Journal of Environmental Protection, 2013, 4, 5764 http://dx.doi.org/10.4236/jep.2013.48A1008 Published Online August 2013 (http://www.scirp.org/journal/jep) 57 An Analytical Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer Glênio A. Gonçalves, Regis S. de Quadros, Daniela Buske* Federal University of Pelotas (UFPel), Department of Mathematics and Statistics (IFM/DME), Pelotas, Brazil. Email: *daniela.buske@ufpel.edu.br Received June 14th, 2013; revised July 16th, 2013; accepted August 3rd, 2013 Copyright © 2013 Glênio A. Gonçalves et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this work we present the solution of the twodimensional advectiondiffusion equation by the GILTT method. The GILTT approach uses, in the series expansion, eigenfunctions given in terms of cosine functions. Here, a different ex pansion for the solution of the advectiondiffusion equation will be explored. In other words, a SturmLiouville problem carrying more information of the original problem is considered, given by Bessel functions. Numerical simulations and comparisons with experimental data are presented. Keywords: Analytical Solution; AdvectionDiffusion Equation; Air Pollution Modeling; Integral Transform; Bessel Functions 1. Introduction The advectiondiffusion equation has long been used to describe the dispersion of contaminants in the atmos phere [1]. Efforts have been made over the years to ob tain analytical solutions of this equation in order to mod eling air pollution. According to [2,3], these solutions are valid for very specialized practical situations, and in ma jority with restrictions on wind and eddy diffusivities vertical profiles [418]. To solve the advectiondiffusion equation for more realistic physical scenario appeared in the literature the ADMM (Advection Diffusion Multi layer Method) approach [19,20], valid for any eddy dif fusivity and wind profile depending on the height. The main idea relies on the discretization of the Atmospheric Boundary Layer (ABL) in a multilayer domain, assuming in each layer that the eddy diffusivity and wind profile take averaged values. The resulting advectiondiffusion equation in each layer is then solved by the Laplace Transform technique. A more general methodology, which skips the multilayer discretisation of the height z appear ing in the ADMM approach, is known in the literature as GILTT (Generalized Integral Laplace Transform Tech nique) approach [2,3,21]. The main idea of this method ology relies on the expansion of the pollutant concentra tion in series of eigenfunctions attained from an auxiliary SturmLiouville problem, replacement of this equation in the advectiondiffusion equation and taking moments. The procedure results a matrix ordinary differential equa tion which is solved analytically by the Laplace Trans form technique. Similar solutions were proposed by [22, 23]. To reach our objective, we begin presenting the solu tion of the twodimensional advectiondiffusion equation in Cartesian geometry by the GILTT approach [2], con sidering that the eddy diffusivity and the vertical wind profile depend on the z variable. Traditionally, the GILTT approach uses as basis eigenfunctions given in terms of cosine functions. Here, another SturmLiouville problem will be considered, carrying more information of the original problem. In this case, the eigenfunctions are given by Bessel functions. Once we construct the general solution, numerical simulations and future per spectives of this methodology are presented. 2. The AdvectionDiffusion Equation and the GILTT Method For a Cartesian coordinate system the advectiondiffu sion equation, using first order closure of turbulence, is written like [24]: xyz cccc uvw txyz ccc KK xxyyzz S (1) *Corresponding author. Copyright © 2013 SciRes. JEP
An Analytical Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer 58 where c denotes the average concentration of a passive (g/m3), u, v, w are the mean wind (m/s) components along the axis x, y and z, respectively and S is the source term. Kx, Ky, Kz are the Cartesian components of eddy diffusivity (m2/s) in the x, y and z directions, respectively. In the first order closure all the information on the turbu lence complexity is contained in the eddy diffusivities. Problem (1) is solved analytically by the 3DGILTT method [3,21,25]. Here, for comparison with experimen tal data we will assume for the advectiondiffusion Equa tion (1): stationary conditions, crosswind integrated con centrations and that the advection is much higher than the diffusion in the xdirection. After the simplifications, let us consider the problem: y z cc uK y zz (2) for 0 < z < h and x > 0, subject to the boundary condi tions of zero flux at the ground and ABL top and a source with emission Q at height Hs ( 0, ys uczQz H at x = 0). Here c represents the crosswind integrated concentration, h is the ABL height, is the eddy dif fusivity variable with the height z ( z Kz), u is the longitudinal wind speed ( zuu), and δ is the Di rac delta function. Problem (2) has a wellknown solution by the GILTT method. Following the works of [2,25] we write the so lution of problem (2) as: 0 , N yn n cxzcx z n (3) where are the eigenfunctions of an associated SturmLiouville problem and nz n cx is the transformed concentration. Traditionally, in the application of the GILTT method, the following auxiliary SturmLiouville problem is cho sen: 20 nnn zz at (4a) 0zh 0 nz at (4b) 0, ,zh which has the solution cos nn zz , where nz are the eigenfunctions and π nnh are the respective eigenvalues. 0,n 1, 2, Here, a different expansion for the solution of the ad vectiondiffusion equation will be explored. In other words, we propose another SturmLiouville problem as the basis generator. The idea of this proposal comes from the fact that the auxiliary problem (4) has the same shape of the ordinary differential equation (relative do z vari able) that appears in the solution of Equation (2) by the method of separation of variables, when the vertical eddy diffusivity is considered constant. This suggests the pos sibility of using an auxiliary problem that appears in the solution of Equation (2) by the method of separation of variables, considering linear vertical eddy diffusivity, z z , given by: 20 nn nn zz z at 0 (5a) zh 0 nz at (5b) 0,zh which has Bessel functions of first specie and order zero as solution 0nn zJ zh , where n 0,1, 2,n are the positive roots of the Bessel func tion of first specie and order one, 1 . Problem (5) car ries more information from the original problem than the previous one. To determine the unknown coefficient n cx we re place Equation (3) in Equation (1). Applying the integral operator , we come out with the result: 0 . h mzz d 00 00 dd hh NN n nnmnmz nn cxuz cxKz zz 0 (6) Using the integration parts technique, we can recast the second integral in Equation (6) as: 00 dd hh nn m zmzmz n zK K zzz z z (6a) and once 0 n mz Kz , the Equation (6) is rewritten as: 00 00 dd hh NN n nnmnmz