The Mathematical Modelling of a Fixed Source of Dust

A mathematical model for the diffusion of dust particles emitted from a fixed source is investigated using the atmospheric diffusion equation. This model poses an initial boundary value problem with a second order linear partial differential equation. The steady state case of this problem when the uniform source is situated at ground level was examined by Sharan et al. [1]. The solution of the unsteady case in closed form for a time dependent source is derived. Two special cases, in which the source function of time is explicitly given and special values of the diffusion parameters are taken, are examined in detail. In the case when diffusion is present only in the vertical direction, it is shown that for small times, the particles spread with a front that travels with the speed of the wind. When diffusion is present only in the direction of the wind, there is no discontinuity front and the particles diffuse slowly into the direction of the wind. The solutions for the special cases considered are examined for large values of time. It is found that the solution approaches that of the corresponding steady state solution of the equation.


Introduction
he mathematical model of particles emitted from a fixed source has been investigated [1][2][3][4][5][6][7][8][9][10][11].The study of transport of such particles by wind in the atmosphere is important because it causes problems to the environment [12][13][14][15].Most industrial establishments have factories with chimneys through which the fumes escape into the atmosphere outside the factory.These fumes diffuse into the surroundings causing pollution and forming a health hazard.In arid lands, as in Oman, strong winds carry dust from the ground and transport it, which can cause damage to roads and also dirt in houses [16][17].
The diffusion of particles emitted from a source in the atmosphere is given by the atmospheric diffusion equation [18].* .
.( : where C is the concentration of the particles after time * t , u is the local velocity of the particles, w is the settling velocity, and D is the stress tensor given in a Cartesian coordinates system by The analytical study by Sharan et al. [1] investigated a steady-state model for low wind speeds where the gravitational force is negligible.The air stream moved with a uniform velocity U in the x  direction.They assumed the presence of diffusion components in three coordinate directions, all of which being linearly proportional to the distance along the wind.They concluded that their result is in reasonable agreement with experimental observations.In this paper, we extend the steady state model by Sharan et al. [1] to include the time variation.We assume that the wind speed is low, the diffusion varies linearly with distance along the wind direction, and gravity is ignored.In section 2, the model of the system is formulated.In section 3, the solution is calculated and examined in detail for some special cases of the diffusion parameters and the time dependence of the source.A general discussion of the solution is also presented in this section.Some concluding remarks are made in section 4.

Formulation of the model
The diffusion of dust particles emitted from a fixed source situated on/or above ground level in the atmosphere is governed by the atmospheric diffusion equation (1).This equation can be written in a Cartesian system of coordinates  Assume that the direction of the wind speed is in the * axis x  and the velocity of dust particles is given by ( , 0, 0 ) ( , , ), U u v w  u (3) where ( , , ) u v w is the velocity of dust particles relative to the local wind speed and ( , 0 , 0 ) U is the wind speed.
If we also assume that the components ,, u v w of the dust particles' velocity are very small in comparison with U , and the variations of concentration in all directions are similar, then the advection term in ( which represents the linearized three-dimensional unsteady atmospheric diffusion equation in the absence of gravitational forces.This equation is solved subject to the following initial and boundary conditions ( , , , ) * * * * ( , , 0, ) 0, C x y t z    (10) where the dust particles emanate from a fixed source at * (0, 0, )

ft
The equation ( 6) and the relevant conditions ( 7) -( 10) can be written in the following dimensionless form ( , ,0, ) 0, C x y t z    (15) where , The system ( 11) -( 15) was solved by integral transform methods to obtain the solution for the concentration in the   ,, x y z plane at any time in closed form for a source of general time dependence [10][11].The solution showed that, as well as the position of the source in the vertical direction, the diffusion parameters ,,    play an important role in the spread of the dust particles in the atmosphere.In this paper, we investigate the influence of diffusion parameters in all directions by studying the following two special cases: (1) vertical diffusion 0, 0

