Analytical Models for the Dispersion of Pollutants in Low Wind Conditions 169
energetic turbulent eddies. In the early stage of plume dispersion from a point source,
Taylor’s (1921) statistical theory of diffusion and dimensional analysis suggest that
the eddy diffusion coeffi cients (K) may be taken as linear functions of downwind distance. Scale analysis of atmospheric advection–diffusion also supports this for near
source dispersion and accordingly, the eddy diffusion coeffi cients are proportional to
Ux (product of mean wind speed and the downwind distance) and can be written as
= α
= β
= γ
,
,
x
y
z
K
Ux K
Ux K
Ux
(6.28)
where α, β, and γ are constants of proportionality and represent the turbulence
parameters. This parameterization accounts for the stability through these turbulent parameters, which essentially represent turbulent intensities (Arya, 1995). Arya
(1995) has also argued that for small travel times (i.e., small distances from the
source) diffusivities can be expressed as linear functions of downwind distance.
Sharan et al. (1996a) formulated a variable K-model for the dispersion of air pollutants in low wind conditions by taking the linear functional form (Equation 6.28)
of eddy diffusivity coeffi cients. After introducing an elevated source of height H s
through x = 0 plane, the analytical solution of modifi ed Equation 6.1 with assumptions
(a) through (c) and (e) in Section 6.3.1, the eddy diffusivity coeffi cients (Equation 6.28)
and boundary conditions:
=
→∞
( , , ) 0, , ,
C x y z
x y z
(6.29a)
∂
−
=
=
∂
0,
0
z
C
K
z
z
(6.29b)
(
)
= δ( )δ −
(0, , )
s
UC y z
Q y z H
(6.29c)
is obtained using the integral transform (Sharan et al., 1996a; Sharan and Yadav,
1998) and given as
+
−
=
+
⎡
⎤
⎣
⎦
π βγ
2
( , , )
2
S
S
z H
z H
Q
C x y z
F
F
U
x
(6.30)
where
(
)
( )
− μ+
+
⎡
⎤
⎛
⎞
+
α
⎢
⎥
= +
+
⎜
⎟
β
γ
⎢
⎥
⎝
⎠
⎣
⎦
1
2
2
2
1
S
s
z H
z H
y
F
x
and
(
)
( )
− μ+
−
⎡
⎤
⎛
⎞
−
α
⎢
⎥
= +
+
⎜
⎟
β
γ
⎢
⎥
⎝
⎠
⎣
⎦
1
2
2
2
1
S
s
z H
z H
y
F
x
in which μ = α
1
2
.
© 2010 by Taylor and Francis Group, LLC
energetic turbulent eddies. In the early stage of plume dispersion from a point source,
Taylor’s (1921) statistical theory of diffusion and dimensional analysis suggest that
the eddy diffusion coeffi cients (K) may be taken as linear functions of downwind distance. Scale analysis of atmospheric advection–diffusion also supports this for near
source dispersion and accordingly, the eddy diffusion coeffi cients are proportional to
Ux (product of mean wind speed and the downwind distance) and can be written as
= α
= β
= γ
,
,
x
y
z
K
Ux K
Ux K
Ux
(6.28)
where α, β, and γ are constants of proportionality and represent the turbulence
parameters. This parameterization accounts for the stability through these turbulent parameters, which essentially represent turbulent intensities (Arya, 1995). Arya
(1995) has also argued that for small travel times (i.e., small distances from the
source) diffusivities can be expressed as linear functions of downwind distance.
Sharan et al. (1996a) formulated a variable K-model for the dispersion of air pollutants in low wind conditions by taking the linear functional form (Equation 6.28)
of eddy diffusivity coeffi cients. After introducing an elevated source of height H s
through x = 0 plane, the analytical solution of modifi ed Equation 6.1 with assumptions
(a) through (c) and (e) in Section 6.3.1, the eddy diffusivity coeffi cients (Equation 6.28)
and boundary conditions:
=
→∞
( , , ) 0, , ,
C x y z
x y z
(6.29a)
∂
−
=
=
∂
0,
0
z
C
K
z
z
(6.29b)
(
)
= δ( )δ −
(0, , )
s
UC y z
Q y z H
(6.29c)
is obtained using the integral transform (Sharan et al., 1996a; Sharan and Yadav,
1998) and given as
+
−
=
+
⎡
⎤
⎣
⎦
π βγ
2
( , , )
2
S
S
z H
z H
Q
C x y z
F
F
U
x
(6.30)
where
(
)
( )
− μ+
+
⎡
⎤
⎛
⎞
+
α
⎢
⎥
= +
+
⎜
⎟
β
γ
⎢
⎥
⎝
⎠
⎣
⎦
1
2
2
2
1
S
s
z H
z H
y
F
x
and
(
)
( )
− μ+
−
⎡
⎤
⎛
⎞
−
α
⎢
⎥
= +
+
⎜
⎟
β
γ
⎢
⎥
⎝
⎠
⎣
⎦
1
2
2
2
1
S
s
z H
z H
y
F
x
in which μ = α
1
2
.
© 2010 by Taylor and Francis Group, LLC