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INTERNATIONAL JOURNAL OF ADVANCED RESEARCH 0976 International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN IN – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) IJARET Volume 4, Issue 6, September – October 2013, pp. 269-277 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) ©IAEME www.jifactor.com TIME – DEPENDENT TWO DIMENSIONAL MATHEMATICAL MODEL OF AIR POLLUTION DUE TO AREA SOURCE WITH SETTLING VELOCITY AND TRANSFORMATION PROCESSES OF PRIMARY AND SECONDARY POLLUTANTS Dr. Lakshminarayanachari.K Associate Professor, Department of Mathematics, Sai Vidya Institute of Technology, Bangalore -560 064, INDIA. ABSTRACT A comprehensive time dependent two dimensional advection-diffusion numerical model for primary pollutants emitted and converted into secondary pollutants for an urban area is presented. This model takes into account the realistic form of variable wind velocity and eddy diffusivity profiles. In this model we consider that the secondary pollutants are formed by means of first order chemical conversion of primary pollutants. This assumption results in a coupled system of partial differential equations of primary and secondary pollutants. This intricate coupled system of mixed initial boundary value problem is solved by using Crank-Nicolson implicit finite difference technique. The effect of time dependent emission of pollutants and the effect of various meteorological parameters on the dispersion of pollutants on concentration contour are analysed extensively. Keywords: Primary and secondary pollutant, Crank-Nicolson method, Dry deposition, Gravitational settling. 1. INTRODUCTION The dispersion of atmospheric contaminant has become a global problem in the recent years due to rapid industrialization and urbanization. The toxic gases and small particles could accumulate in large quantities over urban areas, under certain meteorological conditions. This is one of the serious health hazards in many of the cities in the world. An acute exposure to the elevated levels of particulate air pollution has been associated with the cases of increased cardiopulmonary mortality, hospitalization for respiratory diseases, exacerbation of asthma, decline in lung function, and restricted life activity. Small deficits in lung function, higher risk of chronic respiratory disease and 269 International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME increased mortality have also been associated with chronic exposure to respirable particulate air pollution [1] Epidemiological studies have demonstrated a consistent increased risk for cardiovascular functions in relation to both short- and long-term exposure to the present-day concentrations of ambient particulate matter [2]. Exposure to the fine airborne particulate matter is associated with cardiovascular functions and mortality in older and cardiac patients [3]. Volatile organic compounds (VOCs), which are molecules typically containing 1–18 carbon atoms that readily volatilize from the solid or liquid state, are considered a major source of indoor air pollution and have been associated with various adverse health effects including infection and irritation of respiratory tract, irritation to eyes, allergic skin reaction, bronchitis, and dyspnea [4: 5: 6]. A two dimensional advection-diffusion mathematical model of primary and secondary pollutants of an area source with chemical reaction and gravitational settling is presented [7]. Rudraiah et al. have studied the atmospheric diffusion model of secondary pollutants with settling [8]. Khan and Venkatachalappa et al. have presented a time dependent mathematical model of an air pollutant with instantaneous and delayed removal [9: 10]. The above models are analytical in nature with simple form of wind velocity and eddy diffusivity under restrictive assumptions. Lakshminarayanachari et al. have studied the mathematical model with chemically reactive pollutants without considering the settling velocity on secondary pollutants [11]. Mathematical models are important tools and can play a crucial role in the methodology developed to predict air quality. A numerical model for primary and secondary pollutants in the atmosphere with more realistic wind velocity and eddy diffusivity profiles by considering the various removal mechanisms such as dry deposition, wet deposition and gravitational settling velocity is presented. 