Analytical solutions for the model
where () Hs is the Heaviside function, x and Q are defined in (16), and , tz and The solution (17) specifies the concentration at every point   The contours of the solution (19)  /, dx dt U  i.e. the concentration of particles travels with a speed U away from the source.This is illustrated for two sample values of   0,5 h  in Figures 1 and 2, respectively.The source of particles in this case does not depend on the time.For xt  , the concentration decreases away from the source and spreads further from the source as t increases.For xt  , there are no particles.This situation applies for all values of the height of the source above ground level.For small values of the time t , the proximity of the front at xt  to the source causes the particles to diffuse upwards, (Figures 1,2 has no effect on the distribution of the particles.When the source is situated on the ground, the concentration on the ground is strong but as the height increases, the particles spread over a larger area both horizontally and vertically.When t , the distribution of particles approaches the steady state solution (Figures 1(f) and 2(f)).

  
The solution (17) becomes The contours of the solution (20) in the   , xz plane are illustrated for different values of the time t and decay factor  in Figures 3-6, where ( , , ) ( , , ) / .

C x z t C x z t Q 
We note that the strength of the source in this case depends on the coefficient of decay  as well as on the time.Figures 3 and 4     For large values of the time t , the distribution of the particles converges to the steady state solution (Figures 3   and 4(f)).Figures 5 and 6 show the profiles of the concentration in the space for a fixed value of the time 1 t  and different values of the decay factor  for two different values of the height of the source.For small values of ,  the strength of the source is weak and its ability to push the particles far away from the origin is not strong.When  , the spread of the particles approaches the distribution of particles in case (i).Comparison between the two cases of the function   ft shows that the dependence of the source on the time has an influence on the distribution of the particles in space.This applies to all values of the height (Figures 1 -6).x and Q are defined in (16), t is given in (18), (s)  is the Gamma function, and () gt is the inverse Laplace transform of   F  with respect to the time t , given by The behavior of the concentration in this case depends on the source () ft as represented by () gt in This expression gives the solution of this case in closed form.The model does not depend on height, z , or distance .y It then represents a source fixed along the z -axis at origin and having an infinite length ( 0 z  ).The nature of the solution (21) is studied by using two examples of the function ( ). ft In this case, the solution (21) reduces to 11 ( , ) , ; 1, 1 where   , ax  is the Incomplete Gamma function.
For the special cases       in Figure 8.It is clear from Figure 7 that at every point of the space in the ( , ) xt plane, the presence of particles decreases whenever the diffusion parameter  increases.For a fixed value of the distance , x the concentration increases with time t .Moreover, at a specific time t , the concentration of particles decreases when we move further away from the source.This last situation can happen in real life because as we move further away from the source, there will not have been enough force to carry a mass/ large numbers of particles that far.
The values of  chosen for Figure 8 are made for ease of comparison between the solutions ( 22) and ( 23).If we compare the two figures, we see that the computations for both cases are consistent.

  
The solution for this source is given by  This situation must happen in real life because when the coefficient of the decay increases, the strength of the source increases.When the values of  and  are fixed, then the concentration increases as the time increases for a fixed distance .
x Furthermore, at a specific time , t for fixed strength of the source and longitudinal diffusion, the concentration decreases as we move further away from the source.As  increases, the diffusion steadily increases.For large values of the decay factor  , the concentration approaches the steady state solution.Large values of  result in a strong source and whenever these values increase further the strength of the source converges to the steady state in case (i).For a fixed value of decay factor, at every point in the plane the concentration when 2   is more than that when 10.

 
Comparison between the two cases of () ft in the presence of longitudinal diffusion shows that the force of the source in case (i) is stronger than that for the source in case (ii).But the two will be identical for large values of the decay factor ,  for 0 t  , when the strengths of the two sources become equal.The spread of particles in the horizontal direction in case (ii) is weaker than that in case (i) for 0. t  For large values of the time, the unsteady state solution approaches the steady state results obtained by Sharan et al [1].

