Environmental Engineering Reference
In-Depth Information
!
!
2
y
2
D
y
Q
1
4
t
ðx vtÞ
cðx; y; z; tÞ¼
þ
lt
exp
p
4
3
p
D
x
D
y
D
z
D
x
pt
"
!
!
#
2
2
ð
z
H
Þ
ðz þ HÞ
exp
þ
exp
(16.17)
4
D
z
t
4
D
z
t
(see also: Kathirgamanathan et al.
2002
; Mitsakou et al.
2003
). Overcamp (
1983
)
describes this method for a meteorological model.
16.5
3D Constant Source
Neglect of diffusion in horizontal direction leads to the differential equation:
@
c
@t
¼
@
@y
D
y
@
c
@y
þ
@
@z
D
z
@c
@z
v
@
c
@x
lc
(16.18)
for which the steady state can be reformulated as:
@
c
@x
¼
@
D
y
v
@
c
@y
þ
@
D
z
v
@
c
@z
v
c
(16.19)
@y
@z
From the analytical point of view the differential (
16.19
) is identical to (
16.13
).
Only parameters and variables have different names. For the latter equation, the
solution was already denoted in (
16.12
). Reformulation of the solution in terms of
the new variables and parameters yields:
y
2
D
y
þ
z
2
D
z
Mv
v
4
x
v
x
px
p
cðx; y; zÞ¼
exp
(16.20)
4
D
x
D
y
In fact, (
16.20
) is the steady state solution for a constant source in 3D space. The
product Q
¼ Mv
in the nominator of the first term on the right side represents the
emission rate in unit [mass/time].
Formula (
16.20
) can be modified to account for a source at height H and a no-
flow surface boundary condition along the line
z ¼
0. The procedure, using an
image source, was already described in Sect.
16.4
. In the same manner a steady state
solution for a constant source in 3D is obtained:
!
2
4
3
5
2
vðzHÞ
exp
v
x
(16.21)
4
xD
z
Q
vy
2
4
xD
y
cðx;y; zÞ¼
px
p
exp
exp
!
4
D
y
D
z
2
vðzþHÞ
þ
exp
4
xD
z
