Environmental Engineering Reference
In-Depth Information
The formula (
7.17
) is introduced in the first equation of system (
7.16
):
1
n
1
2
c
a
L
v
@
@x
v
@c
g
d
c
cþ
b
c
cþ
b
¼
@x
a
0
(7.18)
which is a single differential equation of second order. As an equivalent system of
two differential equations one obtains:
@
c
@x
¼ c
0
@c
0
@x
¼
1
n
c
0
a
L
þ
a
g
d
c
cþ
b
c
cþ
b
1
(7.19)
a
L
v
In cases in which dispersive fluxes are small, one obtains a single first order
differential equation:
1
n
@
c
@x
¼
v
g
d
c
cþ
b
c
cþ
b
1
(7.20)
or
n= nð Þ
@
c
@x
¼
c
cþ
b
(7.21)
1
= n
ð Þ
1
¼
ag
with
. Formula (
7.21
) can be re-written as:
1
=ðn
1
Þ
v
d
n= nð Þ
dc ¼ðxx
0
Þ
ð
þ
c
1
(7.22)
For
n ¼
2(
7.22
) can be integrated analytically. One obtains:
2
c
c
in
2
1
ðcÞ
b
ðc
in
Þþ
b
xx
0
¼
cþ
2
b
log
2
b
log
(7.23)
c
in
where
c
in
is the inflow concentration at
x ¼ x
0
. Expression (
7.23
) is an implicit
formula for
c
as function of
x
for given parameters and boundary condition.
Provided a value for
x
is known, the corresponding
c
can be determined by
a zero-finding algorithm. See Sidebar 7.2 for an application using the MATLAB
®
fzero
command.
