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=
=
=
=
+
0
0
0
1
0
0
0
0
1
1
n
j
j
j t
j
eq
n
j
j
j
j
l
C
l
C
l
k
k
l
(19)
The corresponding partial pressure at equilibrium can be also expressed as a function of the initial
partial pressure in each layer:
{ }
=
=
=
=
=
=
+
0
0
0
0
1
0
1..
0
0
0
1
1
n
j
j
j t
j
j
j
eq
n
eq j
n
j
j
j
j
l
k
p
k
l
p
p
l
k
k
l
(20)
Equations (19) and (20) generalize Equation (4) to multilayer materials. It is worth to notice that
they do not require that the layers are initially at equilibrium. However, they assume that the
initial concentration is uniform in each layer. When it is not the case, Equation (19) must be
replaced by a continuous integration over each layer. For general 1D geometries, one gets:
( )


=
=
=
=
+
1
0
1
0
0
1
0
0
0
1
0
1
1
c
j
c
j
c
j
c
j
l
n
x
m
j
j t
j
l
eq
l
l
n
m
m
j
j
j
l
C
x
dx
C
k
x
dx
x
dx
k
(21)
where m=0,1,2 respectively for Cartesian, cylindrical and spherical coordinates.
=
=
0
n
c
j
j
j
l
l
is the
cumulated thickness (starting from the product).
For convenience and according to Equation (17), choosing k
0
=1 leads to identify k
j
to the
partition coefficient between the food product and the packaging layer j.
3.3.2 Transport equations
By assuming a local thermodynamical equilibrium at the interfaces between all layers (including
the food product), the partial pressure is continuous over all layers. As a result, the partial
pressure seems a best choice to implement transport equations in commercial numerical codes.
This choice leads to express the local mass flux in the layer j as a consequence of a gradient in
partial pressure:
= 
= 
= 
2
1
2
1
3
kg m
s
m s
kg m
j
j
j
j
j
j
j
j
C
D
p
p
J
D
x
k
x
x
(22)
If D
j
can be considered uniform in the layer j>1,
=
=
j
j
j
j
j
j
D
D
k
is a new equivalent
transport property. The equivalent local mass balance is accordingly written for any layer j as: