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Editors: Leon P. Berton

Book Description:

Chemical engineering deals with the application of physical science (in

particular chemistry and physics) and mathematics to the process of

converting raw materials or chemicals into more useful or valuable forms.

As well as producing useful materials, chemical engineering is also

concerned with pioneering valuable new materials and techniques; an

important form of research and development with direct applications in

pharmaceuticals, semiconductors, artificial kidneys, oil refineries, solar

panels, clean water, and biocompatible polymers! This book presents new

important research in this explosive field.

Table of Contents:

Chapter Availability

Individual chapters are available for $25 each by sending an email to novascience@earthlink.net. Nova

will provide the chapter for your easy downloading or send it as an email attachment if you prefer.

Preface(pp. vii-xii)

Chapter 1 - Studies on Combustion Characteristics in Micro-Combustor and its Applications; (pp. 1-7)

(Jinsong Hua, Meng Wu and Xuechuan Shan)

Chapter 2 - Evaluation of Mechanical Energy Applied to Powders in Dry Processes and its Application

for Design and Preparation of Functional Particulate Materials; (pp. 71)

(Tomohiro Iwasaki)

Chapter 3 - Fixed Bed Reactors with Gradient Catalyst; (pp. 127-201)

(Valeri M. Khanaev, Elena S. Borisova, and Alexander S. Noskov)

Chapter 4 - Drying Technologies for Chemical and Mineral Processing Applications; (pp. 203-2)

(E. Helland, and T. Muri)

Chapter 5 - Supercritical Carbon-Dioxide Extraction of Essential Oils and Mathematical Modelling on

the Micro-Scale; (pp. 221-249)

(Irena T. Zizovic, Marko D. Stamenic, Aleksandar M. Orlovic, and Dejan U. Skala)

Chapter 6 - Design of Safe Packaging Materials Under Uncertainty; (pp. 251-292)

(Olivier Vitrac and Murielle Hayert)

Chapter 7 - Modelling the Formation And Flow Of Multiphase Droplets In Microfluidics Devices; (pp.

293-3)

(Guoping Lian, Fabien Jousse, Ruth Janes and Mario De Menech)

Chapter 8 - Burning Coal Particle-A Chemical Reactor; (pp. 319-338)

(Vasilije Manovic and Mirko Komatina)

Chapter 9 - Biofuels and Life Cycle Analysis; (pp. 339-353)

(Grigorios Kyriakopoulos)

Chapter 10 - Mass Transfer and Dispersion around a Soluble Solid Particle Immersed in a Packed Bed

of Inert Particles; (pp. 355-381)

(J.M.P.Q. Delgado and M.A. Alves)

Index (pp. 383-395)

Binding: Hardcover

Pub. Date: 2007

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Page 1 sur 2Chemical Engineering Research Trends

03/10/2007https://www.novapublishers.com/catalog/product_info.php?products_id=5174

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1/27

Design of safe packaging materials under uncertainty Olivier Vitrac(1) and Murielle Hayert(2) (1) INRA – UMR 614, Moulin de la Housse, BP 1039, F-51687 Reims cedex 2, FRANCE E-mail: olivier.vitrac@reims.inra.fr (2) ENITIAA – UMR CNRS 6144, La Géraudière, F-44322 Nantes cedex 3, FRANCE E-mail: hayert@enitiaa-nantes.fr

1 Abstract This research work aims at developing simulation methodologies, which help the design of packaging materials by accounting for the possible contamination of the food product in contact by packaging substances. The main originality in the presented approaches consists in considering various possible sources of uncertainties related to i) insufficient scientific knowledge in transport properties (diffusion and partition coefficients, interfacial mass transfer coefficient), in their activation or in product reactions, ii) the confidentiality of manufacturing processes (e.g. composition) and iii) from the final use (final geometry, storage time and temperature). We present a practical treatment for the optimization under uncertainties (type of molecules, initial concentration). To this extent, a progressive modeling based on a range of point estimates (simplified rules) and on probabilistic modeling (sophisticated rules) is developed. The results illustrate on typical cases (either monolayer or multilayer materials, known and poorly known contaminants) the benefits of an iterative process of the quantification of uncertainty effects. Simplified rules based on conservative assumptions generate the results required for a rapid screening of critical points of the contamination of food products. Probabilistic modeling based on continuous distributions is adapted to the dimensionless physical problem and used to enable the assessor to give the decision-maker (client or provider) an increasingly clear picture of the likelihood of acceptable thresholds (e.g. regulatory specific migration limit, toxicological concern) being exceeded.

2 Introduction During the twentieth century, the rise of plastic materials on the market was promoted by the development of specific additives: antioxidants, plasticizers, dyes, etc. and other substances with technological interest: catalysts, chock agent, antistatic agents, etc. Currently, more than 70% of processed food is in contact with plastic materials in northern countries (Han, 2005). For food applications, the substances present in the packaging material are both the intentionally added substances during the different formulation stages (prior polymerization, processing, assembling, printing …) and the possible residues: polymerization residues, degradation products. All these substances are not covalently bounded to the polymer matrix and can be desorbed into food. As acknowledged during the World Summit on sustainable development in Johannesburg in 2002, consumer and environment protection have drawn more and more the attention of public and authorities. In EU, the recent directive proposal named “Registration, Evaluation Authorization and Restriction of Chemicals” (REACH) enforces that the introduction of a new

2/27

chemical substance must follow a risk management decision. More restrictive rules are applied for materials intended to be in contact with food, including an obligation of traceability of processing and formulation of complex end products such as packaging materials (EC, 2004) and risk assessment related to the desorption in food of impurities in starting substances, decomposition and reaction products (proposal of “plastics” superdirective). To comply with such strong requirements, both the European Health and Consumer Protection Directorate General (EC, 2002) and the American Food and Drug Administration encourage the use of mathematical modeling based on the physical description of mass transfer between the food product and its packaging. This chapter describes new methodologies of prediction of the contamination of food products by packaging substances. The objective is to extend the range of applications for which deterministic modeling can be applied and used robustly to demonstrate the compliance of a packaging or of the use of a given substance, to audit a provider or to demonstrate that its product is safer than the one of its competitor. The current limitations refer i) to our partial scientific knowledge about transport properties (e.g. effect of chemical structure, morphology of the polymer), which control the desorption of packaging substances, ii) to the non-disclosure agreements in the formulation and process of materials and finally iii) to other sources of uncertainties including: reaction products generated during processing, fluctuations in storage temperature, food variability, misuse of the packaging material (e.g. reuse, inadequate microwave heating, contact with organic solvents…). The proposed methodologies illustrate how to take into account such sources of uncertainty and to quantify their effects on the prediction of the contamination of food products. Until more experience is gained, it would be premature to give here prescriptive guidance. We identify here the approaches that appear promising for the needs of the chemical, packaging and food industry. The next section reviews the transport models used to predict the contamination from monolayer and multilayer materials. Dimensionless formulations are detailed to identify, which situations are self-similar and generate similar effects to different sources of uncertainty. Section 4 introduced the concept of tiered approaches to refine progressively the characterization of uncertainty and risks. These screening approaches are based on the description of desorption kinetics via a serial association of mass transport resistances, whose complexity is iteratively increased according to the available knowledge. It is shown that this approach is particularly efficient for complex multilayer structures, where the large number of unknown inputs prevents a direct simulation of the complete problem. Section 5 details more sophistically approaches, which were recently proposed by us, to assess probabilistically the contamination of food products. This high-level approach leads to solve simultaneously the transport equations for a large number of combined conditions. An efficient implementation is proposed here and detailed for the assessment of the combined effects of uncertainties on transport properties (diffusion coefficient, mass transport coefficient at the food-packaging interface, partition coefficient). This approach requires an additional modeling step to quantify the relative likelihood or frequencies of different values within a credible range of each input parameter. This contribution is also discussed. The concluding section resituates the uncertainty and risk assessments in its practical industrial context. Finally, the current research needs are identified.

3/27

3 Description of the contamination of food by packaging substances

3.1 Molecular description Thermoplastics are abusively called “plastic materials” and consist in intermingled long polymer chains (typically with molecular mass above 50 kDa). The interminglement and its low relaxation process at room temperature are responsible for the high cohesion and the density of material. The polymer chains can present locally a crystalline morphology (semi-crystalline polymers) or a privileged orientation (oriented polymers). The amorphous phase present two mechanical behaviors, successively glassy and rubbery, on both sides of a characteristic temperature so-called the glass temperature. The reader find further details in reference books on polymers (Painter and Coleman, 1997 ; Sperling, 2006). Conventionally additives are assumed to be located in the amorphous regions of the polymer. Additives such as antioxidants or UV absorbers and residues are present at concentrations below 0.5 % w/w (weight of additive per weight of material). Since the plasticizers aim at increasing the mobility of polymers, they are typically used at concentrations between few and few tens w/w percents. While the amount of desorbed plasticizers does not modify the mobility of the polymer and its microstructure, it is thought that the physical description of mass transport in unplasticized polymers is also valid in plasticized polymers in their normal conditions of use. It is highlighted that the proposed description is not valid for substances showing blooming effects: inhomogeneous dispersion or preferential location at the surface of the plastic material (e.g. lubricant, antistatic agent). At molecular scale and at low concentration, the transport of substances such as additives or residues is similar to the dispersion of non-interacting substances or ghost substances due to thermal agitation. Since the considered substances are larger than voids present in the polymer and present shapes that differ significantly from polymer chains, the translational motions of these substances must be envisioned as a consequence of collective motions involving several polymer segments (Kovarski, 1997; Vitrac and Hayert, 2006). A classical view is a displacement according to a series of activated hops as described in the Eyring transient state theory (Glasstone et al., 1941). At the scale of the diffusant, such translations are performed via successive reorientations which are assessable by dielectric relaxation (Kaji et al., 2003), nuclear magnetic resonance (Heuer, 1996), echo spin relaxation (Kovarski, 1997) or Rayleigh spectroscopy (Fytas et al., 1990). Since on the long term, the trajectory of each substance consists in a collection of uncorrelated displacements, the local mass flux obeys to the macroscopic law of diffusion and is controlled by a macroscopic diffusion coefficient: D with SI units in m2⋅s-1. By contrast to diffusion coefficients of these substances in liquids, the corresponding diffusion coefficients are broadly distributed (up to 10 decades) and vary drastically when the mobility of the polymer is changed or when the size or shape of the diffusing substance is modified. The high molecular mass (noted M) dependence of the diffusion coefficient lead to scaling laws D M α−∝ with 2 6α≤ ≤ for 50 1200M≤ ≤ g⋅mol-1 confirms the predominance of trapping and geometric constraints. The corresponding activation energies are also broadly distributed typically between 40 and 300 kJ⋅mol-1 and confirms the drastic effect of temperature (EC-DG SANCO-D3, 2002). From these considerations, if the temperature is not accurately controlled or if the diffusion coefficient is

4/27

extrapolated from values obtained for similar but different molecules (e.g. from a different antioxidant), a significant uncertainty in the likely D value is expected. Besides the variations in polymer density and in the morphology of the crystalline phase are other possible sources of variation in D (Vitrac et al., 2006).

3.2 Contamination from monolayer materials

3.2.1 Thermodynamic equilibrium In the polymer, molecular diffusion acts macroscopically to smooth concentration differences between internal regions and the external surface, which looses migrant quantities due to a pervious contact with food. At equilibrium, the residual concentration is assumed homogeneous in the packaging material, noted P, and in the food product, noted F. Microscopically, the random walk of molecules is reversible and the molecules can migrate back from F into P. However, since the volume of F is conventionally higher than the volume of the P, the probability of crossing back the F/P interface, when the substance diffuses in the food product, is low. As a result, at equilibrium, the total amount of substance is generally higher in the food product than in the packaging material. This effect can be modulated by the chemical affinity of the substance for both compartments. If both the desorption isotherm of the substance in P and the sorption isotherm of the substance in F are reversible (no hysteresis) and obeys to the Henry law, the ratio of the concentrations between both compartments at equilibrium is a constant independent on the initial concentration in P. This ratio is called partition coefficient, noted K, and is conventionally

defined by the ratio of concentrations noted { }= ,

j eq j P F

C and expressed in mass of substance by

mass of phase j, with j=P or F :

= = ⋅ ⋅F eq P P PP F F Feq

C VK

C V

γ ργ ργ ργ ργ ργ ργ ργ ρ

(1)

where { } = ,j j PFγγγγ , { } ,j j P Fρ = and { } ,j j P FV = are respectively the activity coefficients in j, the density of j and the molar volume of j. By assuming that no reaction and no mass loss between P and F occur, the mass balance between the initial and equilibrium states is given by: ρ ρ ρ =⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅0P P P F F F P P Peq eq tC l C l C l (2)

where lP is the thickness of the packaging material and lF is the characteristic dimension of the food product defined by the ratio between the volume of food, VF, and the surface area of food, A, in contact with the packaging material:

= FFV

lA

(3)

By noting the dilution factor ρρ

⋅=

⋅P P

F F

lL

l, { }

= ,j eq j P F

C are expressed from Equations (1) and (2)

in function of the initial concentration in P, =0P tC , as:

5/27

−

=

=

= + ⋅ = ⋅ +

1

0

0

1 1F Peq t

P Peq t

C CK L

LC C

L K

(4)

3.2.2 Transport equations

3.2.2.1 Phenomenological description The previous physical hypotheses are summarized in Figure 1 by assuming that F is a liquid food and does not interact with the polymer: the constituents of F do not migrate into P (no plasticization or swelling induced by F). Local thermodynamical equilibrium is assumed at the

interface between P and F so that = 32

CK

C at each time.

Figure 1a represents more particularly the interfacial mass flux density (noted j and with units in kg⋅m-2⋅s-1), so-called migration rate, as a result of a mass transport through a serial association of different types of transport resistances with units of a reciprocal velocity s⋅m-1. The corresponding profile and location, where transport resistances are defined, are also depicted in Fig. 1b. RD is related to the resistance to diffusion in P whereas RH is related to several mechanisms (e.g. diffusion, convection), which limit the dispersion of the substance in the food compartment. They are defined by:

−=

−=

j

CCR

j

CCR

H

D

43

21

(5)

where 1 and 4 are arbitrary positions that verify the linear approximation of the concentration profile on both sides of the interface. For complex food or contact between P and F (e.g. imperfect contact), it is highlighted that RH can be an equivalent resistance, which is only used to make the simulated mass flux at the interface matching the real one. By contrast, the thermodynamic resistance, RK, is only phenomenological since positions 2 and 3 are merged. Its effect is however significant, since decreasing the concentration in 2 (without changing the concentration in 3) will increase the slope of the concentration profile and consequently j. From the phenomenological point of view, RK is controlled by K, which acts as refractive index. By analogy with Equation (5), RK may be defined by:

2 3 2(1 )K

C C K CR

j j

− − ⋅= = (6)

It is worth to notice that this resistance is absent (zero) when K=1 and very significant when K1 (high affinity for F) and contribute to reduce overall mass transfer resistance defined by:

6/27

1 4eq D K H

C CR R R R

j

−= = + + (7)

This description is particularly useful to define conditions, which overestimate the desorption rate in food. Several worst cases can be identified:

• RD>0, C2=0, RH=0 (kinetic effects control the desorption, K→∞ is assumed) • RD>0, RK=RH=0 (kinetic effects control the desorption, K=1 is assumed) • RD=RH=0, RK>0 (thermodynamic effects control the desorption)

Other scenarios can be derived by comparing the ratio between RD and RH also known as mass Biot number and defined by:

D F P

H P

R h lBi

R D

⋅= = (8)

where DP is the diffusion coefficient in P and hF is a mass transport coefficient on F side with SI units in m⋅s-1.

3.2.2.2 Mathematical formulation By assuming a one-dimensional transport (side effects are negligible), a uniform initial concentration in P, a constant diffusion coefficient, a constant thickness lP and no change in density due to the desorption of packaging substances, the dimensionless mass transport equation in P is:

2

2*

u u

Fo x

∂ ∂=∂ ∂

(9)

where ,

0

P t x

P t

Cu

C=

= , *P

xx

l= ,

2P

t DFo

l

⋅= are respectively the dimensionless concentration, position

and time (so-called Fourier time). If the temperature is not constant with time, a generalized

Fourier time can be used instead: ( )

02

t

P

D d

Fol

τ τ⋅=∫

. This last representation neglects the effects

of a temperature gradient on D but this approximation is realistic for thin materials such as packaging materials and contact times above several minutes. At the F-P interface (i.e. for x*=1), the combination of Equations (6) and (7) provides a simple expression for the dimensionless boundary condition (BC), known as Robin BC:

( )* 1 ** 1

** x xx

uj Bi K u u

x = →∞=

∂= − = ⋅ ⋅ −∂

(10)

7/27

where 0

* P

P P t

lj j

D Cρ=

=⋅ ⋅

is the dimensionless flux, 3* 1

0x

P t

CK u

C==

⋅ = . Practically,

4*

0x

P t

CK u

C→∞=

⋅ = is assumed to stand for the dimensionless concentration in the bulk, as it would

be measured in F, rather than the concentration far from the interface as it is depicted in Figure 1. Both descriptions are almost equivalent when the concentration is assumed to be homogeneous far from the F-P interface. This situation occurs when the transport property in F is much greater than the transport property in P (case of most of food products) or when a mixing process (e.g. convection) occurs on F side. From these considerations,

*xu

→∞ is derived from the mass balance between P and F:

( ) ( )0 0* * *0 00

1 1 1*

t FoFo Fo

x x xP F Ft

Lu u j d u j d

K C l Kτ τ τ τ

ρ= =

→∞ →∞ →∞=

= + ⋅ ⋅ ⋅ ⋅ = + ⋅⋅ ∫ ∫

(11)

Equation (10) combined with Equation (11) yields the practical form of the BC, written as an integro-differential operator:

( ) ( )0* 1 ** 1 0

* **

FoFo

x xx

uj Bi K u u Bi L j d

xτ τ=

= →∞=

∂= − = ⋅ ⋅ − − ⋅ ⋅ ⋅∂ ∫

(12)

Two extreme cases are reduced from Equation (12) by assuming i) RH=0 (i.e. no external resistance), ii) RK=RH→0 (i.e. Req≈RD). Case i) is inferred by differentiating with time Equation (12) for :

* 1

* 1

**

x

x

u L L uj

Fo K K x=

=

∂ ∂= = −∂ ∂

(13)

By analogy with wave propagation equations, Equation (13) is known as a reflecting boundary condition, where the amount of matter that leaves the interface F-P modifies in return (i.e. after accumulation or “reflection”) the mass transfer resistance at the latter. K/L is the equivalent dimensionless “reflecting distance”, where the quantity K is similar to a dimensionless “absorbing” coefficient (or refractive index). Case ii) corresponds to the condition /K L → ∞ (infinite volume or capacity) in BC (12), that is

* 1 0xu

Fo=

∂→

∂ or the equivalent Dirichlet’s BC:

( ) ( )0* 1 * 1

Fo Fo

x xu u

=

= == (14)

For the left side boundary, x*=0, an impervious is applied:

* 0

0* x

u

x =

∂ =∂

(15)

3.3 Contamination from multilayer materials

8/27

Compact formula are obtained by noting with an index j=0 the food product and j=1..n the n plastic layers of the packaging material. The layer 1 is the layer in contact with food as described in Figure 2. The thickness of each layer is noted lj. By convention, l0 is related to the half thickness of the food product since the product is assumed to be symmetrically in contact with the same material. If it is not the case (e.g. film sheet on a single side), l0 must be replaced by the whole thickness.

3.3.1 Thermodynamic equilibrium

3.3.1.1 Sorption and desorption properties The equilibrium of sorption and desorption in each layer is assumed to be reversible and to obey to Henry law. As a result, an equivalent vapor pressure of the substance in equilibrium with the amount of the substance dispersed locally in each layer j is defined as follow: ( )

� �( )

ρ −⋅ ⋅

= ⋅���

1Pa kg kgHj j j

j j j

k

M

p x k C x (16)

where Hjk and ρ j are respectively the Henry coefficient of the substance in the layer j and the

density of the layer j. M is the molecular mass of the considered substance. It is worth to notice that Hjk (with units in J⋅mol

-1) is also the reciprocal of the solubility of the substance in the layer j.

3.3.1.2 Condition of equilibrium between j1 and j2 Two layers, noted j1 and j2, at a same temperature without external mechanical constraints (i.e. at the same pressure) are at thermodynamical equilibrium when their activity and consequently their partial pressures in desorbable substances are equal:

= ⇔ = =1

2

1 2 1 2

12

/

j jeqj j j jeq eq

jjeq

C kp p K

kC (17)

where 1 2/j j

K is the partition coefficient of the considered substance between j1 and j2. 1j eq

C and

2j eqC are the concentrations at equilibrium respectively in layers j1 and j2.

3.3.1.3 Mass balance considerations By assuming that the considered substance is initially only present in the packaging material and not in the food product, the mass balance between the food product and the packaging material is written in absence of reactions and mass losses as:

�

( )== =

⋅

⋅ ⋅ + ⋅ = ⋅ ⋅

∑ ∑0 0

0 0 0 01 1

j eq

n n

j j j j j jeqeq tj j

k k C

C l C l C lρ ρ ρ (18)

By noticing that the condition of equilibrium enforces{ } 01..

j eqeq j np p

== , the concentration in

food at equilibrium is finally given by:

9/27

ρρ

ρρ

==

=

⋅ ⋅

=

+ ⋅ ⋅

∑

∑

00 01

0

0

0 01

1

nj j

j tj

eq nj j

jj

lC

lC

lk

k l

(19)

The corresponding partial pressure at equilibrium can be also expressed as a function of the initial partial pressure in each layer:

{ }ρρ

ρρ

==

=

=

⋅ ⋅ ⋅

= =

+ ⋅ ⋅

∑

∑

00

0 010

1.. 0

0 01

1

nj j

j tjj

jeq neq j n j j

jj

lkp

k lp p

lk

k l

(20)

Equations (19) and (20) generalize Equation (4) to multilayer materials. It is worth to notice that they do not require that the layers are initially at equilibrium. However, they assume that the initial concentration is uniform in each layer. When it is not the case, Equation (19) must be replaced by a continuous integration over each layer. For general 1D geometries, one gets:

( )ρρ

ρρ

−

−

==

=

⋅ ⋅ ⋅ ⋅ =

⋅ + ⋅ ⋅ ⋅ ⋅

∑ ∫

∑∫ ∫

1

0

1

001

0

0

0 10

1

1

cj

cj

cj

cj

lnx m

j j tj l

eq ll nm m

jjj l

C x dx

C

kx dx x dx

k

(21)

where m=0,1,2 respectively for Cartesian, cylindrical and spherical coordinates. =

= ∑0

nc

jjj

l l is the

cumulated thickness (starting from the product). For convenience and according to Equation (17), choosing k0=1 leads to identify kj to the partition coefficient between the food product and the packaging layer j.

3.3.2 Transport equations By assuming a local thermodynamical equilibrium at the interfaces between all layers (including the food product), the partial pressure is continuous over all layers. As a result, the partial pressure seems a best choice to implement transport equations in commercial numerical codes. This choice leads to express the local mass flux in the layer j as a consequence of a gradient in partial pressure:

� � �

ρρ α

⋅ ⋅ ⋅ ⋅

∂ ⋅ ∂ ∂= − ⋅ ⋅ = − ⋅ = − ⋅∂ ∂ ∂

-2 -1 2 -1 -3kg m s m s kg m

j j jj j j j

j

C D p pJ D

x k x x (22)

If Dj can be considered uniform in the layer j>1, ρ

α δ⋅

= = ⋅j jj j jj

DD

k is a new equivalent

transport property. The equivalent local mass balance is accordingly written for any layer j as:

10/27

( )δ α∂ ∂ ∂ ∂ ⋅ = − ⋅ = ⋅ ⋅ ∂ ∂ ∂ ∂ 1 1m m

j j jm m

p px J x

t x x xx x for j=1..n (23)

Since significant discrepancies may be expected between { }α =1..j j n values and between { }δ =1..j j n values, a dimensionless formulation seems preferable to preserve the numerical stability of the discretization scheme. By analogy with permeation, the reference length scale, lref, is associated to the thickness of the layer with the maximum mass transfer resistance, which is with the lowest { }α =1..j j j nl value. The dimensionless time or Fourier number, Fo, is expressed as:

α ⋅

=2

refj

ref

tFo

l (24)

The dimensionless mass balance equation is finally:

α

δα

∂ ∂ ∂ ⋅ = ⋅ ⋅ ∂ ∂ ∂

* *1*

* **ref

jmj m

j

p px

Fo x xx for j=1..n (25)

with =*ref

xx

land =

0

*eq

pp

p.

At the interface between j=1 and j=0, which is at the position x=0, the boundary condition is written:

ρα τ= = ==

∂ = − ⋅ = ⋅ ⋅ + ⋅ ⋅ ∂

∫0110 0 00 00 0

1t

x x xx

pJ h p J d

x k l (26)

The sign + in front of the cumulated term, τ= ⋅∫ 00

t

xJ d , depicted in Equation (12) comes from the

projection of the flux (vector) on axis x. Indeed, 0x

J=

is counted negatively.

By introducing the dimensionless flux αα α= =

=

∂= − ⋅ = ⋅ ⋅∂

** 1

0 00* 0

1

*

ref

x xref refeqx

lpJ J

x p, one gets the

dimensionless boundary condition:

τ== =

= ⋅ + ⋅ ⋅

∫*

* *00 0

0 0

Fox

x x

p

J Bi L J dk

(27)

where α⋅

= refref

hBi

l is the mass Biot number and =

0

reflL

lis associated to a dilution factor.

An impervious boundary condition is assumed at =* 1x :

=

∂ =∂

*

1

0*x

p

x (28)

11/27

4 Using intervals for worst-case analyzes

4.1 Problem formulation In the equations detailed in section 3 for monolayer materials, the objective functions are either

0, , , , P Fo

F Fo Bi K L CC

=or

max

0P FoC

=such that max

0, , , , P Fo

F Fo Bi K L CC SML

== where SML is a specific migration

limit defined by the regulation or by other considerations. As an example, a packaging industry with appropriate knowledge on the formulation of its material will prefer to estimate

0, , , , P Fo

F Fo Bi K L CC

=, based on its foreseen application (type of contact, contact time) and on a

accurate estimate of 0P Fo

C=

. The final objective is to compare the estimate (possibly

overestimated) of 0

, , , , P FoF Fo Bi K L C

C=

, noted 0

, , , ,

ˆP Fo

FFo Bi K L C

C=

, with the SML. By contrast a food

industry, with a poor knowledge on the packaging material, will prefer to get an estimate

(possibly underestimated) of max

0P FoC

=, noted

max

0

ˆP

FoC

=, by assuming that the contamination is equal

to the SML. In this case, the final objective is to obtain evidence from the provider that he uses

the tested substance at a concentration lower than max

0

ˆP

FoC

=.

Whatever the considered objective, estimating 0

, , , , P FoF Fo Bi K L C

C=

or max

0P FoC

=, it is important to deal

with uncertainties related to in put parameters or in the conditions of use of the packaging material in a way that enforces the confidence that the decision makers have in the estimates. Since this opinion must be shared by the different sides of the decision process – provider/producer, producer/final user, producer/control authority – easy-to-use and transparent methods, which minimizes the risk of putting on the market products are contaminated over a specific limit, are encouraged. In absence of reaction or other sources of contamination than the packaging material, the maximum amount of contaminant in food cannot exceed the initial amount in the packaging material. This situation is a priori the worst of the worst scenarios. It is however more difficult to construct other worst-case scenarios with less stringent conditions. As it is presented for multilayer materials, the number of parameters can be pretty huge and the effects can be antagonist. The physicochemical parameters can be arbitrary classified into molecular transport controlled properties (diffusion coefficients, mass transport coefficients) and thermodynamically controlled properties (partition coefficients, Henry-like coefficient or solubility). How to weight them? As an example, it is not trivial to decide whether the scenario RD=RH=0 and RK>0 (thermodynamically controlled contamination) is safer than the scenario RD>0, RH=RK=0 (transport controlled contamination) for monolayer materials. As a result a robust classification of scenarios is required.

12/27

4.2 Principles Since molecular transport and thermodynamics are physically imbricated, a possible solution consists in separating the contribution of the contamination that is time-dependent (dependent on both transport and thermodynamical properties) from the one that depends only on mass balance and thermodynamical considerations. For monolayer materials, the decomposition of the contamination can be written:

=

−

= = ⋅ + ⋅ ��������������

0

F

1

*

, , , , 0 , , ,

time dependent contributiontime independent contribution

=C

1 1P t

eq

F PFo Bi K L C t Fo Bi K LC C v

K L (29)

where =

= = ⋅ − ⋅

∫0 *1

, , , ,* *

, , , ,, , ,0

1P tF Fo Bi K L C

x Fo Bi K LFo Bi K LF eq

Cv L u dx

C, one notes

≤ ≤*0 1v . As a result, time-dependent effects are negligible when *v →0. An interesting feature to note is that the effects of K and L are not similar on F eqC and

*v . The

effects of K and L on *v is depicted in Figure 3 for different values of transport contributions: 0 ≤ Fo ≤ 2 and Bi. It is shown that the effect on *v of values of K higher than 1 is negligible whatever the value of L. The effect of values of K lower than 1 is significant only for L values, which are close to the upper limit for food packaging applications (>1/20). In most of cases, the mass of food is close or higher than 50 times the mass of its the packaging (L

13/27

estimate FC or max

0P FoC

=. In other words, the question is “which additional information do I need

to switch from a rough estimate towards a more realistic one?”.

4.3 Monolayer materials

4.3.1 Multiattribute value trees The four dimensionless parameters: Fo, Bi, K, L, correspond to 6 independent quantities: D, t, h, lP, lF, K. If the interval on each parameter p is defined by their bounds p and pɶ , the number of different scenarios possibly dependent is 26=64. By assuming that the final geometry of the

packaging material and of the food in contact is well known ( P Pl l≈ ɶ and F Fl l≈ ɶ ) and by noting that the assumption →* 1v does not depend on values of Bi and Fo, the number of independent scenarios finally reduced to 24-23+2=10. All the possible combinations are depicted in Figure 4 and the main typical group of scenarios, noted A, B, C and D, are detailed in Table 1. It is worth to notice that it is not possible to determine whether the scenarios B generate a priori

higher overestimate than the scenarios C. By noting, �B

FC and �C

FC the estimate of the

contamination in F according to scenarios B and C respectively, one gets:

�

�=

−

=

⋅ ⋅ = = + ⋅ ⋅ +

*

*01

0

11 1

C

P tF

B

FP t

C L vC Lv

KC CK L

(30)

The inequality � �C B

F FC C> is obtained only when > +* Kv

L K. Thus, scenario B overestimates

more often FC and surely when K>L (e.g. polyolefins in contact with fatty food). By contrast, scenario C seems more appropriated for glassy polymers in contact with aqueous food (e.g. mineral water). The only condition is that the diffusion coefficient must be enough overestimated.

4.3.2 Application to the estimation of max0P Fo

C=

Too large overestimations of FC (or equivalently too large underestimations of max

0P FoC

=) are only

desirable for rough estimations. In this sense scenarios A and B are particularly relevant since they are defined by analytical solutions. For the group of scenarios A, B and C and D, the safety

margins, defined by �

, ,

k

F

Fk A B C

C

C=

, are: ⋅+ *

1K

L K v,

*

1

v,

�⋅

+

*

*

K v

L K v, �*

*

v

v.

Accordingly to simulated results depicted in Figure 3, Figure 5 presents the estimate of max

0P FoC

=

versus *1 v− according to scenarios A, B, C, D. It is highlighted that *1 v− is similar to an advancement degree of reaction so that the effect of time (or equivalently a more accurate

estimate of *v ) on max

0P FoC

= is read from the left to the right (from 0 to 1). Scenarios A and B

14/27

appear as horizontal lines since their results are independent of kinetic effects. Scenarios C and D lead to increasing curves with *1 v− . Curves predicted by scenario C are always located below the curve predicted by scenarios D. In all subplots in the upper position of Figure 5 (i.e. for similar L and Bi values), scenarios C appear therefore as a single curve very close to the one predicted by scenarios D when K=5. Reciprocally, in all subplots in the bottom position of Figure 5 (i.e. for similar K and L values), scenarios D appear as a single curve above all the others and very close to the curve predicted by scenario C for Bi=104. Abacuses depicted in Figure 5 are particularly valuable to make an audit of a provider with the objective to get a confirmation that a given substance is used in conditions, which ensures a contamination of food lower than a SML. In the current EU regulation, about 400 of the 2750 substances listed in the synoptic document (EC, 2005) are subjected to a regulated SML. For audit purposes or formulation optimization, this value can be replaced by a more severe value (lower value). In the discussion process with the provider, the objective is not to guess the true

0P FoC

= used in

the formulation process (which is confidential) but to guess a value, max

0P FoC

=, confirmed by the

provider, so that max max

0 0 0P P PFo Fo FoC C C

= = =≤ < . With a knowledge poor approach as scenario A, the

risk is to get the inequality max max

0 00

P P PFo FoFo

C C C= =

=< < of little use. The initiated iterative

discussion process with the provider consists in getting significant additional information

(molecular mass, polarity of the molecule…) to fulfill: max

0 0P PFo FoC C

= => . To reach this goal, a

solution consists in precalculating an abacus as depicted in Figure 5 for a large range of conditions before any discussion with the provider. A software has been developed (Decision Tool for Compliance Testing) to generate first a database based on the geometry of the food product and its packaging and on its expected shelf life (Figure 6). Both the provider and the company that requires an audit can share the database on line. The final calculations can thus be performed in real time as additional information is provided (e.g. during a phone talk with the provider).

If the provider accepts to propose a priori a max

0P FoC

=value, an alternative objective consist in

getting the confirmation that the ̂K and/or �*v values derived from max

0P FoC

= and abacuses are

possible overestimates of likely K and/or *v values.