Conclusion
The diffusion of dust particles emitted from a fixed source in the atmosphere in the absence of the settling velocity has been studied mathematically.The study found the solution of the time-dependent diffusion equation in the presence of a point source whose strength is dependent on time.The solution reduces to simpler forms in special cases where the solution can be obtained in explicit expression.In case (1), diffusion in both the longitudinal and latitudinal directions was neglected.The dependence of the distribution of dust on the time variation of the source is investigated for two different functions.When the source is strong for small times, the solution shows a discontinuity.In case (2), the diffusion in the y and z directions was ignored.The profiles of the solutions showed that the concentration of dust particles in the   , xt plane depends on the parameter of diffusion in the direction of the wind.The strong presence of this parameter led to the distribution of the dust in a larger area.The solution approaches the steady state solution of the system.
of the stress tensor in * * * ,, x y z directions, respectively.
and () s  is Dirac's delta function.It is clear from the condition (9) that the source depends on time * t and its strength depends ,, x z t of the domain.It is clear that the time dependence appears only in the amplitude of the concentration and is absent in the exponential dependence.Moreover, the presence of the Heaviside unit function in the amplitude of the solution represents the discontinuity in the solution across the line xt  in the   , xt plane.The solution (17) is illustrated by two examples of the source () ft, which are: (i) Heaviside function () Ht, and (ii) exponential function 1 , 0 t e     .The aim for choosing these specific examples is to examine the effect of the strength of the source as time varies.
(a), (b), (c)).For large values of the time , t the discontinuity at xt  illustrate the profiles of the solution for a fixed value of 10   and different values of the time t with two different heights of the source.It is clear that there is no discontinuity in the distribution of the particles in the ( , ) xz plane whatever the values of the time and height.

Figure 1 .
Figure 1.The isolines of the concentration ( , , ) ( , , ) / C x z t C x z t Q  in the ( , ) xz plane when ( ) ( ) f t H t  and 0 h  , for different values of the time: (a) 0.1 t  , (b) 0.4 t  , (c) 1 t  , (d) 5 t  , (e) 7.5 t  , and (f) 10 t  .Note the precipitation of the particles on the ground as the time increases.Note the position of the characteristic xt  as t is increased.

Figure 2 .
Figure 2. The isolines of the concentration ( , , ) ( , , ) / C x z t C x z t Q  in the ( , ) xz plane when ( ) ( ) f t H t  and 5 h  , for different values of the time: (a) 0.1 t  , (b) 0.4 t  , (c) 1 t  , (d) 5 t  , (e) 7.5 t  , and (f) 10. t  Compare Figures 1 and 2 to notice the influence of increasing the height of an industrial chimney.

Figure 3 .
Figure 3.The isolines of the concentration ( , , ) C x z t in the ( , ) xz plane when ( ) 1

Figure 4 .
Figure 4.The isolines of the concentration ( , , ) C x z t in the ( , ) xz plane when ( ) 1

Figure 5 .
Figure 5.The isolines of the concentration ( , , ) C x z t in the ( , ) xz plane when ( ) 1

Figure 6 .
Figure 6.The isolines of the concentration ( , , ) C x z t in the ( , ) xz plane when ( ) 1

3. 2 .
Case (2): Solution in the case of longitudinal diffusion ( are absent.The solution in this case is given by These asymptotic values confirm the initial and boundary conditions for ( , ) C x t of this case.Another special case occurs when 2,  when the solution (22) reduces to a simpler form.In such a case, the solution (22y is the complementary error function[19].

Figure 7 . 5 ac
Figure 7.The isolines of the concentration ( , ) C x t of case 2(i) in the ( , ) xt plane for the function ( ) ( ) f t H t  for some values of the diffusion parameter: ( ) 1.5 a   , ( )

Figure 8 . 2 
Figure 8.The isolines of the concentration ( , ) C x t in the ( , ) xt plane for the function ( ) ( ) f t H t  for a special case of complementary error function when 2   , in the case when longitudinal diffusion only is present.Compare this with Figure 7 (b).
x is a confluent hypergeometric function[19].The contours of the solution (24) in the( , )   xt plane are presented in Figures 9 and 10 for different values of the decay parameter  and the two specific values of the coefficients of the longitudinal diffusion; quantity of particles at every point of the ( , )xt plane increases whenever the decay factor  increases for a fixed value of longitudinal diffusion .

Figure 9 .
Figure 9.The profile of the concentration ( , ) C x t for a fixed value of 0.25 ( 2)  and different values of the

Figure 10 .
Figure 10.The profile of the concentration ( , ) C x t for a fixed value of 0.05 ( 10) 