Time dependent mathematical model of air pollutants emitted from time-dependent elevated line source into a stable atmospheric boundary layer

A time dependent atmospheric model is represented for chemically reactive primary pollutants emitted from an elevated line source into a stable atmospheric boundary layer, over rough surface terrains. The time dependent model was obtained through an analytical solution from a multiple Inverse Laplace transform of the atmospheric diffusion equation with the parabolic eddy-diffusion coefficient (exchange coefficient) and the wind velocity as functions of vertical height. The pollutants were considered to be of chemically reactive primary pollutants emitted from a time-dependent line source of, step-Function type. In order to facilitate the application of the model, the results for the general situation that includes chemical reaction rate and time dependent line source are incorporated in the model. In this model, the effect of step-function type elevated line source studied near the source. The results obtained in this model are compared with that of the continuous line sources in the previous work. The results obtained in this model are of good agreement with that of the continuous line source.


INTRODUCTION
The study of the atmospheric dispersion of air pollutants from different types of sources received a great deal of attention during the last few decades.The effects of various sources of chemically reactive pollutants are been investigated in the recent studies.In addition, the depletion of pollutants plumes will effect the pollutants concentration especially when the deposition occurs over a long distance (Robson, 1983).An analytical model for air pollutant transport and dispersion from a point source is studied (Donald, 1976).This paper addresses an analytical model of air pollutant dispersion and removal from a step-function type elevated line sources.The removal (washed out) of pollutant depends on the nature of gaseous effluents emitted from the sources and the percentage of water vapors present in the atmosphere.The step-function type of line sources were assumed the series of industries or highways on during certain period of time and not continuous.The results were validated by comparing that of Robson (1987) by chemically nonreactive and time independent case.It is interesting to note that when there is no chemical reaction of primary pollutants, the results of the present model coincide with the results of the model presented by Robson (1987).This paper analyzed unsteady state dispersion of pollutants from a line source into stable atmosphere boundary layer.The quadratic diffusion coefficient and step-function type line source with variable wind profile like constant, constant shear and parabolic wind are *Corresponding author.E-mail: mshekhu123@yahoo.com.incorporated.We adopted multiple Laplace transform and Green's Function technique for the effective solution.In most of the previous models, numerical methods used either fully or partly.In model where the Laplace transform is generally inverted numerically (Robson, 1987).Whereas in this model, the multiple Laplace transform inverted analytically this gives an exact solution.The solutions of this model verified with that of (Robson, 1987) by assuming no delayed removal (chemical reaction rate) and time independent case.

METHODOLOGY
We consider the whole atmospheric boundary layer as subdivided into two layers viz: (i) the surface layer (SL) that is, o z z   0 below the source height and (ii) the layer above SL and up to the inversion layer that is, protected zone Here, H is the height of SL and o z is the source height.The situation analyzed is that the chemically reactive pollutant was emitted from an elevated line source, which is of step-function type.It is assumed that the emission of contaminant are in the gaseous state from a time dependent elevated line source of infinite length into a stable ABL, the variable wind velocity depending on the nature of the surface terrains viz.flat, buildup areas and mountains like elevated regions.The diffusion coefficient (exchange coefficient) was assumed to be of quadratic in nature, the boundary layer assumed to be of stable atmospheric boundary layer, where the mixing of pollutants is limited.To study the variation in the effect of pollutants, the various wind profiles like constant, constant shear and parabolic type are considered.The present work focused on the step-function type of source, where certain industries work during the certain allotted period and then switched off.The pollutant transport governed by the atmospheric advection-diffusion equation with delayed removal is as follows: Where, C is the concentration of pollutants in the atmosphere at any location ( The line source is in the direction of y-axis, which leads the concentration gradient and flux gradient becomes zero along ydirection. The source (S) is of time dependent line source and is assumed to be , suited for stable atmospheric boundary layer, where the penetration of pollutants through the inversion layer does not takes place.
Based on the aforementioned assumptions, the transport and diffusion Equation (1) becomes: Initial and boundary conditions are: R x, z , at t C(x,y,z) (4) Introducing the non-dimensional parameters And (Robson, 1987) where: ; L=the Monin-Obukovlenght; k=Vankarman constant; H=the boundary layer height and n=the constant decided on the stability of boundary layer.
In addition, non-dimensional source term is as follows: Incorporating all the aforementioned non-dimensional parameters in the Equation (2) and dropping out the capitals, we get: Where, α is the non-dimensional chemical reaction.
Initial and boundary conditions are:

Solution
Taking Laplace transforms along t and using initial conditions, we get: Initial and Boundary conditions: Where, Shekhu and Sulochana 105 Again, taking Laplace transform over (8) along x -axis, which assumed of semi-infinite region, then it becomes Where