4.4 Multilayer materials

4.4.1 Main physical approximations Enumerating and classifying all combination of intervals on input parameters for multilayer materials is a much difficult task than for monolayer materials. The objective of this section is to show that a decomposition between known and poorly known is almost always possible and can

15/27

be used to calculate estimates of FC (noted C0 for multilayers) or max

0P FoC

=. The proposed physical

approximations are an extension of the ones for monolayer materials. Thus, as for monolayer materials (see Equation (29)), the contamination from multilayer materials can be decomposed between a time independent and a time dependent contribution:

{ } { } { } { } { }

� { } { } { } { } { }

= = = = = =

= = = = = =

=

⋅�����������������

0.. 0.. 0.. 0.. 0 0..

0.. 0.. 0.. 0.. 0 0..

, , , , , ,

*0 , , , , , ,

time independent contributiontime dependent contribution

j j j j jj n j n j n j n t j n

j j j j jj n j n j n j n t j n

F t h D k l C

eq t h D k l C

C

C v

ρρρρ

ρρρρ (31)

with 0 eqC defined by Equation (19) and

*0 1v≤ ≤ .

Two approximations are particularly relevant: • * 1v → (time independent) • * 1v ≤ (time dependent)

4.4.2 Scheme for the overestimation of the time independent contribution:

0 eqC

If the problem is not time dependent, the contamination does not depend on the arrangement/order of layers and Equation 19 can be partitioned between layers, which present a higher chemical affinity for the migrating substance than for food (i.e. { } 00i i nk k< ≤ ≤ ), and layers, which present a lower chemical affinity (i.e. { } 00i i nk k< ≤ > ). By noting both partitions H and L, a worst-case scenario but nevertheless not too much pessimistic is obtained by choosing:

{ } 0i i Hk k∈ ≈ and { }i i Lk ∈ → ∞ . As a result, an overestimate of Equation (19) for multilayer materials is obtained:

ρ ρρ ρ

ρρ ρρρ ρ

= == =

∈∈ ∈

⋅ ⋅ ⋅ ⋅

= ≤

+ ⋅+ ⋅ ⋅ + ⋅ ⋅

∑ ∑

∑∑ ∑���������

0

0 00 0 0 01 1

0

0 0

0 00 0 0 0

11

eq

n nj j j j

j jt tj j

eqj jj j j j

j j j Hj L j H

C

l lC C

l lC

ll lk k

lk l k l

(32)

If the partition H is empty, the scenario defined by Inequality (32) corresponds to scenario A defined for monolayer materials. Otherwise, Inequality (32) is intermediate between previous scenarios A and B. The estimation can be refined by introducing sub-partitions of H or L with more reliable estimates of { }0 .i i nk < ≤ It is worth to notice that the overestimation defined by Equation (32) by assuming * 1v → seems particularly useful to estimate the maximum contribution to the contamination of food by specialty resins (adhesives, inks…) whose typical thickness is very low.

16/27

4.4.3 Schemes for the overestimation of the time dependent contribution: �*v

4.4.3.1 Effect of the arrangement of layers

The estimation *v depends strongly on the arrangement of the layers. By comparison to monolayer materials with similar initial content and whole thickness, two mechanisms may delay the desorption kinetics:

• a barrier effect (an initially virgin layer is intercalated between the layer formulated with the substance of interest and the food product);

• a reservoir effect (the layer formulated with the substance of interest is located between the food product and an initially virgin layer).

These effects are illustrated on Figure 7 on bilayer structures. These results are subsequently used to define general rules to simulate mass transport in multilayer materials in presence of uncertainty on Henry-like coefficients: ki. In Figure 7, all layers have the same thickness and diffusion coefficient. The dilution factor between the whole packaging material and the food product is 1/25, k0 is set to 1, and Bi → ∞ . The different tested conditions are detailed in Table 2. It is worth to notice that all tested configurations present a same initial amount of substance but do not yield exactly the same concentration in F at equilibrium. For each configuration, *v is calculated by the ratio 0F eqC C .

The barrier effect introduces a typical delay at the beginning of the kinetic: that is an initial period with zero flux at the interface F-P. The delay is in itself not affected by the chemical affinity of the substance for the crossed layer (effect of barrier of diffusion). Only the mass flux at the interface between layers 1 and 2 is affected by the chemical affinity. Thus, increasing k1 reduces the desorption rate (effect of barrier of solubility). By contrast, the reservoir effect appears to be significant only after a certain time. This time is required to “fill” the reservoir before to “empty” it. Since the reservoir is located farther from the exchange surface, the reservoir effect acts mainly by increasing the dimension characteristic of the mass transfer. Its effect varies significantly according to the thickness and the affinity of the substance for the reservoir layer.

4.4.3.2 Proposed approximations to overestimate *v

According to simulated results of Figure 7 and as first approximations, it is stated that the contamination from the bilayer material coded [0 1]-[>1 1] (diffusion and solubility barrier) is overestimated by assuming either:

- a monolayer material equivalent to [0.5 0.5]-[1 1]; - a bilayer material equivalent to [0 1]-[1 1].

The first approximation cannot however be generalized, since it works well only when both layers have similar D values. From this point of view the second approximation, which removes the “solubility barrier” seems more satisfactory. The situation [0 1]-[

17/27

not exist. Thus, the situation [0 1]-[

is tested for a trilayer material presented in Table 3. The equivalent

configurations to overestimate the contribution of each layer to the overall contamination are also detailed. Several approximations are illustrated via the proposed example: overestimation of a

18/27

barrier effect due to layers 1 and 2 (diffusion and solubility), overestimation of a reservoir effect due to layers 2 and 3. The highest value of kj (with k0=1) was retained in all approximations to account for the partitioning between P and F. Besides, it emphasized that the initial configuration is out of the equilibrium (i.e. the partial pressure are not equal at the interfaces between layers). This is generally the case immediately after the assembling of different materials. If the material is expected closer to the equilibrium prior to the contact with food – due to a process at high temperature or to a long storage of the material – mass transport must be simulated successively without contact (impervious BC) and with a contact with food. The results are plotted in Figure 9 by assuming that the contaminations from each layer are independent and by cumulating all overestimations. The individual contribution of each layer to

the overall *v was calculated by the ratio: F FFo eqC C . The overall *v was computed by the

average of all *v calculated individually for each layer. The weights were chosen equal but the final result can be weighted according to the initial amount within each layer. This last result is finally compared with the result obtained by simulating the mass transport in the complete trilayer structure (Figure 9d, 9h and 9l). The overestimated profiles exhibit always a higher slope at the F-P interface than the reference profiles so that the migration rates are always overestimated. It is worth to notice that the proposed approximations do not exaggerate overestimations and remained still consistent with both the true profiles and the true kinetics.

5 Probabilistic estimation of the contamination Worst case analyzes assume generally a combination of several pessimistic of scenarios (see Figure 4) without weighting each overestimation according to their frequency of occurring. As a result, the calculations are performed with a significant safety margin without always controlling reliably the value of the safety margin. The objective of probabilistic modeling is to make possible the calculation of the contamination with a controlled risk that the real contamination exceeds the estimated one. As a result, an overestimate of the real contamination means an estimate, which may be exceeded with a risk lower than 50%. On the opposite, an underestimate will present a risk higher than 50%. In this approach, the effects of both sources of uncertainty and variability and their interactions can be inferred. An efficient adaptation of pseudo-Monte Carlo techniques has been proposed by Vitrac and Hayert (2005) for monolayer materials. A web-based application, so-called MIGRARISK, has been developed and is available on our server (Vitrac, 2006). It has been applied to assess the contamination of 12 packaged food products on the market by ubiquitous contaminants from packaging materials (Vitrac et al., 2006) and to assess the exposure of consumer to styrene from yogurt pots according to household practices (Vitrac and Leblanc, 2006). This section presents first the general principles, which are used to predict the distribution of the contamination of food products by substances originating from monolayer materials according to the distributions of all input parameters detailed in section 3.2 : initial concentration in P, contact time, transport properties. The distribution of each parameter represents indifferently the effects

19/27

of variability or an uncertainty. In the logic of packaging design, the methodology is afterwards illustrated to assess the combined effects on uncertainty on transport properties: DP, hF, K. This situation is particularly interesting since these properties are poorly tabulated in the literature and may be subjected to significant variations according to the conditions of processing of the packaging material (e.g. effect of polymer orientation, crystallization for DP), the conditions of storage (temperature), food composition and structure (for hF and K). It emphasized that this situation is not discussed in the references cited above where, only the uncertainty in DP is considered via probabilistic modeling. The uncertainties on hF and K are considered only via an interval approach.

5.1 Principles for the probabilistic modeling of the contamination from monolayer materials

5.1.1 Decomposition of main physical quantities Each physical quantity, iq , is defined as the product of a scale parameter iq (with a physical unit), and of a random dimensionless contribution, ( )

iqsiq * with a unitary expectation. Random

contributions of independent variables t*, lP*, DP*,*

0P tC

= are distributed respectively to

continuous laws that are defined by experts and controlled by a shape parameter iq

s (Table 4),

noted respectively st, sl, sD, sC0. The scale dimensionless-parameters of Fourier and Biot mass

numbers are accordingly defined by 2P PFo D t l= ⋅ and F P PBi h l D= ⋅ respectively. The

distribution of * * * 2* P PFo Fo Fo D t l= = ⋅ depends non-trivially on sD, st and sl. For complex situations, it was proposed to approximate the distribution of the square root of Fo by a Gamma distribution as an upper limit of its true distribution, which depends only on 2 degree of freedoms: ,a bΓ Γ . From mass balance considerations, it was demonstrated that the proposed

approximations *v is Beta distributed (Vitrac and Hayert, 2005) with parameters: ,a bβ β . When

it is required, the concentration in F can be approximated by a Gamma distribution, which depends only on two parameters: ,

F FC Ca b .

5.1.2 Calculation procedure of: FF CC

bababa ,,,,, ββΓΓ

The values of the statistical parameters, aΓ,bΓ, aβ,bβ, acF,bcF, are tabulated respectively to the variables Fo K , L Bi , st, sl, sD and 0Cs and are calculated within the MIGRARISK software.

Vitrac and Hayert (2005) described the method of calculation of the parameters and principal of the abacuses necessary for the assessment of the migration risk The principles of the method of calculation are taken again briefly hereafter. The elementary steps include i) the calculation of the distribution of the product of 2 independent random variables and ii) the calculation of the distribution of continuous transformation of a random variable.

20/27

The distributions of *Fo and CF* are inferred iteratively as the product of independent quantities YX ⋅ . The distribution of ( )YX ⋅10log is calculated as the product of convolution of the marginal

distributions of ( )X10log and ( )Y10log . Since *v is a continuous regular function of input parameters and in particular of Fo1/2, the distribution of *v , noted ( ) ( )yvpryfv == ** , is calculated as the continuous transformation of a random variable )(* xvy = . Thus, for 0=== BiLK sss (i.e. K, L and Bi are delta distributed) and since *v is a strictly increasing and differentiable function of Fo1/2, ( )yfv * is derived from the distribution of Fo1/2, noted ( ) ( )xFoprxfFo == 2/12/1 :

( ) ( )[ ] ( )y

yvyvfyf

Fov ∂∂

⋅= −−*1*

1** 2/1 (33)

where *1−v is the transformation so that ( )yvx *1−= .

It is worth to notice that for large Bi values and Fo

21/27

by ( )g Fo . It is highlighted that, due mass balance limitations, the transformation that relates Fo to *v is not affine and, in particular, that an apparent large discrepancy in Fo values (for

significantly large Fo values) does not generate a large dispersion in *v values.

The same approach is applied for different Bi values in Figure 11. It shows that the distribution of *v is dependent on the slope of *v with Fo. The results cannot however be used to assess the combined effects on the uncertainty of Fo and Bi. Indeed, in case of doubt on the true Bi value,

Figure 11 generates two distinct distribution of *v , as an example, for a lower value of Bi (e.g.

10Bi = ) and an upper value of Bi (e.g. 100Bi = ). Since the conditional distributions of

10*

Biv

=and

100*

Biv

= have equal weight, the span between extreme predictions are maximized and

no reliable statistics can be derived. This discrepancy is all the higher than the overlapping regions between conditional distributions are smaller. The next paragraph examines how to account for both sources of uncertainties in an efficiently manner.

5.2 Estimations of coupled sources of uncertainty in transport properties

The uncertainty in transport properties DP, K, and hF is related to experimental errors when their values are experimentally assessed, to systematic errors when they are predicted by models, to uncontrolled changes in temperature, to physical simplifications, to other uncontrolled sources of variation (e.g. process conditions, food properties…). When uncertainty does not seem to exist, it may be appropriate to include some to improve the robustness of the predictions. This paragraph analyzes how to combine them. The examples are based on arbitrary distributions of Fo, Bi and K that are plotted in Figure 12.

5.2.1 Combined effect of sD and sh

5.2.1.1 Interest Accounting for the uncertainty in hF, denoted sh, is a particularly interesting feature, since few values of hF values are available in the literature and the physical definition of hF may change according to the simulated conditions. Indeed, from the physical point of view, hF as defined in section 3.2.2 via Equation (10), is a “true” mass transport property only when the food product is a liquid and stirred. In this situation, the bulk concentration in food as measured, CF, is an appropriate estimate of the concentration far from the interface, and hF, which is related to the mass transfer resistance through the boundary layer, can be assessed independently of the dilution factor L. When the food is a solid or a semi-liquid or when the volume of the boundary layer is comparable to the bulk volume, the previous approximation is no more valid and the physical meaning of hF is changed. It is an equivalent transport property, whose value depends on the considered geometry. Practically, its use is however appropriate the addition of an artificial mass transport resistance at the interface, RH, prevents to simulate the mass transport within the food product in contact. In other words, when the parameter hF or its counterpart, the mass Biot

22/27

number, is set appropriately, desorption kinetics can be predicted accurately by modeling only mass transport in P. Such kind of reasoning can be also extended to other situations, where the contact surface area or the composition of the interface (e.g. due to exudation, phase separation…) is not known accurately or may vary with time. All these complex phenomena can be simulated by taking into account uncertainty of hF.

5.2.1.2 Principle It is worth to notice that the dimensionless formulation proposed in section 3.2 does not use directly DP and hF in the transport equations but the dimensionless numbers Fo and Bi. From the statistical point of view, both descriptions are not equivalent since DP and hF are assumed to be independent quantities when Fo and Bi cannot (i.e. DP and hP appear in the expression of Bi). The corresponding random model is:

� ( )

�( )

* *22

*

**

,*

*

D

D h

P PP s

p Fop

F P F P Fs s

P PP

Bi

D t D tFo D Fo Fo

l l

h l h l hBi Bi Bi

D DD

⋅ ⋅= = ⋅ = ⋅ ⋅ ⋅ = = ⋅ = ⋅

(34)

Since ( )

*

DsFo and ( )

*,D hs s

Bi are not independently distributed, the distribution of the resulting

dimensionless concentration ( )* , , ,D hv Fo Bi s s is calculated iteratively (mixture rule) from the marginal distribution ( )* , D iv Fo s Bf :

( ) ( )* , , , * ,0D h D Biv Fo Bi s s v Fo s Bif f f dBi+∞

= ⋅ ⋅∫ (35) where ( )* , Dv Fo s Bif is the distribution of *v calculated for a given Bi value as depicted in Figure

11. Bif is the expected distribution of Bi.

5.2.1.3 Detailed example

The results plotted in Figure 11 ( 0.5Fo = , sD=0.1) are extended to an arbitrary distribution of Bi defined by:

( )( )0,minmax , 10 BiNorm sBi Bi Bi ⋅∼ (36) where the parameter sBi accounts for the uncertainty on both D and h.

23/27

The values of ( )* , Dv Fo s Bif versus the values of ( )* , Dv Fo s Bi and Bi are plotted in Figure 13 for

0.5Bi = , sBi=0.3 and min 5Bi = . For physical consistency, the condition min 1Bi > ensures in particular that the mass transfer resistance is higher in P than in F (see Figure 12). The bivariate probability density function is zero below minBi and its iso-values exhibit a privileged orientation

along the first bisectrix of axes, which confirms that *v values increase with Bi values. The

maximum of probability is obtained close to Fo and Bi .

The combined effects of sD and sBi on ( )* 0.5, 20, ,D hv Fo Bi s s= = are compared in Figure 14 by repeating the analysis performed in Figure 13 for different sBi values. The distributions of Bi corresponding to the tested sBi values are plotted in Figure 12. Results showed both a small shift of the maximum of probability on the right and an increasing span of the distribution of *v . The

first effect is related to the truncation introduced in the distribution of Bi Bi , which moves the expectation towards values higher than 1 for large sBi values. The second effect is related to the

cumulative effect of uncertainty. For the tested condition 0.5Fo = and 20Bi = , the effect of the uncertainty on D is higher than the uncertainty of Bi.

