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1 Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author.

2 PREDICTION OF CHILLING TIMES OF FOODS SUBJECT TO BOTH CONVECTIVE AND EV APORA TIVE COOLING AT THE PRODUCT SURFACE A thesis presented in partial fufilment of the requirements for the degree of Doctor of Philosophy in Biotechnology and Bioprocess Engineering at Massey University. Sawitri (Rojanasaroj) Chuntranuluck 1995

3 ABSTRACT No published chilling time prediction method which covers a wide range of practical conditions, and which can be applied using only simple algebraic calculations for chilling with evaporation at the product surface has been proven accurate. The objective of the present work was to develop and test a simple chilling time prediction method with wide application for situations where significant evaporation as well as convective cooling occurs from the product surface. A numerical method (fmite differences) was used to simulate convection and evaporation at the product surface in cooling of solid products of simple shape (infinite slab, infinite cylinder, and sphere) with constant surface water activity. Semi-log plots relating temperature change to be accomplished to time were linearised by appropriate scale transformations based on the Lewis relationship. The effect of evaporation on cooling rate was measured by considering the slope and intercept of such plots, and comparing these to the slope and intercept that would arise in convection-only cooling. The enhancement of cooling rate due to the evaporative effect depended on six parameters; initial product temperature, cooling medium temperature, Biot number, relative humidity, product shape factor, and surface water activity. Four simple algebraic equations were curved-fitted to the numerically simulated data for predicting temperature-time profiles at centre and mass average positions in the product. The numerically generated results and the simple algebraic equations agreed well with a mean difference close to 0 % for all three shapes, and 95% confidence bounds of about ±3 % for the infinite cylinder, and ±5 % for the infinite slab and the sphere geometries. To test the simple models, chilling experiments were conducted in a controlled air flow tunnel across a range of conditions likely to occur in industrial practice. Tr.ols were conducted using infinite cylinders of a food analogue as an idealised product (with saturated salt solutions percolating over a wet cloth on the product surface to maintain constant surface water activity), and carrots (both peeled and unpeeled) as examples of real food products. ii

4 Measured centre temperatures for both the idealised products and peeled carrots were predicted by the proposed method, assuming a constant surface water activity, within a range of differences which was almost totally explainable by experimental uncertainty. For unpeeled carrots, predictions mode using three different surface water activities in the model (one to represent the initial condition, one to represent the active chilling phase, and one to represent the quasi-equilibrium state at the end of chilling) agreed sufficiently well with experimental centre temperature data for the lack of fit to be largely attributable to experimental uncertainty. No experimental verification for prediction of mass-average temperatures was attempted. The proposed method is recommended for predicting chilling times of food products of infinite slab, infinite cylinder or sphere shapes, across a wide range of commonly occurring chilling conditions provided the product has constant surface water activity. The establishment of bounds on a theoretical basis for limiting the ranges in which surface water activity values are selected for making predictions for products with non-constant surface water activity is proposed, and some guidance on application of these bounds established. Further work to refine the use of these bounds for a range of food products, to consider a wider range of shapes, to test the ability of the proposed method to predict mass-average temperatures is recommended. iii

5 ACKNOWLEDGEMENTS I would like to thank : Professor Andrew C Cleland, Department of Food Technology and Dr. Colin Wells, Department of Agricultural Engineering, Massey University, for their advice, assistance, and patient during the course of this project; Professor Donald Cleland, Department of Process Environmental Technology, Massey University, for his advice in solving some problems with the refrigeration plant plus his comments on this thesis; and Mr. Bruce Collins, Electronics Technician, Mr. Don McLean, Technician, and Mr. John Alger, Technical Officer, for their technical assisstance during the experimental phase of the project. I would like to thank the Royal Thai Government and Kasetsart University, Thailand, for the scholarship & funding. Finally, thank to my family across the ocean for their continual moral and financial support. IV

6 T ABLE OF CONTENTS ABSTRACT ACKNOWLEDGEMENTS TABLE OF CONTENTS APPENDICES LIST OF FIGURES LIST OF TABLES ii IV V x xi XIX CHAPTER PAGE 1 INTRODUCTION 1 2 LITERATURE SURVEY 2. 1 INTRODUCTION 2.2 CHILLING WITHOUT INTERNAL HEAT GENERATION OR EV APORA TION Analytical Solutions Low Biot Number (typically Bi < 0.2) Higher Biot Number (typically Bi > 0.2) Empirical Solutions Numerical Solutions 2.3 CHILLING WITH INTERNAL HEAT GENERATION BUT NO EVAPORATION Description of Internal Heat Generation by Respiration Analysis of Cooling with Internal Heat Generation 2.4 COOLING INCLUDING RADIATION 2.5 COOLING INCLUDING EV APORA TION Moisture Transport Phenomena Modelling of Evaporative Heat Transfer at the Product Surface Evaporation of Mass Transfer Coefficients v

7 2.5.4 Influence of Product Shape Contribution of Evaporation to Chilling Rate Effect of Internal Heat Generation Steady State Models for Moisture Loss with and without Internal Heat Generation Analytical and Empirical Models for Moisture 24 Loss during Chilling Analytical and Empirical Chilling Time Prediction Methods Numerical Chilling Time Prediction Methods Incorporation of Internal Heat Generation in Numerical Prediction Methods 3 PRELIMINARY CONSIDERATIONS 29 4 THEORETICAL MODELLING OF IDEALISED SYSTEMS 4. 1 INTRODUCTION 4.2 THEORETICAL DEVELOPMENT Nonlinear Partial Differential Equations Energy Balance Mass Balance Finite Difference Schemes Energy Transfer Equations Mass Transfer Equations Vapour Pressure Equations Use of the Lewis Relationship 4.3 Computer Program Development and Testing 4.4 Simulation Performed 4.5 Discussion of Simulation Behaviour Steady State (Equilibrium) Condition Active Chilling Period (Unsteady State Condition) VI

8 5 MODEL DEVELOPMENT & RESULT INTRODUCTION DESCRIPTION OF CONVECflON-ONL Y COOLING COMPLICATIONS INTRODUCED BY EVAPORATIVE COOLING DEVELOPMENT OF AN EQUILIBRIUM REFERENCE TEMPERATURE DEVELOPMENT OF A MODIFIED Y VALUE ISOLATION OF THE EV APORA TIVE EFFECf ON COOLING RATE MODEL DEVELOPMENT OF EMPIRICAL EQUATIONS FOR ieva/ico1ly AND jeva/jcofiv The Effects of Product Properties on ievap and j Evap Values The Reference Shape Models for Centre Temperature Position Model Development for iceva/iccofiv Model Development for jceva/jcco1lv Models for Mass-Average Temperature Position Model Development for iaveva/iavco1ly Model Development for javeva/javcofiv Time and Temperature Prediction Equations Quality of Prediction from The Simple Models Relative to The Numerical Simulations Results and Conclusions SUMMARY 66 6 EXPERIMENTAL DESIGN AND METHODS - IDEALISED PRODUCTS 6. 1 INTRODUCTION 6.2 EXPERIMENTAL DESIGN 6.3 EXPERIMENTS Test Samples and Temperature Measurement Probes Maintenance of Constant Surface Water Activity Air Tunnel Air Velocities Relative Humidity 6.4 MEASURING EQUIPMENT Vll

9 6.4.1 Relative Humidity Data Logging Air Velocity 6.5 EXPER NTAL PROCEDURE 6.6 HEAT TRANSFER COEFFICIENTS 6.7 SUMMARY TESTING OF SIMPLE MODEL AGAINST EXPE NT AL 107 DATA FOR IDEALISED MODEL SYSTEMS 7. 1 INTRODUCTION 7.2 EXPE NTAL DESIGN AND RESULTS 7.3 HEAT TRANSFER COEFFICIENTS 7.4 EQUILIBRIUM TEMPERATURE ANALYSIS 7.5 LINEARIZATION OF SEMI-LOG PLOTS 7.6 COOLING WIlli EV APORA TNE EFFECTS 7.7 CHILLING T PREDICTION 7.8 DISCUSSION AND CONCLUSIONS EXPE NTAL DESIGN AND METIIODS FOR EXPERIMENTS 149 WITH A REAL PRODUCT 8.1 INTRODUCTION 8.2 EV APORA TIVE COOLING OF REAL FOOD PRODUCTS Products without Skin Resistance Products with Skin Resistance 8.3 SELECTION OF TEST MATERIAL 8.4 EXPER NT AL METHODS - PEELED CARROT Sample Preparation Weight Loss, Water Content, and Surface Water Activity Measurement Temperature and Relative Humidity Measurement Cooling Trials Heat Transfer Coefficient Measurement viii

10 8.4.6 Experimental Design EXPERIMENTAL METHODS - UNPEELED CARROT Sample Preparation Weight Loss, Water Content, and Surface Water 155 Activity Measurement Experimental Design THERMOPHYSICAL PROPERTIES OF CARROTS HEAT TRANSFER COEFFICIENTS FOR PEELED 156 AND UNPEELED CARROTS 8.8 FINISHING REMARKS TESTING OF THE PROPOSED MODEL AGAINST EXPERIMENTAL 163 DATA FOR A REAL PRODUCT WITHOUT SKIN RESISTANCE 9.1 INTRODUCTION SURFACE WATER ACTIVITY AND EQUILIBRIUM TEMPERA TURE ANALYSIS LINEARIZATION OF SEMI-LOG PLOTS 9.4 COOLING WITH EV APORA TIVE EFFECTS WEIGHT LOSS DISCUSSION AND CONCLUSIONS TESTING OF THE PROPOSED MODEL AGAINST EXPERIMENTAL DATA 186 FOR A REAL PRODUCT WITH SKIN RESISTANCE 10.1 INTRODUCTION SURFACE WATER ACTMTY IN THE QUASI-EQUILIBRIUM STAGE LINEARIZATION OF SEMI-LOG PLOTS f AND j VALVES FOR ACTIVE CHILLING PHASE awl awj COOLING WITH EVAPORATIVE EFFECTS AND CHILLING 190 TIME PREDICTION 10.6 WEIGHT LOSS 192 ix

11 11 DISCUSSION AND CONCLUSION 11.1 DISCUSSION 11.2 CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORK NOTATION 212 REFERENCES 217 APPENDICES 1 COMPUTER PROGRAM USED FOR CHILLING PROCESS SIMULATION AND ANALYSIS 2 SAMPLE OF CALCULATION x

12 LIST OF FIGURES Example plot for convection-only cooling showing the approach of the centre, surface, and mass-average temperatures to the cooling medium temperature. Example plots showing the effect of (aw - H,) on the cooling process. Example semi-plots of cooling by both convection and evaporation with H, = aw Example semi-plots of cooling by both convection and evaporation with Gw < H" aw > H" and aw = H,. Alignment chart to estimate the equilibrium temperature (T q) at aw = 0.6. Alignment chart to estimate the equilibrium temperature (Teq) at aw = 0.8. Alignment chart to estimate the equilibrium temperature (T q) at aw = 1.0. Example plots of In Yc vs Fo for evaporative cooling. Plot of numerically predictedfceva/fccoliy vs Bi for an infinite cylinder. Effect of Bi on the slope ratios of semi-log plots (centre position) at different Ta (predicted by finite difference simulations for an infmite Effect of Ta on the slope ratios of semi-log plots (centre position) at different Bi (predicted by finite difference simulations for an infinite Effect of Ta on the slope ratios of semi-log plots (centre position) at high Bi and different H, (predicted by fmite difference simulations for an infinite Effect of H, on the slope ratios of semi-log plots (centre position) at high Bi and different Gw (predicted by finite difference simulations for an infinite Effect of Ta on the slope ratios of semi-log plots (centre position) at high Bi and different Gw (predicted by fm ite difference simulations for an infinite Effect of Bi on the slope ratios of semi-log plots (centre position) at different Till (predicted by finite difference simulations for an infmite Xl

13 Effect of a.. on the slope ratio of semi-log plots (centre position) at high Bi and different Tifl (predicted by finite difference simulations for an infinite Effect of shape on the slope ratio of semi-log plots (centre position) (predicted by finite difference simulations for an infinite Plot of numerically predicted jceva/jccofl" data for an infmite cylinder vs Bi. Effect of Bi on the intercept ratios of semi-log plots (centre position) at different H, (predicted by fm ite difference simulations for an infinite Effect of Bi on the intercept ratios of semi-log plots (centre position) at different Ta (predicted by finite difference simulations for an infmite Effect of Bi on the intercept ratios of semi-log plots (centre position) at different aw (predicted by finite difference simulations for an infinite Effect of Bi on the intercept ratios of semi-log plots (centre position) at different Tifl (predicted by finite difference simulations for an infmite Effect of H, on the intercept ratios of semi-log plots (centre position) at different Ta (predicted by finite difference simulations for an infinite Effect of Tifl on the intercept ratios of semi-log plots (centre position) at different Ta (predicted by finite difference simulations for an infinite Effect of shape on the intercept ratios of semi-log plots (centre position) (predicted by finite difference simulations for an infinite Plot of numerically predicted jceva/jccofl" data for an infmite cylinder vs Bi. Effect of Bi on the intercept ratios of semi-log plots (mass-average position) at different Ta (predicted by finite difference simulations for an infinite Effect of Bi on the intercept ratios of semi-log plots (mass-average position) xii

14 at different Till (predicted by fmite difference simulations for an infinite Effect of Ta on the intercept ratios of semi-log plots (mass-average position) at different H, (predicted by finite difference simulations for an infinite Effect of shape on the intercept ratios of semi-log plots (mass-average position), (predicted by finite difference simulations for an infmite Shape of cooling curve for centre and mass-average temperature during convective plus evaporative cooling under conditions where the evaporative effect is very small. Shape of cooling curve for centre and mass-average temperature during convective plus evaporative cooling under conditions where the evaporative effect is substantial. Schematic diagram of experimental apparatus installed in the air tunnel showing sample oscillator and wetting agent supply system. The apparatus inside the air tunnel. Example of measured and predicted cylinder temperatures for a run in which the reservoir temperature was too cold. Example of measured and predicted cylinder temperatures for a run in which the reservoir temperature was too hot. Results of an experiment for the large cylinder (with finite difference prediction) in which the reservoir temperature was ideal. Results of an experiment for the small cylinder (with finite difference prediction) in which the reservoir temperature was ideal. Experimental air tunnel used in experiments. Plots of he vs air velocity for heat transfer coefficient measurement runs in which the wetting fluid was water. Plots of he vs air velocity for heat transfer coefficient measurement runs in which the wetting fluid was a saturated solution of NaCl. Plots of he vs air velocity for heat transfer coefficient measurement runs in which the wetting fluid was a saturated solution of KCl. Plots of difference between Teq.p,ed and Teq,exp vs aw for all experimental runs. 130 xiii

15 7.5 Plots of difference between T qpr d and T q.up vs Hr for all experimental runs Plots of difference between T qpr d and T q,ap vs Bi for all experimental runs Plots of difference between T qpr d and T q.up vs va for all experimental runs Plots of difference between T qpr d and T q.up vs Ta for all experimental runs Plots of difference between T qpr d and T q.up vs Till for all experimental runs Typical plot of temperature vs time for cooling without evaporation (T q.up 133 = Ta) 7.11 Typical plot of temperature vs time for cooling with evaporation (T q,ap > 134 Ta and aw < Hr) Typical plot of temperature vs time for cooling with evaporation (T q,ap < 134 Ta and aw > Hr) Typical plot of temperature vs time for cooling with evaporation (T q.up = 135 Ta and aw = Hr) Plot of In Yc,ap vs F 0 using T q.up as the reference temperature for the 135 conditions in Fig Plot of In Yc,ap vs F 0 using T q,ap as the reference temperature for the 136 conditions in Fig Plot of In Yc,ap vs F 0 using T q.up as the reference temperature for the 136 conditions in Fig Comparisons of the % differences in iceva/iccollv between the simple model 137 and the fmite difference model Comparisons of the % differences in jcevo/jccollv between the simple model 137 and the finite difference model Comparisons of the values of iceva/iccollv between the simple model and the 138 experimental results Comparisons of the values of jceva/jccollv between the simple model and the 138 experimental results Plots of % difference in iceva/iccollv values calculated by the simple model 139 compared to experimental values at different Plots of % difference in iceva/hcollv values calculated by the simple model 139 compared to experimental values at different Hr Plots of % difference in iceva/iccollv values calculated by the simple model 140 xiv

16 compared to experimental values at different Ta. Plots of % difference in fceva/fccoftv values calculated by the simple model compared to experimental values at different Tjft. Plots of % difference in fceva,/fccoftv values calculated by the simple model compared to experimental values at different Bi. Plots of % difference in fceva/fccoftv values calculated by the simple model compared to experimental values at different va. Plots of % difference in jceva/jccofiv values calculated by the simple model compared to experimental values at different Ow. Plots of % difference in jeevap/jecoftv values calculated by the simple model compared to experimental values at different Hr. Plots of % difference in jcevap/jccoftv values calculated by the simple model compared to experimental values at different Ta. Plots of % difference in jceva/jccoftv values calculated by the simple model compared to experimental values at different Tjft. Plots of % difference in jceva/jecoftv values calculated by the simple model compared to experimental values at different Bi. Plots of % difference in jceva/jecoftv values calculated by the simple model compared to experimental values at different Va. Comparisons of % differences in time to reach the temperature corresponding to Ye,up = 0.10 between the simple model.ju1d the finite difference model Comparisons of % differences in time to reach the temperature corresponding to Yc,up = 0.35 between the simple model and the finite difference model. Comparisons of % differences in time to reach the temperature corresponding to fe,up = 0.70 between the simple model and the finite difference model. Comparisons of time (hrs.) to reach the temperature corresponding to Ye,up = 0.10 between the simple model and the experimental results. Comparisons of time (hrs.) to reach the temperature corresponding to Ye,up = 0.35 between the simple model and the experimental results xv

17 Comparisons of time (hrs.) to reach the temperature corresponding to Ye,up = 0.70 between the simple model and the experimental results. The best result of comparisons between the simple model, the finite difference with evaporation and without evaporation, and the experimental data. The worst result of comparisons between the simple model, the fmite difference with evaporation and without evaporation, and the experimental data. A typical infinite cylindrical carrot sample used in experimental work. Plot of Nu vs Re for runs with peeled carrots. Plot of percentage difference in he (between calculated results using equation 8.4 and experimental results) for peeled carrots vs Re. Plot of Nu vs Re for runs with unpeeled carrots. Plots of percentage difference in he (between calculated results using equation 8.5 and experimental results) for unpeeled carrots vs Re. Plot of Nu vs Re for runs with peeled & unpeeled carrots. Plot of measured aw and back-calculated aw vs Hr. Plot of difference between Teq"red and Teq,up vs Hr for all experimental runs. Plot of difference between Teq"red and Teq,up vs Bi for all experimental runs Plot of difference between Teq"red and Teq,up vs va for all experimental runs Plot of difference between Teq"red and Teq,up vs Ta for all experimental runs. Plot of difference between Teq"red and Teq,up vs Tilt for all experimental runs Typical plot of temperature vs time for cooling of a peeled carrot without evaporation (Teq,up = Ta) Typical plot of temperature vs time for cooling of a peeled carrot with evaporation at the product surface. Plot of In Ye vs F 0 using Teq,u:p as the reference temperature for the conditions in Fig Plot of % difference in fceva,/fccoltv values calculated by the simple model compared to experimental values vs H,. Plot of % difference in fceva,/fccoltv values calculated by the simple model compared to experimental values vs Ta xvi

18 Plot of % difference in fcevo)fcco"v values calculated by the simple model compared to experimental values vs Ti". Plot of % difference in fcevo)fcco"v values calculated by the simple model compared to experimental values vs Bi. Plot of % difference in fcevo)fccoitv values calculated by the simple model compared to experimental values vs va. Plot of % difference in icevo)icco"v values calculated by the simple model compared to experimental values vs H,.. Plot of % difference in icevo)icco"v values calculated by the simple model compared to experimental values vs Ta. Plot of % difference in icevo)icco"v values calculated by the simple model compared to experimental values vs Ti". Plot of % difference in iceva/icco"v values calculated by the simple model compared to experimental values vs Bi. Plot of % difference in icevo)icco"v values calculated by the simple model compared to experimental values vs va. Comparisons of the values of fcevo)fccoitv between the simple model and the experimental results. Comparisons of the values of jceva/jcco"v between the simple model and the experimental results. Comparisons of time (min.) to reach the centre temperature corresponding to Yc,ap = 0.10 between the simple model and the experimental results. Comparisons of time (min.) to reach the centre temperature corresponding to Yc,ap = 0.35 between the simple model and the experimental results. Comparisons of time (min.) to reach the centre temperature corresponding to Yc,ap = 0.70 between the simple model and the experimental results. Weight loss of peeled carrots vs air velocity. Comparison of the cooling curves of peeled and unpeeled carrot under the same environmental conditions (Runs 15 of Table 9. 1 and 10.1). Plot of aw vs time during a typical chilling experiment in which internal water movement rate to the product cannot always exceed evaporation rate. Relationship between aw back-calculated from Teq,ap using equation (5.4) and XVll