nn cxuz cxKz z 0 (7) which in matrix form reads like: 0FY x Yx (8) Here Yx is the vector whose components are n cx and F = B−1. E; ,nm Bb and are the ma trices whose entries are, respectively: ,nm Ee 0 d h n,mn m bu z and 0 dd d h n n,mm z eK zz . Equation (8) is subject to the initial condition Y(0), which is obtained from the source condition ( 0, ys uczQz H at x = 0) by a similar proce dure leading to 1 00 nms Yc QHB , B−1 being the inverse of matrix B. The transformed problem represented by the Equation (8) is solved analytically following the work [2], by the combined Laplace transform technique and diagonaliza tion of the matrix F 1 FXDX . By this procedure we come out with the result: 10Yx XGxXY , (9) Copyright © 2013 SciRes. JEP
An Analytical Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer 59 where is the diagonal matrix with elements , D is the diagonal matrix of eigenvalues n of the ma trix F, X is the matrix of the respective eigenfunctions and X−1 it is the inverse. Gx ei dx d Therefore, the solution for the concentration given by Equation (3) is now well determined once the vector n cx is known and given by Equation (9). The solution of the problem (2) using in the series ex pansion (3) eigenfunctions given in terms of cosine and Bessel functions will be called here as GILTTC and GILTTB, respectively. 3. Numerical Results The performance of the discussed solution was evaluated against experimental groundlevel concentration using different dispersion experiments available in the litera ture. Below we briefly discuss the Copenhagen, Prairie Grass and Hanford dispersion experiments, which allow us to validate the results encountered by the mentioned solutions. The Copenhagen field campaign took place in the suburbs of Copenhagen in 1978, and is described by [26]. It consisted of tracer released without buoyancy from a tower at a height of 115 m, and collection of tracer sam pling units at the groundlevel positions at the maximum of three crosswind arcs. The sampling units were posi tioned at two to six kilometers from the point of release. The site was mainly residential with a roughness length of the 0.6 m. The meteorological conditions during the dispersion experiments ranged from moderately unstable to convective. Table 1 shows a summary of meteoro logical conditions during the Copenhagen experiments. In the PrairieGrass experiment, according [27], the tracer SO2 was released without buoyancy at a height of 0.46 m, and collected at a height of 1.5 m at five down wind distances (50, 100, 200, 400 and 800 m) at O’Neill, Table 1. The meteorological data observed during the Co penhagen experiment. Exp. u (10 m) (ms−1) u* (ms−1) L (m) w* (ms−1) h (m) 1 3.4 0.36 −37 1.8 1980 2 10.6 0.73 −292 1.8 1920 3 5.0 0.38 −71 1.3 1120 4 4.6 0.38 −133 0.7 390 5 6.7 0.45 −444 0.7 820 6 13.2 1.05 −432 2.0 1300 7 7.6 0.64 −104 2.2 1850 8 9.4 0.69 −56 2.2 810 9 10.5 0.75 −289 1.9 2090 Nebraska in 1956. The Prairie Grass site was quite flat and much smooth with a roughness length of 0.6 cm. Here we consider the experimental data appearing in the paper [28]. Table 2 summaries the meteorological condi tions during the PrairieGrass experiments. The Hanford diffusion experiment was conducted in MayJune, 1983, on a semiarid region of south eastern Washington on generally flat terrain. The detailed de scription of the experiment was provided by [29]. Data were obtained from six dualtracer releases located at 100, 200, 800, 1600 and 3200 m from the