|Advection-diffusion equation; Laplace transform;
Fourier transform; Bessel function
|Environmental problems caused by the huge development and
the big progress in industrial, which cause's a lot of pollutions. The
transport of these pollutants can be adequately described by the
advection–diffusion equation. In the last few years, there has been
increased research interest in searching for analytical solutions for
the advection–diffusion equation (ADE). Therefore, it is possible to
construct a theoretical model for the dispersion from a continuous
point source from an Eulerian perspective, given adequate boundary
and initial conditions and the knowledge of the mean wind velocity
field and of the concentration turbulent fluxes. The exact solution of
the linear advection–dispersion (or diffusion) transport equation for
both transient and steady-state regimes has been obtained .
|The two- and three-dimensional advection–diffusion equation
with spatially variable velocity and diffusion coefficients has been
provided analytically . A mathematical treatment has been proposed
for the ground level concentration of pollutant from the continuously
emitted point source . Essa and El-Otaify studied a mathematical
model for hermitized atmospheric dispersion (self adjoined by itself) in
low winds with eddy diffusivities as linear functions of the downwind
distance . More recently the generalized analytical model describing
the crosswind-integrated concentrations is presented . Also, an
analytical scheme is described to solve the resulting two dimensional
steady-state advection–diffusion equation for horizontal wind speed as
a generalized function of vertical height above the ground and eddy
diffusivity as a function of both downwind distance from the source
and vertical height.
|On the other hand, the literature presents several methods to
analytically solve the partial differential equations governing transport
phenomena [6-10]. For example, the method of separation-of-variables
is one of the oldest and most widely used techniques. Similarly, the
classical Green’s function method can be applied to problems with
source terms and inhomogeneous boundary conditions on finite, Semiinfinite,
and infinite regions [10,11]. Integral transform techniques,
such as the Laplace and Fourier transform methods, employ a
mathematical operator that produces a new function by integrating the
product of an existing function and a kernel function between suitable
|In this study, we obtained a mathematical model for dispersion of
air pollutants in moderated winds by taking into account the diffusion
in vertical height direction and advection along the mean wind. The
eddy diffusivity is assumed to be power law in the vertical length. We
provided analytical solutions to the advection–diffusion equation for
three-dimensional with the physically relevant boundary conditions.
The moderate data collected during the convective conditions. From
nine experiments conducted at Inshas site, Cairo-Egypt , which used
to investigate the analytical solution.
|The dispersion of pollutants in the atmosphere is governed by
the basic atmospheric diffusion equation. Under the assumption of
incompressible flow, atmospheric diffusion equation based on the
Gradient transport theory can be written in the rectangular coordinate
|where C(x, y, z) is the mean concentration of a pollutant (Bq/m3),
(μg/m3) and (ppm); in which t is the time, S and R are the source and
removal terms, respectively; (u, v, w) and (kx, ky, kz) are the components
of wind and diffusivity vectors in x, y and z directions, respectively, in
an Eulerian frame of reference.
|The following assumptions are made in order to simplify equation (1):
|1) Steady -state conditions are considered, i.e., ∂C/∂t = 0
|2) We are going to study Eq. (1) in case, when the components of wind (v, w) tends to zero
|3) Source and removal (physical/chemical) pollutants are ignored
so that S=0 and R=0
|4) Under the moderate to strong winds, the transport due
advection dominates over that due to longitudinal diffusion:
|With the above assumptions, equation (1) reduces to:
|Under the following boundary conditions:
|where δ (…) is Dirac's delta function, and h is the mixing height.
The wind speed u and eddy diffusivity ky, kz is expressed as a
functions of power law of z as:
|Where α ,β ,m,n are turbulence parameters and depend on
|The Analytical Solution
|Eq. (2) can solve analytically as follows:
|Transform the variable x to s by applying the Laplace transform on
Eq. (2) to become
|Again transform the variable y to λ by applying the Fourier
transform on Eq. (8) to become
|Eq. (9) simplified to the form
|Now we will solve the homogeneous equation of Eq. (10) which
takes the form
|Transform then Eq. (11) becomes
|Again transform then Eq. (12) become
|But Eq. (13) is modified Bessel equation which has solution .
|where A and B are constant
|Now the general solution of the non-homogeneous Eq. (10) takes
|where A* and B* are constants
|Apply the boundary condition Eq. (3) on Eq. (16) which become
|Apply the boundary condition Eq. (5) on Eq. (16) which gives
|Substitute B* Eq. (16) in Eq. (17) which gives:
|Apply inverse Fourier and inverse Laplace respectively on Eq. (19)
|The diffusion data for the estimating were gathered during 135I
isotope tracernine experiments in moderate wind with unstable
conditions at Inshas, Cairo. During each run, the tracer was released
from source has height 43 m for twenty four hours working, where
the air samples were collected during half hour at a height 0.7 m.
