174
Air Pollution and Turbulence: Modeling and Applications
6.4.5 MODELS BASED ON CROSSWIND INTEGRATED CONCENTRATIONS
In another approach, the steady-state concentration in three-dimensional domain
can be described by assuming the Gaussian concentration distribution in crosswind
direction and is given as
(
)
2
2
exp
2
( , , )
( , )
2
y
y
y
y
C x y z
C x z
−
σ
=
πσ
(6.37)
where C y is the crosswind integrated concentration and can be obtained from the
equation:
(
)
∂
∂
∂
∂
∂
⎛
⎞
⎛
⎞
=
+
+ δ () δ −
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∂
∂
∂
∂
∂
y
y
y
x
z
s
C
C
C
U
K
K
Q x z H
x
x
x
z
z
(6.38)
Equation 6.38 can also be obtained from Equation 6.1 with assumptions (a)–(c) and
(e) (Section 6.3.1) and integrating it with respect to y from −∞ to ∞. This fact has been
exploited to develop the models for the behavior of dispersion in low wind conditions. Equation 6.38 is a second order two-dimensional partial differential equation
and is relatively easier to solve for the realistic parameterization of eddy diffusivities
and wind speed.
On the basis of this approach, Moreira et al. (2005) have presented a steadystate mathematical model for dispersion of contaminants in low wind conditions in
a fi nite layer. For accounting the spatial dependences of U and K, they have divided
the domain into multi-layers in vertical and downwind directions. In each layer, the
eddy diffusivities and wind speed are assumed to be constant by taking their averaged values. Then the closed form analytical solution of the transformed equation is
obtained by Laplace transform.
It is noticed through the above discussion that the behavior of dispersion of
pollutants in low wind conditions is non-Gaussian. These models can be extended by
relaxing the various assumptions. Most of these models have been evaluated with the
available diffusion data in both stable conditions such as Hanford (Nichola, 1977)
and Idaho National Engineering Laboratories (Sagendorf and Dickson, 1974), and
convective conditions such as IIT diffusion experiment (Singh et al., 1991), EPRI
plume validation experiment (Hudischewskyj and Reynolds, 1983), and Copenhagen
experiment (Gryning, 1981). In addition, uncertainties are associated with the
parameterization of dispersion parameters used in the dispersion models in low wind
conditions.
Diffusion data are required for evaluation of mathematical models and empirically estimating the dispersion or diffusion parameters. The diffusion data available
for the validation of these models in low wind conditions is limited. In the last 45
years, considerable efforts were devoted to conduct the diffusion experiments such
as Prairie Grass (Barad, 1958), Idaho National Engineering Laboratories (Sagendorf
and Dickson, 1974), low wind diffusion experiment in Japan (Adachi and Ohta,
1978), and IIT diffusion experiment (Singh et al., 1991). These experiments differ
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
6.4.5 MODELS BASED ON CROSSWIND INTEGRATED CONCENTRATIONS
In another approach, the steady-state concentration in three-dimensional domain
can be described by assuming the Gaussian concentration distribution in crosswind
direction and is given as
(
)
2
2
exp
2
( , , )
( , )
2
y
y
y
y
C x y z
C x z
−
σ
=
πσ
(6.37)
where C y is the crosswind integrated concentration and can be obtained from the
equation:
(
)
∂
∂
∂
∂
∂
⎛
⎞
⎛
⎞
=
+
+ δ () δ −
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∂
∂
∂
∂
∂
y
y
y
x
z
s
C
C
C
U
K
K
Q x z H
x
x
x
z
z
(6.38)
Equation 6.38 can also be obtained from Equation 6.1 with assumptions (a)–(c) and
(e) (Section 6.3.1) and integrating it with respect to y from −∞ to ∞. This fact has been
exploited to develop the models for the behavior of dispersion in low wind conditions. Equation 6.38 is a second order two-dimensional partial differential equation
and is relatively easier to solve for the realistic parameterization of eddy diffusivities
and wind speed.
On the basis of this approach, Moreira et al. (2005) have presented a steadystate mathematical model for dispersion of contaminants in low wind conditions in
a fi nite layer. For accounting the spatial dependences of U and K, they have divided
the domain into multi-layers in vertical and downwind directions. In each layer, the
eddy diffusivities and wind speed are assumed to be constant by taking their averaged values. Then the closed form analytical solution of the transformed equation is
obtained by Laplace transform.
It is noticed through the above discussion that the behavior of dispersion of
pollutants in low wind conditions is non-Gaussian. These models can be extended by
relaxing the various assumptions. Most of these models have been evaluated with the
available diffusion data in both stable conditions such as Hanford (Nichola, 1977)
and Idaho National Engineering Laboratories (Sagendorf and Dickson, 1974), and
convective conditions such as IIT diffusion experiment (Singh et al., 1991), EPRI
plume validation experiment (Hudischewskyj and Reynolds, 1983), and Copenhagen
experiment (Gryning, 1981). In addition, uncertainties are associated with the
parameterization of dispersion parameters used in the dispersion models in low wind
conditions.
Diffusion data are required for evaluation of mathematical models and empirically estimating the dispersion or diffusion parameters. The diffusion data available
for the validation of these models in low wind conditions is limited. In the last 45
years, considerable efforts were devoted to conduct the diffusion experiments such
as Prairie Grass (Barad, 1958), Idaho National Engineering Laboratories (Sagendorf
and Dickson, 1974), low wind diffusion experiment in Japan (Adachi and Ohta,
1978), and IIT diffusion experiment (Singh et al., 1991). These experiments differ
© 2010 by Taylor and Francis Group, LLC