Environmental Engineering Reference
In-Depth Information
The parameters
m
1
and
m
2
can be obtained by comparison of coefficients in (
5.7
)
and (
5.8
):
m
1
þ
m
2
¼ v=D
m
1
m
2
¼
l
=D
(5.9)
A quadratic equation results, which has the solutions:
p
v
2
1
2
D
m
1
;
2
¼
v
þ
4l
D
(5.10)
Equation
5.8
can now be solved in two steps. First the solution
c
of the equation
@
@x
m
1
c ¼
0
(5.11)
is determined, which is given by
.
C
0
is an integration constant that
is determined below in order to fulfil the boundary conditions. In a second step,
c
is
found as solution of the differential equation
c ¼ C
0
exp
ð
m
1
xÞ
c ¼ c
@
@x
m
2
(5.12)
One obtains a formula for the general solution:
m
2
ðÞC
1
þ C
0
ð
exp
cðxÞ¼
exp
m
1
ðÞ
exp
ð
m
2
x
Þdx
(5.13)
C
0
m
1
m
2
exp
¼ C
1
exp
m
2
ðÞþ
m
1
ðÞ
where
C
1
is the second integration constant. Both
C
0
and
C
1
are determined by the
boundary conditions. The usual condition at the inlet
cðx ¼
0
Þ¼c
in
yields the
condition:
C
1
þ C
0
=
m
1
m
2
ð
Þ ¼ c
in
or
C
1
¼ c
in
C
0
=
m
1
m
2
ð
Þ
. Note that
m
1
and
m
2
, as given by (
5.10
), have opposite signs. As
m
1
is positive, the first term in (
5.13
)
is decreasing with depth, while the second is increasing with depth. For that reason,
the solutions approach infinity for all values
C
0
6¼
0. Vice versa, the function with
C
0
¼
0 is the only solution which guarantees finite concentration for arbitrary high
values of
x
. It is this property which makes the choice
C
0
¼
0 favourable in studies,
where there is no information concerning the downstream boundary condition
(Anderson et al.
1988
; Henderson et al.
1999
). Then the solution simply reads:
cðxÞ¼c
in
exp
m
2
ðÞ
(5.14)
When the second boundary condition requires a vanishing concentration gradi-
ent at depth
L
, i.e.
ð
@c=@x
ÞðLÞ¼
0, the second equation for
C
0
and
C
1
is given by:
