Environmental Engineering Reference
In-Depth Information
The diffusivity
D
is to be estimated based on a measurement within the barrier.
The condition for the best estimate is that the following function
e
is minimized:
X
cðx
fit
Þc
fit
2
e ¼
(10.26)
Like the previous tasks the problem is solved by considering the derivative
@e=@D
, which is required to be zero for the best fit diffusivity:
2
X
cðx
fit
Þc
fit
@c
@D
ðx
fit
Þ¼
@e
@D
¼
0
(10.27)
@c=@D
, which appears in (
10.27
), fulfills a differential equation
that is obtained by differentiation of (
10.25
) with respect to
D
:
The derivative
@
@x
D
@
@x
@c
@D
þ
@
c
¼
0
(10.28)
@x
or, taking again (
10.25
) into account:
@
@x
D
@
@x
@
c
Q
D
¼
(10.29)
@D
In order to test the approach we can work with analytical solutions of (
10.25
) and
(
10.29
). The general solution for
c
is given by:
Q
2
D
x
2
cðxÞ¼
þ C
1
x þ C
0
(10.30)
With boundary conditions
@c
@x
ð
cð
0
Þ¼
1
and
(10.31)
1
Þ¼
0
one obtains:
C
0
¼
1
and
C
1
¼ Q=D
. Analogously, the general solution of
(
10.29
) is given by:
@c
@D
ðxÞ¼
Q
2
D
2
x
2
þ D
1
x þ D
0
(10.32)
As the Dirichlet condition for
c
at the left boundary is independent of
D
one
obtains the boundary conditions:
@c
@D
ð
@x
@c
@
0
Þ¼
0
and
@D
ð
1
Þ¼
0
(10.33)