We collected air samples from 92 m to 184 m around the source in
AEA, Egypt. The study area is flat, dominated by sandy soil with poor
vegetation cover. The air samples collected were analyzed in Radiation
Protection Department, NRC, AEA, Cairo, Egypt using a high volume
air sampler with 220 V/50 Hz bias . Meteorological data have been
provided by the measurements done at 10 m and 60 m.
|For the concentration computations, we require the knowledge of
wind speed, wind direction, source strength, the dispersion parameters,
mixing height and the vertical scale velocity. Wind speeds are greater
than 3 m/s most of the time even at 10 m level. Further the variation
wind direction with time is also visible. Thus in the present study, we
have adopted dispersion parameters for urban terrain which are based
on power law functions. The analytical expressions depend upon
downwind distance, vertical distance and atmospheric stability. The
atmospheric stability has been calculated from Monin-Obukhov length
scale (1/L)  based on friction velocity, temperature, and surface
|Results and Discussion
|The concentration is computed using data collected at vertical
distance of a 30 m multi-level micrometeorological tower. In all
a test runs were conducted for the purpose of computation. The
concentration at a receptor can be computed in the following way:
|Applying formula Eq. (21) which contains eddy diffusivities as
function with power law at y = 0.0 for half hourly averaging.
|As an illustration, results computed from these approaches are
shown in Table 1, for nine typical tests conducted at Ins has site, Cairo-Egypt . This table shows that the observed and predicted
concentrations for 135I using Eq. (20) with power law of eddy diffusivities
and the wind speed are very near to each other of 135I.
|Figure 1 shows the variation of predicted and observed
concentration of 135I with the downwind distance. One gets very good
agreement between observed and predicted concentration.
|Figure 2 shows that the predicted concentrations which are
estimated from Eq. (20) are a factor of two with the observed
|Now, the statistical method is presented and comparison among
analytical, statically and observed results will be offered . The
following standard statistical performance measures that characterize
the agreement between prediction (Cp = Cpred) and observations (Co=Cobs):
|1. Normalized mean square error (NMSE), It is an estimator of
the overall deviations between predicted and observed concentrations.
Smaller values of NMSE indicate a better model performance. It is
|2. Fractional bias (FB): It provides information on the tendency
of the model to overestimate or underestimate the observed
concentrations. The values of FB lie between -2 and +2 andit has a value
of zero for an ideal model. It is expressed as:
|3. Correlation coefficient (R): It describes the degree of association
between predicted and observed concentrations and is given by:
|4. Fraction within a factor of two (FAC2) is defined as:
|FAC2 = fraction of the data for which
|0.5 ≤ (Cp/Co) ≤ 2
|Where σp and σo are the standard deviations of Cp and Co respectively.
Here the over bars indicate the average over all measurements (Nm). A
perfect model would have the following idealized performance: NMSE
= FB = 0 and COR = FAC2 = 1.0
|From the statistical method of Table 2, we find that the predicted
concentrations for 135I lie inside factor of 2 with observed data.
Regarding to NMSE, FB and COR the predicted concentrations for 135I
are better with observed data.
|In this paper, a steady-state three-dimensional mathematical model
for the dispersion of pollutants from a continuously emitting ground
point source in moderated winds is formulated. Besides advection
along the mean wind, the model takes into account the diffusion in
crosswind and vertical directions. The eddy diffusivity and the wind
speed are assumed to be power law in the vertical height z.
|The closed form analytical solution of the proposed problem has
obtained using the methods of Laplace and Fourier transforms.
|In general, the present model is compared with data collected
from nine experiments conducted at Inshas, Cairo (Egypt). One
gets the predicted concentrations are in a best agreement with the
corresponding observation. Moreover, the Statistical results here are in
agreement with the analytical results.
|This work has been completed by the support of Egyptian Atomic Energy
Authority, and the authors thank for this support. The First author’s thank is
extended to all members of Mathematics and theoretical Physics for providing the
experimental data of 135I.
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