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-
=
=
=
+
=

+
1
0
0
1
1
F
P
eq
t
P
P
eq
t
C
C
K
L
L
C
C
L
K
(4)
3.2.2 Transport equations
3.2.2.1
Phenomenological description
The previous physical hypotheses are summarized in Figure 1 by assuming that F is a liquid food
and does not interact with the polymer: the constituents of F do not migrate into P (no
plasticization or swelling induced by F). Local thermodynamical equilibrium is assumed at the
interface between P and F so that
=
3
2
C
K
C
at each time.

Figure 1a represents more particularly the interfacial mass flux density (noted j and with units in
kg
m
-2
s
-1
), so-called migration rate, as a result of a mass transport through a serial association of
different types of transport resistances with units of a reciprocal velocity s
m
-1
. The
corresponding profile and location, where transport resistances are defined, are also depicted in
Fig. 1b. R
D
is related to the resistance to diffusion in P whereas R
H
is related to several
mechanisms (e.g. diffusion, convection), which limit the dispersion of the substance in the food
compartment. They are defined by:

-
=
-
=
j
C
C
R
j
C
C
R
H
D
4
3
2
1
(5)
where 1 and 4 are arbitrary positions that verify the linear approximation of the concentration
profile on both sides of the interface.

For complex food or contact between P and F (e.g. imperfect contact), it is highlighted that R
H
can be an equivalent resistance, which is only used to make the simulated mass flux at the
interface matching the real one. By contrast, the thermodynamic resistance, R
K
, is only
phenomenological since positions 2 and 3 are merged. Its effect is however significant, since
decreasing the concentration in 2 (without changing the concentration in 3) will increase the
slope of the concentration profile and consequently j. From the phenomenological point of view,
R
K
is controlled by K, which acts as refractive index. By analogy with Equation (5), R
K
may be
defined by:
2
3
2
(1
)
K
C
C
K
C
R
j
j
-
-
=
=
(6)
It is worth to notice that this resistance is absent (zero) when K=1 and very significant when
K<<1 (i.e. high affinity for P). By contrast, R
K
is negative when K>>1 (high affinity for F) and
contribute to reduce overall mass transfer resistance defined by: