background image
7/27
where
0
*
P
P
P t
l
j
j
D C
=
=
is the dimensionless flux,
3
* 1
0
x
P t
C
K u
C
=
=
=
. Practically,
4
*
0
x
P t
C
K u
C
=
=
is assumed to stand for the dimensionless concentration in the bulk, as it would
be measured in F, rather than the concentration far from the interface as it is depicted in Figure 1.
Both descriptions are almost equivalent when the concentration is assumed to be homogeneous
far from the F-P interface. This situation occurs when the transport property in F is much greater
than the transport property in P (case of most of food products) or when a mixing process (e.g.
convection) occurs on F side.

From these considerations,
*
x
u
is derived from the mass balance between P and F:
( )
( )
0
0
*
*
*
0
0
0
1
1
1
*
t
Fo
Fo
Fo
x
x
x
P
F
F
t
L
u
u
j
d
u
j
d
K C
l
K
=
=
=
=
+
=
+
(11)
Equation (10) combined with Equation (11) yields the practical form of the BC, written as an
integro-differential operator:
(
)
( )
0
* 1
*
* 1
0
*
*
*
Fo
Fo
x
x
x
u
j
Bi K
u
u
Bi L
j
d
x
=
=
=
= -
=
-
-
(12)

Two extreme cases are reduced from Equation (12) by assuming i) R
H
=0 (i.e. no external
resistance), ii) R
K
=R
H
0 (i.e. R
eq
R
D
). Case i) is inferred by differentiating with time Equation
(12) for :
* 1
* 1
*
*
x
x
u
L
L
u
j
Fo
K
K x
=
=
=
= -
(13)
By analogy with wave propagation equations, Equation (13) is known as a reflecting boundary
condition, where the amount of matter that leaves the interface F-P modifies in return (i.e. after
accumulation or "reflection") the mass transfer resistance at the latter. K/L is the equivalent
dimensionless "reflecting distance", where the quantity K is similar to a dimensionless
"absorbing" coefficient (or refractive index).
Case ii) corresponds to the condition
/
K L
(infinite volume or capacity) in BC (12), that is
* 1
0
x
u
Fo
=
or the equivalent Dirichlet's BC:
( )
(
)
0
* 1
* 1
Fo
Fo
x
x
u
u
=
=
=
=
(14)

For the left side boundary, x*=0, an impervious is applied:
* 0
0
*
x
u
x
=
=
(15)

3.3 Contamination from multilayer materials