134
Air Pollution and Turbulence: Modeling and Applications
the techniques utilized to calculate the “sigmas” as a function of atmospheric stability and the downwind distance from the emission source. Two basic techniques can
be identifi ed as serving this purpose: the fi rst employs adimensional functions built
on the basis of available measurements of turbulent intensity and the second adopts
semiempirical functions for “sigmas” built for each stability class with which atmospheric turbulence has been schematized.
Analytic models can be considered an intermediate stage between K and Gaussian
models. They conserve the simplicity of the latter, in that the concentration fi eld is
described by a simple formula, but, at the same time, they are also able to confront,
in a theoretically correct way, situations in which the wind and turbulent diffusion
coeffi cient vary with height.
5.3.1 EULERIAN APPROACH: K MODELS
Eulerian models are the most suitable for tackling problems of greater complexity, for
example, the dispersion of pollutants over complex terrain or the diffusion of noninert pollutants. They are based on the resolution, on a fi xed spatial-temporal grid, of
the equation of mass conservation of the pollutant chemical species, expressed in
terms of concentration c(x, y, z, t) (Zannetti, 1990):
∂ = − ⋅∇ + ∇ +
∂
2
c
c D c S
t
u
(5.1)
where
u is the wind speed vector of the components u, v, w
D∇ 2 c is the molecular diffusion term (generally neglected), with D the molecular
diffusion coeffi cient
S is the term referring to the source, measuring the emission intensity and representing the pollutant removal kinetic
∇ is the gradient operator
∇ 2 is the Laplacian
In order to resolve Equation 5.1, it is necessary to know the wind fi eld u, something
that is not possible since it is extremely variable in space and time, from the scale of
centimeters to kilometers. Consequently, wind is divided into two parts:
–
u: The so-called ensemble average
u′: The turbulent fl uctuations of wind at mean nil
Thereupon the wind speed is expressed as the sum of the two components, mean
and turbulent:
= + ′
u u u
(5.2a)
The same considerations can be made for c. Therefore:
c c c
= + ′
(5.2b)
The ensemble average refers to the mean value obtained by the repetition of many
experiments in the same meteorological and emission conditions.
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
the techniques utilized to calculate the “sigmas” as a function of atmospheric stability and the downwind distance from the emission source. Two basic techniques can
be identifi ed as serving this purpose: the fi rst employs adimensional functions built
on the basis of available measurements of turbulent intensity and the second adopts
semiempirical functions for “sigmas” built for each stability class with which atmospheric turbulence has been schematized.
Analytic models can be considered an intermediate stage between K and Gaussian
models. They conserve the simplicity of the latter, in that the concentration fi eld is
described by a simple formula, but, at the same time, they are also able to confront,
in a theoretically correct way, situations in which the wind and turbulent diffusion
coeffi cient vary with height.
5.3.1 EULERIAN APPROACH: K MODELS
Eulerian models are the most suitable for tackling problems of greater complexity, for
example, the dispersion of pollutants over complex terrain or the diffusion of noninert pollutants. They are based on the resolution, on a fi xed spatial-temporal grid, of
the equation of mass conservation of the pollutant chemical species, expressed in
terms of concentration c(x, y, z, t) (Zannetti, 1990):
∂ = − ⋅∇ + ∇ +
∂
2
c
c D c S
t
u
(5.1)
where
u is the wind speed vector of the components u, v, w
D∇ 2 c is the molecular diffusion term (generally neglected), with D the molecular
diffusion coeffi cient
S is the term referring to the source, measuring the emission intensity and representing the pollutant removal kinetic
∇ is the gradient operator
∇ 2 is the Laplacian
In order to resolve Equation 5.1, it is necessary to know the wind fi eld u, something
that is not possible since it is extremely variable in space and time, from the scale of
centimeters to kilometers. Consequently, wind is divided into two parts:
–
u: The so-called ensemble average
u′: The turbulent fl uctuations of wind at mean nil
Thereupon the wind speed is expressed as the sum of the two components, mean
and turbulent:
= + ′
u u u
(5.2a)
The same considerations can be made for c. Therefore:
c c c
= + ′
(5.2b)
The ensemble average refers to the mean value obtained by the repetition of many
experiments in the same meteorological and emission conditions.
© 2010 by Taylor and Francis Group, LLC