source during moderately stable to nearneutral conditions. The release height of SF6 was 2 m and average release rate was around 0.3 g/s. The pollutant was collected at a height of 1.5 m. The terrain was considered as an urban terrain with roughness length of 3 cm. The values of ABL pa rameters are given in Table 3. The choice of the turbulent parameterization repre sents a fundamental aspect for pollutant dispersion mod eling. In terms of the convective scaling parameters, the Table 2. The meteorological data observed during the Prai rieGrass experiment. Exp. L (m) h (m) w* (ms−1) u (10 m) (ms−1) Q (gs−1) 1 −9 260 0.84 3.2 82 5 −28 780 1.64 7.0 78 7 −10 1340 2.27 5.1 90 8 −18 1380 1.87 5.4 91 9 −31 550 1.70 8.4 92 10 −11 950 2.01 5.4 92 15 −8 80 0.70 3.8 96 16 −5 1060 2.03 3.6 93 19 −28 650 1.58 7.2 102 20 −62 710 1.92 11.3 102 25 −6 650 1.35 3.2 104 26 −32 900 1.86 7.8 98 27 −30 1280 2.08 7.6 99 30 −39 1560 2.23 8.5 98 43 −16 600 1.66 6.1 99 44 −25 1450 2.20 7.2 101 49 −28 550 1.73 8.0 102 50 −26 750 1.91 8.0 103 51 −40 1880 2.30 8.0 102 61 −38 450 1.65 9.3 102 Copyright © 2013 SciRes. JEP
An Analytical Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer 60 Table 3. The meteorological data observed during the Han ford experiment. Exp. u (2 m) (ms−1) u* (ms−1) L (m) h (m) 1 3.63 0.40 166 325 2 1.42 0.26 44 135 3 2.02 0.27 77 182 4 1.50 0.20 34 104 5 1.41 0.26 59 157 6 1.54 0.30 71 185 vertical eddy diffusivity can be formulated as [30]: 48 1/3 1/3 * 0.2211 e0.0003e z h z Kzz whh h z h (13) while for stable conditions [31]: 0.3 1 13.7 z zhuz Kz (14) where z is height; h is the thickness of the ABL; * is the convective velocity scale; w 54 1Lzh; L is the MoninObukhov length and is the friction velocity. * In our simulations, we use the wind speed profile de scribed by a power law, according [32], u 11 z uz uz (15) where u and 1 u are the mean wind velocity respec tively at the heights z and 1, while z is an exponent that is related to the intensity of turbulence [33]. For the Copenhagen experiment 0.1 and for the Prairie Grass experiment 0.07 . In Tables 46, we present some performances evalua tions of the model for the Copenhagen and PrairieGrass experiments, respectively, using the statistical evaluation procedure described by [34] and defined as: 2 MSEnormalized mean square error , op po CC CC FA2fraction of data%, normalized to 1for 0.5 2, po CC CORcorrelation coefficient , oo pp op CCCC FBfractional bias0.5, op op CC CC FSfractional standard deviations 0.5 , op op where the subscripts o and p refer to observed and pre dicted quantities, respectively, and the over bar indicates an averaged value. The statistical index NMSE repre sents the model values dispersion in respect to data dis persion. The statistical index FB says if the predicted quantities underestimate or overestimate the observed ones. The best results are expected to have values near to zero for the indices NMSE, FB and FS, and near to 1 in the indices COR and FA2. For the Copenhagen experiment the statistical indices of Table 4 point out that a good agreement is obtained between experimental data and the GILTT method for both cosine and Bessel basis, regarding the NMSE, FB and FS values relatively near to zero and COR relatively near to 1. At this point, we can affirm that no significant difference between the models was observed for the high source of the Copenhagen experiment. Table 5 shows the performance of the solution for the PrairieGrass experiment. The statistical indices of the table point out that a reasonable agreement is obtained between experimental data and the GILTT method. It is important to notice that the GILTTB numerically con verges faster than GILTTC (while GILTTB needs 100 eigenvalues, GILTTC needs 300 eigenvalues to reach a Table 4. Statistical indices evaluating the model perform ance using the Copenhagen experiment. Model NMSECOR FA2 FB FS GILTTC N = 1000.05 0.91 1.00 −0.010.14 GILTTB N = 1000.05 0.91 1.00 −0.040.13 Table 5. Statistical indices evaluating the model perform ance using the PrairieGrass experiment. Model NMSECOR FA2 FB FS GILTTC N = 1000.80 0.83 0.64 0.39 0.56 GILTTC N = 2000.23 0.92 0.71 0.06 0.33 GILTTC N = 3000.15 0.95 0.72 −0.010.28 GILTTB N = 100 0.11 0.97 0.71 −0.1 0.23 Table 6. Statistical indices evaluating the model perform ance using the Hanford experiment. Model NMSECOR FA2 FB FS GILTTC N = 300.21 0.91 0.83 −0.16 −0.01 GILTTC N = 600.23 0.91 0.80 −0.20 −0.03 GILTTB N = 300.24 0.91 0.77 −0.20 −0.03 Copyright © 2013 SciRes. JEP
An Analytical Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer 61 similar numerical result). For the Hanford experiment were used the eddy diffu sivity Equation (14) and power wind profile Equation (15) with 0.6 . The statistical indices of Table 6 point out that a good agreement is obtained between experi mental data and models. Again, the GILTTC need more eigenvalues to reach a similar numerical result obtained with the GILTTB. In the following are presented in Tables 79 the nu merical comparisons of the GILTT method results against the experimental data of Copenhagen, PrairieGrass and Hanford experiments. Furthermore, Figures 13 show the observed and pre dicted scatter diagram of crosswind groundlevel con centrations for the three experiments considered in this work. In the graphics the symbol represents the GILTTC, Table 7. Observed and predicted crosswindintegrated con centrations C/Q (10−4 sm−2) at the Copenhagen experiment. Run Distance (m) OBS (10−4 sm−2) GILTTC (10−4 sm−2) GILTTB (10−4 sm−2) 1 1900 6.48 6.86 7.27 1 3700 2.31 3.98 4.11 2 2100 5.38 4.64 4.87 2 4200 2.95 3.06 3.24 3 1900 8.20 8.15 8.45 3 3700 6.22 5.20 5.32 3 5400 4.30 3.99 4.05 4 4000 11.66 9.25 9.30 5 2100 6.72 8.54 8.53 5 4200 5.84 6.73 6.90 5 6100 4.97 5.40 5.51 6 2000 3.96 3.50 3.52 6 4200 2.22 2.51 2.62 6 5900 1.83 1.98 2.05 7 2000 6.70 4.67 4.97 7 4100 3.25 2.76 2.88 7 5300 2.23 2.24 2.31 8 1900 4.16 4.84 4.96 8 3600 2.02 3.28 3.33 8 5300 1.52 2.63 2.65 9 2100 4.58 4.44 4.67 9 4200 3.11 2.92 3.11 9 6000 2.59 2.20 2.31 Table 8. Groundlevel crosswind integrated concentrations (gm−2) measured during the Prairie Grass experiment (first line) and simulated by the GILTTC and GILTTB methods (second and third lines, respectively). Run No.50 m (gm−2) 100 m (gm−2) 200 m (gm−2) 400 m (gm−2) 800 m (gm−2) 1 7.00 5.73 5.62 2.30 3.67 3.62 0.51 1.93 1.93 0.16 0.90 0.90 0.06 0.41 0.41 5 3.30 3.07 2.99 1.80 2.07 2.17 0.81 1.21 1.30 0.29 0.61 0.66 0.09 0.27 0.29 7 4.00 3.02 4.12 2.20 1.93 2.47 1.00 1.07 1.28 0.40 0.52 0.59 0.18 0.23 0.25 8 5.10 3.25 4.46 2.60 2.19 2.92 1.10 1.29 1.59 0.19 0.66 0.77 0.14 0.30 0.33 9 3.70 3.27 2.90 2.20 2.24 2.17 1.00 1.30 1.33 0.41 0.65 0.67 0.13 0.29 0.30 10 4.50 3.56 4.08 1.90 2.23 2.51 0.71 1.21 1.33 0.20 0.58 0.62 0.03 0.25 0.26 15 7.10 5.60 5.59 3.40 3.66 3.66 1.35 2.01 2.01 0.37 1.02 1.02 0.11 0.53 0.53 16 5.00 4.08 4.93 1.80 2.39 2.73 0.48 1.22 1.34 0.10 0.55 0.59 0.02 0.23 0.24 19 4.50 4.09 3.74 2.20 2.77 2.79 0.86 1.60 1.70 0.27 0.80 0.85 0.06 0.36 0.37 20 3.40 2.89 2.55 1.80 2.07 2.05 0.85 1.28 1.35 0.34 0.68 0.73 0.13 0.32 0.33 25 7.90 6.52 6.54 2.70 3.80 4.02 0.75 1.91 2.04 0.30 0.86 0.89 0.06 0.37 0.37 26 3.90 3.33 3.57 2.20 2.28 2.53 1.04 1.35 1.48 0.39 0.69 0.75 0.13 0.31 0.33 27 4.30 2.88 3.80 2.30 2.01 2.66 1.16 1.22 1.50 0.46 0.65 0.75 0.18 0.30 0.33 30 4.20 2.37 3.27 2.30 1.71 2.46 1.11 1.09 1.46 0.40 0.60 0.75 0.10 0.29 0.34 43 5.00 4.22 3.95 2.40 2.70 2.75 1.09 1.48 1.56 0.37 0.70 0.74 0.12 0.31 0.31 44 4.50 2.80 3.87 2.30 1.95 2.68 1.09 1.19 1.51 0.43 0.63 0.75 0.14 0.29 0.33 49 4.30 3.69 3.31 2.40 2.49 2.44 1.16 1.42 1.47 0.45 0.70 0.73 0.15 0.31 0.32 Copyright © 2013 SciRes. JEP
An Analytical Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer 62 Continued 50 4.20 3.53 3.39 2.30 2.37 2.46 0.91 1.37 1.47 0.39 0.68 0.73 0.11 0.31 0.32 51 4.70 2.28 3.61 2.40 1.67 2.68 1.00 1.08 1.60 0.38 0.61 0.83 0.08 0.30 0.37 61 3.50 3.33 3.00 2.10 2.37 2.25 1.14 1.40 1.39 0.53 0.71 0.72 0.20 0.32 0.32 Table 9. Observed and predicted crosswindintegrated con centrations C/Q (10−3 sm−2) at Hanford experiment. Run No. Distance (m) OBS (10−3 sm−2) GILTTC (10−3 sm−2) GILTTB (10−3 sm−2) 1 100 19.5 36.28 38.92 1 200 11.7 22.86 23.65 1 800 3.7 7.43 7.48 1 1600 2.1 4.14 4.15 1 3200 1.3 2.34 2.34 2 100 51.9 82.08 81.74 2 200 36.7 50.11 50.02 2 800 12.9 17.96 17.95 2 1600 9.1 11.00 11.00 2 3200 7.2 6.94 6.93 3 100 27.1 65.82 65.49 3 200 18.1 40.04 39.96 3 800 5.9 13.46 13.46 3 1600 3.3 7.87 7.87 3 3200 1.8 4.73 4.73 4 100 91.8 99.91 99.60 4 200 48.6 63.56 63.47 4 800 20.1 23.70 23.69 4 1600 13.1 14.66 14.66 4 3200 9.2 9.31 9.31 5 100 83.9 78.41 78.06 5 200 42.4 47.09 47.00 5 800 10.5 16.28 16.27 5 1600 8.6 9.79 9.79 5 3200 6.6 6.07 6.07 6 100 88.4 67.05 66.86 6 200 61.1 39.77 39.72 6 800 13.4 13.43 13.42 6 1600 6.2 7.98 7.98 6 3200 3.1 4.89 4.89 Figure 1. Comparison between observed (Co) and predicted (Cp) concentrations (normalized by Q) for the Copenhagen experiment. Lines indicate a factor of two 00.5;2 p CC . Figure 2. Comparison between observed (Co) and predicted (Cp) concentrations for the PrairieGrass experiment. Lines indicate a factor of two 00.5;2 p CC . and lines the GILTTB solution. In all the tables and figures, we considered the data numerically converged for both bases. In this respect, it is important to note that the model simulates quite well the observed concentration for all the cases. The greatest difference between the models is seen for the Prairie Grass experiment. Copyright © 2013 SciRes. JEP
An Analytical Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer 63 Figure 3. Comparison between observed (Co) and predicted (Cp) concentrations (normalized by Q) for the Hanford ex periment. Lines indicate a factor of two 00.5;2 p CC . 4. Conclusions Focusing our attention on the pollution dispersion simu lation in atmosphere, we present an analytical solution in series expansion given by the wellknown GILTT meth od to solve the twodimensional advectiondiffusion equation by the GILTT approach. A SturmLiouville problem carrying more information of the original prob lem, given by Bessel functions, was considered, For the problems discussed, we promptly realize the very good results achieved, under statistical point of view, by the GILTT method when compared with the experi mental data for both cosine and Bessel basis used. For the case of high source no significant difference was ob served between GILTTC and GILTTB. However, for the low source, GILTTB numerically converges faster than GILTTC. We focus our future attention on the direction of the generalization of this solution considering an infi nite boundary layer. 5. Acknowledgements The authors thank CNPq (Conselho Nacional de Desen volvimento Científico e Tecnológico) and FAPERGS (Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul) for the partial financial support of this work. REFERENCES [1] J. H. Seinfeld and S. N. Pandis, “Atmospheric chemistry and physics,” John Wiley & Sons, New York, 1998. [2] D. M. Moreira, M. T. Vilhena, D. Buske and T. Tirabassi, “The StateofArt of the GILTT Method to Simulate Pol lutant Dispersion in the Atmosphere,” Atmospheric Re search, Vol. 92, No. 1, 2009, pp. 117. doi:10.1016/j.atmosres.2008.07.004 [3] D. Buske, M. T. Vilhena, T. Tirabassi and B. Bodmann, “Air Pollution SteadyState Advection Diffusion Equa tion: The General ThreeDimensional Solution,” Journal of Environmental Protection, Vol. 3, No. 2, 2012, pp. 11241134. doi:10.4236/jep.2012.329131 [4] W. Rounds, “Solutions of the TwoDimensional Diffu sion Equation,” Transactions of American Geophysical Union, Vol. 36, 1955, pp. 395405. doi:10.1029/TR036i003p00395 [5] F. B. Smith, “The Diffusion of Smoke from a Continuous Elevated Point Source into a Turbulent Atmosphere,” Journal of Fluid Mechanics, Vol. 2, No. 1, 1957, pp. 49 76. doi:10.1017/S0022112057000737 [6] R. A. Scriven and B. A. Fisher, “The Long Range Trans port of Airborne Material and Its Removal by Deposition and WashoutII. The Effect of Turbulent Diffusion,” At mospheric Environment, Vol. 9, No. 1, 1975, pp. 5969. doi:10.1016/00046981(75)900542 [7] C. Demuth, “A Contribution to the Analytical Steady Solution of the Diffusion Equation for Line Sources,” Atmospheric Environment, Vol. 12, No. 5, 1978, pp. 12551258. doi:10.1016/00046981(78)903992 [8] A. P. van Ulden, “Simple Estimates for Vertical Diffusion from Sources near the Ground,” Atmospheric Environ ment, Vol. 12, No. 11, 1978, pp. 21252129. doi:10.1016/00046981(78)901671 [9] F. T. M. Nieuwstadt, “An Analytical Solution of the TimeDependent, OneDimensional Diffusion Equation in the Atmospheric Boundary Layer,” Atmospheric En vironment, Vol. 14, No. 12, 1980, pp. 13611364. doi:10.1016/00046981(80)901547 [10] F. T. M. Nieuwstadt and B. J. de Haan, “An Analytical Solution of OneDimensional Diffusion Equation in a NonStationary Boundary Layer with an Application to Inversion Rise Fumigation,” Atmospheric Environment, Vol. 15, No. 5, 1981, pp. 845851. doi:10.1016/00046981(81)902894 [11] M. Tagliazucca, T. Nanni and T. Tirabassi, “An Analyti cal Dispersion Model for Sources in the Surface Layer,” NovembreDicembre, Vol. 8, No. 6, 1985, pp. 771781. [12] T. Tirabassi, “Analytical Air Pollution and Diffusion Models,” Water, Air and Soil Pollution, Vol. 47, No. 12, 1989, pp. 1924. doi:10.1007/BF00468993 [13] W. Koch, “A Solution of the TwoDimensional Atmos pheric Diffusion Equation with HeightDependent Diffu sionCoefficient Including GroundLevel Absorption,” Atmospheric Environment, Vol. 23, No. 8, 1989, pp. 17291732. doi:10.1016/00046981(89)900577 [14] T. Tirabassi and U. Rizza, “Applied Dispersion Modelling for GroundLevel Concentrations from Elevated Sources,” Atmospheric Environment, Vol. 28, No. 4, 1994, pp. 611 615. doi:10.1016/13522310(94)90037X [15] M. Sharan, M. P. Singh and A. K. Yadav, “A Mathe matical Model for the Atmospheric Dispersion in Low Copyright © 2013 SciRes. JEP
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