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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 l
j
. By convention, l
0
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), l
0
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:
( )
( )
-
=
1
Pa
kg kg
H
j
j
j
j
j
j
k
M
p
x
k
C
x
(16)
where
H
j
k
and
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
H
j
k
(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 j
1
and j
2
Two layers, noted j
1
and j
2
, 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
1
2
/
j
j
eq
j
j
j
j
eq
eq
j
j
eq
C
k
p
p
K
k
C
(17)
where
1
2
/
j
j
K
is the partition coefficient of the considered substance between j
1
and j
2
.
1
j
eq
C
and
2
j
eq
C
are the concentrations at equilibrium respectively in layers j
1
and j
2
.
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
0
1
1
j
eq
n
n
j
j
j
j
j
j
eq
eq
t
j
j
k k C
C
l
C
l
C
l
(18)
By noticing that the condition of equilibrium enforces
{ }
0
1..
j
eq
eq
j
n
p
p
=
=
, the concentration in
food at equilibrium is finally given by: