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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.