When the wind profile is of constant (n = 0)
The movements of air near the earth's surface retarded by frictional effect proportional to the surface roughness.Thus, the nature of the terrain, the location and density of trees, the location and size of lakes, rivers, hills and building produces different wind velocity gradients in the vertical direction (Warn and Walker, 1967).The wind of constant type may be experienced in case of flat open country, lakes and seas, then the Equation (13) becomes Under the boundary conditions:


The complete solution of equation 16a and b, through Green's function technique Beck et al. (1992) assumed to be of the following form: Under the boundary conditions: And are found to be The arbitrary constant 1 R determined as shown in the Equation (24) by using the boundary conditions on the solution (20a).There of the solution (z) φ a and (z) φ b satisfies the b.c's., at z = 0 and z=1 respectively.Taking inverse Laplace transform using standard tables (Erdelyi et al., 1954) the solution (18) becomes: The complete solution for o z z  is obtained by interchanging z and o z in the above solution.

When the wind profile is of constant shear (n=1)
The wind profile of constant shear is experienced in the metro cities of highly buildup areas.The velocity varies linearly with vertical height then differential equation ( 13) is reduced to the form for quadratic diffusion coefficient.
The solution of Equation 29 through Green's function technique is assumed to be of the following form: Where Under the boundary conditions: This is the Bessel's differential equation of first kind since the order ( ) of Equation 34 is assumed to be of non-integer when it is compared with the generalized Bessel's equation Shekhu and Sulochana 107 (Sherwood and Reed, 1939), and the solutions are found to be Where, An arbitrary constant 2 R is determined by imposing the boundary condition at z = 0 on Equation ( 35) Wronskian determines 2 A with help of the above solutions we get: The solutions of Equation 32 followed by the boundary conditions, which is defined in two regions are follows: As limiting case, where (Beck et al., 1992) as follows: This is the complete solution of the Equation ( 32

When the wind profile is of parabolic type (n = 2)
The wind profile of parabolic type Solutions of Equation ( 48) through Green's function are assumed to be of the form: in the above differential equation then it reduces to the form Taking the transformation on the independent variable in Equation ( 55), we get, This is a Whittaker's differential equation, where the two independent solutions of Equation ( 52) are the Whittaker's Confluent Hyper geometric function as follows: Where, and also Inter-relation between Whittaker's to Bessel's function is derived as follows (Carslaw and Jeager) x F   is the confluent Hyper geometric functions and is defined as By the transformations (62), the solution ( 60) and (61) becomes Then complete solution of the Equation (55) becomes

Shekhu and Sulochana 109
The solution (49) using standard tables, Erdyle et al. (1954), Schaum Series (1950) Beck et al. (1992), and the inverse Laplace transform can be carried out for the solutions assuming Where 3 R can be determined through on of the boundary conditions at z= 0, and is found to be and 3 A is determined though Wronskian and is found to be Then Equation (69) becomes by assuming that the concentration is far from the source, that is, distance Taking inverse Laplace transform along s, we get

RESULTS AND DISCUSSION
This paper analyzed unsteady state dispersion of pollutants from a line source into stable atmosphere boundary layer.The quadratic diffusion coefficient and step-function type line source with variable wind profiles like constant, constant shear and parabolic wind are incorporated.We adopted multiple Laplace transform and Green's Function technique for the effective solution.In most of the previous models, numerical methods used either fully or partly.In model where the Laplace transform is generally inverted numerically (Robson, 1987).Whereas in this model, the multiple Laplace transform inverted analytically this gives an exact solution.The solutions of this model verified with that of (Robson, 1987) by assuming no delayed removal (chemical reaction rate) and time independent case.It is interesting to note that the effect of chemical reaction rte has virtually no impact t the short travel time but has significant effect t large travel time.It is interesting to note that when there is no chemical reaction of pollutant; the results of the present model coincide with the results of the model presented by Sulochana and Moka Shekhu (2009).
The results obtained in this model are illustrated graphically in the Figures 1 to 8. Concentration profiles of that is, at the ground level.(z=0).
study area located near the deep valley and oceans, then the differential Equation (4.3.6)reduced to the form with diffusion coefficient (exchange coefficient) is of quadratic in nature. 0 Figure 1.Ground level concentration for various source heights 9 .0 , 8 .0 , 7 .0 , 6 .0 , 5 .0 , 0       o o o o o o z z z z z z

Figure 2 .Figure 3 .
Figure 2. Ground level concentration for variable chemical reaction rate corresponding to the quadratic diffusion and wind coefficient of constant shear, 0 .0 