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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
,
,
,
tim e dep en dent con tribu tion
tim e in depend ent contrib ution
= C
1
1
P t
eq
F
P
F o B i K L C
t
F o B i K L
C
C
v
K
L
(29)
where
=
=
=
-
0
*
1
,
,
, ,
*
*
,
,
,
,
,
,
,
0
1
P t
F Fo B i K L C
x
F o B i K L
F o B i K L
F eq
C
v
L
u
dx
C
,
one
notes
*
0
1
v
. 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 eq
C
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<1/50). For these
conditions, the effect of K on
*
v is almost non-discernible.

For common applications of packaging material, the uncertainty on K acts on the expected
concentration at equilibrium rather than on
*
v so that:
*
*
,
,
,
,
F o B i K L
F o B i
v
v
. This property will
be used to classify the combination of values of dimensionless input parameters (Fo, Bi, K, L),
which are important to refine the estimate of
F
C or
max
0
P Fo
C
=
. This approach is first discussed for
monolayer materials and finally applied to multilayer materials.

For each dimensionless parameter of the model, two values are possible (they may be equal): a
likely one, noted p , and a safe value, noted p , which leads to overestimate
F
C (or equivalently
to underestimate
max
0
P Fo
C
=
). The main difference between p and p is that only p values are
assumed to be known for all situations. As a result, an estimate of
F
C or
max
0
P Fo
C
=
is always
possible by assuming that all input parameters are equal to their safe values even if the final result
is of little use (e.g. the final contamination is too much overestimated). The difficulty arises in
classifying the minimal substitution of p values by their p values that may improve the most the