Analytical Models for the Dispersion of Pollutants in Low Wind Conditions 159
R is the removal term that represents pollutant removal by chemical reaction,
gravity fallout (settling), etc.
K x , K y , and K z are the eddy diffusivities along x, y, and z directions, respectively
On the left-hand side of Equation 6.1, the fi rst term is the time-dependent term
accounting for nonstationary situations and the remaining three terms describe transport due to advection whereas on the right-hand side, the fi rst three terms represent
the turbulent diffusion. Equation 6.1 forms the basis for most air pollution dispersion
models and allows for the anisotropic diffusion and for the variation of diffusivities
as a function of concentration, time, and spatial coordinates. The justifi cation of
Equation 6.1 for mean concentration is described in Csanady (1973), Seinfeld (1986),
and Arya (1999).
To completely specify a mathematical problem and in order to solve Equation
6.1, one needs to assign the physically relevant boundary and initial conditions
depending on the nature of the physical problem. In general, there are three types of
boundary conditions (Crank, 1976): (1) the Dirichlet type in which the concentration
of the pollutant is prescribed on the boundary, (2) the Neumann type fl ux condition
in which the concentration fl ux normal to the boundary is prescribed, and (3) the
mixed type in which the concentration fl ux across the boundary is proportional to
the difference in the concentrations between the boundary and outside medium. The
constant of proportionality is related to the permeability of the boundary. All these
types of boundary conditions along with their physical interpretation are discussed
in the literature (Crank, 1976; Carslaw and Jaeger, 1986).
The initial conditions to be prescribed are generally expressed in terms of background concentration. Although precise background concentration is normally not
available, one can consider arbitrary functional form in terms of spatial coordinates.
6.3 ANALYTICAL MODELS
Analytical models are not only indispensable tools to predict the concentration of air
pollutants with high mathematical accuracy but are also the easiest way to describe the
unstructured temporal and spatial behavior of the pollutants in the atmosphere. Although
it is not always possible to fi nd an analytical solution for every dispersion model, even
for some particular form of wind velocities and eddy diffusivities. The treatment of
the dispersion of pollutants can be described by two timescales: (1) the duration (t s ) for
which the source is released and (2) travel time (t r ) from the source to the receptor. The
release can be treated as a continuous release if t s > t r , otherwise it is instantaneous.
Accordingly, a continuous release (plume) or instantaneous release (puff) model may be
used for the dispersion of a pollutant emitted from a source (Hanna et al., 1982; Yadav,
1995). These are also often known as the steady- and unsteady- (time-dependent) state
models. Here, we describe the features of the various analytical dispersion models.
6.3.1 GAUSSIAN PLUME MODEL
A simple analytical model leading to the widely used Gaussian model has been
derived by taking the following assumptions in Equation 6.1:
© 2010 by Taylor and Francis Group, LLC
R is the removal term that represents pollutant removal by chemical reaction,
gravity fallout (settling), etc.
K x , K y , and K z are the eddy diffusivities along x, y, and z directions, respectively
On the left-hand side of Equation 6.1, the fi rst term is the time-dependent term
accounting for nonstationary situations and the remaining three terms describe transport due to advection whereas on the right-hand side, the fi rst three terms represent
the turbulent diffusion. Equation 6.1 forms the basis for most air pollution dispersion
models and allows for the anisotropic diffusion and for the variation of diffusivities
as a function of concentration, time, and spatial coordinates. The justifi cation of
Equation 6.1 for mean concentration is described in Csanady (1973), Seinfeld (1986),
and Arya (1999).
To completely specify a mathematical problem and in order to solve Equation
6.1, one needs to assign the physically relevant boundary and initial conditions
depending on the nature of the physical problem. In general, there are three types of
boundary conditions (Crank, 1976): (1) the Dirichlet type in which the concentration
of the pollutant is prescribed on the boundary, (2) the Neumann type fl ux condition
in which the concentration fl ux normal to the boundary is prescribed, and (3) the
mixed type in which the concentration fl ux across the boundary is proportional to
the difference in the concentrations between the boundary and outside medium. The
constant of proportionality is related to the permeability of the boundary. All these
types of boundary conditions along with their physical interpretation are discussed
in the literature (Crank, 1976; Carslaw and Jaeger, 1986).
The initial conditions to be prescribed are generally expressed in terms of background concentration. Although precise background concentration is normally not
available, one can consider arbitrary functional form in terms of spatial coordinates.
6.3 ANALYTICAL MODELS
Analytical models are not only indispensable tools to predict the concentration of air
pollutants with high mathematical accuracy but are also the easiest way to describe the
unstructured temporal and spatial behavior of the pollutants in the atmosphere. Although
it is not always possible to fi nd an analytical solution for every dispersion model, even
for some particular form of wind velocities and eddy diffusivities. The treatment of
the dispersion of pollutants can be described by two timescales: (1) the duration (t s ) for
which the source is released and (2) travel time (t r ) from the source to the receptor. The
release can be treated as a continuous release if t s > t r , otherwise it is instantaneous.
Accordingly, a continuous release (plume) or instantaneous release (puff) model may be
used for the dispersion of a pollutant emitted from a source (Hanna et al., 1982; Yadav,
1995). These are also often known as the steady- and unsteady- (time-dependent) state
models. Here, we describe the features of the various analytical dispersion models.
6.3.1 GAUSSIAN PLUME MODEL
A simple analytical model leading to the widely used Gaussian model has been
derived by taking the following assumptions in Equation 6.1:
© 2010 by Taylor and Francis Group, LLC