Environmental Engineering Reference
In-Depth Information
relevant processes. According to the derivations in Chap. 3 such a situation can be
described by two transport equations:
@
c
A
@t
¼
@
D
@
@x
c
A
vc
A
r
@x
@
c
B
@t
¼
@
D
@
@x
c
B
vc
B
þ r
(4.11)
@x
c
A
and
c
B
denote the concentrations of species
A
and
B
,
D
the dispersivity,
v
the
velocity, and
r
the reaction rates. The system (
4.11
) fulfils the form (
4.10
) with
m ¼
1 and the following functions:
u
f
s
(4.12)
D
@x
c
A
vc
A
D
@x
c
B
vc
B
c
A
c
A
r
þr
10
01
c
¼
¼
¼
¼
Another possible representation is:
u
f
s
D
@x
c
A
D
@x
c
B
v
@x
c
A
r
v
@x
c
B
þ r
10
01
c
A
c
A
¼
¼
¼
¼
c
(4.13)
0. The difference
between both formulations is that in formulation (
4.12
) the flux term f includes
dispersive and advective fluxes, while in formulation (
4.13
) only dispersive fluxes
are included. This difference is also important for the boundary conditions, as is
shown in the following.
For the complete formulation of the mathematical problem initial and boundary
conditions need to be set. Using the terminology introduced above, the initial
condition at time
t
0
is given by the vector equation:
Note that the latter formulation requires the condition
@v=@x ¼
u
x; t
0
ð
Þ¼
u
0
ðxÞ
(4.14)
The boundary condition, valid at locations
x ¼ x
0
and
x ¼ x
n
, is formulated in
MATLAB
®
by
p
þ
q
f
¼
0
(4.15)
The function
p
may to depend on
x
,
t
and
u
, the function
q
on
x
and
t
. Note that
f
is the flux-vector from the differential (
4.10
). At first sight the formulation (
4.15
)
seems rather different from the formulations of boundary conditions given above.
But it turns out that the MATLAB
formulation offers a lot of flexibility and
extended options for all types of conditions. An overview is given in Table
4.1
.
®
