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Air Pollution and Turbulence: Modeling and Applications
to be encouraging. This model can handle even nonhomogeneous meteorological
input as nonhomogeneity is accounted for indirectly by following the trajectory of
the plume.
6.4.1.2 Steady-State Models
a. Steady-state concentration is obtained by integrating the concentration
obtained from the unsteady models with respect to time t from 0 to ∞
∞
=
∫
0
( , , )
( , , , )d
S
C x y z
C x y z t t
(6.20)
By integrating the unsteady solution (Equation 6.13) with respect to time,
the steady-state concentration of a pollutant can be given as
(
)
⎡
⎤
⎛
⎞
⎛
⎞
⎛
⎞
=
−
+
−
⎢
⎥
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
π
⎣
⎦
1
2
1 2
1 2
1 2
1
2
1
1
( , , )
exp
exp
exp
2
2
2
4
x
x
x
x y z
Q
U x
U
U
C x y z
f
f
K
f
K
f
K
K K K
(6.21)
where
(
)
⎡
⎤
+
⎢
⎥
=
+
+
⎢
⎥
⎣
⎦
1 2
2
2
2
1
s
x
y
z
z H
x
y
f
K
K
K
and
(
)
⎡
⎤
−
⎢
⎥
=
+
+
⎢
⎥
⎣
⎦
1 2
2
2
2
2
s
x
y
z
z H
x
y
f
K
K
K
This solution was essentially derived for the fi rst time by Roberts (1923).
The subscript “s” is suppressed in the relation (Equation 6.21).
b. In case (a), it was mentioned that the steady-state solution can be obtained
by integrating the unsteady solution with respect to time from 0 to ∞. This
solution can also be obtained (Sharan et al., 1995b) by taking the steadystate advection diffusion equation:
∂
∂
∂
∂
=
+
+
+ δ ( )δ ( )δ −
∂
∂
∂
∂
2
2
2
2
2
2
C
(
)
x
y
z
s
C
C
C
U
K
K
K
Q x y z H
x
x
y
z
(6.22)
with the boundary conditions:
→
→ ∞
0, as , ,
C
x y z
(6.23a)
∂
−
=
=
∂
0 at
0
z
C
K
z
z
(6.23b)
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