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. CFD MODELLING OF AIR FLOW AND HEAT TRANSFER IN A VENTILATED CARTON QIAN ZOU 1998 CFD MODELLING OF AIR FLOW AND HEAT TRANSFER IN A VENTILATED CARTON A thesis presented in partial fulfilment of the requirements for the Degree of Master of Applied Science in Agricultural Engineering at Massey University Qian Zou 1998 ABSTRACT Forced-air cooling is widely adopted for cooling fresh produce in ventilated packages. Air distribution inside the package is therefore an important factor for efficient cooling, since the heat transfer between product and air is largely affected by air motions. Computational fluid dynamics (CFD) provides a sophisticated but economic tool for modelling air flow. A CFD-based mathematical model was developed to simulate air flow patterns in a ventilated apple carton during precooling. The model took account of both laminar and turbulent situations. In the model, air mass, momentum and energy conservation, as well as energy conservation of apples and packaging materials were described by a set of partial differential equations (PDEs) plus boundary conditions. For the turbulent flow a Lam and Bremhorst Low-Reynolds-number k-E model was introduced to calculate local turbulent eddy viscosity. Two modelling strategies were adopted. In the first approach, the air flow was assumed to be steady-state while the buoyancy force due to natural convention was neglected. Steady­ state Navier-Stokes equations were solved first, and the outputs of fluid velocity were then used as input data to solve energy equations. For the second approach, all transport equations were solved simultaneously with consideration of the effect of natural convection on air flow patterns. All together, three air flow scenarios were considered: steady-state laminar flow, steady-state turbulent flow, and unsteady-state laminar flow. The CFD package PHOENICS (CHAM, UK Ltd) was used to solve the set of PDEs. The curvilinear Body-Fitted Coordinates (BFC) grid system was used for mesh generation. The entire grid system had 19964 cells. Five sets of PHOENICS codes were written for the three different flow situations. An additional PHOENICS programme was also used to calculate the heat transfer coefficients on the carton external surfaces. It took much longer time to reach convergence for unsteady-state laminar flow (91 hours) than for steady-state laminar flow (9-14 hours). ll The predicted flow patterns and temperature profiles were very similar for steady-state laminar and turbulent flows under 0.5 mis inlet velocity. By comparing predictions for steady-state and unsteady-state laminar flows , effects of natural convection were considered negligible in unsteady-state laminar flow. Thus it was reasonable to adopt the programme for steady-state laminar flow instead of unsteady-state laminar flow because of much less computing time in solving steady-state flow. A trial of apple precooling was conducted in which temperature in the centres of apples in various positions were measured. Good agreement between model predictions and experimental data was obtained in most locations, but fairly large errors were found in the apples near carton inlets and outlets. Further work is required to refine the model and to validate air temperature and velocity predictions. ill ACKNOWLEDGEMENTS I would like to thank my chief supervisor, Dr. Linus U . Opara (Institute of Technology and Engineering) and my co-supervisor, Professor Robert McKibbin (Institute of Fundamental Science), and Associate Professor Cliff Studman (Institute of Technology and Engineering) for their advice and assistance during the course of this project. I would also like to express my appreciation to all the technical staff (former Agricultural Engineering Department), especially Mr Gerard Harrigan and Mr Leo Bolter who assisted with much of the experimental work. I would like to acknowledge the following persons for their help and advice: Mr Yun Lu (graduate student, Institute of Technology and Engineering), Mr Phil Etheridge (Institute of Fundamental Science) and Mr David Tanner (PhD student, Department of Food Technology). IV TABLE OF CONTENTS Abstract Acknowledge llI Table of contents IV List of figures Vlll List of tables IX Chapter Introduction 2 Literature review 4 2.1 Modelling methodology 4 2.1 .1 Modelling procedure 5 2.1.2 Types of models 5 2.1 .3 Time discretisation 6 2.1.3 .1 Steady-state models 7 2.1.3.2 Dynamic models 7 2.1.4 Space discretisation 7 2.1.4.1 Zoned models 7 2.1.4.2 Fully Distributed models 8 2.1.5 Model complexity 8 2.2 Principles of Computational Fluid Dynamics 9 2.2.1 Fundamental fluid transport equations 9 2.2.1.1 General transport equations 9 2.2.1.2 Modelling turbulent flow 11 2.2.1.3 Boundary conditions 17 2.2.2 Numerical solution methods 19 2.2.2.1 Finite element method 19 2.2.2.2 Finite volume method 20 2.2.2.3 Computational grids 20 2.2.3 Presentation and verification of CFD results 22 v 2.2.4 Commercial CFD codes 22 2.3 Modelling air flows in controlled environment 23 2.3.1 Modelling air flow patterns in buildings 24 2.3.2 Modelling air flow patterns in refrigerated spaces 28 2.3.3 Modelling air flow patterns in other agricultural areas 29 2.4 Modelling of heat transfer processes during product cooling 31 2.4.1 Product heat conduction model 31 2.4.1.1 Analytical models 32 2.4.1.2 Empirical models 32 2.4.1.3 Numerical models 33 2.4.2 Product heat conduction plus cooling media models 34 3 Research objectives 37 3.1 Summary of literature 37 3.2 Research objectives 38 4 Model development 39 4.1 Introduction 39 4.2 Conservation of air mass (continuity equation) 40 4.3 Conservation of air momentum (equations of motion) 40 4.3.1 General equations 40 4.3.2 Laminar flow 41 4.3.2.1 Unsteady-state laminar flow 41 4.3.2.2 Steady-state laminar flow 43 4.3.3 Turbulent flow 43 4.3.3.1 Unsteady-state turbulent flow 43 4.3.3.2 Steady-state turbulent flow 48 4.4 Conservation of energy (heat transfer equations) 49 4.4.1 Energy equations for air flow 49 4.4.1.1 Laminar flow 49 4.4.1.2 Turbulent flow 50 4.4.2 Energy equation for apple 51 4.4.3 Energy equation for carton and trays 51 4.5 Initial conditions 52 VI 4.5.1 Initial conditions for air 52 4.5.2 Initial conditions for apple and packaging materials 53 4.6 Boundary conditions 53 4.6.1 Wall-type boundary conditions 53 4.6. 1.1 Laminar flow 53 4 .6.1 .2 Turbulent flow 54 4.6.2 Inflow boundary conditions 55 4.6.3 Fixed-pressure boundary conditions 56 4.6.4 Boundary conditions for carton external surfaces 56 4.7 Summary of partial differential equations 56 5 Model implementation 62 5.1 Introduction 62 5.2 Structure of PHOENICS 62 5.3 Discretisation 64 5.3.1 Space discretisation 64 5.3 . 1.1 Configuration of apple carton 64 5.3 .1.2 Grid generation 68 5.3.2 Derivation of discretisation equations 70 5.3 .2.1 Treatment of convection and diffusion terms 79 5.3.2.2 Treatment of source term 88 5.3.2.3 Treatment of unsteady-state situation 89 5.3.2.4 Treatment of diffusion coefficient 90 5.3 .2.5 Discretisation equation 91 5.3 .2.6 Treatment of momentum equations and pressure field 92 5.3 .2.7 Conjugate heat transfer 99 5.4 Solution of discretisation equations 100 5.4.1 Solution procedure 101 5.4.2 Convergence of solution 101 5.4.3 PHOENICS programmes 104 6 Simulation results 107 6.1 Introduction 107 Vil 6.2 Model input data 107 6.2.1 Initial and boundary conditions 107 6.2.2 Relaxation arrangement 109 6.2.3 Thermal properties 109 6.2.4 Time step 109 6.3 Parameters for convergence monitoring 112 6.4 Simulation results and discussion 113 6.4.1 Predicted air flow patterns 113 6.4.2 Temperature predictions 114 6.4.3 Summary 115 7 Model validation 153 7.1 Trial of apple precooling 153 7.2 Comparison of model predictions and measured data 154 8 Conclusion 161 8.1 Conclusion 161 8.2 Future research 162 Nomenclature 163 References 171 Appendix Disk Al PHOENICS programme: steady-state laminar flow (code SLTl) A2 PHOENICS programme: steady-state laminar flow (code SL T2) A3 PHOENICS programme: steady-state turbulent flow (code STTl) A4 PHOENICS programme: steady-state turbulent flow (code STT2) AS PHOENICS programme: unsteady-state laminar flow (code ULT 1) A6 PHOENICS programme: heat transfer coefficient for external carton surfaces (code HTC) Vlll LIST OF TABLES 2.1 Commercial CFD codes 23 2.2 Summary of main types of fluid flow problems that general-purpose CFD codes can solve 24 4.1 Partial differential equations in the generalised form 59 5.1 Calculation of Count- I 00 apple volume and surface area 65 5.2 Values used for specifying the model apple 65 5.3 Original and modified dimensions of carton and tray 70 5.4 Summary of PHOENICS programmes 105 6.1 Input data for initial conditions 108 6.2 Input data for boundary conditions 108 6.3 Relaxation arrangement 109 6.4 Thermal properties 110 6.5 Programme running time and convergence-related parameters 112 7.1 Input data for model validation 155 7.2 Measured and predicted 7 /8-cooling time 156 1X LIST OF FIGURE 2.1 Modelling procedure suitable in the area of refrigeration 5 5.1 Structure of programmes of PHOENICS 63 5.2 TAI Z PACK Count- I 00 apple carton 66 5.3 Arrangement of apples on a tray 66 5.4 Dimensions of TAI Z PACK Count-100 apple carton 67 5.5 Dimensions of the model apple 67 5.6 Relative positions of model apples on a tray 68 5.7 A cell with its four neighbours in the staggered grid 69 5.8 Grid for plane Kl-K3, K9-K11 , Kl 7-Kl 9, K25-K27, K33-K35 , K41-K43 71 5.9 Grid for plane K4, K8 , Kl2, Kl6, K20, K24, K28, K32, K36, K40, K44 72 5.10 Grid for plane K5-K7, K13-K15 , K21-K23 , K29-K31 , K37-K39, K45-K47 73 5.11 Grid for plane JS , J7 , J19, J21 74 5.1 2 Grid for plane J6, J20 74 5.13 Grid for plane JI 2, J 14, J26, J28 75 5.14 Grid for plane Jl3 , J27 75 5.14 Grid for plane Jl-J3 , J9 , Jl 7, J23 76 5.16 Grid for plane J4, J8 , JI 8, J22 76 5.17 Grid for plane J 10, J 16, J24, J30-J32 77 5.18 Grid for plane J 11 , J 15, 125, J29 77 5.19 Grid for plane Il-Il5 78 5.20 Solution procedure 102 6.1 Differences in temperature predicted by using different time steps 1 11 6.2 Predicted steady-state laminar air flow pattern 116 6.3 Predicted steady-state turbulent air flow pattern 118 6.4 Predicted unsteady-state laminar air flow pattern after two-hour cooling 120 6.5 Predicted unsteady-state laminar air flow pattern after four-hour cooling 122 6.6 Predicted unsteady-state laminar air flow pattern after twelve-hour cooling 124 6.7a Predicted air velocity components in the cell (3, 7, 10) 126 6.7b Predicted air velocity components in the cell (8 , 7, 22) 127 6.7c Predicted air velocity components in the cell (3, 7, 42) 6.7d Predicted air velocity components in the cell (3, 14, 6) 6.7e Predicted air velocity components in the cell (8, 14, 26) 6.7f Predicted air velocity components in the cell (3, 14, 38) 6.7g Predicted air velocity components in the cell (3, 21, 10) 6.7h Predicted air velocity components in the cell (8, 21, 22) 6.7i Predicted air velocity components in the cell (3, 21 , 42) 6.7j Predicted air velocity components in the cell (3, 28, 6) 6.7k Predicted air velocity components in the cell (8, 28, 26) 6.71 Predicted air velocity components in the cell (3, 28, 38) 6.8a Predicted air temperature in the cells of bottom layer 6.8b Predicted air temperature in the cells of lower middle layer 6.8c Predicted air temperature in the cells of upper middle layer 6.8d Predicted air temperature in the cells of top layer 6.9a Predicted apple centre temperature in the cells of bottom layer 6.9b Predicted apple centre temperature in the cells of lower middle layer 6.9c Predicted apple centre temperature in the cells of upper middle layer 6.9d Predicted apple centre temperature in the cell s of top layer 6.10 Predicted tray temperature in two cells 6.11 Predicted carton temperature in two cells 6.12 Predicted temperature contours after two-hour cooling for steady-state laminar flow 6.13 Predicted temperature contours after two-hour cooling for steady-state turbulent flow 6.14 Predicted temperature contours after two-hour cooling for unsteady-state laminar flow 7.1 Controlled environmental tunnel 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 145 145 146 146 147 149 151 153 7 .2a Measured and predicted apple centre temperature in the cells of bottom layer 157 7 .2b Measured and predicted apple centre temperature in the cells of lower middle layer 158 x XI 7 .2c Measured and predicted apple centre temperature in the cells of upper middle layer 159 7 .2d Measured and predicted apple centre temperature in the cells of top layer 160