Environmental Engineering Reference
In-Depth Information
The limit solution for
Pe ¼
0 was obtained from the differential equation for no
advection:
@x
D
@
c
@
0 r
@
@x
2
¼
D
c
2
c
@x
l
c ¼
(5.26)
with the general solution:
r
x
v
D
þ C
1
exp
r
x
v
D
cðxÞ¼C
0
exp
(5.27)
In terms of dimensionless parameter and variable this is
cðxÞ¼C
0
exp
Da
2
x
ð
ÞþC
1
exp
ð
Da
2
x
Þ
(5.28)
The free constants
C
0
and
C
1
are again obtained from the boundary conditions.
5.4 Transient Solutions
The analytical solution for the inflow of a front with concentration
c
in
into a region
with concentration
c
0
is given by:
erfc
1
2
erfc
x
vt
2
1
2
exp
vx
D
x
þ
vt
2
p
p
cðx; tÞ¼c
0
exp
ð
l
t
Þ
1
:::
Dt
Dt
þ
(5.29)
erfc
erfc
c
in
2
v
u
2
D
x
x
ut
2
v
þ
u
2
D
x
x
þ
ut
2
þ
exp
p
exp
p
Dt
Dt
l
p
(Wexler
1992
). The solution consists of two parts: the first
describes the decline of the original concentration
c
0
and the second the change of
the inflow concentration
c
in
in the 1D set-up.
In MATLAB
v
2
with
u ¼
þ
4
the solution is to be included in the M-file
'analtrans.m'
, which
was introduced in the previous chapter. The decay parameter
®
l
is added in the input
part of the module:
Then the auxiliary parameter
u
is computed by:
