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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: *D*

*P*

, *h*

*F*

, *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 *D*

*P*

), the conditions of

storage (temperature), food composition and structure (for *h*

*F*

and *K*). It emphasized that this

situation is not discussed in the references cited above where, only the uncertainty in *D*

*P*

is

considered via probabilistic modeling. The uncertainties on *h*

*F*

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,

i

q

, is defined as the product of a scale parameter

i

q

(with a physical

unit), and of a random dimensionless contribution,

( )

i

q

s

i

q

*

with a unitary expectation. Random

contributions of independent variables *t**, *l*

*P*

***, *D*

*P*

**,*

*

0

*P t*

*C*

=

* *are distributed respectively to

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

i

q

s

(Table 4),

noted respectively *s*

*t*

, *s*

*l*

, *s*

*D*

, *s*

*C0*

. The scale dimensionless-parameters of Fourier and Biot mass

numbers are accordingly defined by

2

*P*

*P*

*Fo*

*D*

*t l*

=

and

*F*

*P*

*P*

*Bi*

*h*

*l*

*D*

=

respectively. The

distribution of

*

*

* 2

*

*P*

*P*

*Fo*

*Fo Fo*

*D*

*t l*

=

=

depends non-trivially on *s*

*D*

, *s*

*t*

and *s*

*l*

. 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 b*

. From mass balance considerations, it was demonstrated that the proposed

approximations

*

v

is Beta distributed (Vitrac and Hayert, 2005) with parameters:

,

*a b*

. When

it is required, the concentration in *F* can be approximated by a Gamma distribution, which

depends only on two parameters:

,

*F*

*F*

*C*

*C*

*a*

*b*

.

**5.1.2 Calculation procedure of: **

F

F

C

C

b

a

b

a

b

a

,

,

,

,

,

** **

The values of the statistical parameters*, a*

*,b*

*, a*

*,b*

*, a*

*cF*

*,b*

*cF*

,

are tabulated respectively to the

variables

Fo

K

,

L

Bi

, *s*

*t*

, *s*

*l*

, *s*

*D*

and

0

C

s

and are calculated within the M

IGRARISK

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.