10/27
(
)
=
-
=
1
1
m
m
j
j
j
m
m
p
p
x
J
x
t
x
x
x
x
x
for j=1..n
(23)
Since significant discrepancies may be expected between
{ }
=
1..
j j
n
values and between
{ }
=
1..
j j
n
values, a dimensionless formulation seems preferable to preserve the numerical stability of the
discretization scheme.
By analogy with permeation, the reference length scale, l
ref
, is associated to the thickness of the
layer with the maximum mass transfer resistance, which is with the lowest
{
}
=
1..
j
j j
n
l
value. The
dimensionless time or Fourier number, Fo, is expressed as:
=
2
ref
j
ref
t
Fo
l
(24)
The dimensionless mass balance equation is finally:
=
*
*
1
*
*
*
*
ref
j
m
j
m
j
p
p
x
Fo
x
x
x
for j=1..n
(25)
with
=
*
ref
x
x
l
and
=
0
*
eq
p
p
p
.
At the interface between j=1 and j=0, which is at the position x=0, the boundary condition is
written:
=
=
=
=
= -
=
+
0
1
1
0
0
0
0
0
0
0
1
t
x
x
x
x
p
J
h
p
J
d
x
k
l
(26)
The sign + in front of the cumulated term,
=
0
0
t
x
J
d
, depicted in Equation (12) comes from the
projection of the flux (vector) on axis x. Indeed,
0
x
J
=
is counted negatively.
By introducing the dimensionless flux
=
=
=
= -
=
*
*
1
0
0
0
* 0
1
*
ref
x
x
ref
ref
eq
x
l
p
J
J
x
p
, one gets the
dimensionless boundary condition:
=
=
=
=
+
*
*
*
0
0
0
0
0
Fo
x
x
x
p
J
Bi
L
J
d
k
(27)
where
=
ref
ref
h
Bi
l
is the mass Biot number and
=
0
ref
l
L
l
is associated to a dilution factor.
An impervious boundary condition is assumed at
=
*
1
x
:
=
=
*
1
0
*
x
p
x
(28)