19 Hr for runs with unpeeled carrot. Typical plot of temperature vs time for cooling of an unpeeled carrot without evaporation (Teq,exp = Ta). Typical plot of temperature vs time for cooling of an unpeeled carrot with evaporation at the product surface Plot of In Yc (using Teq,up as the reference temperature) vs Fo for the 204 conditions in Figure Plot of Ow! vs Hr during the active chilling phase Plot of Ow! vs Bi during the active chilling phase Plot of Owj vs Hr during the active chilling phase Plot of Owj vs Bi during the active chilling phase. 206 xviii

20 LIST OF TABLES Percentage differences between results calculated by the proposed curve-fit algebraic equations and results calculated by finite differences for the centre temperature of all three shapes. Percentage differences between results calculated by the proposed curve-fit algebraic equations and results calculated by finite differences for the massaverage temperature of all three shapes. calculate favevap' Equation (5.23) was used to 5.3 Percentage differences between results calculated by the proposed curve-fit 69 algebraic equations and results calculated by finite differences for the massaverage temperature of all three shapes. Equation (5.24) was used to calculate favevap' Ranges of experimental conditions used for model testing. Experimental conditions used for the half-factorial experiment (VII = high velocity, VI = low velocity, v'" = mid-range velocity, L = large size, S = small size). Experimental conditions for the runs used to extend the half-factorial experiment to a plan approximating a central composite design. Experimental conditions for the 'centre point' runs in the central composite experimental design. Water activity of saturated salt solutions used in the experiments. The solubility of salts used in the experiments at different temperature. Conditions used to obtain required relative humidities. Designation of ranges of Biot number in the experimental described in Chapter 6. 95% confidence bounds on he (W m-2 Kl) at the midrange value of Va (2 m -1) s. Experimental data and predicted results for runs using water as the wetting agent Experimental data and predicted results for runs using saturated salt sodium 118 XIX

21 chloride as the wetting agent. 7.5 Experimental data and predicted results for runs using saturated salt potassium chloride as the wetting agent. Differences between results calculated by the finite difference method and results from the experiments (all data) Difference between results calculated by the proposed curve-fitted algebraic equations (simple model) and results from the experiments (all data). Prediction and measured chilling times to Yc ap = 0.10, 0.35, and 0.70 for runs using water as the wetting agent. Prediction and measured chilling times to Yc.ap = 0.10, 0.35, and 0.70 for runs using saturated salt sodium chloride as the wetting agent. Prediction and measured chilling times to Ycpp = 0.10, 0.35, and 0.70 for runs using saturated salt potassium chloride as the wetting agent. Experimental plan for cooling trials with carrots. Experimental data and predicted results for runs using peeled carrot as the product with aw = Predicted and measured chilling times to reach certain centre temperatures which corresponded to Yc,cxp = 0.10, 0.35, and 0.70 for runs using peeled carrot as the product. Differences between results calculated by the finite difference method (assuming aw = 0.977) and results from the experiments (for peeled carrots) Differences between results calculated by the proposed curve-fit equations (simple model) and results from the experiments (for peeled carrots). Experimental data for experiments with unpeeled carrot and comparisons of measured quasi-equilibrium temperature (Teqpp) to predictions of Teq 193 calculated using (A) = 0.977, (B) aw = Hr, (C) aw calculated from equation (10. 1). Comparisons between experimental results and model predictions of fceva/fccollv (at aw = 0.30 and 0. 16) and jceva/jccollv (at awj = 0.977). Comparisons of time (to the temperature that corresponds to Ycpp = 0.10, 0.35, and 0.70) between the experimental results and simple model using I = 0.16 & j = 0.977, awl = 0.30 & awj = 0.977, and Teql'red at (A) = xx

22 0.977, (B) aw = Hr, and (C) aw = 0.792Hr Predicted (using I = 0.30, awj = 0.977, and aw from equation 10.1) and 199 measured chilling time to the time that corresponds to Yc,up = 0.10, 0.35, and 0.70 for an unpeeled carrot Predicted (using awl = 0.16, awj = 0.977, and from equation in 200 equation 10.2) and measured chilling time to the time that corresponds to Yc,up = 0.10, 0.35, and 0.70 for an unpeeled carrot. xxi