13/27
estimate
F
C or
max
0
P Fo
C
=
. 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,
l
P
, l
F
, K. If the interval on each parameter p is defined by their bounds p and p , the number of
different scenarios possibly dependent is 2
6
=64. By assuming that the final geometry of the
packaging material and of the food in contact is well known (
P
P
l
l
and
F
F
l
l
) and by noting
that the assumption
*
1
v
does not depend on values of Bi and Fo, the number of independent
scenarios finally reduced to 2
4
2
3
+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
F
C
and
C
F
C
the estimate of the
contamination in F
according to scenarios B and C respectively, one gets:
=

=
=
=
+
+
*
*
0
1
0
1
1
1
C
P t
F
B
F
P t
C
L v
C
L
v
K
C
C
K
L
(30)
The inequality
C
B
F
F
C
C
>
is obtained only when
>
+
*
K
v
L
K
. Thus, scenario B overestimates
more often
F
C 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
max
0
P Fo
C
=
Too large overestimations of
F
C (or equivalently too large underestimations of
max
0
P Fo
C
=
) 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
F
k A B C
C
C
=
, are:
+
*
1
K
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
0
P Fo
C
=
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
0
P Fo
C
=
is read from the left to the right (from 0 to 1). Scenarios A and B