Assessing food safety of polymer packaging

Preface ......................................................................................................................... 1 1. A Theoretical Approach to Experimental Data ...................................................... 7 1.1 Mass Transfer by Diffusion or Convection: Basic Equations ........................ 7 1.1.1 Diffusion ......................................................................................... 7 1.1.2 Convection ...................................................................................... 7 1.1.3 Analogy With Heat Transfer ........................................................... 7 1.1.4 Basic Equations ............................................................................... 8 1.1.5 Solid-Liquid Interface ...................................................................... 8 1.1.6 Properties of Convection ................................................................. 9 1.1.7 Applications of the boundary conditions ......................................... 9 1.1.8 Note on the Infi nite Value of the Coeffi cient of Convection .......... 10 1.1.9 Partition Factor ............................................................................. 10 1.2 Differential Equation of Diffusion ............................................................. 10 1.3 Methods of Solution with the Separation of Variables ............................... 12 1.4 Solution of the Equation of Diffusion: Sheet of Thickness 2L Immersed in a Liquid of Infi nite Volume and Infi nite Value of the Coeffi cient of Convection .................................................................... 13 1.4.1 Applications of the Equations of Transfer of Substance Into or Out of a Sheet of Thickness 2L with: -L < x < +L .............. 17 1.5 Other Solutions for the Problem of Diffusion: Sheet of Thickness 2L Immersed in a Liquid of Infi nite Volume and Infi nite Value of the Coeffi cient of Convection .................................................................... 19 1.6 Solution of the Equation of Diffusion: Sheet of Thickness 2L Immersed in a Liquid of Finite Volume and Infi nite Value of the Coeffi cient of Convection .......................................................................... 20 ii Assessing Food Safety of Polymer Packages 1.6.1 The Diffusing Substance Enters the Sheet ...................................... 21 1.6.2 The Diffusing Substance Leaves the Sheet ..................................... 23 1.6.3 Note on Equation (1.41) with an Infi nite Volume of Liquid .......... 25 1.7 Solution of the Equation of Diffusion: Sheet of Thickness 2L Immersed in a Liquid of Infi nite Volume and a Finite Value of the Coeffi cient of Convection .................................................................... 26 1.7.1 Notes on the Case Described in Section 1.7 .................................. 28 1.8 Ratio Volume/Area of the Food Package .................................................... 28 1.9 Determination of the Parameters of Diffusion ........................................... 29 1.9.1 Simple Ways of Evaluating the Parameters of Diffusion (Diffusivity) ................................................................................... 29 1.9.2 Using a Numerical Model ............................................................. 30 1.9.3 Using the Appropriate Numerical Model ...................................... 31 1.10 Diffusion through Isotropic Rectangular Parallelepipeds ........................... 38 1.10.1 Diffusion with an Infi nite Value of the Coeffi cient of Convection .... 39 1.10.2 Diffusion with a Finite Value of the Coeffi cient of Convection ........ 40 1.10.3 Approximate Value of the Diffusivity for Small Times and Infi nite h ........................................................................................... 41 1.10.4 Approximate Value of the Diffusivity for Long Times and Infi nite h ........................................................................................... 41 1.11 Case of a Membrane of Thickness L .......................................................... 41 1.11.1 Transport of the Substance through the Membrane for Infi nite h .... 42 1.11.2 Results for the Membrane with Infi nite Value of h on Both Sides .... 43 1.11.3 Results for a Membrane with a Finite Value of h on the Right Side .. 45 1.11.4 Results for a Membrane with Finite Value of h on Both its Sides ..... 47 1.12 Evaluation of the Parameters of Diffusion from the Profi les of Concentration ....................................................................................... 49 1.12.1 Experimental ................................................................................. 50 1.12.2 Theoretical .................................................................................... 50 1.12.3 Results Obtained with the Gradients of Concentration ................. 51 1.13 Conclusions on the Diffusion Process ........................................................ 54 iii Contents 2. Mass Transfer Through Multi-layer Packages Alone ........................................... 57 2.1 Recycling Waste Polymers and Need of a Functional Barrier ..................... 57 2.1.1 Role of the Functional Barrier ....................................................... 57 2.1.2 Mass Transfer Occurring During the Co-extrusion Stage .............. 58 2.1.3 General Problem of Diffusion Through the Layers of the Packaging Alone ........................................................................... 58 2.2 Bi-layer Package: Recycled Polymer-Functional Barrier ............................. 58 2.2.1 Mathematical Treatment of the Process ....................................... 59 2.2.2 Results Obtained with Two Layers of Equal Thicknesses .............. 61 2.3 Bi-layer Package with Various Relative Thicknesses ................................... 67 2.3.1 Mathematical Treatment of the Process ....................................... 67 2.3.2 Results with Two Layers of Different Relative Thicknesses ........... 68 2.4 Three-Layer Packages ................................................................................ 73 2.4.1 Mathematical Treatment of the Process of Matter Transfer ........... 73 2.4.2 Results with Three Layers of Equal and Different Thicknesses ...... 76 2.5 Bi-layer Package with Complex Situations: Different Diffusivities and Factor Coeffi cient Different from One ....................................................... 85 2.5.1 Mathematical Treatment of the Process of Matter Transfer .......... 85 2.5.2 Results with Bi-layer Films in Complex Situations ........................ 87 2.6 Conclusions on Multi-layer Packages ........................................................ 94 3. Process of Co-Extrusion of Multi-Layer Films ..................................................... 99 3.1 Scheme of the Process of Co-Extrusion ...................................................... 99 3.2 Principles of Unidirectional Heat Transfer ............................................... 101 3.2.1 Basic Equations of Heat Transfer by Heat Conduction ............... 101 3.2.2 Heat Convection ......................................................................... 103 3.3 Coupled Heat and Mass Transfer in Bi-Layer Films ................................. 105 3.3.1 Theoretical Treatment of the Transfer of Heat ........................... 105 3.3.2 Theoretical Treatment of the Mass Transfer Coupled with the Heat Transfer ................................................................ 107 3.4 Evaluation of Heat and Mass Transfers in Bi-Layer Films ....................... 108 iv Assessing Food Safety of Polymer Packages 3.4.1 Consideration of the Process of Heat Transfer ............................ 108 3.4.2 Effect of the Value Given to the Coeffi cient of Heat Convection ... 109 3.4.3 Effect of the Thickness of the Film on the Transport of Heat and Matter .................................................................................. 114 3.4.4 Simultaneous Effect of the Thickness of the Film and the Coeffi cient of Convective Heat Transfer ...................................... 118 3.5 Evaluation of Heat and Mass Transfers in Tri-Layer Film ........................ 120 3.5.1 Theoretical Study of Heat and Mass Transfers ............................ 122 3.5.2 Heat and Mass Transfers in a Tri-Layer Film .............................. 122 3.6 Heat and Mass Transfers in Tri-Layer Bottles with a Mould at Constant Temperature on the External Surface ........................................ 123 3.6.1 Theoretical Treatment of the Process .......................................... 123 3.6.2 Heat and Mass Transfers with Heat Conduction through the Mould and Polyethylene Terephthalate (PET) and Heat Convection on the External Surfaces .................................. 125 3.6.3 Results Obtained with the 0.03 cm Thick PET Bottle ................. 126 3.6.4 Results Obtained with a 0.06 cm Thick PET Bottle .................... 130 3.7 Heat and Mass Transfers in Tri-Layer Bottles with a Mould Initially at the Temperature of the Surrounding Atmosphere ................................ 133 3.7.1 Theoretical Treatment of the Process .......................................... 133 3.7.2 Selection of the Values for the Parameters Used for Calculation .. 135 3.7.3 Results Obtained with the Surrounding Atmosphere at 20 °C or 40 °C ............................................................................ 136 3.8 Coupled Mass and Heat Transfers - Conclusions ..................................... 142 4. Mass Transfers Between Food and Packages ..................................................... 145 4.1 General Introduction to the Various Problems ......................................... 145 4.2 Theoretical Treatment ............................................................................. 147 4.2.1 Revision of the Main Parameters and Principles of Diffusion ...... 147 4.2.2 Differential Equation of Diffusion ............................................... 150 4.2.3 The Case of a Sheet of Thickness 2L Immersed in a Liquid of Finite Volume and Infi nite Value of the Coeffi cient of Convection . 150 4.2.4 Case of a Sheet of Thickness 2L Immersed in a Liquid of v Contents Infi nite (or Finite) Volume and a Finite Value of the Coeffi cient of Convection ............................................................ 153 4.2.5 General Conclusions on the Mathematical Treatment ................. 156 4.3 Mass Transfer in Liquid Food from a Single Layer Package ..................... 157 4.3.1 Theoretical for a Single Layer Package in Contact with Liquid Food ................................................................................ 157 4.3.2 Effect of the Coeffi cient of Convective Transfer .......................... 159 4.3.3 Effect of the Ratio of the Volumes of Liquid and Package ل ..........163 4.4 Bi-layer Packages Made of a Recycled and a Virgin Polymer Layer, by Neglecting the Co-extrusion Potential Effect ....................................... 165 4.4.1 Theoretical Treatment with a Bi-layer Package ........................... 165 4.4.2 Results with the Bi-layer Package ................................................ 166 4.5 Mass Transfer from Tri-layer Packages (Recycled Polymer Inserted Between Two Virgin Layers) in Liquid Food ............................................ 171 4.5.1 Theory of the Mass Transfer in Food with the Tri-layer Package ... 171 4.5.2 Results Obtained with the Tri-layer Package in Contact with a Liquid Food ............................................................................. 172 4.6 Effect of the Co-extrusion on the Mass Transfer in Food ......................... 174 4.6.1 Mass Transfer in Food with a Co-extruded Bi-layer Package ...... 174 4.6.2 Mass Transfer in Food with Tri-layer Bottles Co-injected in the Mould whose External Surface is Kept at 8 °C .................. 177 4.6.3 Mass Transfer in Food with Tri-layer Bottles Co-injected in Normal Mould ........................................................................... 181 4.7 Conclusions on the Functional Barrier ..................................................... 183 4.7.1 Interest of a Functional Barrier ................................................... 183 4.7.2 Effect of the Co-extrusion and Co-moulding on the Mass Transfer . 185 4.8 Conclusions on the Diffusion-Convection Process ................................... 185 4.9 Problems Encountered with a Solid Food ................................................ 186 4.9.1 Theoretical Part of the Problem .................................................. 187 4.9.2 Results for the Transfer in the Solid Food ................................... 188 vi Assessing Food Safety of Polymer Packages 5. Active Packages for Food Protection ................................................................. 201 5.1 Process of Transfer with Active Packages ................................................. 201 5.1.1 Passive Packages ......................................................................... 201 5.1.2 Modifi ed Atmosphere Packages for Perception of Freshness ....... 202 5.1.3 Active Packages with Antimicrobial Properties ........................... 203 5.1.4 Applications of Antimicrobial Package in Foods ......................... 205 5.1.5 Testing the Effectiveness of Antimicrobial Packages and Regulatory Issues ........................................................................ 205 5.1.6 Detection Systems ....................................................................... 206 5.2 Active Packages – Theoretical Considerations ......................................... 206 5.2.1 Process of Release and Consumption, and Assumptions ............. 206 5.2.2 Mathematical and Numerical Treatment ..................................... 207 5.3 Results Obtained by Calculation ............................................................. 208 5.3.1 Results Obtained for High Values of R ....................................... 209 5.3.2 Results Obtained with Low Values of R ...................................... 213 5.3.3 Establishment of the Dimensionless Numbers ............................. 217 5.4 Conclusions about the Active Agents ....................................................... 218 6. A Few Common Misconceptions Worth Avoiding ............................................. 221 6.1 Using Equations Based on Infi nite Convective Transfer ........................... 222 6.1.1 The Problem Presented ................................................................ 222 6.1.2 Theoretical Survey ...................................................................... 223 6.1.3 Conclusions Drawn from the Problem ........................................ 224 6.2 Infi nite Thickness of the Film and Infi nite Convective Transfer ................ 229 6.2.1 Description of the Experimental Part .......................................... 229 6.2.2 Theoretical Consideration by the Authors ................................... 230 6.2.3 Conclusions About the Ideas Presented ....................................... 232 6.3 Combination of Semi-infi nite Media and Finite Volume of Liquid ........... 233 6.3.1 Description of the Experimental Part .......................................... 233 6.3.2 Theoretical Part .......................................................................... 234 6.3.2 Tentative Conclusions on the Ideas that have Emerged ............... 234 vii 6.4 Infi nite Rate of Convection in a Finite Volume of Liquid ......................... 235 6.4.1 Principle of the Process ............................................................... 235 6.4.2 Theoretical Development ............................................................ 235 6.4.3 Tentative Conclusions on the Ideas that Emerged ........................ 236 6.5 Double Transfer Process and the Membrane System ................................ 238 6.5.1 Principle of the Double Transfer in Plasticised PVC Immersed in a Liquid .................................................................................. 238 6.5.2 Process of Mass Transfer through a Membrane .......................... 239 6.5.3 Observations on the Assumption of the Membrane .................... 242 6.6 Heat Transfer: Conduction or Convection ............................................... 245 6.6.1 The Problem Considered in the Literature ................................... 245 6.6.2 Recall of the Theory of Mass and Heat Transfers ....................... 246 6.6.3 Conclusions on Convection-Conduction Heat Transfers ............. 248 6.7 Profi les of Concentration in Two Semi-infi nite Media .............................. 250 6.7.1 Description and Study of the Moisan’s Method .......................... 250 6.7.2 Study of the Technique ................................................................ 250 6.7.3 Conclusions on Moisan’s Method ............................................... 251 6.8 Double Transfer of Substances in a Sheet ................................................. 252 6.8.1 Study Carried Out in a First Paper .............................................. 252 6.8.2 Analysis of a Somewhat Similar Study ........................................ 255 6.9 Methodology for Measuring the Reference Diffusivity ............................ 256 6.9.1 Presentation of the Ideas and the Methods .................................. 256 6.9.2 Analysis of the study ................................................................... 257 6.9.3 Conclusions ................................................................................ 258 6.10 Conclusions on the Remarks Made in Chapter 6 ..................................... 259 7. Conclusions ....................................................................................................... 267 Appendix: The First Six Roots ân of â tanâ = R ....................................................... 271