5.2.2 Combined effect of sD and sK The effect of K can be tested similarly by calculating the distribution of

( )( )

1 1

1 * 1* , , ,

K

D K

s

K Lv Fo K s s

K K L

− −

− −

+ ⋅⋅ +

, where the prefactor ( )

1 1

1 * 1

Ks

K L

K K L

− −

− −

+

⋅ + assess the deviation to

the likely equilibrium concentration in F. It is emphasized that sK acts both on the prefactor and *v . For the illustration, the distribution of K was chosen as:

( )( )0,maxmin , 10 KNorm sK K K ⋅∼ (37) To enforce K values to be lower than 1, K was set to 0.1 and Kmax to 1. Typical values of the distribution are plotted in Figure 12. As presented in Equation (35), the marginal distribution of

( )( )

1 1

1 * 1* , , ,

K

D K

s

K Lv Fo K s s

K K L

− −

− −

+ ⋅⋅ +

was calculated from its conditional distributions for particular

K values and integrated over the whole range of K values.

The results corresponding to 0.5Fo = , L=0.05, Bi=20 and different ranges of sD and sK values are plotted in Figure 15. The effect of sK is here higher than the effect of sD. Increasing sK leads to a drastic change in K values towards 1 (Figure 12) so that the distributions of

( )( )

1 1

1 * 1* , , ,

K

D K

s

K Lv Fo K s s

K K L

− −

− −

+ ⋅⋅ +

obtained for low K values tend to resemble those depicted in

Figure 15 for K=1.

24/27

It is worth to notice than the current approach can be iteratively extended to the study of the combined effects of sD, sBi and sK by replacing the conditional distribution ( )* , Dv Fo s Kf in the current

approach by ( )* , ,D Biv Fo s s Kf , where the latter is calculated from Equation (35).

6 Conclusions The prediction of the contamination of food by packaging substances is encouraged by the current regulation in Europe as an alternative to costly and time consuming experimental testing. Recent crises (e.g. isopropylthioxanthone in baby milk in Europe) and new developments (recycled materials, active packaging) have besides heightened interest in the issue of scientific uncertainty, and on how it should be handled in risk assessment and decision-making. As acknowledged in the Working Principles for Risk Analysis recently adopted by the Codex Alimentarius Commission “Constraints, uncertainties and assumptions having an impact on the risk assessment should be explicitly considered at each step in the risk assessment and documented in a transparent manner. Expression of uncertainty or variability in risk estimates may be qualitative or quantitative, but should be quantified to the extent that is scientifically achievable”. (Codex, 2005). This work proposes a general framework to quantify and analyze the effects of uncertainty in mass transport models used to predict the contamination of food products. The applications were focused on the design of packaging materials, but the proposed approach is enough general to be applied to sanitary surveys or controls by enforcement laboratories. Two typical users of these approaches were particularly envisioned i) the producer of finished packaging material, who formulates the material (e.g. a film) and the ii) the packaging filler (food industry), who puts the packaged food product on the market. The first user has a good knowledge on the formulation of its material but an imprecise description of the final use of the material. Its objective is to demonstrate that is product is compliant for most of conditions (contact time, temperature, and type of food). By contrast, the second user is ignorant of the formulation of the material but has a good knowledge on the final use of the material and on possible dependencies between factors that enhance the desorption of packaging substances (temperature, fatty contact…). Its objective is to put on the market on food product, which is at minimum compliant, but also to demonstrate that its product is safer than the comparable product from its competitor. Besides, both users are subjected to scientific uncertainty inherent to insufficiently tabulated transport properties, imprecise extrapolation from semi-empirical models of transport properties. Physical transport models were first extensively reviewed for both monolayer and multilayer materials. They were adimensioned to minimize the number of input parameters and to highlight the physical quantities, whose uncertainty may have similar effect on the prediction. From a phenomenological description of the desorption of packaging substances, it was illustrated how an association of mass transport resistance could be replaced by a simpler association, for which calculations are achievable and whose desorption rates overestimate the “true” desorption rate. The complete approach was described as multi-attribute value trees and applied to both monolayer and multilayer materials. Since this approach is progressive and generate a rapid overview of the parameters and related uncertainty, which affects the predictions, it is particularly appropriate i) for food industry users to audit its providers, ii) for the packaging industry to simulate mass transport in very complex multilayer structures. Besides, since this worst-case

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approach by intervals can be easily implemented, it is progressively introduced in the EU regulation, to which one of us is participating. In addition, this chapter outlines some more sophisticated approaches based on probabilistic modeling. Previous intervals on input parameters and predictions are replaced by continuous distributions. Assessments may always begin by treating all uncertainties qualitatively by intervals, but when those uncertainties appear critical for the outcome they may be analyzed quantitatively. Such refinements make it possible to generate overestimates of the contamination of food products with a controlled risk to be exceeded. The applications are tremendous: prediction of the contamination by a ubiquitous contaminant, by a group of homologous molecules, non by intentionally added molecules, for several thermodynamical or geometrical conditions… As an original illustration, this work illustrates how the combined effects of uncertainty on all transport properties involved in the modeling of the contamination from monolayer materials can be assessed. As a rule of thumb, the amount of effort and detail devoted to analyzing uncertainties should be proportionate to the needs of the problem. A tiered approach as detailed here is recommended. Assessments may begin by point estimates for common values and subsequently using ranges, confidence intervals and distributions. Besides, the user of such methods is encouraged to differentiate variability and uncertainty. Variability refers to variation that exists in the real world, e.g. varying concentrations of substances in packaging materials. Uncertainty is often reducible through research, whereas variability is not. As a result, if a physico-chemical property has a drastic effect on the prediction of the contamination, it may be appropriate to measure it and to share its knowledge in anonymized databases of properties. A EU database of transport properties is available on our server: Safe Food Packaging Portal (Vitrac, 2006).

7 References Codex, 2005. Codex Alimentarius Commission Procedural Manual, 15th Edition. http://www.codexalimentarius.net/procedural_manual.stm. EC (2005), Substances listed in EU directives on plastics in contact with food. Updated to 26 September 2005EC (2004), Regulation (EC) No 1935/2004 of the European Parliament and of the Council of 27 October 2004 on materials and articles intended to come into contact with food and repealing Directives 80/590/EEC and 89/109/EEC/ Official Journal, L 338 of 13.11.2004, 4-14 EC (2002), EU directive 2002/72/EC relating to plastics materials and articles intended to come into contact with foodstuffs. Official Journal, L220 of 15.08.2002, 18-55. EC-DG SANCO-D3 (2002). Conclusions of the thematic network “Evaluation of migration models to be used under directive 90/128/EEC”: SMT–CT98–7513.http://ec.europa.eu/food/food/chemicalsafety/foodcontact/eu_substances_en.pdf. Fytas G., Rizos A., Floudas G., Lodge T. P. (1990). Solvent mobility in polystyrene/Aroclor solutions by depolarized Rayleigh scattering. J. Chem. Phys. 93, 5096-5104. Glasstone S., Laidler K.J., Eyring H. (1941). Theory of rate processes. McGraw-Hill: NewYork, US, 195-197.

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Han J. H. (2005). New Technologies in Food Packaging. In Innovations in Food Packaging. Ed. Han J. H.; Elsevier Adademic Press: San Diego, US, 3-10. Heuer A., Leisen J., Kuebler S. C., Spiess H. W. (1996). Geometry and time scale of the complex rotational dynamics of amorphous polymers at the glass transition by multidimensional nuclear magnetic resonance. J. Chem. Phys. 105, 7088-7096. Kaji H., Miura N., Schmidt-Rohr K. (2003). Rotational motions in atactic poly(acrylonitrile) studied by one- and two-dimensional 15N solid-state NMR and dielectric measurements. Macromolecules 36, 6100-6113. Kovarski A. L. (1997). Molecular Dynamics of Additives in Polymers: New Concepts in Polymer Science. VSP: Utrecht, NL. Painter P. C. , Coleman, M. M. (1997). Fundamentals of Polymer Science: An introductory text – 2nd Edition. CRC Press LLC: Boca Raton, US. Sperling L. H. (2006). Introduction to Physical Polymer Science – 4th Edition. John Wiley & Sons Inc: Hoboken, US. Vitrac O. The safe food packaging portal version 2. http://h29.univ-reims.fr/. Vitrac O., Challe B., Leblanc J.-C., Feigenbaum A.F. (2006). Risk of contamination of packed foods by substances from the plastic contact layer: a generic quantitative methodology at the scale of private households. Accepted in Food Add. Contamin. Vitrac O, Hayert M. (2006). Effect of the distribution of sorption sites on transport diffusivities: a contribution to the transport of medium-weight-molecules in polymeric materials. Accepted in Chem. Eng. Sci. Vitrac O., Hayert M. (2005). Risk assessment of migration from packaging materials into foodstuffs. AIChE J. 51(4), 1080-1095. Vitrac O., Leblanc J.C. (2006). Exposure of consumers to plastic packaging materials: assessment of the contribution of styrene from yoghurt pots. Accepted in Food Additives and Contaminants. Vitrac O., Lézervant J., Feigenbaum A. (2006). Application of decision trees to the robust estimation of diffusion coefficients in polyolefines. J. App. Polymer Sci. 101, 2167–2186. Notations Bi mass Biot number (-) C concentration (kg⋅kg-1) D diffusion coefficient (m2⋅s-1) Fo Fourier number or time (-) h interfacial mass transfer coefficient (m⋅s-1) j mass flux density (kg⋅m-2⋅s-1) k Henry like coefficient or equilibrium constant (J⋅mol-3) kH Henry coefficient (J⋅mol-1) K partition coefficient between F and P (-) l thickness or characteristic dimension (m) L dilution factor (kg⋅kg-1) M molecular mass (g⋅mol-1) p equivalent vapor pressure or partial pressure (Pa) pr probability RD mass transport resistance related to diffusion (s⋅m-1)

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RH mass transport resistance at the interface on F side (s⋅m-1) RK thermodynamical mass transport resistance (s⋅m-1) t time (s) u dimensionless concentration (-) V molar volume (m3) Greek Letters α equivalent transport property (defined in eq. 22) γ activity coefficient ρ density (kg⋅m-3) Special Symbols a, b statistical parameters related to the distribution of q (table 4) BC boundary condition cdf cumulative density function eq equilibrium F food product fq density of probability of q i index j layer index m coordinates index (0=cartesian, 1=cylindrical and 2=spherical) P packaging material pdf probability density function q any physical quantity q* dimensionless value of q

q likely estimate of q ɵq estimated value of q qmax maximum value of q qmin minimum value of q SML specific migration limit sq shape distribution parameter of q

Group of scenarios

number of

scenarios main assumption(s)

A 1 complete extraction (no partitioning and kinetic effect) B 4 partitioning effect and no kinetic effect C 1 kinetic effect and no partitioning D 4 partitioning and kinetic effects

Table 1. Typical group of scenarios for monolayer materials

Code 1 0tC

=

2 0t

C=

k1 k2 Description 0 eqC

[1 1]-[1 1] 1 1 1 1 monolayer (reference) 0.0385 [0 2]-[1 1] 0 2 1 1 barrier effect (half thickness, only related to diffusion) 0.0385 [2 0]-[1 1] 2 0 1 1 reservoir effect (half thickness, only related to diffusion) 0.0385 [0 2]-[2 1] 0 2 2 1 barrier effect (half thickness, related to both diffusion

and poor solubility in the layer in contact) 0.0388

[2 0]-[1 0.5] 2 0 1 0.5 reservoir effect (half thickness, related to both diffusion and a higher solubility)

0.0377

Table 2. Details on tested conditions tested in Figure 7. Concentrations are expressed in arbitrary units. All layers and the food product are assumed to have the same density.

configuration parameter layer 1 layer 2 layer 3

1

n

j jj

l l=∑ 0.2 0.5 0.3

0j FoC

= 1 1 1

Reference

0jk k 3 0.8 2

1

n

j jj

l l=∑ 0.2 0.5 0.3

0j FoC

= 1 0 0

overestimation of the contribution of layer 1

0jk k 5 5 5

1

n

j jj

l l=∑ 0.2 0.5

0j FoC

= 0 1

overestimation of the contribution of layer 2

0jk k 5 5

1

n

j jj

l l=∑ 0.2 0.5

0j FoC

= 0 0.6

overestimation of the contribution of layer 3

0jk k 5 5

Dis

card

ed la

yer

Table 3. Reference conditions and idealized configurations to overestimate the contribution of each layer/. The overall dilution factor between F and P is 1/25. The food product and all layers are assumed to have the same density. The other assumptions are a same diffusion coefficient for all layers and Bi → ∞ . All concentrations are expressed in arbitrary units.

uncertainty factor meaning distribution

log10(DP*) Diffusion coefficient in P Norm(0,sD) t* Contact time Weib(1,st)

*

0P tC

= Initial concentration in P Norm(1,sc0)

lP* Packaging thickness Norm(1,sl) 2/1*Fo Fourier number Gamma(aΓ,bΓ)

*v Dimensionless concentration in P Beta(aβ,bβ) CF* Concentration in F Gamma(aCF, bCF)

Table 4. Distribution for the random contributions *iq . Grayed rows depict dependent quantities.

Figure 1. Phenomenological description of the desorption of packaging substances into food: a) concentration profiles, b) mass flux at the P/F interface through a serial association of 3 resistances.

Figure 2. Indexing of n-multilayers materials in contact with food.

Figure 3. Typical effects of Bi, K, L and Fo on *v .

Figure 4. Tree representations of estimation scenarios for monolayer materials

Figure 5. Predicted

max

0P FoC

= values according to scenarios A (dotted lines), B (dashed lines), C

(dash-dotted lines), D (continuous lines). The first row gathers the combined effects of K and L whereas the second raw gathers the effects of K and Bi.

Figure 6. View of the software "Decision Tool for Testing Compliance". The parameters known by the company, which claims an audit, are listed in the insert on the left whereas the parameters being discussed are located in the right-hand side insert. The horizontal menu bar gives access to the databases and estimators of physicochemical properties. (Vitrac, 2006)

Figure 7. Dimensionless concentration profiles in P (top position) and desorption kinetics (bottom position) for two typical Fo values and 5 typical bilayer structures detailed in Table 2. The configuration coded [1 1]-[1 1] is the reference and is equivalent to a monolayer structure with a similar initial content. The F-P interface is located at x*=0.

Figure 8. a) Details of the principles of iterative approximation for a trilayer material.

b) Equivalent configurations used to overestimate *v with uncertainty on { }1..j j n

k=

values.

Figure 9. Approximation of the contamination from a tri-layer material corresponding to table 3: a-h) concentration profiles (for Fo=10-2 and 0.2), i-l) dimensionless desorption kinetics. The three first columns present the contribution of each layer (from 1 to 3), when it is simulated with reference parameters or worst-case parameters. The last column compares the results obtained by simulating the complete tri-layer material (reference) and by cumulating the individual overestimations of each layer.

Figure 10. Effect of the uncertainty in * ( , , )t D ls s sFo Fo Fo= ⋅ on *v . The curves are plotted for

[ ]0.1,0.5,1,1.5Fo = and ( )* 0.1, 0, 0D t lFo s s s= = = . The other parameters are L=0.05, Bi→∞, K=1. The distributions are normalized so that their maximums are equal.

Figure 11. Effect of Bi of the uncertainty on *v for 0.5Fo = and ( )* 0.1, 0, 0D t lFo s s s= = = . The other parameters are L=0.05, Bi→∞, K=1. The distributions are normalized so that their maximums are equal.

Figure 12. Distributions of a) Fo, b) Bi and c) K used in paragraph 5.2. The distributions are normalized so that their integrals are 1.

Figure 13. Marginal distribution of ( )* 0.5, 0.1Dv Fo s Bif = = when Bi is distributed according to

Equation (36). The surface plot in gray level depicts the bivariate pdf ( )* 0.5, 0.1Dv Fo s Bif = = versus

( )* , Dv Fo s Bi and Bi values. The contour plot depicts the bivariate cumulative density function

( )

*

0.5, 0.10 0

D

v Bi

y Fo s xf dx dy

= =⋅ ⋅∫ ∫ . The projection onto the axis

*v , defined by the marginal

distribution ( )* 0.5, 0.10

Dv Fo s xf dx

+∞

= =⋅∫ , gives the resulting distribution of

( )* 0.5, 0.1, ,D Biv Fo Bi s s= = (Equation 35). The projection onto the axis Bi, defined by the marginal distribution ( )

*

0.5, 0.10

D

v

y Fo s Bif dy

= =⋅∫ gives back the distribution of Bi (Equation 36).

Other parameters are L=0.05 and K=1.

Figure 14. Distributions of ( )* 0.5, 20, ,D Biv Fo Bi s s= = for different sD and sBi values: a-c) probability density functions (pdf), d-f) cumulative probability density functions (cdf). Other parameters are L=0.05 and K=1. All distributions are weighted so that their integrals are 1.

Figure 15. Distributions of ( )

( )1

1 * 1

0.1* 0.5, 0.1, ,

K

D K

s

Lv Fo K s s

K K L

−

− −

+ ⋅ = =⋅ +

for different sD and

sK values: a-c) probability density functions (pdf), d-f) cumulative probability density functions (cdf). Other parameters are L=0.05, Bi=20. All distributions are weighted so that their integrals are 1.