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1
4
eq
D
K
H
C
C
R
R
R
R
j

=
=
+
+
(7)
This description is particularly useful to define conditions, which overestimate the desorption rate
in food. Several worst cases can be identified:
·
R
D
>0, C
2
=0, R
H
=0 (kinetic effects control the desorption, K
is assumed)
·
R
D
>0, R
K
=R
H
=0 (kinetic effects control the desorption, K=1 is assumed)
·
R
D
=R
H
=0, R
K
>0 (thermodynamic effects control the desorption)
Other scenarios can be derived by comparing the ratio between R
D
and R
H
also known as mass
Biot number and defined by:
D
F
P
H
P
R
h
l
Bi
R
D
=
=
(8)
where D
P
is the diffusion coefficient in P and h
F
is a mass transport coefficient on F side with SI
units in m
s
1
.
3.2.2.2
Mathematical formulation
By assuming a onedimensional transport (side effects are negligible), a uniform initial
concentration in P, a constant diffusion coefficient, a constant thickness l
P
and no change in
density due to the desorption of packaging substances, the dimensionless mass transport equation
in P is:
2
2
*
u
u
Fo
x
=
(9)
where
,
0
P t x
P t
C
u
C
=
=
, *
P
x
x
l
=
,
2
P
t D
Fo
l
=
are respectively the dimensionless concentration, position
and time (socalled Fourier time). If the temperature is not constant with time, a generalized
Fourier time can be used instead:
( )
0
2
t
P
D
d
Fo
l
=
. This last representation neglects the effects
of a temperature gradient on D but this approximation is realistic for thin materials such as
packaging materials and contact times above several minutes.
At the FP interface (i.e. for x*=1), the combination of Equations (6) and (7) provides a simple
expression for the dimensionless boundary condition (BC), known as Robin BC:
(
)
* 1
*
* 1
*
*
x
x
x
u
j
Bi K
u
u
x
=
=
= 
=

(10)