Assessing Food Safety of Polymer Packaging.pdf

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Assessing Food Safety of Polymer Packaging

Rapra Technology is a leading international organisation with over 80 years of experience providing technology, information and consultancy on all aspects of rubbers and plastics. In 2006 it became part of The Smithers Group.

Jean-Maurice Vernaud and Iosif-Daniel Rosca

Rapra Technology

Assessing Food Safety of Polymer Packaging

Jean-Maurice Vernaud Iosif-Daniel Rosca

Rapra Technology

17/5/06 1:58:22 pm

Assessing Food Safety of Polymer Packaging

Jean-Maurice Vergnaud Iosif-Daniel Rosca

Smithers Rapra Limited A wholly owned subsidiary of The Smithers Group Shawbury, Shrewsbury, Shropshire, SY4 4NR, United Kingdom Telephone: +44 (0)1939 250383 Fax: +44 (0)1939 251118 http://www.rapra.net

First Published in 2006 by

Smithers Rapra Limited Shawbury, Shrewsbury, Shropshire, SY4 4NR, UK

©2006, Smithers Rapra Limited

All rights reserved. Except as permitted under current legislation no part of this publication may be photocopied, reproduced or distributed in any form or by any means or stored in a database or retrieval system, without the prior permission from the copyright holder. A catalogue record for this book is available from the British Library. Every effort has been made to contact copyright holders of any material reproduced within the text and the authors and publishers apologise if any have been overlooked.

ISBN-13: 098-1-84735-026-8

Typeset by Smithers Rapra Limited Cover printed by Livesey Limited, Shrewsbury, UK Printed and bound by Smithers Rapra Limited

Contents

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 Infinite Value of the Coefficient 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 Infinite Volume and Infinite Value of the Coefficient 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 Infinite Volume and Infinite Value of the Coefficient of Convection .................................................................... 19

1.6

Solution of the Equation of Diffusion: Sheet of Thickness 2L Immersed in a Liquid of Finite Volume and Infinite Value of the Coefficient of Convection .......................................................................... 20 i

Assessing Food Safety of Polymer Packages

1.7

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 Infinite Volume of Liquid .......... 25

Solution of the Equation of Diffusion: Sheet of Thickness 2L Immersed in a Liquid of Infinite Volume and a Finite Value of the Coefficient 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 Infinite Value of the Coefficient of Convection .... 39 1.10.2 Diffusion with a Finite Value of the Coefficient of Convection ........ 40 1.10.3 Approximate Value of the Diffusivity for Small Times and Infinite h ........................................................................................... 41 1.10.4 Approximate Value of the Diffusivity for Long Times and Infinite h ........................................................................................... 41 1.11 Case of a Membrane of Thickness L .......................................................... 41 1.11.1 Transport of the Substance through the Membrane for Infinite h .... 42 1.11.2 Results for the Membrane with Infinite 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 Profiles 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 ii

Contents 2.

Mass Transfer Through Multi-layer Packages Alone ........................................... 57 2.1

2.2

2.3

2.4

2.5

2.6 3.

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

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

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

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

Bi-layer Package with Complex Situations: Different Diffusivities and Factor Coefficient 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

Conclusions on Multi-layer Packages ........................................................ 94

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

3.3

3.4

3.2.1

Basic Equations of Heat Transfer by Heat Conduction ............... 101

3.2.2

Heat Convection ......................................................................... 103

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

Evaluation of Heat and Mass Transfers in Bi-Layer Films ....................... 108 iii

Assessing Food Safety of Polymer Packages

3.5

3.6

3.7

3.8 4.

iv

3.4.1

Consideration of the Process of Heat Transfer ............................ 108

3.4.2

Effect of the Value Given to the Coefficient 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 Coefficient of Convective Heat Transfer ...................................... 118

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

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

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

Coupled Mass and Heat Transfers - Conclusions ..................................... 142

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 Infinite Value of the Coefficient of Convection . 150

4.2.4

Case of a Sheet of Thickness 2L Immersed in a Liquid of

Contents Infinite (or Finite) Volume and a Finite Value of the Coefficient of Convection............................................................ 153 4.2.5 4.3

4.4

4.5

4.6

4.7

General Conclusions on the Mathematical Treatment ................. 156

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 Coefficient of Convective Transfer .......................... 159

4.3.3

Effect of the Ratio of the Volumes of Liquid and Package α ..........163

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

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

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

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 v

Assessing Food Safety of Polymer Packages 5.

Active Packages for Food Protection ................................................................. 201 5.1

5.2

5.3

5.4 6.

5.1.1

Passive Packages ......................................................................... 201

5.1.2

Modified 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

Active Packages – Theoretical Considerations ......................................... 206 5.2.1

Process of Release and Consumption, and Assumptions ............. 206

5.2.2

Mathematical and Numerical Treatment ..................................... 207

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

Conclusions about the Active Agents ....................................................... 218

A Few Common Misconceptions Worth Avoiding ............................................. 221 6.1

6.2

6.3

vi

Process of Transfer with Active Packages ................................................. 201

Using Equations Based on Infinite 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

Infinite Thickness of the Film and Infinite 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

Combination of Semi-infinite 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

Contents 6.4

6.5

6.6

6.7

6.8

6.9

Infinite 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

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

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

Profiles of Concentration in Two Semi-infinite 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

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

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 βn of β tanβ = R ....................................................... 271

vii

Assessing Food Safety of Polymer Packages

viii

Preface

Even if the subject of mass transport controlled by diffusion has been considered in various situations by these two authors over a long period of time, it has been a fascinating but also a ticklish task to write a book with the title: ‘Assessing Food Safety of Polymer Packaging’. The main problems which have arisen come from various sources: i)

Describing the problems of diffusion is highly complex, because various equations are used, which are obtained by a mathematical treatment based on very precise requirements, such as the initial and the boundary conditions, as well as the shape of the material through which the substance diffuses, and still more, without forgetting the fact that this substance enters into the surrounding atmosphere through the process of convection.

ii) The majority of the people using these equations are in general, competent researchers, who are able to manage pretty well, sophisticated apparatus by applying the appropriate techniques. But as a result of this previous competence, and the time necessary to acquire it, little time remains to get the pertinent knowledge on the theoretical treatment of diffusion. iii) It should be noted that if a few excellent books devoted to the subject of diffusion have been launched on the market since 1975, these books are addressed to either theoreticians or at least well-informed researchers about these difficult problems of mass transfer. iv) The problems concerned with diffusion are widely spread over various subjects: •

the release of a permanent gas or the evaporation of a vapour from a solid with the process of drying;

•

the release of a drug from diffusion-controlled release dosage forms in pharmacy;

•

the diffusion of a liquid into and through various solids such as wood (an anisotropic medium) or rubber (provoking eventually a change in dimension); and last but not least; 1

Assessing Food Safety of Polymer Packages •

the release of some additives from the polymer package into the food, with the possibility of inserting some recycled polymer packages through multi-layer systems (that last problem of recycling including a possible inter-diffusion of potential additives through the layers during the stage of co-extrusion, which is mainly controlled by the heat transfer process; this heat transfer process being itself controlled by conduction, for which the laws are similar to those of diffusion through the solid, as well as by convection on the surfaces).

Amongst the books devoted to the process of diffusion the most well-known is ‘The Mathematics of Diffusion’ [1], largely cited everywhere and especially in this present book. This book covers almost all the cases which could exist, theoretically speaking at least. So, they appear in succession and following this order: the infinite or semi-infinite medium, the sheet of finite thickness, and other shapes such as the cylinder and sphere. It should be noted that the trivial cases of a cube or a parallelepiped are not discussed, as they were considered more obvious by the author than perhaps by all the readers. In the same way, the bi- or tri-layer packages made of virgin and recycled polymer layers have not been considered, whether the initial concentration of diffusing substance in the recycled layer is uniform or not. While developing the diffusion through a sheet, the idea of taking 2L for its thickness, with –L0

x = ±L

C = C∞

surfaces

(1.22) 13

Assessing Food Safety of Polymer Packages

Figure 1.2 Scheme of the sheet of thickness 2L, with -L < x < +L. Cin is the uniform concentration of diffusing substance initially in the sheet, and C∞ is the concentration of this substance in the exterior of the sheet as soon as the process starts, with h→∞

As the mid-plane x = 0 of the sheet is a plane of symmetry, there is also the condition at any time: ∂C ∂x

t ≥0

at x = 0

mid-plane

(1.23)

The solution of this problem is given by the general Equation (1.20), obtained by using the method of separation of variables, where the constants are to be determined. It becomes: ∞

(

C x ,t − C∞ = ∑⎡⎣A n sin λ n x + B n cos λ n x⎤⎦.exp −λ 2n Dt n=0

)

(1.24)

The condition at the mid-plane, is written as follows: ∂C x , t ∂x

=0

(1.23?)

and necessitates that the terms in cosines in the derived Equation (1.23) are equal to 0, leading to: An = 0

(1.25)

The boundary condition (1.22), for example x = L and CL,t – C∞ = 0, for t > 0, is fulfilled when: cos λ nL = 0 = cos (2n + 1)

14

π (2n + 1) π and λ n = 2 2L

(1.26)

A Theoretical Approach to Experimental Data The initial condition (1.21) becomes: ∞

Cin − C∞ = ∑ B n ⋅ cos n =0

(2n + 1)πx for -L < x < L 2L

(1.27)

The coefficient Bn is obtained as follows, by multiplying both sides of Equation (1.27) by cos

(2n + 1) πx ∂x , and integrating between -L and +L. Then it becomes: 2L

(C

in

Bn ∫

− C∞ ) +L −L

∫

cos2

+L −L

cos

(2n + 1) πx∂x = ⋅ 2L

+L (2n + 1)πx (2n + 1)πx (2p + 1)πx ∂x + $ + B p ∫ cos ∂x $ (1.28) cos −L 2L 2L 2L

1 1 As cos n ⋅ cos p = ⎡⎣cos ( n + p) + cos ( n − p)⎤⎦ and cos2 n = ⎡⎣1 + cos 2n⎤⎦ 2 2 and applying it to Equation (1.28), all the terms in Bp where p is different from n are equal to 0, while the first term in the right side of Equation (1.28) becomes equal to BnL. Moreover, the integral in the left-hand side of the Equation (1.28) gives: 4L (2n + 1) π which can be written as 4L ⎡−1⎤n sin 2 (2n + 1) π ⎣ ⎦ (2n + 1) π The coefficient Bn is thus equal to: Bn =

n 4 ⎡⎣−1⎤⎦ (2n + 1) π

(1.29)

Finally, the profile of concentration of diffusing substance developed within the sheet of thickness 2L, with -L < x < +L, can be expressed in terms of space x and of time t by the following series: n ⎡ 2n + 1 2 π 2 ⎤ 2n + 1) π ⋅ x 4 ∞ ⎡⎣−1⎤⎦ ( ( ) ⎢ ⋅ exp − ⋅ cos = ⋅∑ D ⋅ t⎥ 2 ⎢ ⎥ Cin − C∞ π n =0 2n + 1 2L 4 L ⎣ ⎦

C x , t − C∞

(1.30)

15

Assessing Food Safety of Polymer Packages The amount of diffusing substance which is transferred into or out of the sheet at time t, Mt, is obtained by integration of the flux of substance through the surface with respect to time: Mt = 2 ⋅

∫

t 0

⎛ ∂C ⎞ D ⋅⎜ ⎟ ⋅ ∂t ⎝ ∂x ⎠x =±L

for x = ±L

(1.31)

⎛ ∂C ⎞ As the gradient of concentration ⎜ ⎟ becomes equal to: ⎝ ∂x ⎠ ⎡ 2n + 1 2 π 2 ⎛ ∂C ⎞ 2 (Cin − C∞ ) ∞ ( ) D ⋅ t⎤⎥ ⎢ ⋅ exp − = ⎜ ⎟ ∑ ⎢ ⎥ L 4L2 ⎝ ∂x ⎠x =±L n =0 ⎣ ⎦

( since ⎡⎣−1⎤⎦ ⋅ sin n

2n + 1) π 2

2n

= ⎡⎣−1⎤⎦ = 1

finally, the amount of diffusing substance transferred at time t, Mt, is given by: Mt =

∞

As the series

∑ n =0

4 (Cin − C∞ ) D L

1

(2n + 1)

2

=

⋅

⎡ 2n + 1 2 π 2 ( ) D ⋅ t⎤⎥ ∂t ⎢ exp − ∫0 ∑ ⎢ ⎥ 4L2 n =0 ⎣ ⎦ t

∞

(1.32)

π2 8

and the amount of diffusing substance which has left or entered the sheet after infinite time, M∞, is given by: M∞ = 2 (Cin − C∞ ) L

(1.33)

the kinetics of transport of the substance into or out of the sheet of thickness 2L is expressed by the following relationship: ⎡ 2n + 1 2 π 2 Mt 8 ∞ 1 ) D ⋅ t⎤⎥ ⎢− ( exp = 1− 2 ∑ ⋅ ⎢ ⎥ M∞ π n =0 (2n + 1)2 4L2 ⎣ ⎦ 16

(1.34)

A Theoretical Approach to Experimental Data

1.4.1 Applications of the Equations of Transfer of Substance Into or Out of a Sheet of Thickness 2L with: -L < x < +L 1. Both the equations for the profiles of concentration (1.30) and for the kinetics of transfer (1.34) can be applied for a sheet of thickness L, with: 0 < x ∞

release of additives in the liquid

(1.6)

CL, t < ∞

absorption of additives by the sheet

(1.6ʹ)

5. A dimensionless number appears in both these equations (1.30 and 1.34), either for the profiles of concentration developed through the thickness of the sheet or for the kinetics of transfer of diffusing substance: D⋅t D⋅t = 4L2 L '2

with a thickness: 2L = Lʹ

(1.35)

6. The problem of release of the additives from a plastic packaging into a liquid of finite volume is of great interest. In this case, the term Ceq is surely better than C∞ in both Equations (1.30) and (1.34) for defining the concentration of the diffusing substance at equilibrium with the liquid. As a matter of fact, whatever the value of the partition factor K, the concentration of the diffusing substance on the surface of the packaging CL,t is in equilibrium at any time with its corresponding value in the liquid; and because of the finite volume of liquid, the concentration in the liquid increases with time up to its final value attained at infinite time (at least theoretically speaking). As described more precisely in Chapter 4, a numerical model is necessary to resolve this interesting and important problem. 7. In fact, it could be said that the only reason to put C∞ in Equation (1.30) is for expressing the value of the concentration on the surface CL,t as well as of the concentration Cx,t, both at infinite time. 8. Equations (1.30) and (1.34) can be applied at any time, and more precisely for any value of the amount of diffusing substance transferred between the sheet and the liquid. 0<

Mt Cin

absorption of diffusing substance by the sheet

(1.6)

C∞ < Cin

release of diffusing substance by the sheet

(1.6´)

1.7.1 Notes on the Case Described in Section 1.7 1

This method of calculation is of great interest to describe the process of drying, but only when the diffusivity is constant and does not depend on the concentration of the diffusing substance in the sheet.

2

In the same way, the surroundings can be considered as infinite provided that the sheet is not confined in a closed volume, because in this case, the concentration of the evaporated substance in the atmosphere, far from remaining constant, would increase regularly as the evaporation proceeds.

3

It should be noted that the rate of convection h is related to the rate of evaporation of the diffusing substance per unit area Ft, by the relationship: Ft = h · CL,t

(1.53)

showing that the rate of evaporation of substance evaporated is proportional to the actual concentration of liquid on the surface of the solid. This concentration of liquid in the polymer, expressed in terms of volume of liquid per total volume of polymer and liquid is lower than 1, while in the pure liquid, the concentration is 1. Of course, when the concentration of the liquid is expressed in terms of mass per volume, the rate of evaporation is also expressed in terms of mass of substance evaporated per unit time and per unit surface of the solid.

1.8 Ratio Volume/Area of the Food Package The most common case encountered in food packages occurs when the volume of food located in the plastic package is finite, and the liquid is not so strongly stirred that the coefficient of convection h would be infinite. It is usual to define the volume of liquid as a fraction of the surface of the package when this food package has the shape of a cube. It gives for a cube of side a: volume a3 a = = 2 area 6⋅a 6 28

(1.54)

A Theoretical Approach to Experimental Data Let us note that this assumption made for a cube is also of reasonable value for a cylindrical bottle, as the area of the packaging for 1 litre (1 dm3) is 5.75 dm2 (instead of 6 for the cubic form). As the volume of the package is proportional to its thickness, with a thickness of 100 μm, for a volume of 1 litre (1 dm3) and the associated area of 6 dm2, the volume of the package is 6 cm3. Thus, the ratio of the volumes and thicknesses shown in Equation (1.43) is either 166 for the cubic form or around 174 for the bottle.

1.9 Determination of the Parameters of Diffusion The parameters of diffusion for a sheet are as follows: mainly the diffusivity (also called coefficient of diffusion), the coefficient of convection at the liquid-solid interface, without forgetting the thickness of the sheet. Surely, the best way to test the accuracy of the values obtained for these parameters is to have the experimental data fitted on the theoretical curve drawn by using these parameters. Two methods can be used for the determination of these parameters: •

Drawing various curves which enable one to determine the values of these parameters.

•

Using a numerical model, taking into account not only the diffusivity but also the coefficient of convection.

1.9.1 Simple Ways of Evaluating the Parameters of Diffusion (Diffusivity) 1.9.1.1 Diffusivity The first way used for evaluating the value of the diffusivity is to express the kinetics of release of the additive in terms of the square root of time. As shown with the Equation (1.39), the value of the slope of the straight line obtained is proportional to 2/L(D/π)0.5 for a sheet of thickness 2L. Only when this curve is a straight line passing through the origin of time (that is obtained when the coefficient of convection h is infinite), does the slope of this straight line give the value of the diffusivity. Theoretically speaking it should be said that the experimental conditions have to meet the following three requirements: •

The ratio of the volume of liquid and of the sheet is very large, e.g., larger than 20;

•

The rate of stirring is so strong that the coefficient of convection is so large that the dimensionless number R is larger than 50 (Figure 1.8); 29

Assessing Food Safety of Polymer Packages •

The amount of diffusing substance released as a fraction of the corresponding value at infinite time (at equilibrium) is lower than 0.5.

Generally, an S-shaped curve similar to those drawn in Figure 1.8, is obtained, proving that the coefficient of convection h is finite. And under these conditions, the value of this parameter h has to be evaluated.

1.9.1.2 Coefficient of convection h An infinite value of the coefficient of convection h can be obtained from the kinetics of release of the additive expressed in terms of time, at the beginning of the process, from the shape of the curve, but a vertical tangent is far from being easy to determine precisely, even by expanding the scale of time. Another easier way consists of expressing the kinetics of additive release in terms of the square root of time and using Equation (1.39). Curves like those shown in Figure 1.8 are helpful in giving a rough idea of the value of this coefficient of convection h.

1.9.1.3 Thickness of the sheet The measure of the thickness should be made accurately, as the time necessary for a given transport controlled by diffusion is proportional to the square of this thickness. Calculations have been carried out for using a sheet in the previous sub-chapters, but nevertheless, the equations and conclusions as well as the curves drawn in the various figures can be used for a cylindrical bottle, for the main and reasonable reason that the radius of this cylinder is far larger than the thickness of the package.

1.9.2 Using a Numerical Model This way of working is surely the easiest one, but only when the model is perfectly built. It is necessary when reading the instructions of the model to find the equation upon which it is based. For example, a model built only by using Equation 1.34 obtained with an infinite value of the coefficient of convection is a very poor model; the accuracy of the value determined for the diffusivity can be appreciated by comparing the kinetics curves obtained either by calculation with the model or by plotting the experimental data. Moreover, the kinetics curves would be expressed preferably either in terms of time or by using the square root of time, and be drawn with an expanded scale of time. 30

A Theoretical Approach to Experimental Data Thus, in the valuable model, the kinetics of release of the additives is expressed by Equation (1.52), while the amount of the additive in the liquid is obtained from the value of the concentration measured at any time t in the liquid, by the obvious relationship: Cliquid, t =

Mt Vliquid

(1.55)

where the volume of liquid is noted in the denominator, and Mt is the amount of additive released at time t. So, the kinetics of release of the additives should be determined by following the increase in the concentration of this additive in the liquid. The method consists of fitting theoretical kinetics with the experimental data, the best theoretical kinetics being obtained with the right value of the dimensionless number R. Of course, the values of the coefficient βn shown in the Equation (1.52) for the theoretical kinetics are necessary, and generally some tables are provided [1-3], but the number of these values is limited, generally to 10 values for the dimensionless number R. While the best way consists of calculating the βn values by using Equation (1.50) associated with some hypothetical values of the dimensionless number R, selected not by the rule of thumb but by using an iterative method, a table is given with a wide range of the R values.

1.9.3 Using the Appropriate Numerical Model The numerical model should take into account the following assumptions: •

the ratio, α, of the volumes of the liquid and of the sheet, is obtained experimentally;

•

the coefficient of convection h is finite;

•

the other experimental data, such as the kinetics and the thickness of the sheet.

In response, the model should be able to draw the theoretical curve which fits the experimental data better, and give the diffusion parameters obtained for this fitting, the diffusivity and the coefficient of convection, as well as the error coefficient evaluating the accuracy of these values as high as possible. The simultaneous effect of the parameters R and α is depicted in Figure 1.9. The curves obtained with the value of R = 100 and either α = 50 (curve 1) or α = 200 (curve 2) are perfectly superimposed, proving that the effect of the ratio of the volumes of liquid and package is negligible when it is larger than 50, as already stated in Section 1.6. On the other hand, the curves 3 and 4 obtained with the same values of 50 (curve 3) or 200 (curve 4) for α, and for the lower value of R = 20, are also superimposed together. However, this 31

Assessing Food Safety of Polymer Packages

Figure 1.9 Kinetics of transfer of the diffusing substance into (or out from) the sheet with a sheet of thickness 2L and a finite value of h, and a finite value of the ratio of volumes defined by α. These master curves are drawn by using the dimensionless numbers Mt/M∞ and D·t/L2 for the co-ordinates. With R = 100 and α = 50 (curve 1), and α = 50 (curve 2); With R = 20 and α = 50 (curve 3), and α = 200 (curve 4)

curve (curve 3 and 4) notably differs from the other one (curves 1 and 2), because of the change in the value of R as well as of h. In order to stress the importance of the effect of the coefficient of convection h and subsequently, that of the dimensionless number R, the following Figures 1.10-1.13 are drawn under the following conditions, while the volume of the liquid is so large that α → ∞: •

Figure 1.10 Obtained when R is infinite, either for the full line and circles.

•

Figure 1.11 S-shaped experimental curve (full line) drawn with D = 10-7 cm2/s and R = 10; curve (dotted line) fitting best the experimental curve, obtained with R infinite, leading to D = 0.73 x 10-7 cm2/s; straight line (full line) tangent to the experimental curve along its linear part, (thus drawn with R infinite) leading to D = 0.92 x 10-7 cm2/s.

•

Figure 1.12 S-shaped experimental curve (full line) drawn with D = 10-7 cm2/s and R = 2; curve (dotted line) fitting best the experimental curve, obtained with R infinite, leading to D = 0.39 x 10-7 cm2/s; straight line (full line) tangent to the experimental curve along its linear part (thus drawn with R infinite), leading to D = 0.61 x 10-7 cm2/s.

32

A Theoretical Approach to Experimental Data

Figure 1.10 Kinetics of transfer of the diffusing substance into (or out from) the sheet with a sheet of thickness 2L and infinite value of h, and infinite value of the ratio of volumes defined by α. These master curves are drawn by using the dimensionless numbers Mt/M∞ and D·t/L2 for the co-ordinates.

•

Figure 1.13 S-shaped experimental curve (full line) drawn with D = 10-7 cm2/s and R = 1; curve (dotted line) fitting best the experimental curve, obtained with R infinite, leading to D = 0.25 x 10-7 cm2/s; straight line (full line) tangent to the experimental curve along its linear part (thus drawn with R infinite), leading to D = 0.44 x 10-7 cm2/s.

These Figures 1.10 to 1.13 lead to some interesting conclusions: i)

Of course, in the hypothetical case when the coefficient h and the dimensionless number R are infinite, or at least very large, the experimental (circles) and the theoretical (full line) are perfectly superimposed, as shown in Figure 1.10.

ii) In the Figures 1.11 to 1.13, when the coefficient h is finite, as well as the dimensionless number R, the experimental S-shaped curve (full line) differs notably from the theoretical curve (dotted line) passing through the coordinates origin. Let us recall that these two curves are calculated and drawn by using the same thickness for the film but different values for the parameters (diffusivity and dimensionless number R). Moreover, the tangent (full line) to the experimental curve along its linear part does not pass through the coordinates origin. 33

Assessing Food Safety of Polymer Packages

Figure 1.11 Kinetics of transfer of the diffusing substance into (or out from) the sheet with a sheet of thickness 2L and a finite value of h, and infinite value of the ratio of volumes defined by α. These master curves are drawn by using the dimensionless numbers Mt/M∞ and the square root of time for the co-ordinates. Full line: experimental S-shaped curve with R = 10 ; D = 10-7 cm2/s; L = 0.1 cm. Dotted line: curve, with infinite R ; L = 0.1 cm ; D = 0.73 x 10-7 cm2/s. Straight line (full line), tangent to the experimental curve with D = 0.92 x 10-7 cm2/s.

iii) Furthermore, the effect of the value given to the parameter h and the dimensionless number R clearly appears by comparing the curves drawn in Figures 1.11 to 1.13. Thus the following statement holds: the lower the coefficient of convection h, as well as the value of the dimensionless number R, the more different the experimental curve is from its theoretical counterpart. iv) Quantitatively, the conclusion given in iii) appears with the values obtained for the diffusivity D when they are calculated by using the equation 1.39: when R = 10, D is lower than the correct value by around 9%, when R = 2, D is lower than the correct value by around 40%, and when R = 1, D is lower than the correct value by around 60%. v) It is necessary to compare the value of the diffusivity obtained by drawing the theoretical curve which gives a closer fit with the experimental curve, in each Figure. 34

A Theoretical Approach to Experimental Data

Figure 1.12 Kinetics of transfer of the diffusing substance into (or out from) the sheet with a sheet of thickness 2L and a finite value of h, and infinite value of the ratio of volumes defined by α. These master curves are drawn by using the dimensionless numbers Mt/M∞ and the square root of time for the co-ordinates. Full line: experimental S-shaped curve with R = 2; D = 10-7 cm2/s; L = 0.1 cm. Dotted line: curve, with infinite R; L = 0.1 cm; D = 0.39 x 10-7 cm2/s. Straight line (full line), tangent to the experimental curve with D = 0.61 x 10-7 cm2/s.

They should be lower than the correct value by 27% in Figure 1.11, 62% in Figure 1.12, and it is only one-quarter of the correct value in Figure 1.13. vi) Thus, there is a dramatic decrease in the value of the diffusivity obtained by calculation from the experimental curves, when using Equation (1.39) instead of Equation (1.52) taking into account the finite values of the dimensionless number R. vii) It is true that in Figure 1.11 with R = 10, drawn with the full scale of the amount of diffusing substance transferred, the three curves do not look very different. But this difference appears more apparent in Figure 1.14 when the scale is reduced for the coordinates. Obviously, this difference between the three curves largely increases from Figures 1.14 to 1.16 where R is 10, 2, or 1, respectively. Let us note that very often, when the amount of the diffusion to be released after infinite time is known, the experimetal curves obtained are those shown in Figures 1.14 to 1.16. 35

Assessing Food Safety of Polymer Packages

Figure 1.13 Kinetics of transfer of the diffusing substance into (or out from) the sheet with a sheet of thickness 2L and a finite value of h, and infinite value of the ratio of volumes defined by α. These master curves are drawn by using the dimensionless numbers Mt/M∞ and the square root of time for the co-ordinates. Full line: experimental S-shaped curve with R = 1; D = 10-7 cm2/s; L = 0.1 cm. Dotted line: curve, with infinite R; L = 0.1 cm; D = 0.25 x 10-7 cm2/s. Straight line (full line), tangent to the experimental curve with D = 0.44 x 10-7 cm2/s

1.9.3.1 Note: Quick Approach to the Value of the Diffusivity When the process of additive release from the sheet into the liquid complies with the following requirements: •

the volume of the liquid is much larger than that of the sheet, with α > 20;

•

the system is so strongly stirred that the coefficient of convection is infinite, or R > 100;

•

the whole kinetics of release is obtained, or at least the amount of additive released at equilibrium is known; in fact it could be the amount which was initially in the sheet, these two facts leading to the value of M∞.

In these cases, and only when these three assumptions are satisfied, can the two equations 1.37 and 1.39 be applied to the time t0.5 at which half the amount of the additive is released, 36

A Theoretical Approach to Experimental Data

Figure 1.14 Kinetics of transfer of the diffusing substance into (or out from) the sheet with a sheet of thickness 2 L and a finite value of h, and infinite value of the ratio of volumes defined by α. These master curves are drawn by using the dimensionless numbers Mt /M∞ and the square root of time for the co-ordinates. Full line: experimental S-shaped curve with R = 10; D = 10-7 cm2/s; L = 0.1 cm. Dotted line: curve, with infinite R; L = 0.1 cm; D = 0.73 x 10-7 cm2/s. Straight line (full line), tangent to the experimental curve with D = 0.92 x 10-7 cm2/s.

Mt/M∞ = 1/2 is thus obtained, L being half the thickness of the sheet: 0.5

Mt 1 2 ⎡ D ⋅ t ⎤ = = ⎢ ⎥ M∞ 2 L ⎣ π ⎦

for

⎡ π2 ⋅ D ⋅ t ⎤ Mt 8 = 1 − 2 exp ⎢− ⎥ M∞ 4L2 ⎦ π ⎣

Mt < 0.5 M∞ for

Mt > 0.5 M∞

(1.37)

(1.39)

From the relationship Equation (1.37), it is found: ⎡ L2 ⎤ D = 0.196 ⋅ ⎢ ⎥ ⎣ t0.5 ⎦

(1.56)

37

Assessing Food Safety of Polymer Packages

Figure 1.15 Kinetics of transfer of the diffusing substance into (or out from) the sheet with a sheet of thickness 2L and a finite value of h, and infinite value of the ratio of volumes defined by α. These master curves are drawn by using the dimensionless numbers Mt/M∞ and the square root of time for the co-ordinates. Full line: experimental S-shaped curve with R = 2; D = 10-7 cm2/s; L = 0.1 cm. Dotted line: curve, with infinite R; L = 0.1 cm; D = 0.39 x 10-7 cm2/s. Straight line (full line), tangent to the experimental curve with D = 0.61 x 10-7 cm2/s

and from the relationship (1.39), we get: ⎡ L2 ⎤ D = 0.192 ⋅ ⎢ ⎥ ⎣ t0.5 ⎦

(1.57)

It clearly appears that the value of the diffusivity D is nearly the same in these two equations, varying only by 2%.

1.10 Diffusion through Isotropic Rectangular Parallelepipeds Rectangular parallelepipeds (a polyhedron with six faces all of which are parallelograms) are sometimes used instead of sheets, making this sub-chapter of practical interest. The sheet has been considered as a rectangular parallelepiped whose one side, the thickness, 38

A Theoretical Approach to Experimental Data

Figure 1.16 Kinetics of transfer of the diffusing substance into (or out from) the sheet with a sheet of thickness 2L and a finite value of h, and infinite value of the ratio of volumes defined by α. These master curves are drawn by using the dimensionless numbers Mt/M∞ and the square root of time for the co-ordinates. Full line: experimental S-shaped curve with R = 1; D = 10-7 cm2/s; L = 0.1 cm. Dotted line: curve, with infinite R; L = 0.1 cm; D = 0.25 x 10-7 cm2/s. Straight line (full line), tangent to the experimental curve with D = 0.44 x 10-7 cm2/s

is very thin as compared with the other two sides, and the effect of the edges on the total matter transfer can be neglected. Some materials are anisotropic, e.g., wood [3] or stretched polymers. In fact, the following equations can also be used for those materials, provided that the right diffusivity associated with each direction is put in the corresponding function. The problem with three axes of diffusion is resolved whatever the value of the coefficient of convection h, finite or infinite.

1.10.1 Diffusion with an Infinite Value of the Coefficient of Convection The amount of diffusing substance transferred through the surface of the parallelepiped after time t, as a fraction of the corresponding amount after infinite time, is expressed in terms of time by the product of the three series obtained for a one-dimensional transport.

39

Assessing Food Safety of Polymer Packages Thus, the following equation is obtained for an infinite coefficient of convection h: M∞ − M t = f (x, a) ⋅ f (y, b) ⋅ f (z, c) M∞

(1.58)

where each function f is given by Equation (1.34) applied to the corresponding axis and length, the three lengths of the parallelepiped being a, b, c, respectively, and x, y, z being the perpendicular axes of diffusion. For example, there is more precisely for f(x, a): f (x, a) =

⎤ ⎡ (2n + 1)2 π2 8 ∞ 1 ⋅ exp D ⋅ t⎥ ⎢− ∑ 2 2 2 4a π n =0 (2n + 1) ⎦ ⎣

(1.59)

the other functions of f being obtained by substituting a for b in f(y, b) and for c in f(z, c).

1.10.2 Diffusion with a Finite Value of the Coefficient of Convection For the finite coefficient of convection on each surface, the equation becomes: M∞ − M t = g(x, h, a) ⋅ g(y, h, b) ⋅ g(z, h, c) M∞

(1.60)

where the function g(x, h, a) is drawn from the Equation (1.52) by replacing the parameters as follows: ⎡ D ⋅ t⎤ 2 ⋅ X2 exp ⎢−β2n 2 ⎥ 2 2 2 a ⎦ ⎣ n =1 βn (βn + X + X) ∞

g(x, h, a) = ∑

(1.52ʹ)

h⋅a D

(1.51ʹ)

with X =

while the βn are now the positive roots of β · tan β = X

(1.50ʹ)

The other functions g(y, h, b) and g(z, h, c) are obtained from the three previous relationships (1.50ʹ) to (1.52ʹ) by substituting the parameters: (a, X, βn) for (b, Y, βn) in g(y, h, b), and for (c, Z, βn) in g(z, h, c), respectively. Of course, the new values of βn in the three functions should be recalculated by using the relationship similar to (1.50ʹ). 40

A Theoretical Approach to Experimental Data

1.10.3 Approximate Value of the Diffusivity for Small Times and Infinite h For an infinite value of the coefficient of convection and low values of time expressed by the ratio Mt/M∞ < 4, the series in Equation (1.38) vanishes, leading to a simple relationship, as shown already in (1.39), applied to the length a instead of L: 0.5

Mt 2 ⎡ D ⋅ t ⎤ = ⎢ ⎥ M∞ a ⎣ π ⎦

for −a < x < +a and h → ∞

(1.39ʹ)

Thus, for the rectangular parallelepiped of sides 2a, 2b and 2c, the following equation is obtained: 0.5

Mt ⎡ 2 2 2⎤ ⎡ D ⋅ t ⎤ = ⎢ + + ⎥⋅⎢ ⎥ M∞ ⎣ a b c ⎦ ⎣ π ⎦

(1.61)

1.10.4 Approximate Value of the Diffusivity for Long Times and Infinite h For long times, e.g., for Mt/M∞ > 0.5, the first term of the series in Equation (1.59) is preponderant compared to the others. And by applying the way of calculation shown in Equation (1.58) to a rectangular parallelepiped of sides 2a, 2b and 2c, the amount of diffusing substance transferred after time t, is expressed by the relationship: ⎡ ⎛ 1 1 1 ⎞ π 2D ⋅ t ⎤ M∞ − M t 512 = 6 ⋅ exp ⎢− ⎜ 2 + 2 + 2 ⎟ ⎥ M∞ π ⎣ ⎝a b c ⎠ 4 ⎦

(1.62)

1.11 Case of a Membrane of Thickness L In spite of the fact that the food package never plays the role of a membrane, this case is still considered in this general chapter. The membrane is widely used as a patch for the transdermal drug delivery because of its advantages [4]. Only the theory of the process is developed, leading to the main equations. The results are expressed in terms of either the kinetics of matter transport or of profiles of concentration of the diffusing substance developed through the thickness of the membrane. Nevertheless, the effect of the value of the coefficient of convection on the surfaces of the membrane is considered (Figure 1.17). 41

Assessing Food Safety of Polymer Packages

Figure 1.17 Scheme of the membrane of thickness L. The concentration of the liquid in contact with each side is Cin on the left and C = ε on the right. At each liquidmembrane interface, the coefficient of convection is either infinite or finite.

1.11.1 Transport of the Substance through the Membrane for Infinite h The membrane is a sheet of thickness L whose sides are maintained at constant concentrations, while the substance is transported from the higher to the lower concentration by diffusion. Thus, the process is controlled by the one-directional transient equation of diffusion through the sheet, as already shown: ∂C ∂2C =D 2 ∂t ∂x

(1.11)

where: C is the concentration of the substance at time t and position x, D is the diffusivity of the substance. When the coefficient of convection is infinite on both sides of the membrane, the concentration on each surface of the membrane is equal to that in the liquid in contact with it, as soon as the process starts. The initial and boundary conditions for a sheet of thickness L playing the role of a membrane are as follows:

42

t=0

00

x=L

CL = ε

A Theoretical Approach to Experimental Data Under these conditions, when the coefficient of convection on the surfaces of the sheet is very large, the concentration of the diffusing substance developed through the thickness of the membrane, obtained by using the method of separation of variables, is expressed in terms of time through the following series: Cx,t C0

= 1−

⎡ x 2 ∞ 1 nπx D ⋅ t⎤ − ∑ ⋅ sin ⋅ exp ⎢−n2 π2 2 ⎥ L π n =1 n L L ⎦ ⎣

(1.65)

The amount of substance, which emerges from the side with the lower concentration, at position L, is expressed by the series: ⎡ M t D ⋅ t L 2L ∞ (−1)n D ⋅ t⎤ = − − 2 ∑ 2 exp ⎢−n2 π2 2 ⎥ 6 π n =1 n C0 L L ⎦ ⎣

(1.66)

Equation (1.65) shows that the concentration of diffusing substance in the membrane increases with time, and the profiles of this substance tend to be linear when the series vanishes. In the same way, from Equation (1.66), the amount of substance emerging from the side of the membrane kept at the lower concentration increases with time as the series vanishes. Thus, after a given period of time the kinetics of the substance leaving the membrane tends to become linear, and the expression of the asymptote is given by: Mt =

DC0 ⎡ L2 ⎤ ⎥ ⎢t − L ⎣ 6D ⎦

(1.67)

the slope of which is: Slope =

M t DC0 = t L

(1.68)

and the intercept on the time-axis is: ti =

L2 6D

(1.69)

1.11.2 Results for the Membrane with Infinite Value of h on Both Sides When the coefficient of convection on the two sides of the membrane is infinite, the Equations (1.65) to (1.69) hold. Calculation is made by using the following values for the parameters: L = 0.05 cm D = 10-8 cm2/s C0 = 50 mg/cm3 h →∞ 43

Assessing Food Safety of Polymer Packages The results are expressed in terms of the kinetics of diffusing substance transferred out of the membrane (Figure 1.18) and of the profiles of concentration developed through the thickness of the sheet (Figure 1.19). They lead to a few conclusions: i)

A typical pattern is shown in Figure 1.18 for the kinetics of transport. Starting at the origin of the two coordinates, the curve increases slowly before tending asymptotically to an asymptote at a time larger than 50 hours.

ii) The profiles of concentration developed through the thickness of the sheet also tend to be linear, and for times greater than 3000 minutes (50 hours) it can be said that the stationary part of the curve is nearly reached. However, it is difficult to define experimentally at which time the profiles of concentration become linear. iii) In the same way, it is not easy to determine with good accuracy the intercept on the time-axis, and the curve needs to be expanded on a larger scale, as shown in Figure 1.18. Thus, from the measurement made on the curve this time seems to be 11.6 hours, while calculation made by using Equation (1.69) and the parameters gives 11.55 hours. The accuracy is quite perfect, but for a rather large value of this time. Let us note that good knowledge of this time is of interest for evaluating an accurate value of the diffusivity.

Figure 1.18 Kinetics of matter leaving the right side of the membrane, when the coefficient of convection is infinite on both sides. L = 0.05 cm; Cin= 50 mg/cm3; D = 10-8 cm2/s; h infinite on both sides.

44

A Theoretical Approach to Experimental Data

Figure 1.19 Profiles of concentration developed through the thickness of the membrane when the coefficient of convection is infinite on both sides. L = 0.05 cm; Cin = 50 mg/cm; D = 10-8 cm2/s; R infinite on both sides.

1.11.3 Results for a Membrane with a Finite Value of h on the Right Side By using a numerical model, the same calculation is made for a finite value of the coefficient of convection h on the right side and infinite value of h on the left side of the membrane, by keeping the other parameters equal to those given in Section 1.11.2: L = 0.05 cm; D = 10-8 cm2/s; C0 = 50 mg/cm3; h = 10-6 cm/s on the right side and R = 5. The results are also expressed in terms of the kinetics of mass transferred out of the membrane (Figure 1.20) and of the profiles of diffusing substance developed through the thickness of the sheet (Figure 1.21). Some conclusions of interest are obtained, especially by comparing the kinetic curves and profiles of concentration with their counterparts obtained with the infinite value of h and R: i)

The kinetics of the diffusing substance transferred out of the membrane in Figure 1.20 look similar, at a rough estimate, to that drawn in Figure 1.18. However, the intercept on the time-axis is much larger, at 15.36 hours. Of course, this greater time results from the fact that the finite convection coefficient on the right side decreases the rate 45

Assessing Food Safety of Polymer Packages

Figure 1.20 Kinetics of matter leaving the right side of the membrane, when the coefficient of convection is infinite on the left side and finite on the right side. L = 0.05 cm; Cin= 50 mg/cm3; D = 10-8 cm2/s; h = 10-6 cm/s on the right side; h infinite on the left side.

Figure 1.21 Profiles of concentration developed through the thickness of the membrane, when the coefficient of convection is infinite on the left side and finite on the right side. L = 0.05 cm; Cin = 50 mg/cm3; D = 10-8 cm2/s; h = 10-6 cm/s on the right side; h infinite on the left side.

46

A Theoretical Approach to Experimental Data of transfer. This fact is also responsible for a lower value of the diffusivity by using Equation (1.69). Thus, neglecting the presence of a finite convection coefficient on the right side gives a shorter wrong value of the diffusivity, as already shown in Section 1.9.3 under other conditions. ii) The profiles of concentration drawn in Figure 1.21 are quite different from those shown in Figure 1.19. This is especially true for times greater than 300 minutes when the diffusing substance emerges out of the right side of the membrane. The concentration on the surface is not 0, but increases constantly up to a limit attained when the stationary state is nearly reached. This fact results from the condition on the right side established by considering that the rate at which the substance leaves the sheet is constantly equal to the rate at which the substance is brought to the surface by internal diffusion; this condition is written with the finite value of the coefficient of convection as follows, which is quite different from Equation (1.3). ⎛ ∂C ⎞ −D ⋅ ⎜ ⎟ = h ⋅ (CL − 0) = h ⋅ CL ⎝ ∂x ⎠ L

right side

(1.70)

iii) The profiles of concentration drawn in Figure 1.21 also show that the concentration on the left side of the membrane is kept at the constant value Cin of the liquid. As soon as the process starts, because of the infinite value of the coefficient of convection h on the left side, the concentration of the diffusing substance on the left surface of the membrane reaches the constant value maintained in the liquid with which it is in contact.

1.11.4 Results for a Membrane with Finite Value of h on Both its Sides The results are obtained by using a numerical model, with the same finite value of the coefficient of convection on both sides, and keeping the other parameters taken in Section 1.11.2: L = 0.05 cm; D = 10-8 cm2/s; Cin = 50 mg/cm3 on the left side; Cout = ε on the right side; h = 10-6 cm/s on both sides and R = 5. In addition to the case Section 1.11.3, the finite coefficient of convection on the left side of the membrane appears, written in a form slightly different from Equation (1.3): ⎛ ∂C ⎞ −D ⎜ ⎟ = h ⋅ (Cin − C0, t ) ⎝ ∂x ⎠ 0

(1.71)

These results are also expressed in terms of the kinetics of mass transferred out of the membrane (Figure 1.22) and of the profiles of diffusing substance developed through the thickness of the sheet (Figure 1.23). 47

Assessing Food Safety of Polymer Packages

Figure 1.22 Kinetics of matter leaving the right side of the membrane, when the coefficient of convection is finite on both sides. L = 0.05 cm; Cin = 50 mg/cm3; D = 10-8 cm2/s; h = 10-6 cm/s on both sides.

Figure 1.23 Profiles of concentration developed through the thickness of the membrane, when the coefficient of convection is finite on both sides. L = 0.05 cm; Cin = 50 mg/cm3; D = 10-8 cm2/s; h = 10-6 cm/s on both sides.

48

A Theoretical Approach to Experimental Data Some interesting conclusions are obtained, especially by comparing the kinetic curves and profiles of concentration with their counterparts obtained with the infinite value of h and R: i)

The kinetics of the diffusing substance leaving the membrane in Figure 1.22 is slightly different from the other two kinetics obtained either with an infinite value of h on both sides (Figure 1.18) or with infinite value of h only on the left side (Figure 1.20). The time at which the asymptote of the curve intercepts the time-axis is longer than for the previous cases, at around 20 hours, resulting from the presence of the low value of h on both sides.

ii) The profiles of concentration drawn in Figure 1.23 are quite different from those shown either in Figure 1.19 with infinite value of h on both sides or in Figure 1.21 with infinite value of h on the left side only. The main difference between the Figures 1.21 and 1.23 results from the presence of the finite value of h on the left side of the membrane in Figure 1.23. In addition to the case shown in Figure 1.21, it takes some time for the concentration of the substance on the left surface to increase up to the value at equilibrium which is maintained at a constant under the stationary condition. This time is around 50 hours, in the same way as in Figure 1.21 when the coefficient of convection is finite only on the right side of the membrane or as in Figure 1.19 when the coefficient of convection is infinite on both sides. iii) It is worth noting that, not only do the profiles of concentration developed through the thickness of the membrane give a fuller insight into the nature of the process of diffusion-convection, but they also enable one to evaluate better the time at which the stationary conditions are attained with the asymptotical tendency of the kinetics curve.

1.12 Evaluation of the Parameters of Diffusion from the Profiles of Concentration A constant drawback with the problems of diffusion comes from the long, time-consuming experiments. The dramatic dilemma arises from the difficulty of approaching the value of the amount of the diffusing substance released after infinite time. In fact, experiments associated with the safety of the consumers can take not only months but years. Of course, a solution exists, not well known by the majority of the workers, which consists of reducing the thickness of the film, according to the dimensionless number D·t/L2 which stands in all the equations of diffusion describing either the kinetics or the profiles of concentration, the time necessary for a given percentage release is proportional to the square of the thickness of the material in contact with the liquid. Nevertheless, if it is not 49

Assessing Food Safety of Polymer Packages difficult to make experiments to follow the kinetics of release of a substance, the time of the operation is often too long. A new method consists of evaluating the profiles of concentration of the diffusing substance developed through the thickness of the sheet during the process of release [5]. Modern equipment has considerably improved the technique first proposed by Moisan [6] a few decades ago.

1.12.1 Experimental In the present case [5], the polymer is a 0.2 cm thick polypropylene (PP) with olive oil as the liquid considered as the diffusing substance. Disks of 3 cm diameter are immersed in the liquid kept at 40 °C for a period not less than 64 days. At intervals, a disk is removed for analysis by using the IR absorbance of oil. The PP disk is put in the analytical beam of an FTIR spectrometer and a virgin PP is used as a reference. After calibration of the olive oil, by drawing the absorbance versus the concentration, quantitative analysis of the olive oil in the PP is made possible. Moreover, greater accuracy is obtained for the position taken through the thickness of the polymer sheet by using a microscope connected with the spectrometer.

1.12.2 Theoretical The equation of one-dimensional diffusion with constant diffusivity is: ∂C ∂2C =D 2 ∂t ∂x

(1.11)

With the initial condition expressing that the polymer sheet is free from olive oil: C = 0 for -L < x < L and t = 0

(1.40)

and the boundary condition expressing that the rate at which the liquid enters the polymer by diffusion is constantly equal to the rate at which the liquid is brought or rather put into contact with the polymer surface by convection and inserted into the polymer: ⎛ ∂C ⎞ −D ⎜ ⎟ = h CL , t − Ceq ⎝ ∂x ⎠

(

)

(1.3)

the general solution for the gradient of concentration of the diffusing substance developed through the thickness of the sheet is given by Equation (1.49), whatever the value of the initial concentration in the polymer sheet, which is 0 in the present case: 50

A Theoretical Approach to Experimental Data ⎛ x⎞ 2R ⋅ cos ⎜βn ⎟ ⎛ C∞ − C x , t D⋅t⎞ ⎝ L⎠ exp ⎜−β2n 2 ⎟ =∑ 2 2 C∞ − Cin n =1 (βn + R + R) ⋅ cos βn L ⎠ ⎝ ∞

(1.49)

where the βn are the positive roots of: β · tan β = R

(1.50)

and the dimensionless number R is given by: R=

h⋅L D

(1.51)

On the other hand, the kinetics of transfer of diffusing substance are expressed in terms of the dimensionless number D·t/L2 by using the dimensionless number Mt/M∞: ⎛ M∞ − M t ∞ 2 ⋅ R2 D⋅t⎞ exp ⎜−β2n 2 ⎟ =∑ 2 2 2 M∞ L ⎠ ⎝ n =1 βn (βn + R + R)

(1.52)

Moreover, the obvious relationship should be added for the value attained at infinite time: M∞ = 2L · C∞

(1.72)

Thus, the values of the three parameters of diffusion, namely, the diffusivity and the coefficient of convection, as well as the amount of liquid absorbed after infinite time, should be determined. Additionally, the value of the concentration C0,t on the surface of the polymer sheet should also be calculated, as it is not possible to determine it experimentally with this apparatus.

1.12.3 Results Obtained with the Gradients of Concentration The profiles of concentration of the diffusing substance developed through half the thickness of the sheet are plotted in Figure 1.24, as they are calculated at different times: 1, 7, 14, 28, 56 and 112 days. Two other results, also evaluated by calculation, are shown in the other Figures. Figure 1.25 shows the kinetics of the increase in the concentration of the diffusing substance on the polymer surface in contact with the liquid, expressed by using dimensionless numbers; the process being symmetrical, only half the sheet is considered. Figure 1.26 represents the kinetics of increase of the amount of the liquid absorbed, expressed by using dimensionless numbers. 51

Assessing Food Safety of Polymer Packages

Figure 1.24 Profiles of concentration of olive oil developed through the thickness of a 0.2 cm thick PP sheet after various times of contact: 1, 7, 14, 28, 56 and 112 days, as they are calculated with the following parameters: L = 0.1 cm; D = 7 × 10–11cm2/s; h = 10–8cm/s. Only a half of the PP sheet of thickness 0.2 cm is shown in the figure, because of the symmetry. Dimensionless numbers are used for the coordinates.

Figure 1.25 Kinetics of increase in the concentration of olive oil on the surface of the 0.2 cm thick PP sheet calculated at various times with the same parameters as in Figure 1.24.

52

A Theoretical Approach to Experimental Data

Figure 1.26 Kinetics of absorption of olive oil by the 0.2 cm thick PP sheet, calculated at the same times and with the same parameters as in Figure 1.24.

From the curves depicted in Figures 1.24-1.26, with the parameters of diffusion D = 7·10–11cm2/s and h = 10–8cm/s, some relevant conclusions are worth mentioning: i)

A relatively good correlation is obtained [5] between the experimental and calculated profiles of concentration developed through the polymer sheet, in spite of the disadvantage of not having the experimental concentration on the surface. Thus, this value C0,t should be extrapolated by calculation from the profiles by using Equation (1.49).

ii) Figure 1.24 clearly shows that the concentration on the surface of the polymer in contact with the liquid increases with time rather slowly, as after 112 days the value is far from reaching equilibrium. This is an obvious and definitive proof for the presence of a convective mass transfer at the liquid-polymer interface. The process is thus definitively controlled either by diffusion through the thickness of the polymer sheet or by convection onto its surface. The value of the dimensionless number R is around 14. iii) The coefficient of convective transfer h plays an important role, considerably reducing the rate of the process of transfer. Moreover, the curve drawn in Figure 1.25, expressing the kinetics of increase in the concentration on the surface with time, is of great interest. It clearly appears that after 112 days, the concentration on the surface C0,t is very close to the value at equilibrium, as the ratio of the concentrations C0,t/C0,∞ reaches a little more than 0.8, making the extrapolation easy to do for infinite time. 53

Assessing Food Safety of Polymer Packages iv) The curve expressing the kinetics of the liquid absorbed in Figure 1.26 is also of great interest, especially on the purpose of didactics. As the amount of liquid absorbed at equilibrium M∞ can be evaluated by using the Equation 1.72 and the corresponding value of the concentration on the surface, C∞, is easily obtained by extrapolation (Figure 1.25), it is possible to draw the kinetic curve of absorption shown in Figure 1.26. v) As shown in Figure 1.26, the amount of liquid absorbed after 112 days as a fraction of the corresponding amount at infinite time, Mt/M∞, is less than 0.25. Thus, for such a low value, it is impossible to extrapolate the value at infinite time, M∞. The advantage of the method based on the profiles of concentration over the usual method consisting of measuring the amount absorbed can clearly be seen. The problem arises because a rather complex apparatus is necessary to do the experiments for measuring these gradients of concentration.

1.13 Conclusions on the Diffusion Process After considering the various parts of the process of the matter transport controlled by diffusion, it seems necessary to recall the most important facts, in the form of a conclusion. Various parameters intervene in the process of diffusion when a film is in contact with a liquid food. First, those concerned with the film itself, with its thickness and the diffusivity of the substance, secondly, the volume of the food. The diffusing substance intervenes through the diffusivity which depends on the diffusing substance-polymer couple, as well as its solubility either in the polymer or in the food; the partition factor results from the presence of the solubility in both these media which limits the concentration of this substance. The rate of stirring plays the major role, because of the presence of the convection stage at the liquid-package interface. The ratio of the volume and package, denoted α, is only of concern when it is lower than 20; it is clear that for a 1 litre bottle, whatever its shape, this ratio α being around 166, its effect becomes negligible except when a very high accuracy is needed. By contrast, the rate of stirring always plays the most important role in the process of transport of the diffusing substance, because of the presence of the convection at the liquid-film interface. This coefficient of convection h is of such great concern that it should always be considered. In fact, it is clear that when h is infinite, the concentration on the film surface reaches the value at equilibrium instantaneously as soon as the process starts. This fact makes the equations expressing either the kinetics (1.34 and 1.39) or the profiles of concentration through the film thickness (1.31) quite oversimplified. 54

A Theoretical Approach to Experimental Data A strong difference also appears between the film playing the role of a package or used as a membrane. On the one hand, a package contains a finite amount of diffusing substance, which decreases during the process; on the other hand, the concentration of the diffusing substance is kept constant on both sides of this membrane, whatever the convection on each side. When it comes to considering the main fact, e.g., the convection, associated with the diffusivity, it should be said that the convection is very important for the food protection as it acts upon the polluting transport as an additional resistance to the diffusion stage. A relevant proof for the presence of this convective transfer at the liquid-package interface is demonstrated by considering the case of diffusion of a liquid such as olive oil through a PP sheet when this polymer sheet is immersed in the liquid. Finally, a numerical model is of interest, if not necessary, to solve the diffusion problems because of the values of the βn, which should be determined through Equation (1.50) for calculating the kinetics of the substance transferred by using Equation (1.52). Some emphasis is put upon the dimensionless numbers D·t/L2 and h·L/D, as well as Mt/M∞ or Mt/Min and Ct/Cin. By using them, the kinetics or the profiles of concentration obtained in typical cases are transformed into master curves which can be used whatever the nature of the diffusing substance-polymer couple and the other parameters, e.g., the dimension of the film and the time, as well as the initial concentration of the substance bound to diffuse.

References 1

J. Crank, The Mathematics of Diffusion, 2nd Edition, Clarendon Press, Oxford, UK, 1975, Chapter 4.

2

J-M. Vergnaud, Controlled Drug Release of Oral Dosage Forms, Ellis Horwood, New York, NY, USA, 1993, Chapters 1, 2 and 3.

3

J-M. Vergnaud, Liquid Transport Processes in Polymeric Materials: Modelling and Industrial Applications, Prentice Hall, Englewood Cliffs, NJ, USA, 1991, Chapters 1 and 13.

4

J-M. Vergnaud and I-D. Rosca, Assessing Bioavailability of Drug Delivery Systems: Mathematical and Numerical Treatment, CRC Press, Boca Raton, FL, USA, 2005, Chapter 4, section 4.1 and Chapter 10.

5

A.M. Riquet, N. Wolff, S. Laoubi, J.M. Vergnaud and A. Feigenbaum, Food Additives & Contaminants, 1998, 15, 6, 690.

6

J.Y. Moisan, European Polymer Journal, 1980, 16, 10, 979. 55

Assessing Food Safety of Polymer Packages

Abbreviations A

half the thickness of the liquid (volume per unit area) in case of finite volume of liquid

a

length of the edge of a cube, considered as a model for the volume/area ratio

α

ratio of the volumes of the liquid and of the sheet, per unit area, in Equation (1.43)

An, Bn, λn

constants used in Equation (1.15)

βn

positive roots of Equation (1.50)

Cx,t, CL,t

concentration of diffusing substance at position x or at position L, at time t.

Cin, C∞, Ceq

concentration of diffusing substance, initially, after infinite time, and at equilibrium with that in the surrounding, liquid or gas respectively

CGS

Centimetre, gram, second system

D

diffusivity (coefficient of diffusion) expressed in cm2/s (square length)/ unit time

Fx,t

flux of matter at position x and time t (mass per unit area and per unit time)

h

coefficient of convection expressed in cm/s

K

partition factor shown in Equation (1.7), dimensionless number

L

half the thickness of the sheet (of thickness 2L)

Mt, M∞

amount of matter transferred by diffusion after time t, infinite time, respectively

PP

polypropylene

qn

non-zero positive roots of Equation (1.42)

S

area of the sheet in contact with the liquid

x, t

coordinate along which the diffusion occurs, and time, respectively

D·t/L2

dimensionless number, expressing time

h·L/D

dimensionless number, expressing the quality of stirring of the liquid

56

2

Mass Transfer Through Multi-layer Packages Alone

2.1 Recycling Waste Polymers and Need of a Functional Barrier Accumulation of waste in the environment has recently stimulated investigations of processes, which can reuse discarded plastics. Since plastics for food packages represent more than 50% of the overall consumption of plastics in Europe, an efficient approach consists of making new packaging materials from recycled plastic packaging. Several techniques are now available. The main difficulty in this field is that discarded plastics may be contaminated by various substances, which may migrate into the food [1]. A solution to this problem could be the incorporation of recycled plastics into multilayer structures, where a layer of virgin polymer is placed between the recycled polymer and the liquid food. The layer of virgin polymer then behaves as a functional barrier, which protects the food from migration of contaminant, at least over a given period of time, which should be known.

2.1.1 Role of the Functional Barrier Of course, the role of this virgin polymer placed between the food and the recycled polymer is of prime importance. As a matter of fact, the contaminant located in the recycled plastics cannot reach the food before a lag time resulting from its diffusion through the functional barrier. This functional barrier is efficient if this lag time is larger, or at least equal, to the intended shelf life of the food. Various parameters determine the efficiency of the functional barrier, such as the level of contaminant originally in the recycled layer, the nature of the polymer-contaminant couple, the presence of a food transfer through the functional barrier, and the thicknesses of the recycled film and of the functional barrier [2-4]. This problem can be very complex, both from a practical and from a theoretical point of view. The functional barrier and the recycled layer can be made of different polymers, so that the diffusivity of contaminants may change in the two layers; moreover, if a layer is made of glassy polymer, the diffusion may be non-Fickian, modifying the mathematical treatment. The concentration of these contaminants is also a relevant factor, as its solubility in these two polymers could be so different that a partition factor could intervene at their interface. Finally, and this 57

Assessing Food Safety of Polymer Packages is not the least stage in the process, if the food enters the polymer with which it is in contact, as proved in studies concerned with plasticised PVC, the concentration of this liquid will also increase the diffusivity of the contaminant [4, 5].

2.1.2 Mass Transfer Occurring During the Co-extrusion Stage During the stage of co-extrusion, both the polymers made of either recycled or virgin materials are melted and put firmly in contact so as to form a bi-layer or a threelayer film, before being cooled down. It is clear that during this stage a fast transfer of contaminant would take place, resulting from a high diffusivity and perhaps some convection effect, but fortunately this strong transfer takes place over a short period of time, as the thin sheet is cooling down rapidly, and this factor may be a counterbalance to the high diffusivity.

2.1.3 General Problem of Diffusion Through the Layers of the Packaging Alone To face the complexity of these problems and resolve them, the various stages will be considered separately in different chapters. In Chapter 2, the diffusion of a contaminant through the polymer layers alone, without liquid is studied. From a practical point of view, this part concerns the behaviour of the packaging before its use as a food container. From the theoretical point of view, the problems of diffusion through the polymers will be examined in depth, and this knowledge should be useful in the following chapters when the packaging is in contact with a food either in liquid or in viscous state. The general conditions of concern which are covered in Chapter 2, are: •

As there is no liquid, both surfaces are in contact with air;

•

As the contaminant is not volatile, the rate of evaporation of this contaminant is assumed to be negligible.

2.2 Bi-layer Package: Recycled Polymer-Functional Barrier The scheme of the bi-layer package is shown in Figure 2.1, which consists of a virgin layer and of a layer made of recycled polymer. 58

Mass Transfer Through Multi-layer Packages Alone

Figure 2.1 Scheme of the bi-layer system with recycled polymer layer on the left and the virgin polymer layer on the right, playing the role of a functional barrier. Cin is the uniform concentration of diffusing substance which is initially in the recycled polymer layer.

2.2.1 Mathematical Treatment of the Process [1, 6] Assumptions The following assumptions are made in order to describe the process precisely: i)

The packaging is a laminate made of two films of the same polymer in perfect contact; there is no resistance to mass transfer at the interface between the two films.

ii) At the beginning of the process, the concentration of contaminant is uniform in the recycled film, while the virgin film is free from contaminant. iii) The transfer of contaminant is controlled by Fickian diffusion. iv) The diffusivity of the contaminant is constant, and it is the same in both films, since they are made of the same polymer. v) The contaminant does not evaporate out of the external surfaces of the packaging. vi) The concentration of contaminant is so low that there is no change in the thickness of each layer resulting from the transfer. vii) There is no contact between the package and food. 59

Assessing Food Safety of Polymer Packages

Mathematical Treatment Following assumptions (iii) and (iv), the transport of contaminant through the two layers is governed by the unidirectional equation of diffusion with a constant of diffusivity: ∂C x , t ∂t

= D⋅

∂2C x , t

(2.1)

∂x 2

where Cx,t is the concentration of contaminant at abscissa x and time t, while D is the constant diffusivity. Following assumptions (ii) and (v), the initial and boundary conditions are expressed by: t=0

t>0

00

x = ±L

A⋅

∂C ∂C = ±D ∂t ∂x

(4.10)

is given as follows, whatever the partition factor [1-3]: Cx,t C∞

⎡ cos(q n x / L) 2 ⋅ (1 + α) D ⋅ t⎤ ⋅ ⋅ exp ⎢−q2n 2 ⎥ 2 2 cos q n L ⎦ 1 + α + α qn ⎣ n =1 ∞

= 1+ ∑

∞ ⎛ q2 ⋅ D ⋅ t ⎞ Mt 2α ⋅ (1 + α) ⋅ exp ⎜− n 2 ⎟ = 1− ∑ 2 2 M∞ L ⎝ ⎠ n =1 1 + α + α ⋅ q n

(4.11)

(4.12)

where qn is the non-zero positive root of: tan qn = –α ⋅ qn

(4.13)

and the ratio of the volumes of liquid and sheet are given either by Equations (4.14) or by (4.15), depending on the value of the partition factor K. 151

Assessing Food Safety of Polymer Packages When the partition factor K is 1, the concentration on the sheet surfaces is constantly equal to the concentration of the diffusing substance in the liquid. When the partition is not equal to 1, the concentration of diffusing substance on both surfaces of the sheet is constantly K times that in the liquid. In this case, the thickness of the liquid is modified, becoming A/K instead of A.

when K = 1

α=

A L

(4.14)

when K ≠ 1

α=

A K⋅L

(4.15)

Some roots of Equation (4.13) are given in various books [1-3] for the values of α corresponding to a few values of the final fractional uptake of the diffusing substance by the sheet.

4.2.3.1 Case of the Diffusing Substance Entering the Sheet At equilibrium, since the total amount of diffusing substance in the sheet and liquid is the same as that initially in the liquid, the matter balance written for this substance gives: A ⋅ C∞ + L ⋅ C∞ = A ⋅ Cin K

(4.16)

by calling C∞ the uniform concentration in the sheet at equilibrium, and C∞/K the corresponding value in the liquid, when the partition factor is K. Obviously when K = 1, the concentrations at equilibrium are similar in the liquid and in the sheet. The amount of diffusing substance at equilibrium in the sheet is given by: M ∞ = 2 ⋅ L ⋅ C∞ =

2 ⋅ A ⋅ Cin 1+ α

(4.17)

and the fractional uptake of the sheet is obtained as follows: M∞ 1 = 2 ⋅ A ⋅ Cin 1 + α

152

(4.18)

Mass Transfers Between Food and Packages

4.2.3.2 The Diffusing Substance Leaves the Sheet The fractional uptake of the diffusing substance is given by the ratio: M∞ α = 2L ⋅ Cin 1 + α

(4.19)

4.2.4 Case of a Sheet of Thickness 2L Immersed in a Liquid of Infinite (or Finite) Volume and a Finite Value of the Coefficient of Convection This case corresponds with the process of diffusion of the diffusing substance through the sheet coupled with evaporation of this substance from the surfaces of the sheet. But it should be said that it also represents the process of a sheet immersed in a liquid when the coefficient of convection is finite. In fact, assuming that the coefficient of convection h is infinite is in all cases totally unreal and unreasonable. With a finite coefficient of convection h, the basic equations are as follows: The one shown already in Equation 1.11, expressing the diffusion through the thickness of the sheet: ∂C ∂2C =D 2 ∂t ∂x

(4.9)

and the other representing the boundary condition with a finite h: −D

∂C = h CL , t − Ceq ∂x

(

)

(4.2)

When the initial condition is given by a uniform concentration of the diffusing substance, expressed by Equation (4.20), there is a solution for the problem: t=0

–L < x < +L

C = Cin

sheet

(4.20)

The solution of this problem with the above initial and boundary conditions is given for the profiles of concentration developed through the thickness of the sheet as follows: ⎛ x⎞ 2R ⋅ cos ⎜βn ⎟ ⎛ C∞ − C x , t D⋅t⎞ ⎝ L⎠ exp ⎜−β2n 2 ⎟ =∑ 2 2 C∞ − Cin n =1 (βn + R + R) ⋅ cos βn L ⎠ ⎝ ∞

(4.21)

153

Assessing Food Safety of Polymer Packages where βn is the positive root of: β ⋅ tan β = R

(4.22)

and the dimensionless number R is given by: R=

h⋅L D

(4.23)

The kinetics of transfer of diffusing substance by using the dimensionless number Mt/M∞ is expressed in terms of the dimensionless number D⋅t/L2 by the following equation: ⎛ M∞ − M t ∞ 2 ⋅ R2 D⋅t⎞ exp ⎜−β2n 2 ⎟ =∑ 2 2 2 M∞ L ⎠ ⎝ n =1 βn (βn + R + R)

(4.24)

Of course, the total amount of substance enters or leaves the sheet, depending on the relative values of the concentrations C∞ and Cin, as shown previously in Equations (1.6) and (1.6´): C∞ > Cin

absorption of diffusing substance by the sheet

(4.25)

C∞ < Cin

release of diffusing substance by the sheet

(4.26)

4.2.4.1 Notes on the Example Described in Section 4.2.4 (1) Process of drying This method of calculation is of great interest in describing the process of drying, but only when the diffusivity is constant and does not depend on the concentration of the diffusing substance in the sheet. In the same way, the surroundings can be considered as infinite provided that the sheet is not confined in a closed volume, because in this case, the concentration of the evaporated substance in the atmosphere, far from remaining constant, would increase regularly as the evaporation proceeds. It should be noticed that the rate of convection, h is related to the rate of evaporation of the diffusing substance per unit area Ft, by the relationship: Ft = h ⋅ CL,t

(4.27)

showing that the rate of evaporation of the substance evaporated is proportional to the actual concentration of liquid on the surface of the solid. This concentration of liquid 154

Mass Transfers Between Food and Packages in the polymer, expressed in terms of volume of liquid per total volume of polymer and liquid is lower than 1, while in the pure liquid, the concentration is 1. (2) Release of the additives from the polymer in the liquid Both Equation (4.21) for the profiles of concentration of the additive developed through the thickness of the polymer sheet and Equation (4.24) for the kinetics of transfer of this additive should be widely used. In fact, they stand for any cases given as follows: •

When drying a polymer sheet (or when a vapour is absorbed by the polymer);

•

When the additives are released from the package into the liquid.

The question of the infinite volume is of great concern, and the answer is as follows: When the ratio of the volumes of liquid and polymer expressed by α in Equation (4.14) is larger than a value, which depends on the accuracy required, the volume of the liquid can be considered as infinite. As already shown in Chapter 1 with Figure 1.6, and more precisely in Figure 1.9, it could be said that a value of α of either 50 or 20 gives similar results with the same kinetics of release. Let us recall that for a package of 1 litre and a thickness of 0.01 cm (100 μm) of the container, the value of α is 166 for a cubic form and 174 for a bottle. Of course, for a thicker bottle, e.g., 0.03 cm, this ratio is 58.

4.2.4.2 Notes on Equations (4.12) and (4.24) with an Infinite Volume of Liquid and Infinite Value of the Coefficient of Convection When the volume of liquid is infinite, α = ∞, the roots of Equation (4.13) relative to the value of qn, are in the form: qn = (n + 0.5)π, and thus Equation (4.12) reduces to Equation (4.28): ⎡ 2n + 1 2 π 2 Mt 8 ∞ 1 ) D ⋅ t⎤⎥ ⎢− ( exp = 1− 2 ∑ ⋅ ⎢ ⎥ M∞ π n =0 (2n + 1)2 4L2 ⎣ ⎦

(4.28)

In the same way, by considering the hypothetical case of the infinite value of the coefficient of convection, the dimensionless number R is infinite, and the βnʹ values are given by βn = (n + 0.5)π so that Equation (4.24) also reduces to Equation (4.28). Let us recall that in the hypothetical case of the infinite value given to the coefficient of convection at the polymer sheet-liquid interface, Equation (4.29) is also of interest.

155

Assessing Food Safety of Polymer Packages 0.5 ∞ ⎡ D ⋅ t ⎤ ⎡ −0.5 n Mt nL ⎤ = 2 ⋅ ⎢ 2 ⎥ ⋅ ⎢π + 2 ⋅ ∑ (−1) ⋅ ierfc ⎥ M∞ ⎣ L ⎦ ⎣ D⋅t ⎦ n =1

(4.29)

where: ierfc(x) is the integral of the error function complementary to the following: erfc(x) = 1 – erf(x) ierfc(x) = π–0.5 ⋅ exp(–x2) – x ⋅ erfc(x) The interesting thing about Equation (4.29) is that it stands for small times. As the series becomes negligible, the kinetics expressed by the ratio of the amounts of substance transferred versus time reduces to: Mt < 0.5 M∞

0.5

Mt 2 ⎡ D ⋅ t ⎤ = ⋅⎢ ⎥ M∞ L ⎣ π ⎦

−L < x < +L

(4.30)

This equation (4.30) shows that a straight line is obtained when plotting the ratio Mt/M∞ versus the square root of time. But, as already stated in Chapter 1, this relationship is of value only when the coefficient of convection is infinite, with a liquid strongly stirred, thus, this fact reduces the interest of this equation in calculating the diffusivity. In fact, Equation (4.30) can be used only as a first approach to obtain an approximate value of the diffusivity. With the same approximation, Equation (4.28) can be useful for determining an approximate value of the diffusivity for long times, as it reduces to the simple Equation (4.31): ⎡ π2 ⋅ D ⋅ t ⎤ Mt Mt 8 > 0.5 = 1 − 2 exp ⎢− 5 ⎥ for 2 M∞ M∞ 4L ⎦ π ⎣

(4.31)

4.2.5 General Conclusions on the Mathematical Treatment As already stated in Chapter 1, devoted to the mathematical treatment of the mass transport from a polymer package into a liquid, when this transfer is controlled by diffusion, the following conclusions can be drawn: 1. The best and unique equation to be used for evaluating the value of the diffusivity from the kinetics of transfer is Equation (4.24), as the rate of stirring is generally not so strong that the coefficient of convective transfer, h could be assumed to be infinite. Moreover, this Equation (4.24) can be used only when the ratio of the volumes of liquid and of the polymer package α is larger than a given value, this value depending on the accuracy required. It has been seen that this value expressed by α should be larger than 50 for the results to be obtained precisely. When the ratio of the volumes α is very low, e.g., lower than 20, a numerical model with finite differences should 156

Mass Transfers Between Food and Packages be used, by taking care that all the facts are accounted for, and especially the ratio of the volumes and the finite coefficient of convection. 2. Equation (4.28), or rather the more simple Equation (4.30), which is mostly appreciated by the users, perhaps because it looks simpler, can be used only from the first approach in order to get an approximative value of the diffusivity. 3. Then, the approximate value of the diffusivity obtained with this simple method shown previously in case 2 with Equation (4.30) can be introduced into Equation (4.24) through the dimensionless number R expressed by the relationship (4.23). Of course, various values of the coefficient h should be tested, by following a trial and error method, and by determining the corresponding values of the βnʹ in Equation (4.22). 4. Finally, when the operations required in the previous sections seems too tedious, the convenient numerical model could be used.

4.3 Mass Transfer in Liquid Food from a Single Layer Package This is the most common case of release of some additives taking place from the polymer package into a liquid food. The process, controlled either by diffusion through the thickness of the package or by convection at the solid-liquid interface, has been described in Chapter 1, as well as in Section 4.2.

4.3.1 Theoretical for a Single Layer Package in Contact with Liquid Food The principle of the process is depicted in Figure 4.1 where the scheme is drawn.

Figure 4.1 Scheme of the package, with the thickness 0 < x < L, showing the symmetry for the diffusion process with respect to x = 0.

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4.3.1.1 Assumptions i)

The thickness of the package is L, with: 0 < x < + L, and there is no mass transfer on its external surface x = 0;

ii) The process is controlled either by diffusion through the thickness of the package or by convection into the liquid food; iii) When the diffusivity is constant and the initial concentration of the diffusing substance is uniform, the mathematical treatment is feasible, leading to an analytical solution. When one of these two requirements is not complied with, a numerical analysis should be made leading to a numerical model with finite differences; iv) When the volume of liquid is larger than that of the package, with α >20, the mathematical treatment is feasible, otherwise, the numerical model should be used. vi) As shown already, because of the convection in the liquid, the concentration of the additive is uniform in the liquid at any time. Thus, the one-dimensional transfer can be considered, per unit area.

4.3.1.2 Mathematical or Numerical Treatment The equation of diffusion through the package is: ∂C ∂2C =D 2 ∂t ∂x

(4.9)

The boundary conditions are written as: ∂C =0 ∂x −D

on the external surface of the package, and as

∂C = h CL , t − Ceq ∂x

(

)

at the package-liquid interface

(4.32)

(4.2)

Depending on the volume of the liquid food, and especially on the ratio of the volumes of the liquid and of the package, α, two ways of calculation are possible. When α >20-50, depending on the accuracy required, the mathematical treatment is feasible, leading to the equation for the kinetics of release of the additive: 158

Mass Transfers Between Food and Packages ⎛ M∞ − M t ∞ D⋅t⎞ 2 ⋅ R2 exp ⎜−β2n 2 ⎟ =∑ 2 2 2 M∞ L ⎠ ⎝ n =1 βn (βn + R + R)

(4.24)

where βn is the positive root of: β ⋅ tan β = R

(4.22)

and the dimensionless number R is given by: R=

h⋅L D

(4.23)

On the other hand, when α < 20, the numerical treatment with finite differences should be applied for determining either the kinetics of release or the profiles of concentration, in order to take into account all the facts, e.g., the finite volume of the liquid and the finite coefficient of convection. It is clear that, because of the symmetry to the plane located at the abscissa x = 0, all the equations written for a thickness 2L of the package with -L < x < +L and a mass transfer on both sides, can be used, and especially Equation 4.24, in spite of the fact that the actual thickness of the package shown in Figure 4.1 is only L. Various studies have been made by considering either the diffusion of the contaminant through the polymer package or its release into the liquid food. The basic question has also been considered with the two possibilities of the transport into the food which can be driven either by diffusion through the solid food or by convection into the liquid food [4].

4.3.2 Effect of the Coefficient of Convective Transfer The results obtained in this case have already been established in Chapter 1. They are expressed in terms of the profiles of concentration of the diffusing substance developed through the thickness of the package (Figure 4.2 and Figure 4.3), and of the kinetics of release of the substance in the liquid food (Figure 4.4 and Figure 4.5). Some emphasis is placed on the value given to the coefficient of convective transfer ‘h’, and thus to the dimensionless number ‘R’, either for the profiles of concentration or for the kinetics of release. Dimensionless numbers are used for the coordinates, and the curves obtained are master curves which can be used whatever the characteristics of the problems. The profiles of concentration of the diffusing substance expanded through the thickness of the package are drawn in Figure 4.2 with the hypothetical infinite value for the number R, and in Figure 4.3 with the realistic value of R = 5. 159

Assessing Food Safety of Polymer Packages

Figure 4.2. Profiles of concentration of the diffusing substance developed through the thickness of the package at different times, for R infinite and infinite volume of food. Time is expressed in terms of the dimensionless time D⋅t/L2.

Figure 4.3. Profiles of concentration of the diffusing substance developed through the thickness L of the package at different times, for R = 5, and a large volume of food with α >20-50. Time is expressed in terms of the dimensionless time D⋅t/L2. 160

Mass Transfers Between Food and Packages

Figure 4.4. Kinetics of transfer of diffusing substance with the package of thickness L, for different values of the dimensionless number R and a large value of the volume of liquid with α >20-50. Dimensionless numbers are used for the co-ordinates: Mt/M∞ and (D⋅t)0.5/L.

Figure 4.5. Kinetics of transfer of diffusing substance with the package of thickness L, for different values of the dimensionless number R and a finite value of the volume of liquid such as α >20-50. Dimensionless numbers are used for the co-ordinates: Mt/M∞ and (D⋅t)0.5/L. The curves are drawn with an expanded scale.

161

Assessing Food Safety of Polymer Packages The kinetics of release of the diffusing substance are drawn in Figure 4.4 for various values of the dimensionless number R, while these same curves are also drawn in Figure 4.5 by using more expanded scales for the coordinates. The following conclusions are worth noting: i) As already stated in Chapter 1, the infinite value given to the coefficient of convection h and to the dimensionless number R is responsible for the following two facts: the concentration of the diffusing substance on the surface in contact with the liquid (x = +L) falls abruptly to 0 instantaneously as the process starts, as shown in Figure 4.2. On the other hand, when the dimensionless number R is finite, the concentration of the diffusing substance on the surface in contact with the liquid falls rather slowly, without reaching the 0 value, even after the dimensionless time D⋅t/L2 is equal to 1. In fact the 0 value can be attained after infinite time only when the volume of the liquid is infinite. For a finite volume of the liquid and the partition factor K of 1, the concentration is the same in the liquid and the polymer, given by: Vp C∞ = Cin Vl + Vp when the concentration is expressed per unit volume of material, and where Vp and V1 are the volumes of the polymer and liquid, respectively. ii) The kinetics of release expressed in terms of the square root of time (Figures 4.4 and 4.5) drawn for various values of the dimensionless number R show precisely the effect of the coefficient of convective transfer at the package-liquid interface. All the curves pass through the origin, but the value of the slope at the beginning of the process increases largely when the value given to the dimensionless number is increased. A straight line is obtained when R is infinite, this fact resulting from the application of the equation 4.30 for short times associated with Mt/M∞ < 0.5. On the other hand, the rate of transfer at the very beginning of the process is infinite with a vertical tangent when time is used on the abscissa. iii) It should be said that the results for the profiles of concentration and for the kinetics, calculated for a volume of liquid infinite, are also of value for a finite volume of the liquid, provided that the ratio of the volumes of liquid and polymer, α, would be larger than, for example, 20 as previously shown [5]. iv) Figures 4.2-4.5 are drawn by using dimensionless numbers, so that they are playing the role of master curves. Thus these curves can be used whatever the values of the various parameters of diffusion intervening in the process, such as the concentration 162

Mass Transfers Between Food and Packages initially located in the polymer (Figures 4.2 and 4.3), the thickness of the package, the diffusivity of the additive, and the time through the dimensionless time. The kinetics of release of the additive in the liquid can also be used (Figures 4.4 and 4.5) whatever the values of the parameters: the amount of the additive initially located in the package, the thickness of the package L, the diffusivity of the additive and the amount of the diffusing substance initially located in the package which is equal to M∞ when the volume of liquid is infinite. This method of calculation enables the user to get a value of the concentration in the liquid when all the parameters are known, for α > 20. v) The effect of the coefficient of convection h, as well as of the dimensionless number R, is so important that it should be necessary to control it, or at least to evaluate its value in any experiments concerned with food packages. Let us recall that the same problem appeared in the pharmaceutical industry in 1984-1988 with the determination of the kinetics of the drug liberation from controlled release dosage forms [6]. Typical apparatus’ were built and tested, and finally normalised, by defining the dimensions of the flask, the way of stirring and the shape of the paddles so as to ensure the same rate of stirring [7].

4.3.3 Effect of the Ratio of the Volumes of Liquid and Package α The effect of the ratio of the volumes of the liquid and of the polymer package, α, as well as that of the coefficient of convection, are considered simultaneously in Figures 4.6 and 4.7. Figure 4.6 shows the profiles of concentration developed through the thickness of the package, with R = 5 and two values of α = 20 and α infinite. Figure 4.7 depicts the kinetics of release of the additive in the liquid in the following three cases: 1 with α = infinite, 2 with α = 20, and 3 with α = 50, and for all them R = 5. Some conclusions are worth noting from these two figures: i)

By comparing the curves drawn in the Figures 4.4 or 4.5 with the curves in Figure 4.7, it clearly appears that the effect of the parameter R on the kinetics is much effective than that of the ratio of the volumes α, even when this last parameter is varied within a wide range: α = 166, for a cube of 1 dm3 (1 litre) and a package thickness of 100 μm. α = 16.6, for a cube of 1 cm3, and a package thickness of 100 μm. α = 50, for a cube of 27 cm3, and a package thickness of 100 μm. 163

Assessing Food Safety of Polymer Packages

Figure 4.6 Profiles of concentration of the diffusing substance developed through the thickness L of the package at different times, for R = 5, and two values of the ratio of the volumes of the food and package α: (full line), α = infinite: (dotted line), α = 20. Time is expressed in terms of the dimensionless time D⋅t/L2.

Figure 4.7 Kinetics of transfer of diffusing substance with the package of thickness L, for different values of α, and with R = 5. Curve 1 with α infinite; curve 2 with α = 20; curve 3 with α = 50. Dimensionless numbers are used for the co-ordinates. 164

Mass Transfers Between Food and Packages ii) The profiles of concentration developed through the thickness of the package (Figure 4.6) are exactly similar for α = infinite and α = 20, with R = 5, when the dimensionless time is lower than 0.4. This time of 0.4 is associated with a value of Mt/M∞ of around 0.5, as shown in Figure 4.7 whatever the value of α, for R = 5. iii) The kinetics of release of the additive drawn for these two values of α = 20 and of α = infinite for R = 5 are nearly similar, and it is necessary to extend the scales of the co-ordinates so as to be able to appreciate the difference. iv) After infinite time, the concentration of the additive remaining in the package as well as the amount of additive released in the liquid are different when α = 20 and α = infinite: For α = 20, there is C∞/Cin = 1/21 when K = 1 and M∞/Min = 20/21 For α = infinite, there is C∞/Cin = 0 and M∞/Min = 1 whatever the value of K. v) This is a confirmation of a previous study concerned with the effect of the volume of liquid as a fraction of the volume of a plasticised PVC [5] on the kinetics of release of the plasticiser in the liquid. It was shown that the ratio of these two volumes, denoted as α, plays a role in the transfer of the plasticiser only when α is lower than 20.

4.4 Bi-layer Packages Made of a Recycled and a Virgin Polymer Layer, by Neglecting the Co-extrusion Potential Effect Because of the interest in recycling old polymer packages into new packages, the consumer’s safety has to be considered and so bi-layer packages must be tested before use. As shown in Figure 4.8, the bi-layer package consists of a recycled polymer layer co-extruded with a virgin polymer layer, while the virgin polymer layer is in contact with the food. As it takes some time for the contaminant potentially located into the recycled polymer to diffuse through both these layers, and especially through the virgin polymer layer, this virgin polymer layer plays the role of a functional barrier. The main problem which stands out is the evaluation of the time of protection of the food offered by this functional barrier.

4.4.1 Theoretical Treatment with a Bi-layer Package The problem is different from the case shown with the single layer package. 165

Assessing Food Safety of Polymer Packages

Figure 4.8 Scheme of the bi-layer system with recycled polymer layer on the left and the virgin polymer layer on the right, playing the role of a functional barrier. Cin is the uniform concentration of the diffusing substance initially in the recycled layer.

4.4.1.1 Assumptions As no analytical solution is obtained from the mathematical treatment, only a numerical method should be used. Thus there are no limitations concerned with the assumptions. Nevertheless, calculations have been made by using a constant diffusivity, and a uniform concentration of the diffusing substance, considered as the contaminant, is assumed to stand initially in the recycled polymer layer. In fact, these assumptions are not necessary, provided that the data are known beforehand. In the same way, one-dimensional diffusion is considered, leading to an easier problem.

4.4.2 Results with the Bi-layer Package The results are expressed in terms of the following figures: Figure 4.9 showing the profiles of concentration of the diffusing substance developed through the thickness of the two layers of the package, when there is no food, and no transfer on both sides of the package. Figure 4.10 showing the profiles of concentration of the diffusing substance developed through the thickness of the two layers of the package of equal thickness, when the food is on the right, at the relative abscissa 1, with the value of the dimensionless number R = 5. 166

Mass Transfers Between Food and Packages

Figure 4.9 Profiles of concentration of diffusing substance developed through the package at various times, expressed in terms of the dimensionless time D⋅t/L2, when there is no contact with the food, and with H = L/2.

Figure 4.10 Profiles of concentration of diffusing substance developed through the package at various times, expressed in terms of the dimensionless time D⋅t/L2, when the package is in contact with the food, with R = 5, and α = 166, and with H = L/2. 167

Assessing Food Safety of Polymer Packages Figure 4.11 depicting the kinetics of release in the food of the diffusing substance initially located in the recycled layer, with a package made of two layers of equal thickness, with various values of dimensionless number R, and a volume of liquid of 1 litre stored in a 0.01 cm thick package, leading to a very large value of α equal to 166. Figure 4.12, describing the profiles of concentration of the diffusing substance developed through the thickness of the two layers of the package, when the thickness of the recycled layer is one-third that of the package, the food being on the right, at the relative abscissa 1, with the value of the dimensionless number R = 5; the volume of liquid is 1 litre, leading to a very large value of α equal to 166. Figure 4.13, depicting the kinetics of release in the food of the diffusing substance initially located in the recycled layer, when the package is made of two layers of different thickness, (the thickness of the recycled layer is only one-third of that of the package), with various values of the dimensionless number R, and a volume of liquid of 1 litre, leading to a very large value of α equal to 166. The Figures 4.9 to 4.13 lead to the following comments: i)

Typical profiles of concentration are drawn in Figures 4.9 and 4.10, as well as in Figure 4.12. Comparison between Figures 4.9 and 4.10 shows the main difference which exists between them. When there is no transfer of substance into the liquid, the gradient on the surfaces is flat, according to Equation (4.32) indicating that there is no transfer. On the other hand, in Figure 4.10, the gradient on the surface is negative, indicating the fact that there is a transfer of substance into the food. Another difference is shown in Figures 4.9, 4.10 and 4.12: when there is no transfer of substance into the food, as shown in Figure 4.9, the profiles are symmetrical from the beginning to the end of the process; the profiles are symmetrical only for short times, when the diffusing substance has not reached the surface in contact with the food (Figure 4.10) or a distance longer than that of the recycled layer (Figure 4.12).

ii) The kinetics of release of the substance in the food drawn in the Figures 4.11 and 4.13 show the important effect of the dimensionless number on the rate of release. A faster release is obtained with the infinite value of R (which corresponds to an infinite value of the coefficient of convection h). iii) The effect of the thickness given to the recycled polymer layer is also of importance on the rate of release in the food. However, it appears obvious that the economic interest should be to have a larger relative thickness than that shown in Figures 4.12 and 4.13. 168

Mass Transfers Between Food and Packages

Figure 4.11 Kinetics of transfer of diffusing substance in the liquid food with α = 166, with the bi-layer system with H = L/2 and different values of the dimensionless number R.

Figure 4.12 Profiles of concentration of diffusing substance developed through the package at various times, expressed in terms of the dimensionless time D⋅t/L2, when the package is in contact with the food, with R = 5, and α = 166, and with H = L/3.

169

Assessing Food Safety of Polymer Packages

Figure 4.13 Kinetics of transfer of diffusing substance in the food with α = 166, with the bi-layer system with H = L/3 and different values of the dimensionless number R.

iv) The effect of the parameters which interfere in this process has been extensively studied in previous papers such as: •

the value of the ratio α of the volumes of liquid and of the package [8],

•

the relative thickness given to the recycled layer as a fraction of the total thickness of the package [9],

•

the value of the coefficient of convection and thus of the dimensionless number R [10].

Moreover, general theoretical results [11-13] have been published in order to pave the way to the further studies to be carried out by researchers. v) As dimensionless numbers are used either for the time with D⋅t/L2 or for the concentration in the package Ct/Cin, as well as for the amount of substance released in the food Mt/M∞, master curves are obtained, which can be employed whatever the values of the parameters, provided that they are known.

170

Mass Transfers Between Food and Packages

4.5 Mass Transfer from Tri-layer Packages (Recycled Polymer Inserted Between Two Virgin Layers) in Liquid Food This case is of great interest in the sense that the recycled polymer layer being inserted between two virgin polymer layers, the protection against contamination is ensured on both sides of the film, or inside and outside of the bottle. The problem is considered in this section by neglecting the effect of the mass transfer taking place during the stage of co-extrusion. The scheme of the process is shown in Figure 4.14.

4.5.1 Theory of the Mass Transfer in Food with the Tri-layer Package The process of transfer of the diffusing substance, initially located in the recycled layer, is controlled either by diffusion through the thickness of the three layers of the package or by convection at the polymer-liquid interface. In the present case, as shown in Figure 4.14, the initial concentration of the contaminant in the recycled polymer layer is uniform. However, this assumption is not mandatory, as the model can take into account any shape for this initial profile. As already stated, when the food is in a liquid state, the convection of the contaminant is so strong and so fast compared

Figure 4.14 Scheme of the package made of a tri-layer system where the recycled layer is inserted between two virgin layers. Cin is the uniform concentration of the diffusing substance initially in the recycled layer.

171

Assessing Food Safety of Polymer Packages to the slow rate of diffusion through the polymer, that the concentration in the liquid is taken as uniform at any time. Moreover, the radius of the bottles is much larger than the thickness of the package. For these two reasons, a one-dimensional diffusion can be considered. In the same way as for the bi-layer film in contact with the liquid food, there is no analytical solution for the problem. It should be resolved by using a numerical treatment with finite differences.

4.5.2 Results Obtained with the Tri-layer Package in Contact with a Liquid Food The results obtained either for a film or for a bottle are expressed in terms of profiles of concentration of the diffusing substance developed through the thickness of the package, and of the kinetics of transfer into the liquid food. Figure 4.15 shows the profiles of concentration of the contaminant initially located in the recycled layer, as they develop through the tri-layer package when it is in contact with the liquid food. The total thickness of the package is 0.03 cm and the volume of liquid is 1 litre. Figure 4.16 shows the kinetics of release of the contaminant in the food, for various values of the dimensionless number R which characterises the convection in the food. The volume of liquid is 1 litre and the thickness of the package is 0.03 cm. Some comments of interest are drawn from these two figures: i) As there is no transfer at the external surface of the bottle, the gradients of concentration of the contaminant in Figure 4.15 are flat next to this surface, according to the relationship (4.32): ∂C = 0 external surface at x = 0 ∂x

(4.32)

On the contrary, the gradients of concentration on the internal surface of the package in contact with the liquid food, exhibit a negative slope, according to Equation (4.2): −D

∂C = h CL , t − Ceq ∂x

(

)

(4.2)

ii) As it takes some time for the contaminant initially located in the recycled layer to diffuse through the virgin polymer layer in contact with the food (at the abscissa L in Figure 4.14), the virgin layer in contact with the food plays the role of a functional barrier. In the present case, the time expressed in terms of the dimensionless number D⋅t/L2 is around 0.012. 172

Mass Transfers Between Food and Packages

Figure 4.15 Profiles of concentration of the diffusing substance developed at various times (dimensionless time D⋅t/L2) when the three layers have the relative thicknesses shown in the figure. The package is in contact with a liquid food with α = 55.3 and R = 5.

Figure 4.16 Kinetics of transfer of diffusing substance in the food of large volume with α = 55.3, associated with the case shown in Figure 4.15 with various values of R. 173

Assessing Food Safety of Polymer Packages iii) The virgin layer located on the external surface of the bottle (between 0 and L1) is acting in two ways: First, it protects the consumer’s hand from contamination when the contaminant does not evaporate. Secondly, this layer acts as a reservoir in which a part of the contaminant penetrates and thus is stored. Resulting from this fact, as there is no transfer on the external surface, the concentration of the contaminant increases in this layer, and becomes larger than that in the recycled layer, at a time between 0.03 and 0.05. Thus, finally, this external layer could be also considered as the external part of the functional barrier. iv) The kinetics of release of the contaminant in the food shown in Figure 4.16 for various values of the dimensionless number R, are typical. They look like those obtained for the bi-layer system in Figure 4.13. In the same way, the effect of the coefficient of convection at the package-liquid interface is of prime importance. As dimensionless numbers are used either for time and the amount of contaminant transferred, these curves are master curves which can be used in any case, provided that the parameters are known. v) The effect of the relative thicknesses of the three layers has been deeply studied intensively in various papers [14-18], while an overview of these results was given [19]

4.6 Effect of the Co-extrusion on the Mass Transfer in Food 4.6.1 Mass Transfer in Food with a Co-extruded Bi-layer Package 4.6.1.1 Heat Transfer and Mass Transfer During the Stage of Co-extrusion The scheme of the process of co-extrusion is shown in Figure 3.1, where the two layers after their co-extrusion into a single package at 300 °C are cooled down by heat convection in the surrounding atmosphere at room temperature. According to the temperature-time histories (Figure 3.2) drawn when the heat convection is either high (ht = 0.5 cal/cm2⋅s) or low (ht = 0.05 cal/cm2⋅s), the temperature through the 0.03 cm thick package falls rapidly during this cooling period to room temperature after 0.4 s in the first case and after 0.6 s in the second case. The profiles of concentration of the diffusing substance developed during the same cooling stage shows that they are slightly different in the two following cases: when h = 0.5 with L = 0.03 cm (Figure 3.5) and when h = 0.05 with L = 0.03 cm (Figure 3.6). From these figures, the main results are: i)

A profile of concentration of the contaminant is developed during the process of coextrusion, up to the short time of 0.1 s, whatever the value of the coefficient of heat transfer h.

ii) The profile of concentration shows that the diffusing substance reaches the relative abscissa 0.04 with h = 0.5, and around 0.05 with h = 0.05. 174

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4.6.1.2 Mass Transfer from the Co-extruded Bi-layer into the Liquid Food The scheme of the process is shown in Figure 4.17, where the mass transfer is controlled either by diffusion through the thickness of the bi-layer package or by convection into the liquid, while there is no transfer on the external surface of the package. In Figure 4.18, are drawn the profiles of concentration of the contaminant developed through the thickness of the package (0.03 cm) when it is in contact with the liquid food (1 litre, expressed by the thickness of 1.66 cm), with the dimensionless number R = 5. The times are expressed in terms of the dimensionless number D⋅T/L2. Figure 4.19 shows the kinetics of transfer of the contaminant in the food, under the same conditions as shown in Figure 4.18 (L = 0.03 cm; 1 litre liquid), for various values of the coefficient of convection defined by the dimensionless number R. Some conclusions of interest can be drawn from these two figures: i) As shown in Figure 4.18, at time 0 when the package is put in contact with the liquid food, the profile of concentration is not uniform in the recycled layer, and the virgin layer is not free from contaminant. However, this profile of concentration at time 0 differs slightly from the profile taken when the effect of the co-extrusion is neglected.

Figure 4.17. Scheme of the bi-layer package made of a recycled layer and a virgin layer in contact with liquid food, by taking into account the transfer of the diffusing substance during the co-extrusion stage.

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Figure 4.18. Profiles of concentration of the diffusing substance developed through the bi-layer package in contact with the liquid food of finite volume with α = 55.3, for various values of the dimensionless time D⋅t/L2, with R = 5. The initial concentration at time 0 is that attained at the end of the co-extrusion stage.

Figure 4.19 Kinetics of transfer of diffusing substance in the liquid food of volume such as α = 55.3, in the case of the bi-layer package shown in Figure 4.18, with various values of the dimensionless number R. Dimensionless numbers are used in the co-ordinates.

176

Mass Transfers Between Food and Packages ii) From this time 0, the contaminant diffuses through the thickness of the package, leading to the profiles of concentration shown in Figure 4.18. The slope of these gradients is equal to 0 on the external surface where there is no transfer, on the contrary, this slope is negative on the internal surface in contact with the liquid food, this last fact being associated with a transfer of the contaminant into the food. As the dimensionless number R = 5, is finite, the concentration of the contaminant on the internal surface does not fall to 0. iii) There is an axis of symmetry (x/L = 0.5, and Ct/Cin = 0.5) until the diffusing substance has not reached the external surface. iv) The kinetics of release of the contaminant into the liquid food drawn in Figure 4.19 shows the importance of the effect of the coefficient of convection at the packageliquid interface, through the dimensionless number R. v) The virgin layer (0.015 cm thick) plays the role of a functional barrier, the food being fully protected over a time around 0.125, expressed in terms of the dimensionless number D⋅t/L2. In fact, the following main results appear as highly relevant: 1. The effect of the mass transfer which has taken place during the stage of coextrusion is not significant; 2. The effect of the coefficient of heat transfer during the co-extrusion stage is negligible. 3. Of course, these results are of value only if the temperature-dependent diffusivity follows the law selected in Chapter 3. vi) As dimensionless numbers are used in both Figures 4.18 and 4.19, master curves are obtained which can be used whatever the operational conditions.

4.6.2 Mass Transfer in Food with Tri-layer Bottles Co-injected in the Mould whose External Surface is Kept at 8 °C After their preparation, the tri-layer bottles, made of three layers co-injected in the mould, are put in contact with the liquid food, according to the scheme in Figure 4.20. As shown in this Figure, at time 0, the contaminant is located not only in the recycled polymer layer but also in the borders of the two virgin polymer layers. This profile of concentration has been developed during the cooling stage of the bottle in the mould. The thickness of the bottle is 0.03 cm, each layer having the same thickness of 0.01 cm. 177

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Figure 4.20. Scheme of the tri-layer package, in contact with the liquid food, obtained after the stage of co-extrusion.

There is no transfer of the contaminant on the external surface of the bottle, at x = 0, while a mass transfer takes place into the food on the internal surface at x = 3L. The operational conditions described in Chapter 3 as shown in Figure 3.21, are summarised: Figure 3.21 shows the profiles of concentration of the contaminant developed through the thickness of the bottle cooled in the mould whose external surface is kept at 8 °C. Thickness of the polyethylene terephthalate (PET) bottle = 0.03 cm; thickness of the mould = 1.5 cm; temperature of the PET at injection = 280 °C; time of residence of the PET bottle in the mould = 1 s. Figure 4.21 shows the profiles of concentration of the contaminant when the bottle is put in contact with the liquid food. The thickness of the bottle being 0.03 cm and the volume of the liquid 1 litre (thickness of 1.66 cm with one-dimensional transport), α = 55.3. Figure 4.22 shows the kinetics of release of the contaminant into the liquid food, for various values of the dimensionless number R (which characterises the convection). Figure 4.23, where the kinetics of release of the contaminant into the liquid food shown in Figure 4.22, are represented with a larger scale. Some results of concern are worth noting from Figures 4.21-4.23: i)

The profiles drawn in Figure 4.21 give a fuller insight into the nature of the process either during the stage of moulding or the stage over which the bottle is in contact with the food.

178

Mass Transfers Between Food and Packages

Figure 4.21. Profiles of concentration of the diffusing substance developed through the thickness of the tri-layer package when it is in contact with a liquid food of volume such as α = 55.3, with the dimensionless number R = 5. At time 0 for the dimensionless time D⋅t/L2, the profile of concentration is that attained at the end of the stage of co-extrusion.

Figure 4.22. Kinetics of transfer of diffusing substance in the liquid food of finite volume, such as α = 55.3, with various values of the dimensionless number R. The kinetic curves correspond with the case shown in Figure 4.21.

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Figure 4.23. Kinetics of transfer of diffusing substance in the liquid food of finite volume such as α = 55.3, with various values of the dimensionless number R. The kinetic curves correspond to the case shown in Figure 4.21, with a larger scale than in Figure 4.22.

The main point appears here. When the bottle is extracted from the mould, a profile of concentration has been developed for the contaminant. This profile is taken at time 0 when the bottle is put in contact with the food, at the beginning of the stage of release into the food. It is clear that the profile of concentration at time 0 slightly differs from the profile defined when the stage of heating and co-extrusion is neglected (with square angles). ii) The process of release of the contaminant is thus controlled either by diffusion through the three layers of the bottle or by convection into the liquid (on the right at abscissa x = L). iii) As there is no mass transfer through the external surface, as shown in Figure 4.20, the slope of the gradients of concentration is equal to 0 in Figure 4.21. On the contrary, resulting from a contaminant transfer into the food, the slope of the gradients of concentration is negative on the internal surface of the bottle in contact with the food. iv) As it takes some time (around 0.01 for the dimensionless time D⋅t/L2) for the contaminant to diffuse through the virgin polymer layer in contact with the food, this layer acts upon 180

Mass Transfers Between Food and Packages the pollution process as a functional barrier. Nevertheless, as a significant part of the contaminant is stored in the other virgin layer in contact with the external surface of the bottle, this layer plays the role of a reservoir, very efficient at the beginning of the process, for dimensionless times between 0.04 and 0.1. For this reason, it can be said that the external layer contributes to the protection, not only for the consumer’s hand but also for the food as it takes part to the time of food protection. v) The kinetics of release of the contaminant into the liquid food shown in Figure 4.22 exhibit the effect of the value of the rate of convection at the polymer-liquid interface, characterised by the dimensionless number R. These kinetics of release shown in Figure 4.23 by using an extended scale for the co-ordinates, allows the determination of the time of full protection.

4.6.3 Mass Transfer in Food with Tri-layer Bottles Co-injected in Normal Mould This is the case where the mould is initially at room temperature, in the surrounding air at 40 °C, with a convection heat transfer on the external surface of the mould. The PET layers are injected in the mould, by following the operational conditions described in Figure 3.30. The bottle components, initially at 280 °C have been previously injected in the mould, where they are cooled down for 1 second. The thickness of the mould is 1.5 cm. In Figure 4.24, the kinetics of release of the contaminant into the food are shown for various values of the dimensionless number R. In Figure 4.25, the kinetics of release of the contaminant in the food shown in Figure 4.24, are extended with a large scale. Some interesting results appear from these curves: i)

An important fact is shown in Figure 4.26, where the profiles of concentration of the contaminant are drawn at the end of the stage of cooling, associated with the various cases of co-extrusion and of co-moulding calculated in Chapter 3. In Figure 4.26 all the profiles of concentration of the contaminant obtained at the end of the stage of cooling collected in the various cases shown in Figures 3.21, 3.24, 3.27 and 3.30, are altogether nearly similar.

ii) Thus it is obvious that the profiles of concentration developed by the contaminant in Figure 4.21 are of value in the present case. 181

Assessing Food Safety of Polymer Packages

Figure 4.24. Kinetics of transfer of a diffusing substance in the liquid food of finite volume such as α = 55.3, with various values of the dimensionless number R. The initial profile of concentration of the diffusing substance is that attained at the end of the co-moulding stage described in Figure 3.30 for the three-layer bottle. Dimensionless numbers are used for the co-ordinates.

Figure 4.25. Kinetics of transfer of a diffusing substance in the liquid food of finite volume such as α = 55.3, with various values of the dimensionless number R, with expanded scale. The initial profile of concentration of the diffusing substance is that attained at the end of the co-moulding stage described in Figure 3.30 for the three-layer bottle. Dimensionless numbers are used for the co-ordinates. 182

Mass Transfers Between Food and Packages

Figure 4.26. Profiles of concentration attained by the diffusing substance at the end of the stage of co-moulding with three-layer bottles in various cases: 1: Figure 3.21; 2: Figure 3.24; 3: Figure 3.27; 4: Figure 3.30.

iii) Finally, the kinetics of release of the contaminant into the food for various values of the dimensionless numbers drawn in Figures 4.24 and 4.25 are similar to those drawn in Figures 4.22 and 4.23.

4.7 Conclusions on the Functional Barrier In terms of conclusions about the effect of the functional barrier, two facts of importance can be looked upon, either by estimating the time of protection provided by the presence of this virgin polymer layer or by considering the decrease in this time of protection resulting from the advancement of the profiles of concentration of the contaminant which takes place during the process of co-extrusion or co-moulding.

4.7.1 Interest of a Functional Barrier The interest of the functional barrier is obvious, as it appears in the various figures depicting the kinetics of release of the contaminant in the liquid food. In each case, there is a time 183

Assessing Food Safety of Polymer Packages of full protection. Moreover, after this time, the amount of contaminant in the liquid food (as well as the concentration) increases rather slowly, bringing another partial protection. On the contrary, without a functional barrier, the kinetics of release of the contaminant into the liquid food exhibits a high rate, especially at the beginning of the process. Quantitative results are given in the following three Tables (Tables 4.1, 4.2 and 4.3).

Table 4.1 Contamination versus dimensionless times for various values of R Mono-layer Mt/Min R 0.0001 0.001 0.01 0.1 -4 -3 -2 1.02 × 10 1.08 × 10 0.1281 1 1.00 × 10 -5 -4 -3 2.11 × 10 2.37 × 10 3.50 × 10-2 5 2.01 × 10 10 1.00 × 10-5 1.08 × 10-4 1.28 × 10-3 2.22 × 10-2 50 2.00 × 10-6 2.35 × 10-5 3.49 × 10-4 1.10 × 10-2 ∞ 7.86 × 10-5 7.93 × 10-3 7.50 × 10-7

Table 4.2 Contamination versus dimensionless times for various values of R Bi-layer Mt/Min R 0.0001 0.001 0.01 0.1 -2 -2 -2 1 2.04 × 10 3.48 × 10 7.49 × 10 0.2458 -2 -2 -2 2.47 × 10 4.72 × 10 0.1325 5 1.57 × 10 -2 -2 -2 2.22 × 10 4.13 × 10 0.1135 10 1.44 × 10 -2 -2 -2 50 1.27 × 10 1.92 × 10 3.50 × 10 9.59 × 10-2 ∞ 1.80 × 10-2 3.30 × 10-2 9.10 × 10-2 1.19 × 10-2

Table 4.3 Contamination versus dimensionless times for various values of R Tri-layer Mt/Min R 0.0001 0.001 0.01 0.1 -3 -2 -2 1.72 × 10 3.92 × 10 0.1656 1 9.79 × 10 -3 -2 -2 1.18 × 10 2.32 × 10 7.18 × 10-2 5 7.40 × 10 10 6.72 × 10-3 1.05 × 10-2 1.98 × 10-2 5.80 × 10-2 50 5.76 × 10-3 8.74 × 10-2 1.60 × 10-2 4.53 × 10-2 ∞ 8.02 × 10-3 1.47 × 10-2 4.18 × 10-2 5.31 × 10-3

184

Mass Transfers Between Food and Packages They provide information on the time (expressed in terms of the dimensionless time D⋅t/L2, L being the total thickness of the package or bottle) necessary for given values of the amount of contaminant (expressed in terms of the amount of contaminant in the liquid food as a fraction of the amount initially in the recycled polymer layer, Mt/Min) to be released in the food, with various values of the dimensionless number R defining the convective effect at the package-liquid interface. The Tables 4.1, 4.2 and 4.3 show the results obtained: With the mono-layer, and α = 166, as shown in Figures 4.4 and 4.5 With the bi-layer, and α =166, as shown in Figure 4.11 With the tri-layer, and α = 55.3, as shown in Figures 4.22 and 4.24 From these tables, the effect of the convective transfer at the package-liquid food definitely appears in a quantitative manner.

4.7.2 Effect of the Co-extrusion and Co-moulding on the Mass Transfer From the calculations made by using a law of temperature-dependency of the diffusivity, as far as this law was representative of the fact, the following results clearly appear: •

The time of residence of the bottle in the mould is short, at around 1 second, but the heat transfer is strong. This fact results from the high thermal conductivity of the metal with respect to the lower one of the polymer.

•

Even if the diffusivity of a contaminant in PET is high at a temperature of 280 °C, the temperature decreases rather quickly, so that the transfer of the contaminant is small.

•

It is not necessary to cool down the mould, moreover, the effect of the value of the temperature in the surrounding atmosphere ranging from 20 to 40 °C is negligible.

4.8 Conclusions on the Diffusion-Convection Process Finally, in terms of conclusions on the process of pollution in a liquid food, the striking points can be noted: i)

Because of the convective transfer through the liquid, the concentration of any contaminant at any time is uniform. 185

Assessing Food Safety of Polymer Packages ii)

And thus, the contaminant transfer from the package into the liquid food can be considered as a one-dimensional transfer, taking place per unit area, the volumes being reduced to lengths, whatever the shape of the flask or bottle.

iii) The process of transfer is controlled either by one-dimensional transient diffusion through the thickness of the polymer package or convective transfer at the polymerliquid interface. iv) The ratio of the volumes of liquid and package, α, on the transfer is negligible, when this number is larger than 20-40, depending on the desired accuracy. The value of α is equal to 166 for a litre bottle in cubic form with a thickness of 100 μm. v)

The effect of the convective transfer is of great importance. The dimensionless number R defined by the ratio h⋅L/D, where L is the total thickness of the polymer package in contact with the liquid food on one side, expresses the effect of the convective transfer. As the rate of convective transfer h plays such an important role in the contaminant transfer, it should be necessary for the food package problems which concern the public health, to follow the process which was defined for pharmaceutical applications. The pharmacists are obliged to control precisely the conditions of stirring when they evaluate the kinetics of drug release from controlled release dosage forms by using standardised tests [6, 7].

vi) The figures drawn by using dimensionless numbers are master curves which can be used, whatever the data. They are: Mt/M∞, Ct/Cin, D⋅t/L2, h⋅L/D. vii) Recycling polymer packages in new packages is made possible by using a virgin polymer layer in contact with the food, which acts upon the process as a functional barrier. viii) The decrease in temperature during the process of co-extrusion for a film or comoulding for a bottle is fast, whatever the technique used. Thus the effect on the change in the profiles, of concentration of the contaminant at the interface between the layers is so low that it becomes negligible, as far as the law expressing the temperature-dependency of the diffusivity is correct. ix) The data shown either in the tables of Section 4.7 as well as in Figures 4.5, 4.11 and 4.22 can be of help for selecting the type of package, which is necessary to comply with the requirements desired for food protection.

4.9 Problems Encountered with a Solid Food There are two different cases when the food is in a solid state: the one when the solid food is made of grains separated from each other by a gas; the other when the solid is 186

Mass Transfers Between Food and Packages homogeneous as, for example, yoghurts or butter. Only the second case is considered here, the food being in a gel state, however homogeneous and isotropic. Thus the transfer of any additive through the solid food is controlled by diffusion. The problem of diffusion through a solid food stored into a bottle has already been studied and resolved through various papers. The first when the bottle is made of a virgin layer [20], the second by considering a bottle made of a bi-layer system with a recycled polymer layer and a virgin polymer layer in contact with the food [21], while the third is devoted to the effect of the relative thicknesses of the two layers of the bi-layer bottle [22].

4.9.1 Theoretical Part of the Problem 4.9.1.1 Assumptions The following assumptions are made to define the process: i)

The food is either in a solid state or in a gel state, however isotropic and homogeneous.

ii) The process of transfer into, and through, the solid food is controlled by transient diffusion. iii) Because of the bottle shape, the diffusion is both radial and longitudinal either through the solid food or through the thickness of the bottle. iv) A perfect contact is obtained at the bottle-food interface. v) The diffusivity is constant either in the bottle or in the solid food. vi) The partition factor is 1 at the food-bottle interface.

4.9.1.2 Mathematical Treatment The equation of diffusion taking into account the radial and the longitudinal transport is: ⎡ ∂2C ∂2C 1 ∂C ⎤ ∂C = D ⋅⎢ 2 + 2 + ⋅ ⎥ r ∂r ⎦ ∂t ∂r ⎣ ∂z

(4.33)

where z represents the longitudinal axis and r the radial axis, C is the local concentration of the diffusing substance which is a function of time and of position defined by the two co-ordinates r and z, and D is the constant diffusivity. 187

Assessing Food Safety of Polymer Packages An analytical solution is found when the values of the diffusivity are the same in the polymer and in the food [3, 4], but it seems that no solution exists when the diffusivity of a diffusing substance in the food is different from that in the polymer. The initial conditions in the polymer bottle and in the solid food are: t=0

C = Cin in the polymer of the bottle C = 0 in the food

(4.34)

The boundary conditions are: At the external surface of the bottle: t>0

⎛ ∂C ⎞ ⎛ ∂C ⎞ ⎜ ⎟ = 0 and ⎜ ⎟ = 0 ⎝ ∂h ⎠ ⎝ ∂r ⎠

(4.35)

meaning that the contaminant does not diffuse out of the bottle. At the food-bottle interface, where the concentrations are equal in the polymer and the food: ⎛ ∂C ⎞ ⎛ ∂C ⎞ Dp ⋅ ⎜ ⎟ = Df ⋅ ⎜ ⎟ ⎝ ∂r ⎠p ⎝ ∂r ⎠f

(4.36)

⎛ ∂C ⎞ ⎛ ∂C ⎞ Dp ⋅ ⎜ ⎟ = Df ⋅ ⎜ ⎟ ⎝ ∂h ⎠p ⎝ ∂h ⎠f

(4.36´)

meaning that the rate of transfer of the diffusing substance is the same on both sides of the food-bottle interface. Moreover, Equation (4.33) holds either for the polymer bottle or for the food located in it. As the diffusivity of the contaminant (diffusing substance) is generally much larger in the food than in the polymer of the bottle, no simple analytical solution is obtained. Thus, the problem is resolved by using a numerical method based on finite differences. It takes into account the radial and longitudinal diffusion, leading to an axis of symmetry along the Z-axis.

4.9.2 Results for the Transfer in the Solid Food The results are expressed in terms of kinetics of release of diffusing substance in the solid food and of profiles of concentration of this diffusing substance developed in various

188

Mass Transfers Between Food and Packages places of the food. These profiles of concentration are calculated and drawn in two sections of the food and polymer: the one which is a cross section perpendicular to the Z-axis of the bottle at position A1A3, the other which is a radial section of radius 2 cm (D1D3) in Figure 4.27. The scheme of the process is depicted in Figure 4.27, with a quarter of the bottle. The scheme is not drawn to scale. The radial and longitudinal diffusion are shown. The following other figures are considered: Figure 4.28 represents the kinetics of release of the diffusing substance into the solid food, for three values of the diffusivity in the food selected within a large range, while the diffusivity is kept constant in the polymer of the bottle.

Figure 4.27. Scheme of the one-layer bottle in contact with a solid food.

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Figure 4.28. Kinetics of transfer of the diffusing substance into the solid food, for various values of the diffusivity in the solid food expressed in cm2/s. The diffusivity in the bottle is 10–8 cm2/s. The dimensions of the bottle are: radius = 4 cm and height = 20 cm, with a thickness of the bottle of 0.03 cm.

Figure 4.29. Profiles of concentration of the diffusing substance developed through the plane section perpendicular to the Z-axis at position A1A3, and especially in the polymer thickness denoted A2A3 with Dp = 10–8 cm2/s and Df =10-6 cm2/s. 190

Mass Transfers Between Food and Packages Figure 4.29 represents the profiles of concentration of the diffusing substance developed through the plane section perpendicular to the Z-axis at the position A1A3, and especially in the polymer thickness noted A2A3 when the diffusivity is 10–8 cm2/s in the polymer and 10-6 cm2/s in the food. Figure 4.30 represents the profiles of concentration of the diffusing substance developed through the plane section perpendicular to the Z-axis at the position A1A3, and especially in the food thickness noted A1A2 when the diffusivity is 10–8 cm2/s in the polymer and 10–6 cm2/s in the food. Figure 4.31 represents the profiles of concentration of the diffusing substance developed through the cylindrical section parallel with the Z-axis of radius 2 cm, taken at position D1D3, and especially in the polymer thickness denoted D2D3 when the diffusivity is 10–8 cm2/s in the polymer and 10–6 cm2/s in the food. Figure 4.32 represents the profiles of concentration of the diffusing substance developed through the cylindrical section parallel with the Z-axis of radius 2 cm, taken at position D1D3, and especially in the food thickness noted D1D2, when the diffusivity is 10–8 cm2/s in the polymer and 10–6 cm2/s in the food.

Figure 4.30. Profiles of concentration of the diffusing substance developed through the plane section perpendicular to the Z-axis at the position A1A3, and especially in the food thickness noted A1A2 when Dp = 10–8 cm2/s and Df = 10–6 cm2/s. 191

Assessing Food Safety of Polymer Packages

Figure 4.31. Profiles of concentration of the diffusing substance developed through the cylindrical section parallel with the Z-axis of radius 2 cm, taken at position D1D3, and especially in the polymer thickness denoted D2D3 when Dp = 10–8 cm2/s and Df = 10–6 cm2/s.

Figure 4.32. Profiles of concentration of the diffusing substance developed through the cylindrical section parallel with the Z-axis of radius 2 cm, taken at position D1D3, and especially in the food thickness denoted D1D2, when Dp = 10–8 cm2/s and Df = 10–6 cm2/s. 192

Mass Transfers Between Food and Packages Figure 4.33 represents the profiles of concentration of the diffusing substance developed through the plane section perpendicular to the Z-axis at the position A1A3, and especially in the polymer thickness denoted A2A3, when the diffusivity is 10–8 cm2/s in the polymer and 10–4 cm2/s in the food. Figure 4.34 represents the profiles of concentration of the diffusing substance developed through the plane section perpendicular to the Z-axis at the position A1A3, and especially in the food thickness noted A1A2 when the diffusivity is 10–8 cm2/s in the polymer and 10–4 cm2/s in the food. Figure 4.35 represents the profiles of concentration of the diffusing substance developed through the cylindrical section parallel with the Z-axis of radius 2 cm, taken at position D1D3, and especially in the polymer thickness noted D2D3 when the diffusivity is 10–8 cm2/s in the polymer and 10–4 cm2/s in the food. Figure 4.36 represents the profiles of concentration of the diffusing substance developed through the cylindrical section parallel with the Z-axis of radius 2 cm, taken at position D1D3, and especially in the food thickness denoted D1D2, when the diffusivity is 10–8 cm2/s in the polymer and 10–4 cm2/s in the food.

Figure 4.33. Profiles of concentration of the diffusing substance developed through the plane section perpendicular to the Z-axis at the position A1A3, and especially in the polymer thickness denoted A2A3, when Dp = 10–8 cm2/s and Df = 10–4 cm2/s. 193

Assessing Food Safety of Polymer Packages

Figure 4.34. Profiles of concentration of the diffusing substance developed through the plane section perpendicular to the Z-axis at the position A1A3, and especially in the food thickness denoted A1A2 when Dp = 10–8 cm2/s and Df = 10–4 cm2/s.

Figure 4.35. Profiles of concentration of the diffusing substance developed through the cylindrical section parallel with the Z-axis of radius 2 cm, taken at position D1D3, in the polymer thickness denoted D2D3 when Dp = 10–8 cm2/s and Df = 10–4 cm2/s. 194

Mass Transfers Between Food and Packages

Figure 4.36. Profiles of concentration of the diffusing substance developed through the cylindrical section parallel with the Z-axis of radius 2 cm, taken at position D1D3, and especially in the food thickness denoted D1D2, when Dp = 10–8 cm2/s and Df = 10–4 cm2/s.

The characteristics are shown in Table 4.4

Table 4.4 Characteristics of the problem used for calculation Bottle

Internal radius = 4 cm; height = 20 cm, thickness = 0.03 cm, α = 61.35

Diffusivity in the polymer

Dp = 10–8 cm2/s

Diffusivity in the food

Df = 10–6 cm2/s and Df = 10–4 cm2/s

C∞/Cin = 0.0163

D2C2 = 2 cm

A1B2 = 4 cm

From Figures 4.28 to 4.30, the following comments are worth noting: i)

As shown in Figure 4.28, the effect of the rate of transfer of the diffusing substance through the food, expressed by the diffusivity, is of importance. The obvious statement holds: the larger the diffusivity of the substance in the food, the faster the kinetics of release. Nevertheless, the transfer is controlled either by diffusion through the polymer 195

Assessing Food Safety of Polymer Packages or through the food. In the present case, the diffusivity of the substance through the polymer plays the role of a limiting factor. ii) The kinetics of release of the diffusing substance in the food is also expressed in terms of change in the concentration of this substance in various places in the polymer and in the food. Because of the perfect contact between the food and the bottle, the concentration of the diffusing substance is the same on both sides of the food-bottle interface. Obviously, the concentration of the diffusing substance decreases in the polymer as it increases in the food. iii) After a long time, infinity, theoretically speaking, the concentration in the food tends to a uniform value, which is inversely proportional to the ratio α of the volumes of food and polymer. In the present case, this concentration, expressed in terms of the ratio Cf,∞/Cin being equal to 1/1 + α is 0.0163 as the partition factor K is taken as 1. iv) The effect of the value of the diffusivity of the substance in the food is of importance, as shown by comparing the curves in the Figure 4.28 for the kinetics, and the profiles of concentration developed at various times either in Figures 4.29-4.32 when the diffusivity in the food is Df = 10–6 cm2/s or in Figures 4.33-4.36 obtained with Df = 10–4 cm2/s. v) From the first approach, the profiles of concentration of the diffusing substance developed through the thickness of the bottle look similar, but a closer study enables one to appreciate the differences. For example, after the short dimensionless time of 0.01, the relative concentration at the radial polymer-food interface A2 is lower than 0.1, while it is larger than 0.1 at the same interface taken at position D2 between the food and the bottom of the bottle. In the same way, the profiles developed through either the radial or the flat cross-section in Figures 4.29 and 4.31 exhibit other differences. vi) Of course, because of Equation (4.36) and (4.36ʹ), expressing the equality of the rate of transfer through the polymer and the food, especially at their interface, the slope of the profiles of concentration being inversely proportional to the corresponding diffusivity, is considerably higher in the polymer than in the food (without considering the change in scale taken in these figures for the food and polymer). vii) The profiles of concentration developed in the food either through the flat crosssection A1A2 or through the radial section D1D2 are quite different. For example, at the dimensionless time of 5, the substance has diffused across about 3 cm, reaching the position 1 (Figure 4.30) and about the same position along the Z-axis (Figure 4.32). However, the strong difference appears, as the concentration after this time of 5 remains the same along the Z-axis. Thus at a distance of nearly 4-5 cm from 196

Mass Transfers Between Food and Packages the bottom of the bottle, the effect of the transfer from the bottom becomes of low significance. In other words, at this distance from the bottom, the radial diffusion is becoming more and more important compared to the longitudinal diffusion as the distance at which it is taken away from the bottom is larger. viii) The effect of the value of the diffusivity in the food appears with the Figures 4.334.36, by comparing them with the corresponding Figures 4.29-4.32. At first glance, these figures look similar, two at a time when evaluated and drawn at the same places. However, the profiles are quite different, depending on the value of the diffusivity of the food. For example, a dimensionless time of 2 is enough for the profiles to become flat with Df = 10–4 cm2/s while it is more than 20, with Df = 10–6 cm2/s. Another strong difference is worth noting, with the profiles of concentration developed through the food. For example, the profiles obtained with Df = 10–4 cm2/s (Figure 4.34) are not so steep as the corresponding ones shown in Figure 4.30 obtained with Df = 10–6 cm2/s. ix) The effect of the longitudinal transfer from the bottom of the bottle on the profiles of concentration is nearly the same, whatever the diffusivity in the food. Thus, as shown in Figure 4.36 and Figure 4.32, at a distance between 3 and 4 cm from the bottom, the transfer seems to be radial only, leading to the same concentration along the Z-axis.

References 1.

J. Crank, The Mathematics of Diffusion, 2nd Edition, Clarendon Press, Oxford, UK, 1975, Chapter 4.

2.

J-M. Vergnaud, Controlled Drug Release of Oral Dosage Forms, Ellis Horwood, New York, USA, 1993, Chapters 1, 2, and 3.

3.

J-M. Vergnaud, Liquid Transport Processes in Polymeric Materials: Modelling and Industrial Applications, Prentice Hall, Englewood Cliffs, NJ, USA, 1991, Chapters 1, and 13.

4.

S. Laoubi and J-M.Vergnaud, Polymers and Polymer Composites, 1996, 4, 6, 397.

5.

D. Messadi and J-M. Vergnaud, Journal of Applied Polymer Science, 1981, 26, 7, 2315.

6.

J-M. Vergnaud and I-D. Rosca, Assessing Bioavailability of Drug Delivery Systems: Mathematical and Numerical Treatment, CRC Press, Boca Raton, FL, USA, 2005, Chapter 4, section 4.1, and Chapter 10. 197

Assessing Food Safety of Polymer Packages 7.

M. Siewert, European Journal of Drug Metabolism and Pharmacokinetics, 1993, 18, 1, 7.

8.

S. Laoubi and J-M. Vergnaud, Food Additives and Contaminants, 1996, 13, 3, 293.

9.

S. Laoubi, A. Feigenbaum and J-M. Vergnaud, Packaging Technology and Science, 1995, 8, 5, 249.

10. S. Laoubi and J-M. Vergnaud, Packaging Technology and Science, 1995, 8, 2, 97. 11. S. Laoubi, A. Feigenbaum and J-M. Vergnaud, Packaging Technology and Science, 1995, 8, 1, 17. 12. A. Feigenbaum, S. Laoubi and J-M. Vergnaud, Journal of Applied Polymer Science, 1997, 66, 3, 597. 13. S. Laoubi and J-M.Vergnaud, Food Additives and Contaminants, 1997, 14, 641. 14. S. Laoubi and J-M. Vergnaud, Journal of Polymer Engineering, 1996-1997, 16, 1-2, 25. 15. A.L. Perou, S. Laoubi and J-M. Vergnaud, Computational and Theoretical Polymer Science, 1998, 8, 3-4, 331. 16. A.L. Perou, S. Laoubi and J-M. Vergnaud, Journal of Applied Polymer Science, 1999, 73, 10, 1939. 17. A.L. Perou, S. Laoubi and J-M. Vergnaud, Advances in Colloid and Interface Science, 1999, 81, 1, 19. 18. A.L. Perou and J-M. Vergnaud, Journal of Polymer Engineering, 1997, 17, 5, 349. 19. J-M. Vergnaud, Advances in Colloid and Interface Science, 1998, 78, 3, 267. 20. I-D. Rosca and J-M. Vergnaud, Plastics, Rubber, and Composites Processing and Applications, 1997, 26, 235. 21. I-D. Rosca and J-M. Vergnaud, Polymer Recycling, 1999, 3, 2, 131. 22. I-D. Rosca and J-M. Vergnaud, Journal of Applied Polymer Science, 1997, 66, 7, 1291. 198

Mass Transfers Between Food and Packages

Abbreviations A

half the thickness of the liquid (volume per unit area) in case of finite volume of liquid

a

length of the edge of a cube, considered as a model for the volume/area ratio

α

ratio of the volumes of the liquid and of the sheet, per unit area, in Equation (4.14)

βn

positive roots of the Equation (4.22). Table is given in the Appendix

Cx,t, CL,t

concentration of diffusing substance at position x, on the side at position L, at time t, respectively

Cin, C∞, Ceq

concentration of diffusing substance, initially, and after infinite time, respectively, and at equilibrium with that in the surrounding, liquid or gas

CGS

Centimetre, gram, second system of physical units

D

diffusivity (coefficient of diffusion) expressed in: cm2/s, square length/ unit time

Fx,t

flux of matter at position x and time t (mass per unit area and per unit time)

h

coefficient of convection expressed in cm/s

K

partition factor shown in Equation (4.15), dimensionless number

L

half the thickness of the sheet (of thickness 2L); thickness of the package

Mt, M∞

amount of matter transferred by diffusion after time t, infinite time, respectively

n

integer

PET

polyethylene terephthalate

PVC

polyvinyl chloride

qn

non-zero positive roots of Equation (4.13) 199

Assessing Food Safety of Polymer Packages S

area of the sheet in contact with the liquid

x, t

coordinate along which the diffusion occurs, and time, respectively

D⋅t/L2

dimensionless number, expressing time

h⋅L/D

dimensionless number, expressing the quality of stirring of the liquid

200

5

Active Packages for Food Protection

5.1 Process of Transfer with Active Packages 5.1.1 Passive Packages Red wines and some aged cheeses are just about the only packaged foods that get better as they get older. Beyond this well known fact, virtually all food products deteriorate over time. As a result, packaging researchers are developing technology to slow that deterioration and, in some cases, to use the package in actively improving food quality. Semi-rigid plastic and flexible packages are taking over an increasing share of the packaging market from glass, jars and metal cans; demand for glass, cans, and paper-board is stalled, while plastic and flexible package use is growing. This trend is driven both by cost and functionality. Using ketchup, as an example: it is very oxygen-sensitive, and therefore it must have a very high oxygen barrier material in order for it not to darken and solidify. Unfortunately although glass ‘fits the bill’, for oxygen permeability, people like ketchup in a squeezable bottle, a function which cannot be obtained with glass. Oxygen permeability is an issue with plastic bottles, too. Oxygen gets in and makes the product go bad with common plastics, whereas this does not happen with glass. On the other hand, plastic takes up less room, weighs less, and does not break, this quality being particularly attractive for beer sales at sporting events and concerts, where plastic bottles are often thrown away. The can also suffers from consumers’ impressions about canned food, including that the interior coating does not look good, and also thinking that, that fact affects the flavour, of course, it is not true, but the perception is there [1]. Another disadvantage of the can is the customer’s perception of it – they compare cans with flexible, polyethylene terephthalate (PET) containers, and the can does not seem to be very modern. In other words, new packages spark new interest. Consumers reap benefits from all these package developments in terms of user-friendly containers, such as longer product shelf life, and convenient ready-to-eat foods. Everybody keeps looking for the elusive perfect material. Until recently, the emphasis has been on passive barriers, which just sit there and act as a barrier between the environment and the product. Such materials are often mixed so 201

Assessing Food Safety of Polymer Packages as to take advantage of the final desired properties - if a film looks like it is one layer, in fact it might be five to nine layers of different plastics. An example is flexible high-barrier materials that greatly reduce the rate of oxygen transfer to the food, such as squeezable ketchup bottles, the first breakthrough type consisting of two layers of polypropylene attached by tie layers to an inner barrier polymer layer of ethylene vinyl alcohol. Other passive materials, such as plasticised PVC, slow moisture loss while letting oxygen pass through - this property is useful for products like red meat, which needs oxygen to maintain a bright colour generally associated with quality by consumers - at the same time, the moisture barrier prevents the meat from drying out. Oxygen and moisture are not the only substances that must be kept on the appropriate side of a package, since flavour and aroma barriers become more necessary. So packages are being developed to make sure that the good flavours are kept in and the bad flavours out. These materials can be polyester or oriented polypropylene metallised with a thin coat of aluminium. There are numerous other possible packaging combinations. Materials such as PET, polyamides and polypropylene are coated with silicon or aluminium oxide to create barriers for oxygen and organics. Clay-polyimide nanocomposites have been evaluated as barrier materials for oxygen, carbon dioxide and water vapour. Other food packages include polyethylene naphthalate (PEN) and PEN-PET blends as high-barrier films or rigid containers. As a matter of conclusion, there are a large number of materials as well as the ability to combine them in a lot of different ways to produce passive barriers of all kinds, but the next step is bound to be the active packages.

5.1.2 Modified Atmosphere Packages for Perception of Freshness People want to consume more products that are perceived as fresh, and in answer to this desire, perhaps, some active packages may go to far in terms of what they imply about the nature of their contents. Oxygen and carbon dioxide levels in modified atmosphere packages (MAP) change as a function of the respiration rate of the produce, temperature, the characteristics of the film and especially the O2 and CO2 permeability coefficients of the package materials [2]. High moisture content pasta placed in a carbon dioxide-nitrogen atmosphere within a moisture-barrier package has an extended shelf life to about six weeks - such packages include packets containing iron-based compounds, which rust and thus absorb oxygen out of the package. 202

Active Packages for Food Protection The modified atmosphere packages, with high oxygen and carbon dioxide permeability, give salads or other sorts of vegetables two weeks of shelf life. The package also has to be breathable, because the product continues to respire, emitting gases, and when these gases build up inside, they would spoil the produce, so they have to permeate through the package. On the other hand, the packages may contain potassium permanganate adsorbed onto silica to absorb ethylene and retard ripening. In conclusion, as food products become more complex, so must their packages [1].

5.1.3 Active Packages with Antimicrobial Properties On the whole, an active package works in concert with the food product and its environment to produce a desired effect. While the passive package simply provides a barrier able to protect the product, the active package plays an active role in maintaining or even improving the quality of the enclosed food. The possible permutations on this theme are to some extent endless. In fact, one option of using active packaging has been to provide an increased margin of safety and quality. The following generation of food packaging includes materials with antimicrobial properties. These package technologies can play a role in extending the shelf-life of foods and reducing the risk from pathogens. Antimicrobial polymers may be useful in other contact applications as well. An antimicrobial package is a form of active package, as it interacts with the product or the headspace between the package and the food system, to obtain a desired outcome [3]. Likewise, antimicrobial food packaging acts to reduce, inhibit or retard the growth of microorganisms that may be present in the packed food or the packaging material itself [4].

5.1.3.1 Types of Antimicrobial Packages These packages can take several forms including [4]: i)

addition of sachets or pads containing volatile antimicrobial agents into packages;

ii) incorporation of volatile and non-volatile antimicrobial agents directly into polymers; iii) coating or adsorbing antimicrobials onto polymer surfaces; iv) immobilisation of antimicrobials to polymers by ion or covalent linkages; v) use of polymers that are inherently antimicrobial. 203

Assessing Food Safety of Polymer Packages

5.1.3.2 Addition of Pads Containing Antimicrobial Agents to Packages The most successful commercial application of this type of package has been sachets that are either enclosed loose or attached to the interior of a package. The following three forms have predominated: oxygen absorbers, moisture absorbers and ethanol vapour generators. Oxygen and moisture absorbers have been used in bakery, pasta, and meat packaging to prevent oxidation and water condensation. If oxygen absorbers are not exactly antimicrobial, reduction in oxygen inhibits the growth of aerobes and moulds, in the same way as moisture absorbers [5]. Ethanol vapour generators consist of ethanol absorbed or encapsulated in carrier materials, enclosed in polymer packets which let the ethanol permeate the selective barrier and thus be released into the headspace within the package.

5.1.3.3 Incorporation of Antimicrobial Agents Directly into the Polymers Incorporation of antimicrobials into polymers has been commercially applied in drug and pesticide delivery, surgical implants and other biomedical devices. Few food-related applications have been commercialised, but the number of articles and patents indicate that research on this subject has more than doubled in the last five years. Numerous potential antimicrobials have been considered. Allyl isothiocyanate, an antimicrobial extracted from plants, has been approved as an additive in Japan - diffusing as a vapour it can extend the shelf life of meat, fish and cheese. Derived agents such as benzoic acid, sodium benzoate, sorbic acid, and propionic acid, some of them having been used in edible coatings. Zeolites have also been incorporated in packages to release ions, or enzymes able to release antimicrobials as hydrogen peroxide.

5.1.3.4 Coating or Adsorbing Antimicrobials to the Polymer Surface Antimicrobials that cannot tolerate the temperature used in the processing of packaging are often coated onto the material after forming or are added to cast films. Examples include nisin, an antibiotic, coated onto low-density polyethylene film in a methylcellulose carrier [4]. 204

Active Packages for Food Protection

5.1.3.5 Immobilisation of Antimicrobials by Ionic or Covalent Linkages to Polymers This type of immobilisation requires the presence of functional groups on both the antimicrobial and the polymer, these groups being spacer molecules linking the polymer surface to the bioactive agent. The enzyme naringinase has been immobilised in a polymer that could be potentially be used as a liner inside a grapefruit carton, and the flavonones responsible for bitterness in citrus products are broken down. When the juice is stored in contact with the film, the enzyme hydrolyses the bitter compounds, making the beverage taste sweeter over time [1, 4].

5.1.3.6 Use of Polymers that are Inherently Antimicrobial Antimicrobial packages have also been developed. As surface growth of microorganisms is one of the leading causes of food spoilage, a package system that allows for slow release of an antimicrobial agent into the food could significantly increase the shelf life and so improve the quality of a large variety of food [1, 4].

5.1.4 Applications of Antimicrobial Package in Foods Antimicrobial polymers can be used in several food applications, including packaging. The first use is to promote safety and thus extend the shelf-life by reducing the rate of growth of specific microorganisms by direct contact of the package with the surface of solid foods, such as meat, or in the bulk of liquids as milk. Secondly, these antimicrobial packages could be self-sterilising, reducing the potential for recontamination of the products and simplify the treatment required to eliminate the product contamination.

5.1.5 Testing the Effectiveness of Antimicrobial Packages and Regulatory Issues There are a variety of official test methods to determine the resistance of plastic materials to microbial degradation, but there is no agreement upon standard methods to determine their effectiveness [4]. Food packaging is highly regulated around the world and development projects on antimicrobial packages must take these regulations into consideration. At this time, antimicrobials in food packaging that may migrate to food are considered to be food additives and must meet the food additives standards. 205

Assessing Food Safety of Polymer Packages

5.1.6 Detection Systems Despite all these efforts, microorganisms continue to be found in food. Researchers are working on a detection system for bacterial toxins and pathogens in food. The idea being that this system using cells engineered to react with their surroundings, would promote colour changes in the presence of pathogens in food. Also used to alert consumers to problems are time-temperature indicators, which can show whether the enclosed frozen food product has been mishandled during shipping or storage.

5.2 Active Packages – Theoretical Considerations 5.2.1 Process of Release and Consumption, and Assumptions The process of release and consumption of the agent is described by considering the three stages in succession: 1. The package consists of two main parts: the usual polymer package, which is impermeable to the agent, and a polymer sheet called linen which contains the active agent with a uniform concentration. 2. The agent diffuses through the linen and is released into the food. 3. The agent reacts with the microorganisms, and a part of this agent is consumed. The following assumptions are made in order to simplify the problem: i) In spite of the fact that the package is a bottle, or is bottle-shaped, there is a monodirectional transfer of the agent through the linen and into the food. This assumption can be made as the thickness of the polymer is very low as compared with the radius of the bottle. Moreover the ratio of the volume of the food and the surface of the package V/S = a/6 for a cube, and thus for a litre of food, the thickness of the food is 1.66 cm. ii) The kinetics of release of the agent is controlled either by diffusion through the polymer linen or by convection into the food. iii) The kinetics of consumption of the agent by the microorganisms located in the food is described by a first-order reaction with respect to the concentration of the agent [6]. 206

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5.2.2 Mathematical and Numerical Treatment The transfer of the agent through the polymer linen is expressed by the one-dimensional equation of diffusion with constant diffusivity: ∂C ∂2C = D⋅ 2 ∂t ∂x

(5.1)

and at the linen-food interface, the boundary conditions apply: −D ⋅

∂C = h ⋅ (CL , t − C f , t ) ∂x

(5.2)

meaning that the rate at which the agent enters the food is constantly equal to that at which it is brought to the linen surface by diffusion, where h is the coefficient of convection into the food and D is the diffusivity of the agent through the linen. On the other surface of the linen, there is no transfer of the agent: ∂C =0 ∂x

(5.3)

The rate of consumption of the agent by the microorganisms in the food is given by: −

∂C = K ⋅ Cf ,t ∂t

(5.4)

in the homogeneous food phase, where the concentration Cf,t is uniform. When a reaction takes place, the amount of the agent located in the food Y is given by the relationship: dY dM = − K ⋅ Yt dt dt

(5.5)

where M is the amount of the agent delivered in the food by the linen at time t. The kinetics of release of the agent out of the linen, controlled either by diffusion through the thickness of the linen or by the convection at the linen-food interface, is expressed by the following relationship: 207

Assessing Food Safety of Polymer Packages ⎛ β2 ⋅ D ⋅ t ⎞ M∞ − M t ∞ 2R 2 exp ⎜− n 2 ⎟ =∑ 2 2 2 M∞ L ⎝ ⎠ n =1 βn (βn + R + R)

(5.6)

where βn is the positive root of: β ⋅ tan β = R

(5.7)

and R is given by: R=

h⋅L D

(5.8)

The letters given for the various parameters are defined at the end of Chapter 5.

5.3 Results Obtained by Calculation The scheme of the system, representing the process, is shown in Figure 5.1, with the impermeable layer, the linen of thickness L1 = X2 – X1 containing the active agent with the initial concentration Cin, and the food. As already proved in Chapter 1, resulting from the fact that the internal convective transfer through the liquid is fast enough, and at least much larger than the rate of release of the agent into the food, the concentration of this active agent as well as that of the microorganisms in the liquid food is constantly uniform.

Figure 5.1 Scheme of the system: impermeable polymer (1); linen (2); liquid food (3). 208

Active Packages for Food Protection

Figure 5.2 Kinetics of release of the agent in the food when there is no consumption of the agent (expressed through the dimensionless number D⋅t/L2), with various values of R = h⋅L/D = 1; 2; 20; 100; K = 0; L = 0.1 cm and Lf = 1.66 cm (1 litre food)

The kinetics of release of the active agent into the liquid free from any microorganisms is shown in Figure 5.2 for a wide range of values of the dimensionless numbers R. Other dimensionless numbers are also used: D⋅t/L2 for time, and the ratio Cf/Cin of the concentration of the agent in the food at any time as a fraction of the initial concentration in the linen. The problem is similar to that of the release of a drug from a dosage form whose release is controlled by diffusion [7]. No analytical solution exists, and the problem should be resolved by a numerical treatment. Time is expressed in terms of finite increments Δt, with the integer j such that t = j⋅Δt. Starting with j = 0, which corresponds with the initial conditions, calculation is made at j = 1 for either the amount of active agent released out of the linen at that time or for the proportion which is consumed by the microorganisms, by following the Equations (5.4) to (5.6), and so on, for the following values of j.

5.3.1 Results Obtained for High Values of R The dimensionless number, sometimes called the Sherwood number, is taken at R = 100, by considering two values of the components of R, e.g., the thickness of the linen and the coefficient of convection h. With R = 100, the process of release is controlled by diffusion through the thickness of the linen, as the coefficient of convection is rather high. 209

Assessing Food Safety of Polymer Packages Two ways are followed to obtain this value of R = 100, by varying simultaneously the thickness of the linen and the value of the coefficient of convection, as shown in Table 5.1.

K (/s)

Table 5.1 Values of the parameters used for calculation 0 10–8 10–7 10–6 10–5 10–4 10–3

SN

0

0.01

0.1

1

10

100

1000

Figures 5.3 and 5.5

SN

0

0.0025

0.025

0.25

2.5

25

250

Figures 5.4 and 5.6

The following figures are considered: Figure 5.3 depicting the kinetics of the active agent remaining free in the liquid, with the process controlled by diffusion and the values: L = 0.1 cm; D = 10–8 cm2/s; h = 10–5 cm/s, for various values of the killing rate constant K and of the Savoie number (SN). Figure 5.4 showing the kinetics of the active agent remaining free in the liquid, with the diffusion-controlled process and the values: L = 0.05 cm; D = 10–8 cm2/s; h = 2⋅10–5 cm/s, for various values of the killing rate constant K and of the SN. Figure 5.5 representing the kinetics of the agent consumed by the microorganisms in food with the diffusion-controlled process for various values of K and SN, with L = 0.1 cm; D = 10–8 cm2/s; h = 10–5 cm/s. Figure 5.6 Kinetics of the agent consumed in food with the process controlled by diffusion, with Sherwood number 100, for various values of K and SN, with L = 0.05 cm; D = 10–8 cm2/s; h = 2⋅10–5 cm/s. From these figures, some comments are worth considering: i)

The effect of the value of the SN, (K⋅L2)/D, which is proportional to the rate constant of the kinetics of consumption of the agent is of considerable importance. Curve 1 in the Figures 5.3-5.6 is obtained with no consumption reaction (K = 0). It can also be seen that for low values of K, e.g., less than 10–8/s, the kinetics of the agent consumed and thus the complementary kinetics of the agent remaining free in the food are nearly similar (curve 2) to those obtained with K = 0, over a time of 1,000 hours. The corresponding values of the SN are 0.01 with the conditions of the Figures 5.3 and 5.5 and 0.0025 with the conditions of the Figures 5.4 and 5.6. In both these cases, the active agent can be considered entirely free, since a small amount is consumed.

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Figure 5.3. Kinetics of the agent remaining free in food with the process controlled by diffusion, with a Sherwood number of 100, for various values of K and SN: K⋅L2/D. L = 0.1 cm; D = 10–8 cm2/s; h = 10–5 cm/s; Lf = 1.66 cm (1 litre food). 1: K = 0; 2: K = 10–8; 3: K = 10–7; 4: K = 10–6; 5: K = 10–5; 6: K = 10–4; 7: K = 10–3/s

Figure 5.4. Kinetics of the agent remaining free in food with the process controlled by diffusion, with a Sherwood number of 100, for various values of K and SN: K⋅L2/D. L = 0.05 cm; D = 10–8 cm2/s; h = 2⋅10–5 cm/s; Lf = 1.66 cm (1 litre food). 1: K = 0; 2: K = 10–8; 3: K = 10–7; 4: K = 10–6; 5: K = 10–5; 6: K = 10–4; 7: K = 10–3/s

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Figure 5.5. Kinetics of the agent consumed in food with the process controlled by diffusion, with a Sherwood number of 100, for various values of K and SN: K⋅L2/D. L = 0.1 cm; D = 10–8 cm2/s; h = 10–5 cm/s; Lf = 1.66 cm (1 litre food). 1: K = 0; 2: K = 10–8; 3: K = 10–7; 4: K = 10–6; 5: K = 10–5; 6: K = 10–4; 7: K = 10–3/s

Figure 5.6. Kinetics of the agent consumed in food with the process controlled by diffusion, with a Sherwood number of 100, for various values of K and Savoie number: K⋅L2/D. L = 0.05 cm; D = 10–8 cm2/s; h = 2⋅10–5 cm/s; Lf = 1.66 cm (1 litre food). 1: K = 0; 2: K = 10–8; 3: K = 10–7; 4: K = 10–6; 5: K = 10–5; 6: K = 10–4; 7: K = 10–3/s 212

Active Packages for Food Protection ii) For larger values of the consumption rate constant and of the SN, the shape of the kinetics is quite modified. Thus, the statement holds: that the higher the value of K, the faster the kinetics of consumption and the lower the amount of the agent remaining free in the liquid food. For high values of K, e.g., larger than10–4/s, the active agent is quickly consumed after a time which does not exceed 200-400 hours depending on the thickness of the linen. iii) For intermediate values of the rate constant K, typical curves are obtained. On the one hand, the curves expressing the kinetics of the drug remaining free, pass through a maximum whose position (time and height) depends on the value of K. On the other hand, the kinetics of consumption exhibit a S-shape which is clearly shown as curve 4 obtained with K = 10–6/s and SN = 1 under the conditions of Figure 5.5 and SN = 0.25 with those of Figure 5.6. iv) The relative effect of the thickness of the linen is also of interest. Whatever the value of the coefficient of convection, a thicker linen is responsible for faster kinetics of the agent release in the liquid, as well as for the agent consumed.

5.3.2 Results Obtained with Low Values of R The values of the dimensionless number R is taken at 1, by considering the two values of the components of R, the thickness of the linen and the coefficient of convection h. With such a low value, the process of the agent release from the linen is controlled by convection at the linen-liquid food interface. The values of the Moûtiers number (MN) are shown in Table 5.2.

K (/s)

Table 5.2 Values of the parameters used for calculation 0 10–8 10–7 10–6 10–5 10–4 10–3

MN

0

0.01

0.1

1

10

100

1000

Figures 5.7 and 5.9

MN

0

0.0025

0.025

0.25

2.5

25

250

Figures 5.8 and 5.10

The following figures are considered: Figure 5.7 depicting the kinetics of the active agent remaining free in the liquid, with the process controlled by diffusion and the values: L = 0.1 cm; D = 10–8 cm2/s; h =10–7 cm/s, for various values of the killing rate constant K and of the MN. 213

Assessing Food Safety of Polymer Packages Figure 5.8 showing the kinetics of the active agent remaining free in the liquid, with the diffusion-controlled process and the values: L = 0.05 cm; D =10–8 cm2/s; h = 2⋅10–7 cm/s, for various values of the killing rate constant K and of the MN. Figure 5.9 representing the kinetics of the agent consumed by the microorganisms in food with the diffusion-controlled process for various values of K and of the MN, with L = 0.1 cm; D = 10–8 cm2/s; h =10–7 cm/s. Figure 5.10 Kinetics of the agent consumed in food with the process controlled by diffusion, with Sherwood number 1, for various values of K and of the MN, with L = 0.05 cm; D = 10–8 cm2/s; h = 2⋅10–7 cm/s. The following conclusions can be drawn from Figures 5.7-5.10: i)

For low values of the rate constant of the reaction K (equal or lower than 10–8/s) and MN lower than either 0.01 in Figures 5.7 and 5.9 (curve 1 and 2) or 0.0025 in Figures 5.8 and 5.10, the active agent remains almost free in the liquid food. Obviously, there is no consumption of this active agent when K = 0.

ii) For large values of the rate constant of the reaction K (larger than 10–4/s), with the corresponding values of the MN of either 100 in the case of the Figures 5.7 and 5.9 (curves 6 and 7) or 25 in the case of the Figures 5.8 and 5.10 (curves 6 and 7), the active agent reacts and thus disappears as soon as it is released in the food. iii) For the intermediate values of the rate constant of the reaction K, and with the corresponding values of the MN located between 0.0025 and 25, the curves expressing the kinetics of the concentration of the active agent remaining free in the food pass through a maximum, as shown in Figures 5.7 and 5.8. The position of this maximum varies with the value of the MN, the time at which it is obtained, as well as its height decreases as the rate constant K is increased. On the other hand, in a complementary way, the kinetics of consumption of the active agent increases with the value of the rate constant K, exhibiting a kind of S-shaped curve. iv) The thickness of the linen plays an important role on the process of release of the active agent in the food. Obviously, the kinetics of release of the active agent in the food increases with the thickness of the linen, as shown by comparing the curves in Figures 5.7 and 5.8. In the same way, the kinetics of consumption of the active agent also increases with the thickness of the linen. 214

Active Packages for Food Protection

Figure 5.7. Kinetics of the agent remaining free in food with the process controlled by diffusion, with a Sherwood number of 1, for various values of K and SN: K⋅L2/D. L = 0.1 cm; D = 10–8 cm2/s; h = 10–7 cm/s; Lf = 1.66 cm (1 litre food). 1: K = 0; 2: K = 10–8; 3: K = 10–7; 4: K = 10–6; 5: K = 10–5; 6: K = 10–4; 7: K = 10–3/s

Figure 5.8. Kinetics of the agent remaining free in food with the process controlled by diffusion, with a Sherwood number of 1, for various values of K and SN: K⋅L2/D. L = 0.05 cm; D = 10–8 cm2/s; h = 2⋅10–7 cm/s; Lf = 1.66 cm (1 litre food). 1: K = 0; 2: K = 10–8; 3: K = 10–7; 4: K = 10–6; 5: K = 10–5; 6: K = 10–4; 7: K = 10–3/s

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Figure 5.9. Kinetics of the agent consumed in food with the process controlled by diffusion, with a Sherwood number of 1, for various values of K and SN: K⋅L2/D. L = 0.1 cm; D = 10–8 cm2/s; h = 10–7 cm/s; Lf = 1.66 cm (1 litre food). 1: K = 0; 2: K = 10–8; 3: K = 10–7; 4: K = 10–6; 5: K = 10–5; 6: K = 10–4; 7: K = 10–3/s

Figure 5.10. Kinetics of the agent consumed in food with the process controlled by diffusion, with a Sherwood number of 1, for various values of K and SN: K⋅L2/D. L = 0.05 cm; D = 10–8 cm2/s; h = 2⋅10–7 cm/s; Lf = 1.66 cm (1 litre food). 1: K = 0; 2: K = 10–8; 3: K = 10–7; 4: K = 10–6; 5: K = 10–5; 6: K = 10–4; 7: K = 10–3/s

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5.3.3 Establishment of the Dimensionless Numbers On the whole, there are two ways to establish a dimensionless number [8]. The one, which is very simple but also hazardous, consists of building a fraction where the numerator has the same dimension as the denominator. The other, which is more consistent, is obtained by equilibrating the two opposite acting forces which are facing each other, and simplifying afterwards. According to the second method [9, 10] selected, let us recall that the dimensionless number R is obtained as follows: Equation (5.2) can be rewritten in the form: D ⋅ Cp Lp

= h ⋅ (C p − C f )

(5.9)

leading to the well-known relationship expressing the dimensionless number R: h ⋅ Lp D

=

Cp Cp − Cf

=R

(5.10)

where the right side-member is dimensionless, being the ratio of two concentrations of the active agent. In the same way, the other new dimensionless numbers are established in the following way: When the process is controlled by diffusion: ∂C ∂M = −D ⋅ p = K ⋅ M = K ⋅ C f ⋅ L f ∂t ∂x

(5.11)

which can be rewritten in the simple manner: D ⋅ Cp Lp

= K ⋅ Cf ⋅ L f

(5.12)

This equation leads to the dimensionless number, the SN: K ⋅ L2p D

= SN

(5.13)

217

Assessing Food Safety of Polymer Packages since the ratio of the concentrations is inversely proportional to the volumes: Cf L p = Cp L f

(5.14)

When the process is controlled by convection at the polymer-liquid interface, with a low value of the coefficient of convection, the limiting relationship is obtained by writing that the rate of the agent released in the food is equal to its rate of consumption by the microorganisms: h ⋅ (C p − C f ) = K ⋅ C f ⋅ L f

(5.15)

which leads to the dimensionless number, the MN: K ⋅ Lp h

= MN

(5.16)

by taking into account relationship (5.14).

5.4 Conclusions about the Active Agents It is difficult to predict what the future for the development of the active agents will be. By considering what was declared by very well informed authors [1], there are potential applications for the active agents to achieve long-term food protection. Certainly, it is true that protecting food for a few days is one thing, but there is a quite another objective, more attractive, to be able to keep the same food under safety conditions over a period of time exceeding one or two weeks. However, a problem appears with the various qualifications of the specialists necessitated by the work. There should be the specialists in polymers and polymer additives, the people responsible for the food, the experts in packaging, but also the bacteriologists as well as the analysts, but fortunately, the experts in chemical engineering could unite all those people, and the first step is the introduction of the dimensionless numbers. The dimensionless number allows those people to work independently, nevertheless having in mind the same objective.

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References 1.

S.L. Wilkinson, Chemical and Engineering News, 1998, 76, 24, 26.

2.

T. Al-Ati and J.H. Hotchkiss, The Role of Film Permeselectivity in MAP (private review paper).

3.

A.L. Brody, E.R. Strupinsky and L.R. Kline, Active Packaging for Food Applications, Lancaster, PA, Technomic Publishing Co, 2001.

4.

P. Appendini and J.H. Hotchkiss, Review of Antimicrobial Food Packaging (Private review paper).

5.

Active Food Packaging, Ed., M. Rooney, Blackie Academic & Professional, Glasgow, UK, 1995.

6.

I-D. Rosca and J-M. Vergnaud, Pharmaceutical Sciences, 1995, 1, 391.

7.

J-M. Vergnaud and I-D. Rosca, Assessing Bioavailability of Drug Delivery Systems: Mathematical and Numerical Treatment, CRC Press, Boca Raton, FL, USA, 2005.

8.

W.H. McAdams, Heat Transmission, 3rd Edition, McGraw-Hill, New York, NY, USA, 1954.

9.

M. El Kouali, M. Salouhi, F. Labidi, M. El Brouzi and J-M.Vergnaud, Plastics, Rubber and Composites, 2003, 32, 3, 127.

10. M. El Kouali, M. Salouhi, F. Labidi, M. El Brouzi and J-M. Vergnaud, Polymers and Polymer Composites, 2003, 11, 4, 301.

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Abbreviations a

side of a reference cube of volume V and area S

βns

positive roots of the Equation (5.7)

C

concentration of the active agent

Cin

initial concentration of the active agent in the linen

Cf

concentration of the active agent in the food at any time

Cf/Cin

dimensionless number, for the concentration of the agent in the food

D

diffusivity (cm2/s)

D⋅t/L2

dimensionless number, for time

h

coefficient of convective transfer at the linen-liquid interface (cm/s)

K

rate constant of the first-order bactericidal reaction (/s)

L

thickness of the linen (cm)

Lf

thickness of the liquid food, with one-dimensional transfer (10/6 cm for 1 litre)

MAP

Modified atmosphere packages

Mt, M∞

amount of active agent released at time t, infinite time, from the linen (Equation (5.6))

MN

Moûtiers number, dimensionless number K⋅L/h

PEN

Polyethylene napthalate

PET

Polyethylene terephthalate

R

Sherwood number, dimensionless number h⋅L/D

S

area of the reference cube of side a

SN

Savoie number, dimensionless number K⋅L2/D

V

volume of the reference cube of side a

x

position

Y

amount of active agent in the bottle

t

time

220

6

A Few Common Misconceptions Worth Avoiding

It has been written in June 2005 [1] that from a recent survey, one-third of scientists admitted to questionable practices. ‘Okay, so the game’s up - scientists, it turns out, are neither perfect nor perfectly ethical. Scientists’ flaws include engaging in a wide range of questionable practices’. Without looking at the cases of gross scientific misconducts, such as fabricating or plagiarising results, that can end up in the headlines, they are many other behaviours that can compromise the integrity of research and the interest of the results finally obtained. Among the top 10 behaviours and the percentage of respondents admitting to them are the following: •

Failing to present data that contradict one’s own previous research (6%)

•

Overlooking others use of flawed data or questionable interpretation of data (12.5%)

•

Changing the design, methodology, or results of a study in response to pressure from a funding source (15.5%)

•

Inappropriately assigning authorship credit (10%)

•

Dropping data points from an analysis based on a gut feeling that they were inaccurate (15.3%)

•

Keeping inadequate records related to research projects (27.5%)

•

Refusing to give one’s own data to other people who might interpret them differently, in order to avoid any discussion on this interpretation.

It was also said that ‘the modern scientist faces intense competition, and is further burdened by difficult, sometimes unreasonable regulatory, social, and managerial demands. This mix of pressures creates many possibilities for the compromise of scientific integrity that extends well beyond the official definition of research misconduct, which is fabrication, falsification, or plagiarism in proposing, performing, or reviewing research results’. It was also added that ‘chemistry journal editors are seeing a growing number of cases in which authors are trying to publish essentially the same manuscript in different journals, a practice known as duplicate submission, or self-plagiarism’. 221

Assessing Food Safety of Polymer Packages Another reason stands out, especially when the analysis of the data needs some difficult theoretical treatment. It is easy to understand that as science is spreading out on quite different sides, people in research have to become specialised. Thus, there are people able to obtain good data by using either costly or new apparatus, as well as other people who will be able only to treat these results theoretically. The researcher is necessary because he gets the data, but the theoretician can improve these data by evaluating accurate values of the parameters and perhaps finding a general scheme for the process. In fact, in a few words, this conclusion can be seen to hold true: we cannot have the researcher without the theoretician. A few misinterpretations are thus considered in this chapter, which are made either by using an inadequate equation resulting from a wrong or poor theoretical approach, or by building an inaccurate model for describing the process.

6.1 Using Equations Based on Infinite Convective Transfer 6.1.1 The Problem Presented This is the most common case when the researcher uses an equation, or some mathematical model found in the literature, which is based on the rough assumption that the coefficient of convective transfer at the interface between the polymer film and the liquid is infinite. Of course, as already stated in this book, and not only in Chapter 1 (the mathematical treatment of the diffusion), the assumption of the infinite coefficient of convection corresponds with the extreme case of evaporation of the diffusing substance behaving like a permanent gas; and in this case the rate of evaporation of the gas, not a vapour, is so high that it may be considered as infinite. In all the other cases, in the surrounding air, the rate of evaporation of the vapour is finite, and furthermore, the same thing happens in a liquid where the diffusing substance has not the same ability to escape as rapidly as in the surounding air. A typical example is found in a recent paper [2], concerned with the kinetics of sorption of various alcohols in a variety of polyethylene terephthalates (PET). Not only is the programme utilised in the work, based on the infinite coefficient of convection at the liquid-polymer interface, but the kinetics provided by the software [3] were drawn by way of nearly two straight lines, the one diverting slightly from the ordinate axis and remaining very close to it, and the other parallel with the abscissa axis, being practically recorded at right angle. No reader is able to use the data resulting from these figures, so as either to confirm the result provided by the software or to try to discuss them. 222

A Few Common Misconceptions Worth Avoiding

6.1.2 Theoretical Survey Recalling the equations shown in Chapter 1 devoted to the theoretical basis of the diffusion process, there are the following equations to consider: •

The case of a finite value of the coefficient of convection at the sheet-liquid interface. The main equations for a sheet of thickness 2L in contact on both sides with the liquid are as follows, with: -L < x < +L The solution of the problem, whatever the partition factor, is given as follows: The kinetics of transfer of diffusing substance, by using the dimensionless number Mt/M∞, is expressed in terms of the dimensionless number D⋅t/L2 by the following Equation (1.52): ⎛ M∞ − M t ∞ D⋅t⎞ 2 ⋅ R2 exp ⎜−β2n 2 ⎟ =∑ 2 2 2 M∞ L ⎠ ⎝ n =1 βn (βn + R + R)

(6.1)

where the βn are the positive roots of Equation (1.50): β ⋅ tan β = R

(6.2)

while the dimensionless number R is given by the relationship (1.51): R= •

h⋅L D

(6.3)

The case of the highly optimistic hypothetical assumption of an infinite value of the coefficient of convection. It has been shown that for the infinite value of the coefficient of convection h and of the dimensionless number R, Equation (6.1) reduces to Equation (6.4): ⎡ 2n + 1 2 π 2 Mt 8 ∞ 1 ) D ⋅ t⎤⎥ ⎢− ( exp = 1− 2 ∑ ⋅ ⎢ ⎥ M∞ π n =0 (2n + 1)2 4L2 ⎣ ⎦

(6.4)

In order to point out the error from using such a software based on the infinite value of the coefficient of convective transfer of the substance at the polymer-liquid interface, the kinetics of the absorption (or desorption) of the same substance are calculated by using the same operational conditions, except for the value of the dimensionless number R which is taken either as finite or infinite. Moreover, the curves are expressed in different ways by selecting two different scales for the time put in the abscissa. 223

Assessing Food Safety of Polymer Packages The kinetics are shown in the following six figures for different values of R selected within the 10-2 range, and for R infinite, as well as for different values of the thickness of the film, so as to compare them. These sheets are in contact with the liquid on both sides. Figure 6.1 Kinetics calculated with R = 10, L = 0.01 cm, D = 10-7 cm2/s, and drawn, the one up to 500 minutes, the other only up to 25 minutes, for the scale of time (full line). Kinetics calculated with the same parameters, except for R which is infinite (dotted line). Figure 6.2 Kinetics calculated with R = 5, L = 0.01 cm, D = 10-7 cm2/s, and drawn, the one up to 500 minutes, the other only up to 25 minutes, for the scale of time (full line). Kinetics calculated with the same parameters, except for R which is infinite (dotted line). Figure 6.3 Kinetics calculated with R = 2, L = 0.01 cm, D = 10-7 cm2/s, and drawn, the one up to 500 minutes, the other only up to 25 minutes, for the scale of time (full line). Kinetics calculated with the same parameters, except for R which is infinite (dotted line). Figure 6.4 Kinetics calculated with R = 10, L = 0.03 cm, D = 10-7 cm2/s, drawn, the one up to 4000 minutes, the other only up to 100 minutes, for the scale of time (full line). Kinetics calculated with the same parameters, except for R which is infinite (dotted line). Figure 6.5 Kinetics calculated with R = 5, L = 0.03 cm, D = 10-7 cm2/s, and drawn, the one up to 4000 minutes, the other only up to 100 minutes, for the scale of time (full line). Kinetics calculated with the same parameters, except for R which is infinite (dotted line). Figure 6.6 Kinetics calculated with R = 2, L = 0.03 cm, D = 10-7 cm2/s, and drawn, the one up to 4000 minutes, the other only up to 100 minutes, for the scale of time (full line). Kinetics calculated with the same parameters, except for R which is infinite (dotted line).

6.1.3 Conclusions Drawn from the Problem From Figures 6.1 to 6.6, by considering either the effect of the value given to the dimensionless number R or the effect of the scale of time selected in the drawing, the following comments are worth noting: i)

On the whole, the effect of the scale of time selected in the curves expressing the kinetics of transfer of the diffusing substance appears clearly on these six figures. With the thickness of 0.01 cm, when this scale of time is very large, up to 500 minutes, the two curves obtained with the two different values of the dimensionless number, e.g., R finite or R infinite, are quite well superimposed, as shown in Figures 6.1-6.3. By contrast, these

224

A Few Common Misconceptions Worth Avoiding

Figure 6.1. Kinetics of transfer of the diffusing substance calculated with D = 10-7 cm2/s and L = 0.01 cm either with R = 10 (full line) or infinite R (dotted line). Two scales of time are used, one with 500 minutes, the other with 25 minutes.

Figure 6.2. Kinetics of transfer of the diffusing substance calculated with D = 10-7 cm2/s and L = 0.01 cm either with R = 5 (full line) or infinite R (dotted line). Two scales of time are used, one with 500 minutes, the other with 25 minutes.

225

Assessing Food Safety of Polymer Packages

Figure 6.3. Kinetics of transfer of the diffusing substance calculated with D = 10-7 cm2/s and L = 0.01 cm either with R = 2 (full line) or infinite R (dotted line). Two scales of time are used, one with 500 minutes, the other with 25 minutes.

Figure 6.4. Kinetics of transfer of the diffusing substance calculated with D = 10-7 cm2/s and L = 0.03 cm either with R = 10 (full line) or infinite R (dotted line). Two scales of time are used, one with 4000 minutes, the other with 100 minutes. 226

A Few Common Misconceptions Worth Avoiding

Figure 6.5. Kinetics of transfer of the diffusing substance calculated with D = 10-7 cm2/s and L = 0.03 cm either with R = 5 (full line) or infinite R (dotted line). Two scales of time are used, one with 4000 minutes, the other with 100 minutes.

Figure 6.6. Kinetics of transfer of the diffusing substance calculated with D = 10-7 cm2/s and L = 0.03 cm either with R = 2 (full line) or infinite R (dotted line). Two scales of time are used, one with 4000 minutes, the other with 100 minutes.

227

Assessing Food Safety of Polymer Packages same kinetic curves drawn with the other scale of 25 minutes, are expressed through two curves, which are distinctly separated. Thus, the following statement holds true: using a too large a scale of time can lead to deceptive information and wrong results. This consequence remains whatever the value of the dimensionless number R. Similar results are obtained when the thickness is 0.03 cm (Figures 6.4-6.6) with a longer time. ii) The effect of the rate of stirring is shown in the six figures, by comparing the kinetics shown as the value of R and subsequently of the coefficient of convective transfer at the polymer-liquid interface which can measure the effect of this rate of stirring. iii) Of course, resulting from the dimensionless number D⋅t/L2, by keeping the value of the diffusivity constant, the time necessary for a given transfer is proportional to the square of the thickness of the sheet. Thus, for a thickness of 0.03 cm, the times are almost ten times larger than for the thinner thickness of 0.01 cm, for the same value of R. This fact explains the larger time scale, nearly ten times longer, used in Figures 6.4-6.6. iv) An important fact also appears, resulting from the value given to R, either infinite or finite. In each of the Figures 6.1-6.6, the kinetics obtained with these two values of R (finite or infinite) are superimposed on each other, by taking the larger scale of 500 minutes when L = 0.01 cm, and of 4000 minutes when L = 0.03 cm. But, if the kinetics drawn with R finite are obtained by taking D = 10-7 cm2/s, the other kinetics drawn with R infinite should be obtained with a much lower value of the diffusivity expressed by the essential relationship: D (with infinite R) = a·D (with finite R) this coefficient ‘a’ being lower than 1. Moreover, the value of ‘a’ largely depends on the values given to the other two parameters, e.g., R, and the thickness of the sheet L, as shown in Table 6.1.

L, cm 0.01 0.03

Table 6.1 Values of a with the values given to R and L R a R a R 10 0.73 5 0.59 2 10 0.73 5 0.59 2

a 0.38 0.38

v) In other words, to conclude, two successive stages take place in the process of transfer of the diffusing substance from the polymer into the liquid: the diffusion of the substance through the thickness of the sheet, followed by the convection of the substance through the polymer sheet-liquid interface. Thus, assuming that R is infinite means that the

228

A Few Common Misconceptions Worth Avoiding second stage is eliminated, which leads to a faster rate of transfer than in the case of a finite value of R. Finally, the same kinetics curves obtained with either an infinite or a finite value of R implies that the value of the diffusivity should be larger when the value of R is finite. But the worst is coming with the fact that from one experiment to another, it is probable, that the conditions for stirring are different, and finally that the values of the dimensionless number R are kept constant. The process of stirring is highly complex, and the coefficient of convection h, and in the same way R, largely depends on various factors, and amongst them are: the nature of the liquid with this viscosity and wettability, the volume and form of the flask (the apparent viscosity of a liquid varies with the dimensions of the flask in which it is stirred), and the nature of the rotating paddle. It is worth noting that the systems of flask and rotating paddle have been extensively studied in pharmaceutical applications in order to evaluate the capability of the in vitro dissolution tests which are strictly standardised [4, 5]. So, what has been true for pharmaceutical applications (for the purpose of comparing precisely the kinetics of dissolution of various dosage forms) would also be true in food safety applications (for determining the time of food safety allowed by a package as precisely as possible).

6.2 Infinite Thickness of the Film and Infinite Convective Transfer [6] 6.2.1 Description of the Experimental Part The materials and methods used for evaluating the matter transfer are presented next. Three-layer films made of high-impact polystyrene (HIPS) are co-extruded in such a way that the contaminated layer is inserted between the two virgin layers. The contaminant is either toluene or chlorobenzene, which is previously inserted into the contaminated layer by absorption. The films are cut and sealed to pouches of 2 dm2 inner surface, and the pouches are filled with 20 ml food simulant made of 50% ethanol in water at 40 °C. The concentration of the contaminant in this liquid is determined by gas chromatography at various times over 76 days, for different values of the thickness of the barrier B varying between 0 and 0.02 cm and keeping the thickness of the contaminated layer constant at 0.015 cm. From these dimensions, it appears that the thickness of the liquid is very low, and that no stirring is possible in it. The values of the diffusivity for each simulant are given at 40 °C as well as 200 °C: Toluene

D = 1.25 x 10–12 cm2/s

D = 1 x 10-6 cm2/s

Chlorobenzene

D = 1.72 x 10–12 cm2/s

D = 8.1 x 10-6 cm2/s 229

Assessing Food Safety of Polymer Packages

6.2.2 Theoretical Consideration by the Authors [6] In this paper [6], a ‘physico-mathematical model describing the migration across functional barrier layers into foodstuffs’ is described. The system appears as follows: a polymer barrier B of thickness ‘b’ initially free from contaminant is surrounded by a semi-infinite matrix 1, with a concentration C1 of diffusing substance, considered as a contaminant and by another semi-infinite matrix 2, whose potential pollutant concentration is C2, while the diffusivity of the diffusing substance is called in these three media: D1, D2, DB, respectively. The diffusion being mono-directional, and perpendicular to the planes separating the media, the equation of diffusion is: ∂C ∂2C = D⋅ 2 ∂t ∂x

(6.5)

where C, the concentration of the diffusing substance, varies with time and space. By following, these assumptions, the initial condition may be written as: t= 0

x>0

C=0

(6.6)

C = Cin

(6.7)

while the boundary condition is: t>0

x=0

The solution of the problem of diffusion from a semi-infinite medium into another semiinfinite medium, is generally expressed in terms of the error-function complement [7]: C x , t = Cin ⋅ erfc

x 2(D ⋅ t)0.5

(6.8)

and the kinetics of the matter transfer through the two semi-infinite media is obtained, by evaluating the derivative with respect to space x of the concentration of the substance at position 0 where the two media are separated, and by integrating this derivative with respect to time. The following equations are obtained. For the derivative with respect to space at position 0: ⎛ ∂C ⎞ Cin ⎜ ⎟ = ⎝ ∂x ⎠x =0 (π ⋅ D ⋅ t)0.5

230

(6.9)

A Few Common Misconceptions Worth Avoiding and for the kinetics of the matter transferred through the two semi-infinite media: 0.5

⎛D⋅ t⎞ M t = 2 ⋅ Cin ⋅ ⎜ ⎟ ⎝ π ⎠

(6.10)

Equation 6.10 is the basis of the model used in the paper [6]. At that time, the mathematical treatment is correct, provided that the assumption of an infinite thickness for the recycled layer and for a volume of liquid infinity is accepted. By applying to the barrier polymer B, of thickness b, the equation expressing the amount of substance transferred through a membrane under stationary state, it appears: Mt =

DB ⋅ C1 ⎛ b2 ⎞ ⎟ ⎜t − b ⎝ 6.DB ⎠

(6.11)

by assuming that the concentration C1 is kept constant and the concentration C2 is maintained very low, if not equal to 0. But Equation (6.11) is transformed in such a way by the authors [6] that the amount transferred into the matrix 2 (food of semi-infinite medium) becomes at the time t = ϑ: M θ = 0.039 ⋅ b ⋅ C1/ B

(6.12)

where b is the thickness of the polymer barrier B, and C1/B is the concentration at the matrix 1-barrier B interface. After this transformation, the kinetics of the diffusing substance released in the food (medium 2 semi-infinite) is thus expressed as follows by the authors [6]: ⎤ ⎡ ⎛D ⎞ M t = C1 ⋅ ⎢0.039 ⋅ b + 2 ⎜ P ⎟ (t0.5 − θ0.5)⎥ ⎝ π ⎠ ⎦ ⎣

(6.13)

in which Dp is not specified, but which could be assumed to be that of the barrier B. The comments about this method are given, as much as possible, in Section 6.2.3. 231

Assessing Food Safety of Polymer Packages

6.2.3 Conclusions About the Ideas Presented [6] There is more than one idea in the paper, although all of them are not followed by a clear development and application in terms of profiles of concentration or of kinetics: i)

The first idea is concerned with the possible values of the diffusivity in the three media, with the following: D1>>DB