20/27
The distributions of
*
Fo and C
F
* are inferred iteratively as the product of independent quantities
Y
X
. The distribution of
(
)
Y
X
10
log
is calculated as the product of convolution of the marginal
distributions of
( )
X
10
log
and
( )
Y
10
log
. Since
*
v
is a continuous regular function of input
parameters and in particular of Fo
1/2
, the distribution of
*
v
, noted
( )
(
)
y
v
pr
y
f
v
=
=
*
*
, is calculated
as the continuous transformation of a random variable
)
(
* x
v
y
=
. Thus, for
0
=
=
=
Bi
L
K
s
s
s
(i.e.
K, L and Bi are delta distributed) and since
*
v
is a strictly increasing and differentiable function
of Fo
1/2
,
( )
y
f
v
*
is derived from the distribution of Fo
1/2
, noted
( )
(
)
x
Fo
pr
x
f
Fo
=
=
2
/
1
2
/
1
:
( )
( )
[ ]
( )
y
y
v
y
v
f
y
f
Fo
v
=


*
1
*
1
*
*
2
/
1
(33)
where
*
1

v
is the transformation so that
( )
y
v
x
*
1

=
.
It is worth to notice that for large Bi values and Fo<<1 values,
( )
y
y
v

*
1
is a constant and
*
v
and
Fo
1/2
are distributed similarly with a right tail. For large Fo values (i.e. Fo >1),
( )
y
y
v

*
1
increases
drastically and
*
v
is left tailed.
The general distribution of the joint probability density function (pdf) of
*
v
corresponding to
Bi
L
K
Fo
Bi
L
K
Fo
Fo
v
Bi
L
K
Fo
Fo
v
s
s
s
b
a
b
s
s
s
b
a
a
,
,
,
*
*
,
,
,
,
,
,
,
,
,
2
/
1
2
/
1
2
/
1
2
/
1
is calculated as a mixture of
Beta distributions. The distribution parameters,
F
F
C
C
b
a
b
a
b
a
,
,
,
,
,
, are calculated from a mean
square approximation of the corresponding calculated pdfs.
5.1.3 Examples
Typical distributions of
*
v
in simple cases where the uncertainty in only related to the value of
the diffusion coefficient D
P
are plotted in Figures 10 and 11. As a result, Fo* and D
P
* are log
normally distributed, such that log
10
(Fo*) have zero mean and a standard deviation equal to s
D
.
Such a kind of distribution reproduces generally pretty well the uncertainty in D values as
assessed in databases (Vitrac et al., 2006). In addition, according to the Arrhenius relationship, it
is emphasized that an uncontrolled temperature during the storage, which would fluctuate
normally around a mean value, would also lead to lognormally distributed diffusion coefficients.
Figure 10 plots the distributions of Fo corresponding to s
D
=0.2 and different values of Fo . Each
value of the distribution of Fo yields a dimensionless contamination value
*
v
with a relative
weight controlled by its probability of occurrence. The final result is a distribution of
*
v
corresponding to a value of Fo and a value of s
D
. By noticing that the conventional
dimensionless desorption kinetic corresponds to
( )
*
v
g Fo
=
, the uncertainty in Fo can also be
interpreted as an uncertainty envelope, whose the likely value (50
th
percentile) is given