls output subroutine. Determines matrix coefficients used in first sweep in ADI finite difference algorithm. Determines matrix coefficients used in second sweep in ADI finite difference algorithm. Solves Thomas Algorithm for inversion of tri- diagonal matrix. Table C.1 (cont'd). SUBROUTINE PFIND SUBROUTINE READING SUBROUTINE OUTPUT 307 Finds values for thermal properties required for ADI finite difference calculations from the property values determined in CONSPR. Writes input data to output file. Writes resulting temperature and quality retention values to output file. 308 Table C.2 Computer Code for Two Dimensional Transient Heat Conduction and Quality Retention Program. PROGRAM FREEZE c***************************************************************** c***************************************************************** c Residual Shelf-life Program c by c Elaine Scott c 1985 C***************************************************************** This program calculates the temperature and quality distri- bution histories of a two dimensional frozen food product subject to fluctuating ambient temperatures during storage below 0C. 0 0 0 0 Input parameters include unfrozen product density, thermal conductivity and specific heat. The initial freezing temperature or molecular weight of solids is required to predict these values for the frozen food product. 0 0 0 0 Boundary conditions are assumed to be convective, requiring an input of the ambient temperature as a function of time, and the convective heat transfer coeffient. The initial condition must be a known function of position. 0 0 0 0 c***************************************************************** parameter(maxp-20,maxd-lOl,maxc-51,maxs-201) integer model double precision wf0,ms,dp,kp,cp,t0,ds character title*20,ttlfil*4,filynl*l,filyn2*l,filyn*l,fildat*12, &inpdat*12 logical itmode common/mod/model,/itm/itmode,/tt1/tit1e,ttlfil,/profil/prpfil, &/datfil/fildat,inpdat,kindat,/prop/wf0,ms,dp,kp,cp,t0,/d/ds 309 Table C.2 (cont'd). C Set ITMODE - .FALSE. if running batch. ITMODE - .false. IF(ITMODE)THEN write(5,1000) 1000 format('l',72('*'),/,'0',t23,'Residual Shelf-life Program',/,'0', &t35,'by',/,'0',t30,'E1aine Scott',/,'0',t24,'Michigan State', &' University',/,'0',t30,'January l986',/,'0',72('*')) WRITE(5,100) 100 FORMAT('O','Program Menu:',//,' ',' 1. Product properties (<0C)’ &,/,' ',' 2. Temperature distribution history: known Ta and h', &/,' ',' 3. Temp. & qual. dist. histories: exact kinetic prop.', &/,' ',' 4. Temp. & qual. dist. hist.: random kinetic prop.', &/,'0',' Ta - Ambient temp.:h - Surface heat trans. coef.', &//,' ','Selection? ',$) ENDIF READ(5,10)mode1 10 FORMAT(Il) IF(ITMODE) write(5,200) 200 format(' ',/,' ','Product: ',$) READ(5,20)TITLE IF(ITMODE) write(5,300) 300 format(' ',/,' ','Key word for data files; 4 Characters: ',$) READ(5,20)TTLFIL 20 FORMAT(A) if(model.eq.l)then filynl - 'n' else if(itmode) write(5,400) 400 format(' ',/,' ','Are product properties approximations',/,' ',2x, &'with temperature stored on file? (y/n) ',$) read(5,20)filynl if(itmode) write(5,500) 500 format(' ',/,' ','Are input initial and boundary conditions',/,' ' &,2x,'and geometrical dimensions stored on file? (y/n) ',$) read(5,20)fi1yn2 if(model.ge.3) then if(itmode) write(5,600) 600 format(' ',/,' ','Are the kinetic properties stored on file? ', &' (y/n) ',$) read(5,20)filyn3 endif endif if(filynl.eq.'n'.or.fi1yn1.eq.'N')then call proper CALL CONSPR endif if(model.ne.1)then if(filyn2.eq.'n'.or.filyn2.eq.'N')then call inputl endif if(model.ge.3)then if(filyn3.eq.'n'.or.fi1yn3.eq.'N')then call input2 Table c See c See c See c See c See c See 310 C.2 (cont'd). endif endif call soln endif end SUBROUTINE PROPER Appendix B. DOUBLE PRECISION FUNCTION MOIST(X) Appendix B. DOUBLE PRECISION FUNCTION DENS(X) Appendix B. DOUBLE PRECISION FUNCTION KI(X) Appendix B. DOUBLE PRECISION FUNCTION CONDUC(X) Appendix B. DOUBLE PRECISION FUNCTION SPHEAT(X) Appendix B. 311 Table C.2 (cont'd). SUBROUTINE CONSPR c See Appendix B. SUBROUTINE INTEGR(thi,tlow,avgdp,avgkp,avgcp,ncase) c See Appendix B. BLOCK DATA CONST 0 See Appendix B. SUBROUTINE INPUTl This subroutine provides the input for the boundary condi- tions on the product for the case where the ambient temper- ature and surface heat tranfer coeffient are known and assumed to be constant over a given storage period. 0000 Input varibles include, initial product temperature, sym- metry of boundary conditions, number of constant temperature storage periods, length of storage period, and surface heat transfer coefficient. 0000 The variables used in this subroutine are: 0 parameter(maxp-ZO) integer per,symx,symy,stepx,stepy,unit,shape,ixt,iyt,cyn double precision ti,temp(maxp),stor(maxp),htc(maxp,4),tunit(maxp), &lx0,lx,1y character yn*1,tit1e*20,ttlfil*4,fildat*12,inpdat*12 logical itmode common /ttl/tit1e,ttlfil,/mod/model,/itm/itmode,/datfi1/fildat, &inpdat,kindat Table 50 60 70 80 90 100 110 312 C.2 (cont'd). S ave if(itmode)go to 10 read*,ti,shape,1x0,1x,1y,per do i - 1,per read*,temp(i),unit,stor(i) read*,htc(i,1),htc(i,2),htc(i,3),htc(i,4) if(unit.eq.l)then tunit(i)-3600.0d0 else tunit(i)-86400.0d0 endif enddo go to 500 write(6,20) format('l',72('-'),/,'0',27x,'Storage Conditions',/,'0',72('-')) write(6,40) format(' ',/,' ','Initial product temperature (C): ',$) read*,ti write(6,50) format('O','Enter product geometry: ',/,' ',5x,'1 - slab',/,' ',5x &,'2 - cylinder') read*,shape if(shape.eq.1)then write(6,60) format(' ',/,' ','Enter dimensions for slab:',/,' ',5x, & 'width - ',$) read*,lx 1x0 - 0.0d0 write(6,70) format(' ',5x,'height or length - ',$) read*,ly 1y0 - 0.0d0 else write(6,80) format(' ',/,' ','Enter dimensions for cylinder:',//, & ' ',5x,'Is the cylinder hollow? (0-No,l-Yes) ',$) read*,cyn if(cyn.eq.1)then write(6,90) format(' ',5x,'Inner radius (m)- ',$) read*,le else 1x0 - 0.0d0 endif write(6,100) format(' ',5x,'Outer radius (m)- ',$) read*,lx write(6,110) format(' ',5x,'Length of cylinder (m)- ',$) read*,ly endif if(shape.eq.1)then 313 Table C.2 (cont'd). write(6,120) 120 format(' ',/,' ','Indicate whether the heat transfer ', & 'coefficients are',/,' ',' the same on opposite ends of the ', & 'slab:',/,' ',5x,'in the x-direction (width)? (0-No,1-Yes) ',$) read*,symx write(6,130) 130 format(' ',5x,'in the y-direction (height or length)? ', & ' (0-No,1-Yes) ',$) read*,symy else symx - 0 write(6,145) 145 format(' ','Are the heat transfer coefficients the same ', & /,' ',5x,'on opposite ends of the cylinder? (0-No,1-Yes) ',$) read*,symy endif write(6,160) 160 format(' ',/,' ','Enter number of constant temp. storage ', &'periods: ',$) read*,per write(6,170) 170 format(' ',/,' ','Are these values correct? (y/n) ',$) read(5,180)yn 180 format(a) if(yn.ne.'y'.and.yn.ne.'Y')goto 30 c input boundary conditions for each storage period do i-1,per 190 write(6,200)i 200 format(' ',/,' ','Enter data for period ',i3,':',/, & ' ',5x,'Storage temperature (C): ',$) read*,temp(i) 210 write(6,220) 220 format(' ',5x,'Enter units for storage temp.:',/,' ',7x, & '1- hours',/,' ',7x,'2- days') read*,unit if(unit.le.0.and.unit.gt.2)then print*,'Try again!’ goto 210 endif if(unit.eq.l)then tunit(i)-3600.0d0 else tunit(i)-86400.0d0 endif 230 write(6,240) 240 format(' ',/,' ',5x,'Length of storage period: ',$) read*,stor(i) if(stor(i).lt.0)then print*,'Try again!’ goto 230 endif write(6,250) 250 format('O',5x,'Enter surface heat transfer coef. (W/m“2C):') if(shape.eq.1)then 314 Table C.2 (cont'd). 260 270 280 290 300 310 320 330 write(6,260) format(' ',9x,'at side 1 along width of slab - ',$) read*,htc(i,l) else if(cyn.eq.1)then write(6,270) format(' ',9x,'along inner radius of cylinder - ',$) read*,htc(i,l) else htc(i,1) - 0.0d0 endif endif if(symx.ne.1.)then if(shape.eq.1)then write(6,280) format(' ',9x,'at side 2 along width of slab - ',$) else write(6,290) format(' ',9x,'along outer radius of cylinder - ',$) endif read*,htc(i,3) else htc(i,3) - 0.0d0 endif if(shape.eq.1)then write(6,300) format(' ',9x,'at side 1 along height or length of slab - ',$) else write(6,310) format(' ',9x,'at side 1 along length of cylinder - ',$) endif read*,htc(i,2) if(symy.ne.1.)then if(shape.eq.1)then write(6,320) format(' ',9x,'at side 2 along height or length of slab - ',$) else write(6,330) format(' ',9x,'at side 2 along length of cylinder - ',$) endif read*,htc(i,4) else htc(i,4) - 0.0d0 endif write(6,170) read(5,180)yn if(yn.ne.'y'.and.yn.ne.'Y')goto 190 enddo c input geometry and size 500 ti - ti+273.150d0 if(symx.eq.1.and.shape.eq.l)Lx - Lx*0.50d0 if(symy.eq.1)Ly - Ly*0.50d0 do i - 1,per temp(i) - temp(i)+273.150d0 stor(i) - stor(i)*tunit(i) 315 Table C.2 (cont'd). enddo if((lx-le).ge.0.40d0)then ixt - 20 stepx - 5 go to 510 endif if(lx.lt.0.02d0)then ixt - 2 stepx - 1 go to 510 endif do i-1,5 if((lx-le).lt.i*0.040d0)then ixt - 1*4 stepx - i go to 510 endif enddo 510 if(ly.ge.0.40d0)then iyt - 20 stepy - 5 go to 520 endif if(ly.lt.0.02d0)then iyt - 2 stepy - 1 go to 520 endif do i-1,5 if(ly.lt.i*0.040d0)then iyt - 1*4 stepy - i go to 520 endif enddo write(inpdat,530)ttlfil,'inp.dat' 530 format(' ',a,a) open(unit-12,name-inpdat(l:l2),type-'new',carriagecontrol-'list') write(12,*)ti,shape,1x0,1x,1y write(12,*)symx,symy,cyn,per do i - 1,per write(12,*)temp(i),stor(i),tunit(i) write(12,*)htc(i,1),htc(i,2),htc(i,3),htc(i,4) enddo write(12,*)ixt,stepx,iyt,stepy close(unit-12) return end 316 Table C.2 (cont'd). SUBROUTINE INPUT2 Kinetic properties to determine quality loss in a food product are entered in this subroutine. A file titled 'TTLFILkin.dat' containing the kinetic properties is created. This file is re- opened in the solution, and it may be reused again in subsequent runs. 00000 integer model double precision q0,tref,ea,vea,vq0 character title*20,ttlfil*4,fildat*12,inpdat*12,kindat*12 logical itmode common /ttl/title,ttlfil,/mod/model,/itm/itmode, &/datfil/fildat,inpdat,kindat o Read batch file data (if itmode - .false.) if(itmode)go to 1 read*,q0,tref,ea if(model.eq.4)then read*,vea,vq0 endif go to 30 o Read interactive input 1 write(5,2) 2 format('l',72('-'),/,'0',t23,'Reference Shelf-life data',/,'O', &72('-')) 10 write(5,100) 100 format(' ',/,' ','Enter reference shelf-life (days) : ',$) read*,qO write(5,200) 200 format(' ',/,' ','Enter reference temperature for reference', &' shelf-life (C) : ',$) read*,tref write(5,300) 300 format(' ',/,' ','Enter activation energy constant (kJ/mole) : ' &.$) read*,ea if(model.eq.4)then write(5,400) 400 format(' ',/,' ','Enter standard deviation of activation ', &'energy const. (kJ/mole) : ',$) read*,vea write(5,500) 500 format(' ',/,' ','Enter standard deviation of ref. shelf-life', &' (daYS) : ',$) read*,qu endif 317 Table C.2 (cont'd). write(5,550) 550 format(' ',/,' ','Are these values correct? (y/n) ',$) read(5,20)yn 20 format(a) if(yn.eq.'n'.or.yn.eq.'N')go to 10 30 tref - tref+273.150d0 ea - ea*1000.0d0 if(model.eq.4)then vea - (vea*1000.0d0)**2.0d0 vq0 - vq0*vq0 endif write(kindat,600)ttlfil,'kin.dat' 600 format(' ’,a,a) open(unit-12,name-kindat(l:12),type-'new',carriagecontrol-'list') write(12,700)q0,tref,ea 700 format(' ',2x,f8.l,2x,f7.2,2x,f13.0) if(model.eq.4)then write(12,800)vea,vq0 800 format(' ',2(2x,e11.3)) endif close(unit-l2) return end c********************************************************************* c********************************************************************* SUBROUTINE SOLN Parameter(maxd-101,maxc-51,maxs-201,maxx-31,maxp-20, &tol-l.0d0,r-8.3140d0) Integer per,shape,ixt,iyt,symx,symy,stepx,stepy,bc,xy,ix,iy, &cyn,ynavg,ixy,ixpl,iyp1 Double Precision dt2,dens,conduc,spheat,ds, &htflx(maxp,4),a(maxx),b(maxx),c(maxx),d(maxx), &txy(maxx),pi,hxy(4,3),bcxy(4,3),qual(maxx,maxx),dq(2), &time,ptime,tavg,qavg,vq,ftime,t(maxx,maxx,3) c Declare all variables in common blocks. Double Precision wf0,ms,dp,kp,cp,t0, &denst(maxd),densc(maxd),condt(maxc),condc(maxc),spht(maxs), & sphc(maxs), &ti,temp(maxp),stor(maxp),htc(maxp,4),tunit(maxp), &lx,1x0,ly,dx,dy, &ea,q0,vea,vq0,tref, &tl,th,avgd,avgk,avgc 318 Table C.2 (cont'd). character title*20,ttlfil*4,fildat*12,inpdat*12,kindat*12 Common /prop/wf0,ms,dp,kp,cp,t0, &/conp/denst,densc,condt,condc,spht,sphc, &/nconp/nsd,nsc,nss, &/bound/per,ti,temp,stor,htc,tunit &/geom/shape,1x,1x0,ly,dx,dy,symx,symy,cyn,ixpl,iyp1,stepy, &/shelf/ea,q0,vea,vq0,tref, &/mod/model,/itm/itmode, &/datfil/fildat,inpdat,kindat, &/ttl/title,ttlfil, &/pavg/tl,th,avgd,avgk,avgc,ynavg 8 ave o Read in geometry, dimensions, boundary conditions and initial condition write(inpdat,100)ttlfil,'inp.dat' 100 format(' ',a,a) open(unit-l2,name-inpdat(l:12),type-'old',carriagecontrol-'list') read(12,*)ti,shape,lx0,lx,ly read(12,*)symx,symy,cyn,per do i - 1,per read(12,*)temp(i),stor(i),tunit(i) read(12,*)htc(i,1),htc(i,2),htc(i,3),htc(i,4) enddo read(12,*)ixt,stepx,iyt,stepy close(unit-12) c Read in constant property assumptions and associated temperature ranges c Read in constant property assumptions write(fildat,110)'karlprp.dat' 110 format(' ',a) open(unit-12,name-fildat(l:12),type-'old',carriagecontrol-'list') read(12,*)wf0,t0,ms read(12,*)dp,kp,cp read(12,*)nsd,nsc,nss read(12,*)tl,th,avgd,avgk,avgc,ynavg do i - 1,nsd read(12,*)denst(i),densc(i) enddo do i - 1,nsc read(12,*)condt(i),condc(i) enddo do i - 1,nss read(12,*)spht(i),sphc(i) enddo close(unit-12) 319 Table C.2 (cont'd). C C Read in kinetic data if(model.ge.3)then write(kindat,100)ttlfil,'kin.dat' open(unit-12,name-kindat,type-'old',carriagecontrol-'list') read(12,*)q0,tref,ea if(model.eq.4)then read(12,*)vea,vq0 else vea - 0.0d0 vq0 - 0.0d0 endif close(unit-12) endif Set imposed heat flux equal to 0; user may change if desired. do i - 1,per do k - 1,4 htflx(i,k) - 0.0d0 enddo enddo dx - (lx-lx0)/dfloat(ixt) dy - ly/dfloat(iyt) ixpl - ixt+l iypl - iyt+l pi - dacos(-1.0d0) do i - 1,ixpl do j - l,iypl do k - 1,3 t(i,j,k) - ti enddo if(model.ge.3)qual(i,j) - q0*86400.0d0 enddo enddo tavg - ti do k - 1,3 if(shape.eq.1)then hxy(l,k) - htc(1,1)*dy hxy(2,k) - htc(1,2)*dx hxy(3,k) - htc(l,3)*dy hxy(4,k) - htc(l,4)*dx bcxy(l,k) - (htc(l,l)*temp(1)+htflx(l,l))*dy bcxy(2,k) - (htc(l,2)*temp(1)+htflx(l,2))*dx bcxy(3,k) - (htc(l,3)*temp(l)+htflx(1,3))*dy bcxy(4,k) - (htc(l,4)*temp(l)+htf1x(1,4))*dx else hxy(l,k) - htc(l,1)*dy*2.0d0*pi*lx0 hxy(2,k) - htc(l,2)*pi*dx hxy(3,k) - htc(1,3)*dy*2.0d0*pi*lx hxy(4,k) - htc(l,4)*pi*dx bcxy(l,k) - (htc(l,1)*temp(l)+htflx(l,l))*dy*2.0d0*pi*lx0 bcxy(2,k) - (htc(l,2)*temp(l)+htf1x(l,2))*pi*dx bcxy(3,k) - (htc(l,3)*temp(1)+htflx(l,3))*dy*2.0d0*pi*lx bcxy(4,k) - (htc(l,4)*temp(l)+htflx(l,4))*pi*dx endif 320 Table C.2 (cont'd). enddo if(model.ge.3)then qavg - q0*86400.0d0 if(model.eq.4)then dq(l) - 0.0d0 dq(2) - 0.0d0 vq0 - vq0*86400.0d0 vq - vq0 else vq - 0.0d0 endif else qavg - 0.0d0 vq - 0.0d0 endif time - 0.0d0 ptime - 0.0d0 ftime - 0.0dO count - 0 nptime - 0 nccc - 0 nprint - 0 if(model.lt.3)then headtq - 1 else headtq - 2 endif call headng nprint - 0 call output(t,time,ptime,l,nprint,tavg,qual,qavg,vq) ntime - l cit***********************m**************************************** do 500 ii - 1,per ptime - 0.0d0 if(ii.ne.l)then time - time+stor(ii-1) if(nccc.ne.0)then nprint - 1 call output(t,time,ptime,ii,nprint,tavg,qual,qavg,vq) endif endif dt2 - 600.0d0 nptime - 0 if(per.ne.l)then do k - 2,3 if(shape.eq.1)then ta - temp(ii) hxy(l,k) - htc(ii,1)*dy hxy(2,k) - htc(ii,2)*dx hxy(3,k) - htc(ii,3)*dy hxy(4,k) - htc(ii,4)*dx bcxy(l,k) - (htc(ii,l)*temp(ii)+htflx(ii,l))*dy bcxy(2,k) - (htc(ii,2)*temp(ii)+htflx(ii,2))*dx bcxy(3,k) - (htc(ii,3)*temp(ii)+htflx(ii,3))*dy 321 Table C.2 (cont'd). bcxy(4,k) - (htc(ii,4)*temp(ii)+htflx(ii,4))*dx else hxy(l,k) - htc(ii,l)*dy*2.0d0*pi*lx0 hxy(2,k) - htc(ii,2)*pi*dx hxy(3,k) - htc(ii,3)*dy*2.0d0*pi*lx hxy(4,k) - htc(ii,4)*pi*dx bcxy(l,k) - (htc(ii,l)*temp(ii)+htflx(ii,l))*dy*2.0d0*pi*lx0 bcxy(2,k) - (htc(ii,2)*temp(ii)+htflx(ii,2))*pi*dx bcxy(3,k) - (htc(ii,3)*temp(ii)+htflx(ii,3))*dy*2.0d0*pi*lx bcxy(4,k) - (htc(ii,4)*temp(ii)+htflx(ii,4))*pi*dx endif enddo endif c c Check if time is > length of storage period 200 if(ptime.ge.stor(ii))go to 500 c Check if temp. is close to ambient temperature eabs - abs(t(l,l,3)-temp(ii)) do i - 1,ixpl do j - l,iypl if(i.ne.1.and.j.ne.1)then eeabs - abs(t(i,j,3)-temp(ii)) if(eabs.lt.eeabs)eabs - eeabs endif enddo enddo if(eabs.lt.tol)go to 300 ptime - ptime+dt2 if(ptime.gt.stor(ii))then kjtime - ptime-dt2 ptime - stor(ii) dt2 - ptime-kjtime endif 0 First sweep xy - 0 itpl - ixpl do j - 1,iyp1 ixy-J if(j.eq.l)bc - -l if(j.eq.iypl)bc - 1 if(j.ne.1.and.j.ne.iyp1)bc - 0 call coef1(t,ixy,xy,hxy,bcxy,dt2,bc,a,b,c,d) call thomal(a,b,c,d,itpl,txy) 322 Table C.2 (cont'd). do i - l,ixp1 t(i.J.2) - txy(i) enddo enddo c Next sweep in the Y-direction xy - 1 itpl - iypl do j - l,ixpl ixy - J if(j.eq.l)bc - -1 if(j.eq.ixpl)bc - l if(j.ne.l.and.j.ne.ixpl)bc - 0 call coef2(t,ixy,xy,hxy,bcxy,dt2,bc,a,b,c,d) call thomal(a,b,c,d,iypl,txy) do i - 1,iypl t(j,i,3) - txy(i) enddo enddo tavg - 0.0d0 do i - l,ixpl do j - 1,iyp1 tavg - t(i,j,3)+tavg enddo enddo tavg - tan/(ixpl*iyp1) C c Find quality distribution and adjust time step if(model.ge.3)then vsum - 0 qavg - 0 do i - l,ixpl do j - 1,iyp1 eex - 1.0dO/t(i,j,3)-l.0dO/tref d1 - exp(-(ea/r*eex)) qual(i,j) - qua1(i,j)-d1*dt2 qavg - ang+qual(i,j) if(model.eq.4)then dqdea -dl*eex*dt2/r vsum - vsum+dqdea endif enddo enddo c Find mass average quality and variance qavg - qavg/(ixpl*iypl) if(model.eq.4)then dqdea - vsum/(ixp1*iyp1) dq(2) - dq(1)+dqdea 323 Table C.2 (cont'd). vq - vq0+dq(2)**2*vea d(1(1) - dQ(2) endif endif ftime - ftime+dt2 c if(ftime.ge.0.25d0*stor(ii))then nccc - nccc+l if(nccc.eq.6)then c printout nprint - 2 call output(t,time,ptime,ii,nprint,tavg,qua1,qavg,vq) ftime - 0.0d0 nccc - 0 endif c initial t for next time step do i - 1,ixpl do j - 1,iypl t(i,J,1)-t(1,j,3) enddo enddo do i - 1,4 hxy<1.1> - hxy<1.3> bcxy(i,1) - bcxy(i,3) enddo ntime - ntime+l nptime - nptime+l goto 200 c end of finite difference calculations c ********************************************************* c set product temp. - ambient temperature; determine quality 300 tavg - 0.0d0 do i - 1,ixp1 do j - 1,iyp1 do k - 1,3 t(1.J.k) - temp(ii) enddo enddo enddo tavg - temp(ii) nx-l dt2 - (stor(ii)-ptime)/nx do ij - l,nx ptime - ptime+dt if(model.ge.3)then 324 Table C.2 (cont'd). eex - l.0d0/temp(ii)-l.0d0/tref dl - exp(-(ea/r*eex)) qavg - 0.0d0 do i - l,ixpl do j - l,iypl qua1(i,j) - qual(i,j)-dl*dt2 qavg - qavg+qual(i,j) enddo enddo qavg-ang/(ixp1*iypl) if(model.eq.4)then dqdea-dl*eex*dt/r dq(2)-dq(1)+dqdea vq~vq0+dq(2)**2*vea dQ(l)-dq(2) endif endif ntime - ntime+l nptime - nptime+l count-0 enddo do i - 1,ixpl do j - 1,iyp1 t(1.J.1) - t(1.j.3) enddo enddo do i - 1,4 hxy(i,l) - hxy(i,3) bcxy(i,l) - bcxy(i,3) enddo if(ii.eq.per.and.nccc.ne.0)then nprint - 3 call output(t,time,ptime,ii,nprint,tavg,qual,qavg,vq) endif 500 continue return end c********************************************************************* c********************************************************************* SUBROUTINE COEFl(t,ixy,xy,hxy,bcxy,dt,bc,a,b,c,d) Parameter(maxd-101,maxc-51,maxs-201,maxx-31) integer bc,it,itpl,ixy,shape,symx,symy,ixpl,iypl,stepx,stepy, &nsd,nsc,nss,cyn,xy,ynavg Double precision pi,beta,nu,omega,gama,hxy(4,3),bcxy(4,3), 325 Table C.2 (cont'd). &a(maxx),b(maxx),c(maxx),d(maxx),dax2,dbx2,dcx2,dax,day,dbxy, &dcy,dcx,dd,ck(maxx,8),csd(maxx,8),dt,t(maxx,maxx,3),hxy2,hxy4, &bxy2,bxy4 c Declare all variables in common blocks. Double Precision wf0,ms,dp,kp,cp,t0, &denst(maxd),densc(maxd),condt(maxc),condc(maxc),spht(maxs), & sphc(maxs), &lx,lx0,1y,dx,dy, &tl,th,avgd,avgk,avgc common /prop/wf0,ms,dp,kp,cp,t0, &/nconp/nsd,nsc,nss, &/conp/denst,densc,condt,condc,spht,sphc, &/geom/shape,lx,1x0,ly,dx,dy,symx,symy,cyn,ixpl,iyp1,stepy, &/pavg/tl,th,avgd,avgk,avgc,ynavg pi - dacos(-1.0d0) ix - ixpl-l c Weighting function for ADI finite difference method. c Modified Crank-Nicolson Method c l. Weighting coefficients for d2T/dx2; c a. at time t: beta - 0.50d0 c b. at time t+1/2*dt and t+dt nu - 0.50d0 c 2. Weighting coefficients for dT/dt; c a. at time t: omega - -1.0d0 gama - 1.0d0 itpl - ixpl c Find product property values for each y-value for constant x (ix). call pfind(t,ixy,xy,itpl,dt,bc,ck,csd) c 1st boundary point if(bc.eq.-l)then if(shape.eq.1)then hxy2 - hxy(2,l) bxy2 - bcxy(2,1) else hxy2 - hxy(2,l)*(lx0+0.25d0*dx) 326 Table C.2 (cont'd). bxy2 - bcxy(2,l)*(1x0+0.25d0*dx) endif a(1) - 0.0d0 c(l) - nu*ck(l,3) b(l) - -nu*hxy(1,2)*0.50d0-gama*csd(1,3)-c(1) dcy - -ck(1,4) dcx - -beta*ck(1,3) dbxy - (beta*hxy(l,1)+hxy2)*0.50d0+omega*csd(l,3)-dcy-dcx dd - —(nu*bcxy(1,2)+beta*bcxy(l,l)+bxy2)*0.50d0 d(1) - dbxy*t(l,ixy,1)+dcy*t(l,ixy+1,l)+dcx*t(2,ixy,1)+dd endif if(bc.eq.0)then c(l) - nu*ck(l,3) b(l) - -nu*hxy(1,2)-gama*(csd(l,2)+csd(l,3))-c(l) day - -ck(l,2) dcy - -ck(l,4) dcx - -beta*ck(l,3) dbxy - beta*hxy(l,l)+omega*(csd(l,2)+csd(1,3))-day-dcy-dcx dd - ~nu*bcxy(l,2)-beta*bcxy(1,1) d(1) - day*t(l,ixy-l,1)+dbxy*t(l,ixy,l)+dcy*t(l,ixy+l,l)+ & dcx*t(2,ixy,l)+dd endif if(bc.eq.1)then if(shape.eq.1)then hxy4 - hxy(4,1) bxy4 - bcxy(4,1) else hxy4 - hxy(4,l)*(lx0+0.25d0*dx) bxy4 - bcxy(4,l)*(lx0+0.25d0*dx) endif c(l) - nu*ck(l,3) b(l) - -nu*hxy(1,2)*0.50d0-gama*csd(1,2)-c(l) day - -ck(l,2) dcx - -beta*ck(l,3) dbxy - (beta*hxy(1,l)+hxy4)*0.50d0+omega*csd(l,2)-day-dcx dd - -(nu*bcxy(l,2)+beta*bcxy(l,l)+bxy4)*0.50d0 d(1) - day*t(l,ixy-1,l)+dbxy*t(l,ixy,1)+dcx*t(2,ixy,1)+dd endif c Interior Points do i - 2,ix if(bc.eq.-1)then if(shape.eq.1)then hxy2 - hxy(2,1) bxy2 bcxy(2,1) else hxy2 - hxy(2,1)*2.0d0*(lx0+dfloat(i-l)*dx) bxy2 - bcxy(2,1)*2.0d0*(lx0+dfloat(i-l)*dx) endif a(i) - nu*ck(i,l) c(i) - nu*ck(i,3) b(i) - -gama*(csd(i,3)+csd(i,4))—a(i)-c(i) Table C.2 & 327 (cont'd). dax - -beta*ck(i,1) dcy - -ck(i,4) dcx - ~beta*ck(i,3) dbxy - hxy2+omega*(csd(i,3)+csd(i,4))-dax-dcy-dcx dd - -bxy2 d(i) - dax*t(i-l,ixy,1)+dbxy*t(i,ixy,l)+dcy*t(i,ixy+1,l) +dcx*t(i+l,ixy,l)+dd endif if(bc.eq.0)then & & a(i) - nu*ck(i,1) c(i) - nu*ck(i,3) b(i) - -gama*(csd(i,1)+csd(i,2)+csd(i,3)+csd(i,4))-a(i)-c(i) dax - -beta*ck(i,1) day - -ck(i,2) dcy - -ck(i,4) dcx - -beta*ck(i,3) dbxy - omega*(csd(i,l)+csd(i,2)+csd(i,3)+csd(i,4))-dax-day-dcy -dcx d(i) - dax*t(i-l,ixy,l)+day*t(i,ixy-l,1)+dbxy*t(i,ixy,1)+ dcy*t(i,ixy+1,1)+dcx*t(i+l,ixy,1) endif if(bc.eq.1)then & if(shape.eq.1)then hxy4 - hxy(4,1) bxy4 - bcxy(4,1) else hxy4 - hxy(4,1)*2.0d0*(lx0+(i-1)*dx) bxy4 - bcxy(4,1)*2.0d0*(lx0+(i-l)*dx) endif a(i) - nu*ck(i,l) c(i) - nu*ck(i,3) b(i) - -gama*(csd(i,1)+csd(i,2))-a(i)-c(i) dax - -beta*ck(i,1) day - -ck(i,2) dcx - -beta*ck(i,3) dbxy - hxy4+omega*(csd(i,l)+csd(i,2))-dax-day-dcx dd - -bxy4 d(i) - dax*t(i-1,ixy,l)+day*t(i,ixy-l,1)+dbxy*t(i,ixy,1) +dcx*t(i+l,ixy,l)+dd endif enddo C c 2nd Boundary if(bc.eq.-1)then if(shape.eq.1)then hxy2 - hxy(2,1) bxy2 - bcxy(2,1) else hxy2 - hxy(2,l)*(lx-0.25d0*dx) bxy2 - bcxy(2,l)*(lx-0.25d0*dx) endif a(ixpl) - nu*ck(ixpl,l) b(ixpl) - -nu*hxy(3,2)*0.50d0-gama*csd(ixpl,4)-a(ixpl) 328 Table C.2 (cont'd). dax - -beta*ck(ixp1,1) dcy - -ck(ixpl,4) dbxy - (beta*hxy(3,l)+hxy2)*0.50d0+omega*csd(ixpl,4)-dax-dcy dd - o(nu*bcxy(3,2)+beta*bcxy(3,l)+bxy2)*0.50d0 d(ixpl) - dax*t(ix,ixy,l)+dbxy*t(ixp1,ixy,1)+ & dcy*t(ixpl,ixy+l,1)+dd endif if(bc.eq.0)then a(ixpl) - nu*ck(ixp1,l) b(ixpl) - -nu*hxy(3,2)-gama*(csd(ixp1,l)+csd(ixp1,4))-a(ixpl) dax - -beta*ck(ixpl,1) day - —ck(ixp1,2) dcy - -ck(ixpl,4) dbxy - beta*hxy(3,l)+omega*(csd(ixpl,l)+csd(ixpl,4))-dax-day-dcy dd - -nu*bcxy(3,2)-beta*bcxy(3,l) d(ixpl) - dax*t(ix,ixy,1)+day*t(ixp1,ixy-l,l)+dbxy*t(ixpl,ixy,l) & +dcy*t(ixpl,ixy+1,l)+dd endif if(bc.eq.1)then if(shape.eq.1)then hxy4 - hxy(4,1) bxy4 - bcxy(4,l) else hxy4 - hxy(4,l)*(lx-0.25d0*dx) bxy4 - bcxy(4,1)*(lx-0.25d0*dx) endif a(ixpl) - nu*ck(ixp1,1) b(ixpl) - -nu*hxy(3,2)*0.50d0—gama*csd(ixpl,l)-a(ixpl) dax - -beta*ck(ixpl,l) day - -ck(ixpl,2) dbxy - (beta*hxy(3,l)+hxy4)*0.50d0+omega*csd(ixpl,l)-dax-day dd - -(nu*bcxy(3,2)+beta*bcxy(3,1)+bxy4)*0.50d0 d(ixpl) - dax*t(ix,ixy,l)+day*t(ixpl,ixy-1,1)+dbxy*t(ixpl,ixy,1) & +dd endif return end C********************************************************************* c********************************************************************* SUBROUTINE COEF2(t,ixy,xy,hxy,bcxy,dt,bc,a,b,c,d) c This is the preliminary version of the subroutine which finds the coef c for the second sweep in the Y-direction. Parameter(maxd-101,maxc-Sl,maxs-201,maxx-3l) integer bc,iy,itpl,xy,ixy,nsd,nsc,nss,shape,symx,symy,ixpl, &iyp1,stepx,stepy,cyn,ynavg 329 Table C.2 (cont'd). Double precision pi,betax,betay,nuy,nux,omega,gama,hxy(4,3), &bcxy(4,3),a(maxx),b(maxx),c(maxx),d(maxx),dax2,dbx2,dcx2,dax, &day,dbxy,dcy,dcx,dd,ck(maxx,8),csd(maxx,8),t(maxx,maxx,3), &hxy2a,hxy2b,hxy4a,hxy4b,bxy2a,bxy2b,bxy4a,bxy4b c Declare all variables in common blocks. Double Precision wf0,ms,dp,kp,cp,t0, &denst(maxd),densc(maxd),condt(maxc),condc(maxc),spht(maxs), & sphc(maxs), &lx,lx0,ly,dx,dy, &tl,th,avgd,avgk,avgc common /prop/wf0,ms,dp,kp,cp,t0, &/conp/denst,densc,condt,condc,spht,sphc, &/nconp/nsd,nsc,nss, &/geom/shape,lx,lx0,ly,dx,dy,symx,symy,cyn,ixpl,iyp1,stepy, &/pavg/tl,th,avgd,avgk,avgc,ynavg pi - dacos(-l.0d0) c Weighting function for ADI finite difference method. c Modified Crank-Nicolson Method c 1. Weighting coefficients for d2T/dx2 and d2T/dy2; c a. at time t: betax - 0.50d0 betay - 0.50d0 c b. at time t+1/2*dt and t+dt nux - 0.50d0 nuy - 0.50d0 c 2. Weighting coefficients for dT/dt; c a. at time t: omega - -l.0d0 gama - 1.0d0 iy - iypl-1 c Find product property values for each y-value for constant x (ix). call pfind(t,ixy,xy,iypl,dt,bc,ck,csd) c lst boundary point a(1) - 0.0d0 if(bc.eq.-1)then if(shape.eq.1)then hxy2a - hxy(2,1) hxy2b - hxy(2,3) 330 Table C.2 (cont'd). bxy2a - bcxy(2,1) bxy2b - bcxy(2,3) else hxy2a - hxy(2,1)*(lx0+0.25d0*dx) hxy2b - hxy(2,3)*(1x0+0.25d0*dx) bxy2a - bcxy(2,l)*(lx0+0.25d0*dx) bxy2b - bcxy(2,3)*(lx0+0.25d0*dx) endif c(l) - nuy*ck(1,8) b(l) - -nuy*hxy2b*0.50d0-gama*csd(l,7)-c(l) dcx2 - —nux*ck(l,7) dbx2 - nux*hxy(l,2)*0.50d0-dcx2 & & & dcy - -betay*ck(l,4) dcx - —betax*ck(1,3) dbxy - (betax*hxy(l,1)+betay*hxy2a)*0.50d0+omega*csd(l,3)-dcy -dcx dd - -(nux*bcxy(1,2)+betax*bcxy(l,1)+nuy*bxy2b+ betay*bxy2a)*0.50d0 d(l) - dbx2*t(ixy,l,2)+dcx2*t(ixy+l,l,2)+dbxy*t(ixy,1,l)+ dcy*t(ixy,2,l)+dcx*t(ixy+l,1,l)+dd endif if(bc.eq.0)then if(shape.eq.1)then hxy2a - hxy(2,1) hxy2b - hxy(2,3) bxy2a - bcxy(2,1) bxy2b - bcxy(2,3) else hxy2a hxy(2,l)*2.0d0*(lx0+(ixy—l)*dx) hxy2b hxy(2,3)*2.0d0*(1x0+(ixy-l)*dx) bxy2a bcxy(2,l)*2.0d0*(lx0+(ixy-l)*dx) bxy2b - bcxy(2,3)*2.0d0*(lx0+(ixy-l)*dx) endif c(l) - b(l) - nuy*ck(l,8) -nuy*hxy2b-gama*(csd(1,7)+csd(l,8))-c(l) dax2 - -nux*ck(1,5) dcx2 - -nux*ck(l,7) dbx2 - -dax2-dcx2 dax - -betax*ck(l,1) dcy - -betay*ck(1,4) dcx - -betax*ck(l,3) dbxy - betay*hxy2a+omega*(csd(l,3)+csd(1,4))-dax-dcy-dcx dd - -nuy*bxy2b-betay*bxy2a d(1) - dax2*t(ixy-1,1,2)+dbx2*t(ixy,l,2)+dcx2*t(ixy+l,l,2)+ & dax*t(ixy-1,l,1)+dbxy*t(ixy,l,l)+dcy*t(ixy,2,1)+ & dcx*t(ixy+1,1,1)+dd endif if(bc.eq.1)then if(shape.eq.1)then hxy2a - hxy(2,1) hxy2b - hxy(2,3) bxy2a - bcxy(2,1) bxy2b - bcxy(2,3) else hxy2a - hxy(2,l)*(lx-0.25d0*dx) hxy2b hxy(2,3)*(lx-0.25d0*dx) Table & 331 C.2 (cont'd). bxy2a - bcxy(2,1)*(lx-0.25d0*dx) bxy2b - bcxy(2,3)*(lx-0.25d0*dx) endif c(l) - nuy*ck(1,8) b(l) - -nuy*hxy2b*0.SOdO-gama*csd(l,8)-c(l) dax2 - -nux*ck(1,5) dbx2 - nux*hxy(3,2)*0.50d0-dax2 dax - -betax*ck(l,l) dcy - -betay*ck(1,4) dbxy - (betax*hxy(3,l)+betay*hxy2a)*0.50d0+omega*csd(1,4)-dax -dcy dd - -(nux*bcxy(3,2)+betax*bcxy(3,1)+nuy*bxy2b+ & betay*bxy2a)*0.5d0 & d(1) - dax2*t(ixy-l,l,2)+dbx2*t(ixy,1,2)+dax*t(ixy-1,l,1)+ dbxy*t(ixy,l,l)+dcy*t(ixy,2,l)+dd endif c Int & & & & erior Points do i - 2,iy if(bc.eq.-l)then a(i) - nuy*ck(i,6) c(i) - nuy*ck(i,8) b(i) - -gama*(csd(i,6)+csd(i,7))-a(i)-c(i) dcx2 - -nux*ck(i,7) dbx2 - nux*hxy(1,2)-dcx2 day - -betay*ck(i,2) dcy - -betay*ck(i,4) dcx - -betax*ck(i,3) dbxy - betax*hxy(1,l)+omega*(csd(i,2)+csd(i,3))-day-dcy-dcx dd - -nux*bcxy(1,2)-betax*bcxy(1,l) d(i) - dbx2*t(ixy,i,2)+dcx2*t(ixy+l,i,2)+day*t(ixy,i-l,1)+ dbxy*t(ixy,i,1)+dcy*t(ixy,i+l,l)+dcx*t(ixy+1,i,1)+dd endif if(bc.eq.0)then a(i) - nuy*ck(i,6) c(i) - nuy*ck(i,8) b(i) - -gama*(csd(i,5)+csd(i,6)+csd(i,7)+csd(i,8))-a(i)-c(i) dax2 - -nux*ck(i,5) dcx2 - -nux*ck(i,7) dbx2 - -dax2-dcx2 dax - -betax*ck(i,l) day - -betay*ck(i,2) dcy - -betay*ck(i,4) dcx - ~betax*ck(i,3) dbxy - omega*(csd(i,1)+csd(i,2)+csd(i,3)+csd(i,4))-dax-day -dcy-dcx d(i) - dax2*t(ixy-l,i,2)+dbx2*t(ixy,i,2)+dcx2*t(ixy+1,i,2)+ dax*t(ixy—l,i,l)+day*t(ixy,i-1,l)+dbxy*t(ixy,i,l)+ dcy*t(ixy,i+1,l)+dcx*t(ixy+1,i,1) endif if(bc.eq.1)then a(i) - nuy*ck(i,6) 332 Table C.2 (cont'd). c(i) - nuy*ck(i,8) b(i) - -gama*(csd(i,5)+csd(i,8))-a(i)-c(i) dax2 - -nux*ck(i,5) dbx2 - nux*hxy(3,2)-dax2 dax - -betax*ck(i,1) day - -betay*ck(i,2) dcy - -betay*ck(i,4) dbxy - betax*hxy(3,l)+omega*(csd(i,1)+csd(i,4))-dax-day-dcy dd - -nux*bcxy(3,2)-betax*bcxy(3,1) d(i) - dax2*t(ixy-1,i,2)+dbx2*t(ixy,i,2)+dax*t(ixy-l,i,1)+ & day*t(ixy,i-1,1)+dbxy*t(ixy,i,1)+dcy*t(ixy,i+l,l)+dd endif enddo c 2nd Boundary c(iypl) - 0.0d0 if(bc.eq.-1)then if(shape.eq.1)then hxy4a - hxy(4,1) hxy4b - hxy(4,3) bxy4a - bcxy(4,l) bxy4b - bcxy(4,3) else hxy4a - hxy(4,1)*(lx+0.25d0*dx) hxy4b - hxy(4,3)*(lx+0.25d0*dx) bxy4a - bcxy(4,1)*(1x+0.25d0*dx) bxy4b - bcxy(4,3)*(lx+0.25d0*dx) endif a(iypl) - nuy*ck(iyp1,6) b(iypl) - -nuy*hxy4b*0.50d0-gama*csd(iypl,6)-a(iypl) dcx2 - -nux*ck(iyp1,7) dbx2 - nux*hxy(1,2)*0.50d0-dcx2 day - -betay*ck(iyp1,2) dcx - -betax*ck(iyp1,3) dbxy - (betax*hxy(1,l)+betay*hxy4a)*0.50d0+omega*csd(iyp1,2)- & day-dcx dd - —(nux*bcxy(l,2)+betax*bcxy(1,1)+nuy*bxy4b+ & betay*bxy4a)*0.50d0 d(iypl) - dbx2*t(ixy,iyp1,2)+dcx2*t(ixy+l,iypl,2)+ & day*t(ixy,iypl-l,1)+dbxy*t(ixy,iypl,1)+dcx*t(ixy+l,iypl,l)+dd endif if(bc.eq.0)then if(shape.eq.1)then hxy4a - hxy(4,1) hxy4b - hxy(4,3) bxy4a - bcxy(4,l) bxy4b - bcxy(4,3) else hxy4a - hxy(4,l)*2.0d0*(lx0+(ixy-1)*dx) hxy4b - hxy(4,3)*2.0d0*(lx0+(ixy-l)*dx) bxy4a - bcxy(4,l)*2.0d0*(lx0+(ixy-1)*dx) bxy4b bcxy(4,3)*2.0d0*(lx0+(ixy-1)*dx) 333 Table C.2 (cont'd). endif a(iypl) - nuy*ck(iyp1,6) b(iypl) - -nuy*hxy4b-gama*(csd(iypl,5)+csd(iyp1,6))-a(iypl) dax2 - -nux*ck(iyp1,5) dcx2 - -nux*ck(iypl,7) dbx2 - -dax2-dcx2 dax - -betax*ck(iyp1,1) day - -betay*ck(iypl,2) dcx - -betax*ck(iypl,3) dbxy - betay*hxy4a+omega*(csd(iypl,l)+csd(iyp1,2))-dax-day- & dcx dd - -nuy*bxy4b-betay*bxy4a d(iypl) - dax2*t(ixy-1,iypl,2)+dbx2*t(ixy,iyp1,2)+ & dcx2*t(ixy+l,iypl,2)+dax*t(ixy-1,iyp1,l)+day*t(ixy,iy,l)+ & dbxy*t(ixy,iyp1,1)+dcx*t(ixy+l,iypl,l)+dd endif if(bc.eq.1)then if(shape.eq.1)then hxy4a - hxy(4,1) hxy4b - hxy(4,3) bxy4a - bcxy(4,l) bxy4a - bcxy(4,3) else hxy4a - hxy(4,1)*(lx-0.25d0*dx) hxy4b - hxy(4,3)*(lx-0.25d0*dx) bxy4a - bcxy(4,1)*(lx-0.25d0*dx) bxy4a - bcxy(4,3)*(lx-0.25d0*dx) endif a(iypl) - nuy*ck(iyp1,6) b(iypl) - -nuy*hxy4b*0.50d0-gama*csd(iypl,5)-a(iypl) dax2 - —nux*ck(iyp1,5) dbx2 - nux*hxy(3,2)*0.50d0-dax2 dax - -betax*ck(iyp1,l) day - -betay*ck(iypl,2) dbxy - (betax*hxy(3,1)+betay*hxy4a)*0.50d0+omega*csd(iypl,l)- & dax-day dd - -(nux*bcxy(3,2)+betax*bcxy(3,1)+nuy*bxy4b+ & betay*bxy4a)*0.50d0 d(iypl) - dax2*t(ixy-l,iyp1,2)+dbx2*t(ixy,iypl,2)+ & dax*t(ixy-1,iyp1,l)+day*t(ixy,iy,l)+dbxy*t(ixy,iypl,1)+dd endif return end C *32******************************************************************* c ********************************************************************* SUBROUTINE THOMAL(a,b,c,d,itpl,t) Parameter(maxx - 31) 334 Table C.2 (cont'd). integer itpl double precision a(maxx),b(maxx),c(maxx),d(maxx),cc(maxx), &dd(maxx),t(maxx) CC(1) - C(1)/b(1) dd(1) - d(1)/b(1) do i - 2,itp1 ii - i-l cc(i) - c(i)/(b(i)-a(i)*cc(ii)) dd(i) - (d(i)-a(i)*dd(11))/(b(i)-a(i)*CC(ii)) enddo t(itpl) - dd(itp1) do i - 2,itpl ii - itpl-1+1 t(ii) - dd(ii)-cc(ii)*t(ii+l) enddo return end c>9:~k*******************m*****mm********************************** C7k*1:****************mmm*******m*********************** SUBROUTINE PFIND(t,ixy,xy,itpl,dt,bc,ck,csd) Parameter (maxd-lOl,maxc-51,maxs-201,maxx-31) Integer nsd,nsc,nss,k1,k2,k3,k4,k5,k6,k7,k8,sd2,sd4,sd6,sd8,xy, &nc(24),itp1,bc,ixy,shape,cyn,symx,symy,ixpl,iypl,stepy,ynavg Double Precision pi,dt,dti,dxi,dyi,voll,vol2,tavgk(8), &tavgsd(8),ck(maxx,8),dc(8),spc(8),csd(maxx,8), &dens,conduc,spheat,ark(3),volsd(2),t(maxx,maxx,3) c Declare all variables in common blocks. Double Precision wf0,ms,dp,kp,cp,t0, &denst(maxd),densc(maxd),condt(maxc),condc(maxc),spht(maxs), & sphc(maxs), &ds, &lx,lx0,1y,dx,dy, &tl,th,avgd,avgk,avgc Common /prop/wf0,ms,dp,kp,cp,t0, &/conp/denst,densc,condt,condc,spht,sphc, &/nconp/nsd,nsc,nss, &/d/ds &/geom/shape,1x,1x0,ly,dx,dy,symx,symy,cyn,ixpl,iyp1,stepy &/pavg/tl,th,avgd,avgk,avgc,ynavg External dens,conduc,spheat 335 Table C.2 (cont'd). pi - dacos(-l.0d0) dti - 1000.0d0/dt dxi - 1.0d0/dx dyi - 1.0d0/dy voll - dx*dy*0.25d0 vol2 - pi*dx*dy*0.5d0 if(xy.eq.0)then kkxy - 4 kkl - 4 kk2 - 8 kk3 - 12 else kkxy - 8 kkl - 8 kk2 - 16 kk3 - 24 endif do 100 i - 1,itpl if(ynavg.eq.l)then do ii - 1,8 ck(i,ii) - avgk dc(ii) - avgd spc(ii) - avgc enddo go to 40 endif do kk - 1,kk3 nc(kk) - 0 if(kk.le.kk1)then tavgk(kk) - 0.0d0 tavgsd(kk) - 0.0d0 endif enddo if(xy.eq.0)then if(i.ne.l)tavgk(l) - 0.50*(t(i,ixy,l)+t(i-1,ixy,l)) if(i.ne.1xpl)tavgk(3) - 0.50*(t(i,ixy,l)+t(i+l,ixy,1)) if(ixy.ne.1)tavgk(2) - 0.50*(t(i,ixy,1)+t(i,ixy-l,l)) if(ixy.ne.iyp1)tavgk(4) - 0.50*(t(i,ixy,l)+t(i,ixy+l,l)) if(i.ne.1.and.ixy.ne.1)then tavgsd(l) - 0.0625*(9.0*t(i,ixy,1)+3.0*t(i-l,ixy,l)+ & 3.0*t(i,ixy-l,1)+t(i-l,ixy-l,1)) endif if(i.ne.ixp1.and.ixy.ne.l)then tavgsd(2) - 0.0625*(9.0*t(i,ixy,l)+3.0*t(i+l,ixy,l)+ & 3.0*t(i,ixy-l,1)+t(i+1,ixy-l,1)) endif if(i.ne.ixp1.and.ixy.ne.iypl)then tavgsd(3) - 0.0625*(9.0*t(i,ixy,1)+3.0*t(i+l,ixy,l)+ 336 Table C.2 (cont'd). & 3.0*t(i,ixy+1,1)+t(i+l,ixy+l,1)) endif if(i.ne.1.and.ixy.ne.iyp1)then tavgsd(4) - 0.0625*(9.0*t(i,ixy,l)+3.0*t(i-l,ixy,l)+ & 3.0*t(i,ixy+1,1)+t(i-l,ixy+1,1)) endif else if(ixy.ne.1)then tavgk(l) - 0.50*(t(ixy,i,l)+t(ixy-l,i,1)) tavgk(S) - 0.50*(t(ixy,i,l)+t(ixy-1,i,l)) endif if(ixy.ne.ixp1)then tavgk(3) - 0.50*(t(ixy,i,l)+t(ixy+l,i,l)) tavgk(7) - 0.50*(t(ixy,i,1)+t(ixy+l,i,1)) endif if(i.ne.l)then tavgk(2) - 0.50*(t(ixy,i,l)+t(ixy,i-1,1) tavgk(6) - 0.50*(t(ixy,i,1)+t(ixy,i-l 1) endif if(i.ne.iyp1)then tavgk(4) - 0.50*(t(ixy,i,1)+t(ixy,i+l,l)) tavgk(8) - 0.50*(t(ixy,i,l)+t(ixy,i+l,1)) endif if(i.ne.l.and.ixy.ne.1)then tavgsd(l) - 0.0625*(9.0*t(ixy,i,l)+3.0*t(ixy-l,i,1)+ ) ) & 3.0*t(ixy,i-l,l)+t(ixy-l,i-l,1)) tavgsd(5) - 0.0625*(9.0*t(ixy,i,l)+3.0*t(ixy-1,i,l)+ & 3.0*t(ixy,i-l,1)+t(ixy-l,i-1,l)) endif if(i.ne.1.and.ixy.ne.ixpl)then tavgsd(2) - 0.0625*(9.0*t(ixy,i,l)+3.0*t(ixy+1,i,1)+ & 3.0*t(ixy,i-l,l)+t(ixy+1,i-l,1)) tavgsd(6) - 0.0625*(9.0*t(ixy,i,l)+3.0*t(ixy+1,i,l)+ & 3.0*t(ixy,i-1,1)+t(ixy+l,i-1,1)) endif if(i.ne.iypl.and.ixy.ne.ixpl)then tavgsd(3) - 0.0625*(9.0*t(ixy,i,1)+3.0*t(ixy+l,i,1)+ & 3.0*t(ixy,i+l,1)+t(ixy+l,i+l,l)) tavgsd(7) - 0.0625*(9.0*t(ixy,i,l)+3.0*t(ixy+l,i,1)+ & 3.0*t(ixy,i+l,1)+t(ixy+l,i+l,1)) endif if(i.ne.iyp1.and.ixy.ne.l)then tavgsd(4) - 0.0625*(9.0*t(ixy,i,1)+3.0*t(ixy-l,i,1)+ & 3.0*t(ixy,i+l,l)+t(ixy-l,i+1,l)) tavgsd(8) - 0.0625*(9.0*t(ixy,i,1)+3.0*t(ixy-1,i,1)+ & 3.0*t(ixy,i+1,l)+t(ixy-l,i+l,1)) endif endif c********************************************************************* c Find Product Properties Cooresponding to Averaged Temperatures c First find appropriate thermal conductivity value 337 Table C.2 (cont'd). do 20 kk - 1,kkxy if(tavgk(kk).eq.0)go to 20 if(tavgk(kk).ge.t0)then ck(i,kk) - kp else if(tavgk(kk).ge.t0-4.0d0)then ck(i,kk) - conduc(tavgk(kk)) else do j - 2,nsc if(nc(kk).eq.1)go to 20 if(tavgk(kk).1e.condt(j))then ck(i,kk) - condc(j-l) nc(kk) - 1 endif enddo endif endif 20 continue C ****************************m************************************** 0 Next find appropriate density and specific heat value do 30 kk - 1,kkxy if(tavgsd(kk).eq.0)go to 30 if(tavgsd(kk).ge.t0)then dc(kk) - dp spc(kk) - cp nc(kk+kk1) - l nc(kk+kk2) - 1 else if(tavgsd(kk).ge.t0-4.0d0)then dc(kk) - dens(tavgsd(kk)) spc(kk) - spheat(tavgsd(kk)) nc(kk+kkl) - l nc(kk+kk2) - 1 else do j - 2,nsd if(nc(kk+kkl).eq.l)go to 25 if(tavgsd(kk).le.denst(j))then dc(kk) - densc(j-l) nc(kk+kk1) - 1 endif enddo 25 do j - 2,nss if(nc(kk+kk2).eq.1)go to 30 if(tavgsd(kk).1e.spht(j))then spc(kk) - sphc(j-l) nc(kk+kk2) - 1 endif enddo endif endif 30 continue 338 Table C.2 (cont'd). C********************************************************************* c Determine surface area for conductive heat transfer c Areas for sweep in the X-direction 40 if(shape.eq.l.and.xy.eq.0)then ark(2) - dx if(i.eq.l.or.i.eq.itp1)ark(2) - dx*0.50d0 if(bc.ne.-1.and.bc.ne.l)ark(l) - dy if(bc.eq.-l.or.bc.eq.l)ark(1) - dy*0.50d0 ark(3) - ark(l) endif if(shape.eq.2.and.xy.eq.0)then if(i.eq.l)ark(2) - pi*dx*(1x0+0.25d0*dx) if(i.eq.itpl)ark(2) - pi*dx*(lx-0.25d0*dx) if(i.ne.l.and.i.ne.itpl)ark(2) - pi*dx*(lx0+(i-l)*dx) ark(l) - 2.0d0*pi*dy*(lx0+((i)-l.50d0)*dx) ark(3) - 2.0d0*pi*dy*(1x0+((i)-0.50d0)*dx) if(bc.eq.-l.or.bc.eq.l)then ark(l) - 0.50d0*ark(l) ark(3) - 0.50d0*ark(3) endif endif c Areas for sweep in the Y-direction if(shape.eq.l.and.xy.eq.l)then ark(2) - dx if(bc.eq.l.or.bc.eq.-1)ark(2) - dx*0.50d0 if(i.ne.1.and.i.ne.itpl)ark(1) - dy if(i.eq.l.or.i.eq.itpl)ark(l) - dy*0.50d0 ark(3) - ark(l) endif if(shape.eq.2.and.xy.eq.1)then if(bc.eq.-1)ark(2) - pi*dx*(lx0+0.250d0*dx) if(bc.eq.1)ark(2) - pi*dx*(lx0-0.250d0*dx) if(bc.eq.0)ark(2) - pi*dx*(lx0+((ixy)-l.0d0)*dx) if(bc.ne.-l)ark(1) - 2.0d0*pi*dy*(lx0+((ixy)-l.5d0)*dx) if(bc.ne.1)ark(3) - 2.0d0*pi*dy*(lx0+((ixy)-0.5d0)*dx) if(i.eq.l.or.i.eq.itpl)then ark(l) - ark(l)*0.50d0 ark(3) - ark(3)*0.50d0 endif endif c Determine Volume Elements for use with Specific Volume if(shape.eq.1)then do jj - 1,2 volsd(jj) - voll*dti enddo endif if(shape.eq.2.and.xy.eq.0)then 339 Table C.2 (cont'd). if(i.ne.1)volsd(1) - v012*(1x0+((i)-1.250d0)*dx)*dti if(i.ne.itpl)volsd(2) - v012*(lx0+((i)-0.750d0)*dx) & *dti endif if(shape.eq.2.and.xy.eq.1)then if(bc.ne.-l)volsd(1) - v012*(1x0+((ixy)-1.250d0)*dx) & *dti if(bc.ne.1)volsd(2) - v012*(lx0+((ixy)-0.750d0)*dx) & *dti endif C********************************************************************* c Multiply thermal conductivity values by areas, and density*specific heat c by volumes c X - sweep if(xy.eq.0)then if(i.ne.1)then ck(i,1) - ck(i,1)*ark(l)*dxi if(bc.ne.-l)then csd(i,1) - dc(l)*spc(1)*volsd(l) endif if(bc.ne.1)then csd(i,4) - dc(4)*spc(4)*volsd(1) endif endif if(i.ne.itp1)then ck(i,3) - ck(i,3)*ark(3)*dxi if(bc.ne.-1)then csd(i,2) - dc(2)*spc(2)*volsd(2) endif if(bc.ne.1)then csd(i,3) - dc(3)*spc(3)*volsd(2) endif endif if(bc.ne.-l)then ck(i,2) - ck(i,2)*ark(2)*dyi endif if(bc.ne.1)then ck(i,4) - ck(i,4)*ark(2)*dyi endif endif c Y - Sweep if(xy.eq.l)then if(bc.ne.-l)then ck(i,1) - ck(i,1)*ark(1)*dxi ck(i,5) - ck(i,5)*ark(l)*dxi if(i.ne.1)then csd(i,1) - dc(l)*spc(1)*volsd(l) csd(i,5) - dc(5)*spc(5)*volsd(l) 340 Table C.2 (cont'd). endif if(i.ne.itpl)then csd(i,4) - dc(4)*spc(4)*volsd(2) csd(i,8) - dc(8)*spc(8)*volsd(2) endif endif if(bc.ne.1)then ck(i,3) - ck(i,3)*ark(3)*dxi ck(i,7) - ck(i,7)*ark(3)*dxi if(i.ne.1)then csd(i,2) - dc(2)*spc(2)*volsd(2) csd(i,6) - dc(6)*spc(6)*volsd(2) endif if(i.ne.itpl)then csd(i,3) - dc(3)*spc(3)*volsd(l) csd(i,7) - dc(7)*spc(7)*volsd(l) endif endif if(i.ne.1)then ck(i,2) - ck(i,2)*ark(2)*dyi ck(i,6) - ck(i,6)*ark(2)*dyi endif if(i.ne.itpl)then ck(i,4) - ck(i,4)*ark(2)*dyi ck(i,8) - ck(i,8)*ark(2)*dyi endif endif 100 continue return end c********************************************************************* c********************************************************************* SUBROUTINE HEADNG(nprint,headtq) parameter(maxp-ZO) integer per,shape,model,sym,sstep,m,eend,day,dead,cyn c Declare all variables in common blocks. Double Precision wf0,ms,dp,kp,cp,t0, &ti,temp(maxp),stor(maxp),htc(maxp,4),tunit(maxp), &lx,lx0,ly,dx,dy, &ea,q0,vea,vq0,tref character title*20,ttlfil*4,outfil*12 Common /prop/wf0,ms,dp,kp,cp,t0, &/bound/per,ti,temp,stor,htc,tunit, &/geom/shape,lx,1x0,ly,dx,dy,symx,symy,cyn,ixpl,iypl,stepy, 341 Table C.2 (cont'd). &/shelf/ea,q0,vea,vq0,tref, &/mod/model, &/ttl/title,ttlfil write(outfil,1000)ttlfil,'out.dat' 1000 format(' ',a,a) open(unit-12,name-outfil(l:12),type-'new',carriagecontrol-'list') write(12,l)title l format(' ',///,3x,'Title: ',a20,/3x,' ----- ',//,l4x,'Input Para', +'meters',/,l4x,l6('-')//) if(model.ge.3)then write(12,3) 3 format(' ','Kinetic Parameters') write(12,4)q0 format(' ',/,' ',2x,'Reference shelf-life (days) ............ ', &f7.l) abcd-tref-273.150d0 write(12,5)abcd 5 format(' ',2x,'Reference temperature (C) .............. ',f6.2) abcd-ea/1000.0d0 write(12,6)abcd 6 format(' ',2x,'Activation energy constant (kJ/mole)...',f8.2) abcd—vq0**0.50d0 if(model.eq.4)then write(12,8)abcd 8 format(' ',2x,'St. dev. of ref. shelf-life (days) ..... ',f6.2) abcd-vea**0.50d0/1000.0d0 write(12,9)abcd D 9 format(' ',2x,'St. dev. of ea (kj/mole) ............... ',f6.2) endif endif write(12,10) 10 format(' ',/,' ','Unfrozen Product Properties',/) abcd-wf0*100.0d0 write(12,11)abcd 11 format(' ',2x,'Moisture content (%) ................... ',f6.2) abcd-t0-273.150d0 write(12,12)abcd 12 format(' ',2x,'Initial freezing temperature (C) ....... ',f6.2) write(12,13)ms 13 format(' ',2x,'Molecular weight of solids (kg/mole)...',f8.2) write(12,14)dp 14 format(' ',2x,'Unfrozen product density (kg/m33) ...... ',f8.2) write(12,15)kp 15 format(' ',2x,'Thermal conductivity (W/mK) ............ ',f6.3) write(12,16)cp 16 format(' ',2x,'Specific heat (kJ/kgK) ................. ',f7.3) abcd-ti-273.150d0 write(12,17)abcd 17 format(' ',/,' ','Initial Condition:',/,' ',2x,'Product temp.’ +,' (C) at time-0 ........... ',f6.2) c product geometry if(symy.eq.1)ly - ly*2.0d0 342 Table C.2 (cont'd). if(shape.eq.1)then if(symx.eq.1)lx - lx*2.0d0 write(12,18)lx,ly 18 format( ' ',/,' ','Slab Geometry:',/,' ',2x,'width', & ': x direction (m)',17('.'),f9.5,/,' ',2x,'height or length', & ': y direction (m)',6('.'),f9.5) else if(lx0.eq.0.0)then write(12,20)lx,ly 20 format(' ',/,' ','Cyclindrical Geometry:',/,' ',2x,'radius (m)', & 29('.'),f9.5,/,' ',2x,'height or length: y direction (m)', & 6('.'),f9.5) else write(12,22)lx0,1x,ly 22 format(' ',/,' ','Cyclindrical Geometry:',/,' ',2x,'inner radius & '(m)',23('.'),f9.5,/,' ',2x,'outer radius (m)',23('.'),f9.5, & /,' ',2x,'height or length: y direction (m)',6('.'),f9.5) endif endif c boundary conditions do 50 i-1,per write(12,24)i 24 format(' ',/,' ','Boundary Conditions for Period ',12,':',/) abcd-stor(i)/tunit(i) if(tunit(i).eq.3600.0d0)then write(12,25)abcd 25 format(' ',2x,'Storage time(hours) .................... ',f7.2) else write(12,26)abcd 26 format(’ ',2x,'Storage time (days) .................... ',f7.2) endif abcd-temp(i)-273.150d0 write(12,27)abcd 27 format(' ',2x,'Storage temperature (C) ................ ',f6.2,/) write(12,28) 28 format(' ',2x,'Convective heat transfer coeff. (W/m“2K):') if(shape.eq.1)then write(12,30)htc(i,l) 30 format(' ',4x,'in the x-direction: ',/,' ',6x, & 'at x - 0.0 ........................ ',f7.2) if(symx.eq.0)then write(12,32)lx,htc(i,3) 32 format(' ',6x,'at x - ',f9.5,' .................. ',f7.2) endif else if(cyn.eq.1)then write(12,34)lx0,htc(i,1) 34 format(' ',4x,'in the radial direction: ',/,' ',6x, & 'at x - ',f9.5,' (inner radius)','....',f7.2) endif write(12,36)1x,htc(i,3) 36 format(' ',6x,'at x - ',f9.5,' (outer radius)','....',f7.2) endif 343 Table C.2 (cont'd). write(12,38)htc(i,2) 38 format(' ',4x,'in the y-direction: ',/,' ',6x, & 'at y - 0.0 ........................ ',f7.2) if(symx.eq.0)then write(12,40)ly,htc(i,4) 40 format(' ',6x,'at y - ',f9.5,' .................. ',f7.2) endif 50 continue return end C uh?******************************************************************* SUBROUTINE OUTPUT(t,time,ptime,ii,nprint,tavg,qual,qavg,vq) parameter(maxx - 31) integer ixpl,iypl,ii,model,per,shape,symx,symy,stepy,cyn double precision t(maxx,maxx,3),time,ptime,1x,1x0,ly,dx,dy, &abc(10),qual(maxx,maxx),tavg,qavg,vq,ea,q0,vea,vq0,tref,hr,phr, &ttime,pttime,day,pday character title*20,ttlfil*4,outdat*15,hhll*29,hh22*21 common/ttl/title,tt1fil, &/geom/shape,lx,lx0,ly,dx,dy,symx,symy,cyn,ixpl,iypl,stepy, &/shelf/ea,q0,vea,vq0,tref, &/mod/mode1 C NPRINT - 0 if printing initial conditions at the beginning of first C storage period. C NPRINT - 1 if printing temperature distribution and/or quality C distributions at the end/beginning of itermentent storage C periods. C NPRINT - 2 If printing temperature distribution and/or quality C distributions at itermentent times during a storage period. C NPRINT - 3 if printing temperature distribution and/or quality C distributions at the end of last storage period. write(12,20)title 20 format(///,10x,'Product - ',a,//) ttime - time+ptime day - ttime/86400.0d0 hr - ttime/3600.d0 pday - ptime/84600.0d0 phr - ptime/3600.0d0 if(nprint.eq.0)then write(12,30) 30 format(le,'These are the initial conditions at the beginning ', & 'of the ',/,13x,'first storage period.',/) 344 Table C.2 (cont'd). else if(nprint.eq.l)then write(12,40) & 'Total elapsed time .................... ',day,'days',hr,'hrs', & 'Beginning of storage period .................. ',ii, & 'End of storage period ........................ ',ii-l 40 format(le,a,f6.2,a,f8.2,a,/,2(10x,a,i3,/)) else if(nprint.eq.2)then write(12,50) & 'Total elapsed time ............... ',day,'days',hr,'hrs -', & ttime,'sec', & 'Storage Period ............................... ',ii, & 'Elapsed time from beginning of', & ' storage period ..................... ',pday,'days',phr,'hrs' 50 format(le,a,f6.2,a,f8.2,a,f8.1,a,/,10x,a,13,2(/,10x,a),f6.2,a, &f8.2,a,/) else if(nprint.eq.3)then write(12,60) & 'Total elapsed time .................... ',day,'days',hr,'hrs', & 'End of last storage period ................... ',ii 60 format(le,a,f6.2,a,f8.2,a,/,10x,a,i3,/) endif if(shape.eq.1)then if(symx.eq.l.and.symy.eq.0)then write(12,70) 70 format(le,'Note: Distribution is symmetrical in the x direction;' & /,l3x,'results are shown for half-thickness only.'/) else if(symx.eq.1.and.symy.eq.l)then write(12,80) 80 format(le,'Note: Distribution is symmetrical in both ', & 'x and y dimensions;',/,l6x,'results are shown for ', & 'half-thicknesses only.'/) endif endif if(shape.eq.2)then if(cyn.eq.0.and.symy.eq.1)then write(12,90) 90 format(le,'Note: Distribution is symmetrical in both ', & 'radial and y dimensions;',/,l6x,'results are shown for ', & 'half-diameter and half-thickness only.'/) else if(cyn.eq.1.and.symy.eq.l)then write(12,100) 100 format(le,'Note: Distribution is symmetrical in the y dir', & 'ection;'/,16x,'resu1ts are shown for half-thickness only.'/) endif endif hh22-'DISTRIBUTION HISTORY' if(model.eq.2)then hh11-' TEMPERATURE (C) ' else hhll-'TEMPERATURE (C) & QUALITY (%)' endif write(12,110)hhll,hh22 110 format(/,29x,a,/,33x,a,/,29x,29('-'),/) Table 120 130 140 150 160 170 180 190 200 345 C.2 (cont'd). write(12,120) format(37x,'y—position (m)',/,1lx,'x-position (m)|') if(iypl.eq.3)then nstep - 3 else nstep - 5 endif do i - 1,nstep abc(i)-(i-1)*stepy*dy enddo if(iypl.eq.3)then write(12,130)abc(l),abc(2),abc(3) write(12,150) else write(12,140)abc(l),abc(2),abc(3),abc(4),abc(5) write(12,160) endif format(25x,'|'3(f8.4)) format(25x,'|'5(f8.4)) format(le,42('—')) format(le,58('-')) if(model.ge.3)then c8-100.0d0/(86400.0d0*q0) do i - 1,ixpl do j - l,iypl if(qual(i,j).lt.0.0d0)dead-l enddo enddo endif stepx - (ixpl-l)/4 do i - 1,nstep do j - 1,nstep abc(j) - t((i-1)*stepx+1.(J-1)*stepy+l,3)-273.15d0 if(model.ge.3)then abc(j+nstep)-qual((i-1)*stepx+1,(j—1)*stepy+l)*c8 endif enddo if(nstep.eq.3)then write(12,170)(i-1)*stepx*dx,abc(l),abc(2),abc(3) else write(12,180)(i-l)*stepx*dx,abc(l),abc(2),abc(3),abc(4), abc(5) endif format(l6x,f8.4,' |',3(f7.2,1x),'C') format(l6x,f8.4,' |',5(f7.2,1x),'C') if(model.eq.3)then if(nstep.eq.3)then write(12,190)abc(4),abc(5),abc(6) else write(12,200)abc(6),abc(7),abc(8),abc(9),abc(10) endif format(25x,’|',3(f7.2,1x),'%') format(25x,’|',6(f7.2,1x),'%') endif 346 Table C.2 (cont'd). enddo if(iypl.eq.3)then write(12,150) else write(12,160) endif write(12,210)tavg-273.15 210 format(///,10x,'Average temperature (C) - ',f7.2) if(model.ge.3)then write(12,220)qavg*100.0d0/(86400.0d0*q0) 220 format(/,10x,'Average quality (% ref. quality) - ',f7.2) if(model.eq.4)write(12,230)(vq)**0.50d0*c8 230 format(/,10x,'Average st. dev. of quality ', & '(% of ref. quality) - ',e7.1) endif if(dead.eq.1)then write(12,240) 240 format(/,10x,'She1f-life was exceeded at some point on body.') endif if(nprint.eq.3)then write(12,250)'End of date file.’ 250 format(/,10x,a,i3,a) close(unit-12) endif return end APPENDIX D APPENDIX D SURFACE HEAT TRANSFER COEFFICIENT ESTIMATION PROGRAM The program used to estimate surface heat transfer coefficients using the sequential regularization procedure, discussed in Chapter 3, is presented here. An outline of the program is given in Table D.l, and the listing for the program, written in Fortran 77 for a Vax 11/750 is given in Table D.2. 347 348 Table D.l Description of Surface Heat Transfer Coefficient Estimation Program. Subroutine Title PROGRAM FREEZE SUBROUTINE PROPER DOUBLE PRECISION FUNCTION MOIST(X) DOUBLE PRECISION FUNCTION DENS(X) DOUBLE PRECISION FUNCTION KI(X) DOUBLE PRECISION FUNCTION CONDUC(X) DOUBLE PRECISION FUNCTION SPHEAT(X) SUBROUTINE CONSPR SUBROUTINE INTEGR BLOCK DATA CONST SUBROUTINE INPUTl SUBROUTINE SOLN SUBROUTINE COEFF SUBROUTINE PFIND SUBROUTINE OUTPUT SUBROUTINE SIMUL SUBROUTINE RAND Description Main program; contains program menu. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. See Table B.1. Allows interactive input of ambient and internal product temperature measurements, and product geometry. Writes output to data file. Computes surface heat flux and surface heat transfer coefficients as a function of time using the sequential reqularization procedure. Calls output subroutine. Determines matrix coefficients used in first sweep in ADI finite difference algorithm. See Table B.1. Writes estimated surface heat flux and surface heat transfer coefficients to output data file. Matrix inversion subroutine. Generates normally distributed random numbers used in determining optimal parameters. 349 Table D.2 Computer Code Listing for Surface Heat Transfer Coefficient Estimation Program. PROGRAM IHCPlD c***************************************************************** c***************************************************************** C C C C Surface Heat Transfer Coefficient Estimation Program by Elaine Scott 1986 c***************************************************************** 0 0 0 0 0 0 0 0 0 0 0 0 0 0 This program estimates the surface heat transfer coefficient as a function of time during frozen food storage. The program assumes one dimensional heat transfer, with a convective boun- dary condition at x - 0, and an insulated boundary at x - L. Input property parameters include unfrozen product density, thermal conductivity and specific heat. The initial freezing temperature or molecular weight of solids also is required to predict these values for the frozen food product. Boundary conditions are found from internal temperature meas— urements and known ambient temperatures. The input data file includes product geometry; number of nodes; number of time steps for both data points and for finite difference calulations; number of thermocouples and location; and ambient temperature and thermocouple reading at each time step. c***************************************************************** parameter(maxt-250,maxm-41,maxd—lOl,maxc=51,maxs—201) integer model,ynavg double precision wf0,ms,dp,kp,cp,t0, &ds, &th,tl,avgd,avgk,avgc character title*40,ttlfil*10,filyn1*l,filyn2*l,filyn*1,fildat*16, &inpdat*l6 common/mod/model, &/ttl/title,ttlfil, 350 Table D.2 (cont'd). &/Pr0P/Wf0.ms.dp.kp.CP.t0. &/d/ds, &/pavg/th,tl,avgd,avgk,avgc,ynavg write(5,1000) 1000 format('l',72('*'),/,'0',t20,'Heat Transfer Coefficient ', &'Estimation',/,'0',t35,'by',/,'0',t30,'E1aine Scott',/,'0',t24, &'Michigan State',’ University',/,'0',t32,'May 1986',/,'0',72('*')) WRITE(5,100) 100 FORMAT('O','Program Menu:',//,' ',' 1. Product properties (<0C)’ &,/,' ',' 2. Estimate heat transfer coefficient h', &/,'0',' h - Surface heat trans. coef.', &//,' ','Selection? ') READ(5,10)mode1 10 FORMAT(Il) write(5,300) 300 format(' ',/,' ','Key word for data files; 6 Characters: ') READ(5,20)TTLFIL 20 FORMAT(A) if(model.eq.l)then filynl - 'n' else write(5,400) 400 format(' ',/,' ','Are product properties approximations',/,' ',2x, &'with temperature stored on file? (y/n) ') read(5,20)filynl write(5,500) 500 format(' ',/,' ','Are input initial and boundary conditions',/,' ' &,2x,'and geometrical dimensions stored on file? (y/n) ') read(5,20)filyn2 endif if(filynl.eq.'n'.or.filyn1.eq.'N')then call proper CALL CONSPR endif if(model.ne.1)then if(filyn2.eq.'n'.or.filyn2.eq.'N')then call inputl endif call soln endif end SUBROUTINE PROPER c See Appendix B. 351 Table D.2 (cont'd) DOUBLE PRECISION FUNCTION MOIST(X) c See Appendix B. DOUBLE PRECISION FUNCTION DENS(X) c See Appendix B. DOUBLE PRECISION FUNCTION KI(X) c See Appendix B. DOUBLE PRECISION FUNCTION CONDUC(X) c See Appendix B. DOUBLE PRECISION FUNCTION SPHEAT(X) c See Appendix B. SUBROUTINE CONSPR c See Appendix B. SUBROUTINE INTEGR(thi,tlow,avgdp,avgkp,avgcp,ncase) c See Appendix B. 352 Table D.2 (cont'd) BLOCK DATA CONST c See Appendix B. SUBROUTINE INPUTl This subroutine provides the input for the boundary condi- tions on the product for the case where the ambient temper- ature and surface heat tranfer coeffient are known and assumed to be constant over a given storage period. 0000 Input varibles include, initial product temperature, sym- metry of boundary conditions, number of constant temperature storage periods, length of storage period, and surface heat transfer coefficient. 0000 parameter(maxt - 250,maxtc - 10) integer shape,mpli,ntime,ntc,ndt double precision delt,tamb(maxt),tc(maxt,maxtc),xtc(maxtc),L character yn*l,title*40,ttlfil*10,fildat*l6,inpdat*l6 common /ttl/tit1e,ttlfil, &/datfil/fildat,inpdat save write(6,l) l format(' ',/,' ','Product: ') READ(5,2)TITLE 2 format(a) c input geometry and size 5 write(6,10) 10 format('l','Geometry',/,' ',8('-'),/'0','Enter product geometry: ' &,/,' ',5x,’1 - slab',/,' ',5x,'2 - cylinder',/,' ',5x, &'3 - sphere') read*,shape if(shape.eq.1)then write(6,20) 20 format(' ',/,' ','Enter dimensions for slab;',/,' ',5x, &'thickness in direction of heat transfer (m) : ') read*,l else if(shape.eq.2)then write(6,30) 353 Table D.2 (cont'd) 30 format(' ',/,' ','Enter dimensions for cylinder',/, &' ',5x,'radius (m) : ') read *,1 else write(6,40) 40 format(' ',/,' ','Enter dimensions for sphere (m)',/, &' ',5x,'radius (m) : ') read *,1 endif endif write(5,50) 50 format(' ',/,' ','Enter total number of nodes in F.D. ', &'calculations : ') read*,mpli write(5,60) 60 format('O','Storage Conditions',/,' ',18('-')) write(6,70) 70 format('O',/,' ','Enter total number of temperature ', &'measurements ',/,' ','(include time - 0) : ') read*,ntime write(6,80) 80 format(' ',/,' ','Enter time increment between each ', &'temperature measurement (sec) : ') read*,delt write(6,85) 85 format(' ',/,' ','Enter number of time steps per temperature', &' measurement interval ',/' ','(for F.D. calculations) : ') read*,ndt write(6,90) 90 format(' ',/,' ','Enter total number of thermocouples : ') read*,ntc write(6,100) 100 format(' ',/,' ','Are these values correct? (y/n) ') read(5,110)yn 110 format(a) if(yn.ne.'y'.and.yn.ne.'Y')goto 5 write(6,120) 120 format(' ',/,' ','Enter location of each thermocouple.’/) do itc - 1,ntc write(6,130)itc 130 format(' ','Location of T.C. No. ',12,'(m) : ') read*,xtc(itc) enddo write(6,140) 140 format(' ',/,' ',’Enter ambient temp. and each thermocouple ', &'measurement (C) ',/,' ','for every time step.',/) do it - 1,ntime write(6,150)it-l 150 format(' ','Time step:',i3,5x,'tamb- ') read*,tamb(it) do itc - 1,ntc write(6,160)itc 160 format(' ','T.C.(',i2,')- ') read*,tc(it,itc) enddo enddo 354 Table D.2 (cont'd) do i - 1,ntime if(tamb(i).ne.0.)then tamb(i) - tamb(i)+273.150d0 endif do itc - 1,ntc if(tc(i,itc).ne.0.)then tc(i,itc) - tc(i,itc)+273.150d0 endif enddo enddo write(inpdat,170)ttlfil,'ihcpinp.dat' 170 format(' ',a,a) open(unit-l2,name-inpdat(l:16),type-‘new',carriagecontrol-'list') write(12,*)title write(12,*)shape,L,mpli write(12,*)ntime,de1t,ndt write(12,*)ntc do i - 1,ntc write(12,*)xtc(i) enddo do i - 1,ntime write(12,*)tamb(i) do itc - 1,ntc write(12,*)tc(i,itc) enddo enddo close(unit-l2) return end SUBROUTINE soln parameter(maxd - 101, maxc - 51, maxs - 201, iterat - l) parameter(maxm-21,maxtc-6,maxr—15,maxrpl-l6,maxt—250,r-8.3140d0) parameter(maxrdt - 500) integer shape,mpli,ntime,ntc,rfts,mx,ynavg,mistc,irfts, &rfti,ntci,yn double precision a(maxm),b(maxm),c(maxm),da(maxm),db(maxm), &dc(maxm),dz(maxm),dt(maxm),cc(maxm),ddt(maxm),ddx(maxm), &t(maxm,maxrdt),x(maxm,maxrdt),q1(maxrpl),q2(maxt),qlstar, &xk(maxtc,maxr),ds,htc,xmat(maxr,maxr,maxtc),xtx(maxr,maxrpl), &h0(maxr,maxr),hl(maxr,maxr),h2(maxr,maxr),sum,sum2, &tt(maxr,maxtc),xx(maxr,maxtc),xmatt(maxr,maxr,maxtc),qq(maxr), &sumhl,sumh2,tstar(maxm,maxrdt),ratiox,ttime,sumhtc,htca,shtc c Declare variables in common blocks double precision wf0,ms,dp,kp,cp,t0, &l,dxi, 355 Table D.2 (cont'd) &DENST(maxd),DENSC(maxd),CONDT(maxc),CONDC(maxc),SPHT(maxs), &SPHC(maxs), &delt, &tamb(maxt),tc(maxt,maxtc),xtc(maxtc), &tl,th,avgd,avgk,avgc, &alpha,w0,wl,w2 &sumtl,sumt2,ctl,ct2,tal,ta2 character title*40,ttlfil*10,fildat*16,inpdat*l6 common/ttl/title,ttlfil, &/geom/shape,l,dxi, &/datfil/fildat,inpdat, &/NCONSTP/NSD,NSC,NSS, &/CONSTP/DENST,DENSC,CONDT,CONDC,SPHT,SPHC, &/pr0p/wf0.ms.dp.kp.CP.t0./d/ds. &/meas/delt,ntime,ntc,ndt fi/measZ/tamb,tc,xtc, &/pavg/tl,th,avgd,avgk,avgc,ynavg, &/ihcp/irfts,alpha,w0,wl,w2 save c Read in boundary and initial conditions write(inpdat,600)ttlfil,'.dat' 600 format(' ',a,a) open(unit-12,name-inpdat(1:l6),type-'old',carriagecontrol-'list') read(12,*)tit1e read(12,*)shape,l,mp1i read(12,*)ntime,delt,ndt read(12,*)ntc read(12,*)xtc(1),xtc(2),xtc(3) read(12,*)xtc(4),xtc(5),xtc(6) do i - 1,ntime read(12,*)tamb(i),tc(i,1),tc(i,2),tc(i,3),tc(i,4),tc(i,5),tc(i,6) if(tamb(i).ne.0.0)then tamb(i) - tamb(i)+273.15d0 endif sum - 0.0d0 sumx - 0.0d0 do j - 1,ntc if(tc(i,j).ne.0.0)then tc(i,j) - tc(i,j)+273 15d0 endif sum - sum +tc(i,j) sumx - sumx+ xtc(j) enddo enddo close(unit-l2) write(5,11) 11 Format(' ','Average ambient temperature values? (l-y,0-n) ') 356 Table D.2 (cont'd) read(5,*)yn if(yn.eq.l)then sumtl - 0 sumt2 - 0 ctl - 0 ct2 - 0 do i - 1,ntime if(tamb(i).gt.258.15)then sumtl - sumtl+tamb(i) ctl - ctl+l else sumt2 - sumt2+tamb(i) ct2 - ct2+l endif enddo tal - sumtl/ctl ta2 - sumt2/ct2 do i - 1,ntime if(tamb(i).gt.258.15)then tamb(i) - tal else tamb(i) - ta2 endif enddo endif c Read in constant property assumptions WRITE(FILDAT,310)'ld_t1cPRP.DAT' 310 FORMAT(' ',a) OPEN(UNIT-l2,NAME-FILDAT(1:16),TYPE-'OLD',CARRIAGECONTROL-‘LIST') READ(12,*)WFO,T0,MS READ(12,*)DP,KP,CP READ(12,*)NSD,NSC,NSS READ(12,*)tl,th,avgd,avgk,avgc,ynavg DO I-1,NSD READ(12,*)DENST(I),DENSC(I) ENDDO DO I-1,NSC READ(12,*)CONDT(I),CONDC(I) ENDDO DO I-1,NSS READ(12,*)SPHT(I),SPHC(I) ENDDO CLOSE(UNIT-l2) c Check for bad T.C. do i - 1,ntime mistc - 0 sum - 0.0d0 do itc - 1,ntc sum - sum + tc(i,itc) if(tc(i,itc).eq.0.0)then mistc - mistc + 1 endif 357 Table D.2 (cont'd) enddo do itc - 1,ntc if(tc(i,itc).eq.0.0)then tc(i,itc) - sum/(ntc-mistc) endif enddo if(tamb(i).eq.0.0)then if(i.eq.l)then ii - l 4 if(tamb(ii+1).ne.0.0)then do jj - 1,ii tamb(jj) - tamb(ii+l) enddo else ii - ii+1 go to 4 endif else ii - i 6 if(tamb(ii+l).ne.0.0)then tamb(i) - (tamb(i-l)+tamb(ii+l))/2.0d0 else ii - ii+l go to 6 endif endif endif enddo c Regularization components write(6,3) 3 format(' ',/,' ','Enter alpha,w0,wl,w2: ',$) read*,alpha,w0,wl,w2 write(6,2) 2 format(' ',/,' ','Enter number of future time steps- ',$) read*,irfts rfts - irfts dti - delt/ndt mi - mpli-l dxi - L/mi ndtt - rfts*ndt ntci - ntc do i - l,rfts q1(i) - 0.0d0 do j - l,rfts if(i.ne.j)then h0(i,j) - 0.0d0 else h0(i,j) - 1.0d0 endif 358 Table D.2 (cont'd) O hl(i,j) - 0.0d0 h2(i,j) - 0.0d0 enddo if(i.lt.rfts)then hl(i,i) - -1.0d0 h1(i,i+l) - 1.0d0 endif if(i.lt.rfts-1)then h2(i,i) - 1.0d0 h2(i,i+1) - -2.0d0 h2(i,i+2) - 1.0d0 endif enddo htc - 0.0d0 Assume value for ql*: Use q1* - 0.0 qlstar - 0.0d0 call output(0,q1,htc,l) finite difference solution do 100 it - 1,ntime-1 q2(it) - 0.0d0 Assume value for ql*: Use ql* - ql(1) if(it.gt.1)then call output(l,ql,htc,it,0.) endif Initialize Tstar and set sensitivity coefficients equal to xero. if(it.eq.l)then sum - 0.d0 ntci - ntc do itc - 1,ntc if(tc(l,itc).eq.0.0d0.and.ntc.ne.l)then ntci - ntc-l else sum - sum+tc(l,itc) endif enddo do i - 1,mpli Tstar(i,1) - sum/ntci t(i,l) - sum/ntci x(i,l) - 0.0d0 enddo if(tamb(l).eq.0.0)tamb(l) - tamb(2) endif Table D.2 359 (cont'd) c thomas algorithm c find coefficients for thomas algorithm call coeff(t,dti,mpli,a,b,c,da,db,dc) do ir - l,rfts do idt - 1,ndt 9' 9‘ indt - ndt*(ir-l)+idt if(ir.eq.1)then dt(l) - db(1)*tstar(l,indt)+dc(l)*tstar(2,indt) -qlstar*0.50d0 else dt(l) - db(l)*tstar(l,indt)+dc(l)*tstar(2,indt)-qlstar endif if(idt.eq.l)then dz(1) - db(1)*x(1,indt)+dc(l)*x(2,indt)-0.5d0 else dz(1) - db(1)*x(1,indt)+dc(1)*x(2,indt)-l.0d0 endif do i - 2,mpli-1 dt(i) - da(i)*tstar(i-1,indt)+db(i)*tstar(i,indt) +dc(i)*tstar(i+1,indt) dz(i) - da(i)*x(i-1,indt)+db(i)*x(i,indt) +dc(i)*x(1+1,indt) enddo if(ir.eq.1)then dt(mpli) - da(mpli)*tstar(mpli-l,indt)+db(mpli) *tstar(mpli,indt)-0.Sd0*(q2(it)+q2(it+l)) else dt(mpli) - da(mpli)*tstar(mpli-l,indt)+db(mpli) *tstar(mpli,indt)-q2(it+l) endif dz(mpli) - da(mpli)*x(mpli-l,indt)+db(mpli)*x(mpli,indt) 00(1) - C(1)/b(1) ddt(l) - dt(1)/b(l) ddX(1) - dZ(1)/b(1) do k - 2,mpli kk - k-l cc(k) - c(k)/(b(k)-a(k)*cc(kk)) ddt(k) - (dt(k)-a(k)*ddt(kk))/(b(k)-a(k)*cc(kk)) ddx(k) - (dz(k)-a(k)*ddx(kk))/(b(k)-a(k)*cc(kk)) enddo tstar(mpli,indt+1) - ddt(mpli) x(mpli,indt+l) - ddx(mpli) do k - 2,mpli kk - mpli-k+l tstar(kk,indt+l) - ddt(kk)-cc(kk)*tstar(kk+l,indt+1) x(kk,indt+l) - ddx(kk)-cc(kk)*x(kk+1,indt+l) enddo enddo enddo 360 Table D.2 (cont'd) do itc - 1,ntc do i - 2,mpli if(xtc(itc).lt.(i-1)*dxi)then mx - i ratiox - (xtc(itc)-(i-2)*dxi)/dxi go to 5 endif enddo 5 do ir - 2,rfts+1 irr - (ir-l)*ndt+l tt(ir-1,itc) - tstar(mx-1,irr)+ & ratiox*(tstar(mx,irr)-tstar(mx-1,irr)) xx(ir,itc) - x(mx-l,irr)+ratiox*(x(mx,irr)-x(mx-1,irr)) enddo enddo c ........................................................... c Calculate q from measured temperatures do itc - 1,ntc do ii - l,rfts do jj - 1,ii xmat(ii,jj,itc) - xx(ii-jj+2,itc) enddo enddo enddo do itc - 1,ntc do ii - l,rfts do jj - l,rfts xmatt(ii,jj,itc) - xmat(jj,ii,itc) enddo enddo enddo do ii - l,rfts sum2 - 0.0d0 do jj - l,rfts+1 sum - 0.0d0 sumhl - 0.0d0 sumh2 - 0.0d0 do itc - 1,ntc do j - l,rfts if(jj.le.rfts)then sum - sum + xmatt(ii,j,itc)*xmat(j,jj,itc) else sum2 - sum2 + (tc(it+j,itc)-tt(j,itc)) & *xmatt(ii,j,itc) endif if(itc.eq.1)then sumhl - sumhl + hl(ii,j)*hl(jj.j) sumh2 - sumh2 + h2(11,j)*h2(Jj.j) endif enddo enddo if(jj.le.rfts)then xtx(ii,jj) - sum + alpha*(w0*h0(ii,jj)+wl*sumhl+w2*sumh2) 361 Table D.2 (cont'd) else xtx(ii,jj) - sum2 endif enddo enddo c Call Gauss elimination subroutine call simul(xtx,qq,rfts,l) do ir - l,rfts ql(ir+l) - qlstar+qq(ir) enddo c Knowing ql, reevaluate d(1) and repeat back substition to find c t. do ir - 1,ndt if(it.eq.l.or.ir.ne.1)then dt(l) - db(l)*t(l,ir)+dc(l)*t(2,ir)-q1(2) else if(ir.eq.1)then dt(l) - db(1)*t(l,ir)+dc(l)*t(2,ir)-0.5d0*ql(l) & -0.5d0*q1(2) endif endif do i - 2,mi dt(i) - da(i)*t(i-l,ir)+db(i)*t(i,ir)+dc(i)*t(i+l,ir) enddo if(ir.eq.1)then dt(mpli) - da(mpli)*t(mi,ir)+db(mpli)*t(mpli,ir) & -0.5d0*(q2(it)+q2(it+l)) else dt(mpli) - da(mpli)*t(mi,ir)+db(mpli)*t(mpli,ir) & -q2(it+l) endif ddt(1)-dt(1)/b(l) do k-2,mpli kk-k—l ddt(k)-(dt(k)-a(k)*ddt(kk))/(b(k)-a(k)*cc(kk)) enddo t(mpli,1+ir)-ddt(mpli) do k-2,mpli kk-mi-k+2 t(kk,l+ir)-ddt(kk)-cc(kk)*t(kk+l,l+ir) enddo enddo C 0 Estimate the heat transfer coefficient from q and t. sum - 0.0d0 do i - l,ndt+l if(i.eq.l.or.i.eq.ndt+1)then sum - sum+0.50d0*t(l,i) 362 Table D.2 (cont'd) else sum - sum+t(l,i) endif enddo sum - sum/ndt htc - ql(2)/((tamb(it+l)+tamb(it))*0.50d0-sum) c end of finite difference calculations c ***********************************************'k********* c Initialize t and ql ql(1) - ql(2) if(it.eq.ntime-l)call output(l,ql,htc,ntime) do i - 1,mpli t(i,1) - t(i,ndt+1) tstar(i,l) - t(i,ndt+1) enddo if(it.ge.ntime-irfts)then rfts - rfts-l endif 100 continue c printout call output(2,q1,htc,it) return end SUBROUTINE COEFF(t,dti,mpli,a,b,c,da,db,dc) parameter(maxm-Zl,maxp-lO,maxd—lOl,maxc=51,maxs=201,maxr-15) parameter(maxrdt - 500) integer shape,mi,mpli,ii,nsd,nsc,nss,ynavg double precision beta,nu,omega,gama, &aar,ar(maxm),arl(maxm),area,avgl,avg2,ck(maxm), &csd(maxm,2),a(maxm),b(maxm),c(maxm),da(maxm),db(maxm),dc(maxm), &t(maxm,maxrdt),z(maxm,maxrdt),dti,pi,dxx,dtt c Declare variables in common statements 363 Table D.2 (cont'd) double precision l,dxi, &wf0,ms,dp,kp,cp,t0, &DENST(maxd),DENSC(maxd),CONDT(maxc),CONDC(maxc),SPHT(maxs), &SPHC(maxs),ds, &th,t1,avgd,avgk,avgc common/geom/shape,l,dxi, &/pr0p/wf0.ms.dp.kp.cp.t0. &/CONSTP/DENST,DENSC,CONDT,CONDC,SPHT,SPHC, &/NCONSTP/NSD,NSC,NSS, &/d/ds, &/pavg/tl,th,avgd,avgk,avgc,ynavg pi - dacos(-1.0d0) c weighting functions for finite difference method c modified crank-nicolson method c weight. coeff. for d2t/dx2 c for time t: beta-0.50d0 c for time t+l: nu-0.50d0 c weight. coeff. for dt/dt c for time t: omega--l.0d0 c for time t+1: gama-1.0d0 mi - mpli-l dxx-l.0d0/dxi dtt-l.0d0/(2.0d0*dti) if(shape.eq.2)then aar-2.0d0*pi else if(shape.eq.3)then aar-4.0d0*pi endif endif do 10 i-l,mpli c slab if(shape.eq.1)then ar(i)-1.0d0 arl(i)-l.0d0 else c cylinder 364 Table D.2 (cont'd) if(shape.eq.2)then ar(i)-aar*(i-1)*dxi arl(i)-ar(i)+aar*dxi/2.0d0 else c sphere ar(i)-aar*((i-l)*dxi)**2.0d0 arl(i)-aar*((i-1)*dxi+dxi/2.0d0)**2.0d0 endif endif 10 continue CALL PFIND(T,Mi,CK,CSD,dti,dxi) c ********************************************************** c lst boundary point AVGl - (AR(l)+AR1(l))*0.50d0 a(1) - 0.0d0 c(l) - nu*dxx*CK(1)*arl(l) b(l) - -gama*CSD(l,l)*avg1-c(l) da(l) - 0.0d0 dc(l) - -beta*dxx*CK(l)*ar1(l) db(l) - omega*CSD(l,l)*avg1-dc(l) c ********************************************************** c interior points do i-2,mi AVGl - (AR(I)+AR1(I))*0.50dO AVG2 - (AR(I)+AR1(I-1))*0.50d0 a(i) - nu*dxx*CK(I-1)*arl(i-1) c(i) - nu*dxx*CK(I)*arl(i) b(i) - -gama*(CSD(I,1)*avg2+CSD(I,2)*avgl)-a(i)-c(i) da(i) - -beta*dxx*CK(I-l)*arl(i-l) dc(i) - -beta*dxx*CK(I)*arl(i) db(i) - omega*(CSD(I,l)*avg2+CSD(I,2)*avgl)-da(i)-dc(i) enddo c *********************************************************** c 2nd boundary point AVG2 - (AR(mpli)+AR1(Mi))*0.50d0 365 Table D.2 (cont'd) c(mpli) - 0.0d0 a(mpli) - nu*dxx*CK(Mi)*arl(mi) b(mpli) - -gama*CSD(mpli,2)*avg2-a(mpli) dc(mpli) - 0.0d0 da(mpli) - -beta*dxx*CK(Mi)*arl(mi) db(mpli) - omega*CSD(mpli,2)*avg2-da(mpli) return end SUBROUTINE PFIND(T,Mi,CK,CSPD,dti,dx) C See Appendix A SUBROUTINE OUTPUT(nprint,ql,htc,it) parameter(maxp-lO,maxm-Zl,maxr-lS,maxt-250,maxtc-6) integer shape,ntime,ntc,irfts double precision ql(maxr),htc,ttc(6),ttamb c Declare variables in common statements double precision wf0,ms,dp,kp,cp,t0, &l,dxi, &delt, &tamb(maxt),tc(maxt,maxtc),xtc(maxtc), &alpha,w0,wl,w2 character title*40,ttlfil*10,outfil*20,hhll*30,hh22*21 common/ttl/title,ttlfil, &/Pr0P/wf0.ms.dp.kp.cp.t0. &/geom/shape,l,dxi, &/meas/delt,ntime,ntc,ndt &/meas2/tamb,tc,xtc, &/ihcp/irfts,alpha,w0,wl,w2 C NPRINT - 0 if printing input parameters and headings. C NPRINT - 1 if printing estimated heat flux and heat C transfer coefficient. 366 Table D.2 (cont'd) C NPRINT - 2 if printing end of file. IF(NPRINT.EQ.0)THEN GO TO 1100 ELSE if(nprint.eq.l)then go to 1200 else go to 1300 endif endif 1100 write(outfil,1000)ttlfil,'out.dat' 1000 format(' ',a,a) open(unit-12,name-outfil(1:20),type-'new',carriagecontrol-'list') write(12,10)title 10 format(///,3x,'Tit1e: ',a20,/3x,' ----- ',//,14x,'Input Para', +'meters',/,14x,l6('-')//) write(12,20) 20 format(/'Unfrozen Product Properties',/) abcd-wf0*100.0d0 write(12,30)abcd 30 format(2x,'Moisture content (%) ................... ',f6.2) abcd-t0-273.150d0 write(12,40)abcd 40 format(2x,'Initial freezing temperature (C) ....... ',f6.2) write(12,50)ms 50 format(2x,'Molecular weight of solids (kg/mole)...',f8.2) write(12,60)dp 60 format(2x,'Unfrozen product density (kg/mA3) ...... ',f8.2) write(12,70)kp 70 format(2x,'Thermal conductivity (W/mK) ............ ',f6.3) write(12,80)cp 80 format(2x,'Specific heat (kJ/kgK) ................. ',f7.3) sum - 0 do i - 1,ntc sum - sum+tc(l,i) enddo abcd - sum/ntc-273.150d0 write(12,90)abcd 90 format(/'Initial Condition:', & /2x,'Avg. Product temp. (C) at time-0 ...... ',f7.2) c product geometry if(shape.eq.1)then write(12,100)l 100 format( /'Slab Geometry:', + /2x,'thickness (m) ......................... ',f10.6) else if(shape.eq.2)then write(12,110)l 110 format(/'Cyclindrical Geometry:', + /2x,'radius (m) ............................. ',f10.6) else write(12,120)l 367 Table D.2 (cont'd) 120 format(/'Spherical Geometry:', + /2x,'radius (m) ............................. ',f10.6) endif endif c input conditions write(12,130)ntime 130 format(/2x,'Total no. of measurements .............. ',13) write(12,140)de1t*(ntime-l) 140 format(2x,'Total time (sec) ....................... ',f10.2) tothrs - delt*(ntime-l)/3600.0d0 write(12,150)tothrs 150 format(2x,'Total time (hrs) ........................ ',f7.3) write(12,160)delt 160 format(2x,'Time step for IHCP solution (sec) ...... ',f10.2) write(12,162)delt/ndt 162 format(2x,'Time step for FD solution (sec) ........ ',f10.2) write(12,168)dxi 168 format(2x,'Position step for FD solution (sec) ....',f7.4) write(12,170)ntc 170 format(2x,'Total no. of thermal couples ........... ',i3) do itc - 1,ntc write(12,180)itc,xtc(itc) 180 format(4x,'Location of T.C.(',12,') ............. ',f7.4) enddo write(l2,l82)irfts,a1pha,w0,w1,w2 182 format(2x,'IHCP regularization solution parameters:',/, & 4x,'No. of future time steps ............. ',12,/, & 4x,'Regu1arization parameter (alpha) ..... ',e7.l,/, & 4x,'Weighting coefficient - 0th order ....',f5.3,/, & 4x,'Weighting coefficient - lst order ....',f5.3,/, & 4x,'Weighting coefficient - 2nd order ....',f5.3) write(12,190)tit1e 190 format(/////'Title- ',a,/) hh22-‘TRANSFER COEFFICIENTS' hhll-‘ESTIMATED HEAT FLUXES AND HEAT' write(12,200)hhll,hh22 200 format(/18x,a,/,22x,a,/,18x,29('-'),/) write(12,210) 210 format('lNo.| Tamb | TC(1) | TC(2) | TC(3) | TC(4) | TC(S) |', a ' TC(6) | q | h |'/72('-')) c Printout T.C. temperature measurements 1200 ttamb - tamb(it) - 273.150d0 do itc - 1,ntc ttc(itc) - tc(it,itc) - 273.150d0 enddo write(12,220)it,ttamb,ttc(l),ttc(2),ttc(3), & ttc(4),ttc(5),ttc(6),q1(l),htc 368 Table D.2 (cont'd) 220 format(i3,lx,f6.l,lx,6(f7.2,1x),f6.1,lx,f6.2) re turn c Printout end line 1300 230 c*** c*** c*** c*** c*** c*** c*** c*** 30 c*** c*** c*** write(12,230) format(72('-')) close(unit-12) return end SUBROUTINE SIMUL(ASTORE,X,N,INDIC) parameter(maxr - 15,maxr1 - 16) double precision AIJCK, DETER, EPS, PIVOT, PIVOTI double precision A(maxr,maxr1) double precision IROW(maxr),JCOL(maxr),JORD(maxr) double precision ASTORE(maxr,maxrl),X(maxr) INITIALIZE PARAMETERS MAX - N IF(INDIC.GE.0) MAX - N+l EPS - 1.0D-20 STORE THE ARRAY SINCE INVERTED IN PLACE DO J-1,MAX DO I-1,N A(I,J) - ASTORE(I,J) enddo enddo FOR DEBUGING PURPOSES, PRINTOUT THE "A" MATRIX BEGIN THE ELIMINATION PROCEDURE DETER - 1.0DO DO 130 K-1,N KMl - K-l SEARCH FOR THE PIVOT ELEMENT PIVOT - 0.0D0 369 Table D.2 (cont'd) DO 80 I-1,N DO 70 J-1,N c*** C*** SCAN IROW AND JCOL ARRAYS FOR INVALID PIVOT SUBSCRIPTS c*** IF(K.EQ.1) GO TO 60 DO ISCAN-1,KM1 DO JSCAN-1,KM1 IF(I.EQ.IROW(ISCAN)) GO TO 70 IF(J.EQ.JCOL(JSCAN)) GO TO 70 enddo enddo 6O IF(DABS(A(I,J)).LE.DABS(PIVOT)) GO TO 70 PIVOT - A(I,J) IROW(K) - I JCOL(K) - J 70 CONTINUE 80 CONTINUE c*** C*** INSURE THE SELECTED PIVOT IS LARGER THAN EPS c*** IF(DABS(PIVOT).GT.EPS)GO TO 90 c WRITE(5,2000) PIVOT c 2000 FORMAT(' ','VALUE FOR PIVOT IS TOO SMALL, SUBROUTINE TERMINATED.') c WRITE(1,2000) PIVOT RETURN C*** C*** UPDATE THE DETERMINANT VALUE C*** 90 IROWK - IROW(K) JCOLK - JCOL(K) DETER - DETER*PIVOT c*** C*** NORMALIZE PIVOT ROW ELEMENT c*** PIVOTI - 1.0D0/PIVOT DO J-l,MAX A(IROWK,J) - A(IROWK,J)*PIVOTI enddo c*** C*** CARRY-OUT ELIMINATION AND DEVELOP INVERSE c*** A(IROWK,JCOLK) - PIVOTI DO I-l,N AIJCK - -A(I,JCOLK) IF(I.EQ.IROWK) GO TO 120 A(I,JCOLK) - AIJCK*PIVOTI DO J-1,MAX IF(J.NE.JCOLK) A(I,J) - A(I,J) + AIJCK*A(IROWK,J) enddo 120 enddo 130 CONTINUE c*** C*** ORDER THE SOLUTION VALUES (IF ANY) AND CREATE JORD ARRAY c*** Table 140 370 D.2 (cont'd) DO 140 I-l,N IROWI - IROW(I) JCOLI - JCOL(I) JORD(IROWI) - JCOLI IF(INDIC.GE.O) X(JCOLI) - SNCL(A(IROWI,MAX)) CONTINUE RETURN END SUBROUTINE RAND(tc,ntime,ntc) Parameter(maxt-250,maxtc-6) integer ntime,ntc,seedi*4 Double Precision tc(maxt,maxtc),u(maxt),a0,al,b1,b2,r,e,t,ak c Subroutine to generate normally distributed random numbers. 10 20 30 write(5,1) format(' ','Enter seed no. for this run:') read(5,*)seedi a0 - 2.30753 a1 - 0.27061 bl - 0.99299 b2 - 0.04481 do itc - 1,ntc do i - 1,ntime u(i) - ran(seedi) enddo do j - 1,ntime r - u(j) if(r-0.5d0)10,10,20 ak - 1.0d0 go to 30 ak - -1.0d0 go to 30 t - (log(l.0/(r*r)))**0.5 e - t-(a0+al*t)/(l.0+b1*t+b2*t*t) tc(j,itc) - tc(j,itc) + ak*e*0.27 enddo enddo return end APPENDIX E APPENDIX E RESULTS FROM SECOND AND THIRD EXPERIMENTAL TEST REPETITIONS This appendix contains the results from the second and third experimental test repetitions. 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(3°) eJnioJedwei 379 0¢ .Aow 00090 0000:50 00moaxm 0:0 £003 gnaw H0%0A 0Hnsoa Eouu muazm0m Hmu:0aau0ax0 ou 00:09800 :oHuSHom Hmowu0adz H0:on:0EHQ 0:0 00.0 0:3000 A0505 0EF 9. an ...N m: m o L '— Ir I? L b n F L L b 0.0.?! 2; 07$ {2-3 “2; filo .310: n x; .E> . x.._ n x 00 AoonE\>>v 0000 080:0...» ~00: u x; E 2.5 u .j 52m 3 8056:: Ton»... E; 21% 6.0.3- a: ...uno 6:0... .... .95: .25. . 90$- n .95: 65... 3.9.... As 83 A5 Bad xv 5:200 _oo_..0Eaz xv cozgom BotoEaz AE 00.0 n xv 33000 cmaxm .m>< AE 000.0 .I.. xv 3.2000 .390 .m>< I (3°) eanoJedLuei 381 00 .Aom 00080 00000050 00monxm 039 £HH3 swam 00%04 0HQHMH 80:0 00H300m 000:05Hu0nx0 00 00umasoo :OHHSHom H00HH0sfiz H0:ofim:0EHa 038 00.0 0usmwm A0595 0E... Nb 0.x Wm 0 L h # L AE 000.0 E x .E 000.0 a xv 3.3000 0090 .m>< I AE mud n. z .E 00.0 n. xv 3.3000 coax“. .m>< To 1 AE 000.0 I a .5 000.0 E xv 0003.00 80.00502 ll ... . I 0 00 AE «N0 I z .E 2.0 n. Xv 0003.00 30.3832 .... AE mud n .06 0.x0l> 000.0 000000.. 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