Environmental Engineering Reference
In-Depth Information
C
0
m
1
m
1
m
2
C
1
m
2
exp
m
2
ðÞþ
exp
m
1
ðÞ¼
0
(5.15)
which leads to the solution:
c
in
m
2
m
2
m
1
ð
Þ
exp
m
2
ðÞ
C
0
¼
m
1
exp
m
1
ðÞ
m
2
exp
m
2
ðÞ
(5.16)
c
in
m
1
exp
m
1
ðÞ
C
1
¼
m
1
exp
m
1
ðÞ
m
2
exp
m
2
ðÞ
.
This will be done in the next sub-chapter. Here, we want to point out that the given
procedure can be applied to obtain the solution for different types of boundary
conditions. In all cases the free constants
C
0
and
C
1
in a formula for the general
solution, like in (
5.13
), have to be determined to fulfil the conditions. For Dirichlet
boundary conditions on both sides
cð
With these formulae the solution is complete to be computed in MATLAB
®
0
Þ¼c
in
and
cðLÞ¼c
0
one obtains:
c
in
m
2
m
1
ð
Þ
exp
m
2
ðÞ
C
0
m
1
m
2
C
0
¼
C
1
¼ c
in
(5.17)
exp
m
1
ðÞ
exp
m
2
ðÞ
Higher order decay usually is much more difficult to handle than decay of first
order. In order to tackle more complex formulations MATLAB
®
offers the possi-
bility to use numerical methods for ordinary differential equations. Such methods
are treated in Chap. 9.
5.3 Dimensionless Formulation
In dimensionless form the solution (
5.13
) can be written as:
exp
c
ð
x
Þ
c
in
¼ C
0
exp
C
0
ðÞþ
m
1
1
ðÞ
m
2
(5.18)
with dimensionless depth
x ¼ x=L;
m
1
¼
m
1
L;
m
2
¼
m
2
L
and one integration con-
stant
C
0
.
The parameters m
1
and m
2
can be expressed as function of the dimensionless
P´clet number
Pe ¼ vL=D
(see Chap. 3.5) and the dimensionless second
ohler
1
number
Da
2
¼
l
L
2
Damk
=D
:
€
1
Gerhard Damk
ohler (1908-1944), German chemist.
€
