Environmental Engineering Reference
In-Depth Information
The concentration distribution
c
, given by (
16.12
), is a solution of the differential
equation:
@c
@t
¼
@
@x
D
x
@
c
@x
þ
@
@y
D
y
@
c
@y
v
@c
@x
lc
(16.13)
The formula (
16.12
) can be derived from the statistical formulation (
16.9
) with
the help of the relationship between diffusivities as transport parameters on one side
and standard deviations as statistical characteristics on the other side:
p
2
D
x
t
s
x
¼
2
D
y
p
, already encountered above. Various more complex models are
based on more general relationships between these two types of parameters. Smith
(
1989
) gives an overview and more details on this topic.
and
s
y
¼
16.3
2D Constant Line Source
The 2D steady state analytical solution, describing the effect of a constant line
source on an infinitely extended plane, is given by:
K
0
v
2
x
2
4
D
x
þ
v
2
y
2
4
D
x
D
y
c
0
Q
xv
2
D
x
cðx; yÞ¼
p
D
x
D
y
exp
(16.14)
2
p
(see: Fried
1975
;Bear
1976
) with
K
0
being the modified Bessel function of the
second kind and zero order.
The formula can easily be implemented in MATLAB
. The Bessel function of
®
second kind is reached by the command
where
nu
is the order and
Z
the array of arguments.
16.4
3D Instanteneous Source
The effect of diffusion, advection in a constant unidirectional flow field, and decay
is given by the analytical solution:
!
!
2
M
ðx vtÞ
y
2
D
y
þ
z
2
D
z
1
4
t
cðx; y; z; tÞ¼
exp
þ
lt
p
4
3
p
D
x
D
y
D
z
D
x
pt
(16.15)
(see also: Kinzelbach
1987
, Wexler
1992
), which can be derived in analogy to
(
16.7
) and (
16.12
). There are many applications of this equation, mainly for
pollution spreading in the atmosphere. Richter and Seppelt (
2004
) apply the