2. MODEL DEVELOPMENT The physical problem consists of an area source which is spread out over the surface of the city with finite down wind and infinite cross wind dimensions. We assume that the pollutants are emitted at a constant rate from uniformly distributed area source. The major source being vehicular exhausts due to traffic flow and all other minor sources are aggregated. The vertical height extends up to mixing height 624 meters above which pollutants do not rise due to the temperature profile of the atmosphere. We have considered the source region within the urban centre (0 ≤ x ≤ l) which extends up to l = 6 km from the origin and source free region beyond l (l ≤ x ≤ X0). We compute the concentration distribution till the desired down wind distance X0 = 12 km i.e. 0 ≤ x ≤ X0. We have considered two layers, surface layer and planetary boundary layer to evaluate the wind velocity as accurate as possible. The physical description of the model is shown schematically in figure 1. Fig 1. Physical Layout of the Model 270 International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 2.1 PRIMARY POLLUTANT The basic governing equation of primary pollutant ∂C p ∂C p ∂ ∂C p ∂t +U ( z) ∂x = Kz ( z) ∂z ∂z ( ) − k + K wp C p , (1) where C p = C p ( x, z , t ) is the ambient mean concentration of pollutant species, U is the mean wind speed in x-direction, K z is the turbulent eddy diffusivity in z-direction, k wp is the first order rainout/washout coefficient of primary pollutant C p and k is the first order chemical reaction rate coefficient of primary pollutant C p . We assume that the region of interest is free from pollution at the beginning of the emission. Thus the initial conditions are Cp = 0 at t = 0, 0 ≤ x ≤ X0 and 0 ≤ z ≤ H, (2) Cp = 0 at x = 0, 0 ≤ z ≤ H and ∀ t > 0. (3) The air pollutants are being emitted at a steady rate from the ground level and are removed from the atmosphere by ground absorption ∂C p Kz = VdpC p − Q at z = 0, 0 < x ≤ l ∂z ∀t > 0 , (4) = VdpC p at z = 0, l < x ≤ X 0 ∂C p Kz = 0 at z = H , 0 < x ≤ X 0 , ∀t > 0 , (5) ∂z where Q is the emission rate of primary pollutant species, l is the source length in the downwind direction, Vdp is the dry deposition velocity, X0 is the length of desired domain of interest in the wind direction and H is the mixing height. 2.2 SECONDARY POLLUTANT The basic governing equation for the secondary pollutant Cs is ∂Cs ∂C ∂ ∂C ∂C s + U ( z ) s = K z ( z ) s + Ws + Vg kCs − K ws Cs . (6) ∂t ∂x ∂z ∂z ∂z The appropriate initial and boundary conditions on Cs are : Cs = 0 at t = 0, for 0 ≤ x ≤ X0 and 0 ≤ z ≤ H, (7) Cs = 0 at x = 0, for 0 ≤ z ≤ H and ∀t > 0. (8) Since there is no direct source for secondary pollutants, we have 271 International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME ∂C s Kz + Ws Cs = Vds Cs at z = 0, 0 ≤ x ≤ X 0 , ∀t > 0 , (9) ∂z ∂Cs Kz = 0 at z = H , ∀t > 0 , (10) ∂z where Kws is the first order wet deposition coefficient of secondary pollutants, Vg is the mass ratio of secondary particulate species to the primary gaseous species which is being converted, Ws is the gravitational settling velocity and H is mixing height. 3. METEOROLOGICAL PARAMETERS In order to solve the equations (1) and (6), it is essential to know the profiles of eddy diffusivity and wind speed for various atmospheric stability conditions. It is assumed that the surface layer terminates at z = 0.1k (U∗ f ) for neutral stability condition, where k is the Karman’s Constant ≈ 0.4, f is the Coriolis parameter and U* is the friction velocity. For stable case, the surface layer extends up to z = 6L, where L is the Monin-Obukhov stability length parameter. The following wind velocity profiles are used. In the surface layer, logarithmic profiles are used for neutral stability with z < 0.1k (U∗ f ) , ie U∗ Z + Z0 U= ln (within surface layer ) . (11) k Z0 z U∗ Z + Z0 α For stable flow with 0 < < 1, U= l n + z . (12) L K Z0 L z U∗ Z + Z0 For stable flow with 1 < < 6, U= l n + 5.2 . (13) L K Z0 In the planetary boundary layer, above the surface layer, power law scheme has been used p Z − Z SL ( U = U g − U SL ) + U SL , (14) Z m − Z SL where Ug is the geostrophic wind, ZSL is the top of the surface layer, USL is wind at ZSL, Zm is the mixing height H and p is an exponent which depends up on the atmospheric stability. 0.2 for neutral condition We have used, p = 0.35 for slightly stable flow 0.5 for stable flow . 272 International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME The following eddy – diffusivity profiles are used for the entire boundary layer (surface layer and planetary boundary layer) K z = 0.4 U*Ze−4 Z / H , for neutral case (15) kU ∗ Z −bη Z −1/ 2 U KZ = e , for stable case b = 0.91, η = µ and µ = ∗ . (16) 0.74 + 4.7 Z / L L fL U* is the friction velocity and f is the Coriolis parameter (= 10-4). 4. NUMERICAL SOLUTION We have used Crank-Nicolson implicit finite difference method for the solution of the equations (1) and (6). The derivatives are replaced by the arithmetic average of its finite difference approximations th th at the n and ( n + 1) time steps. Then equation (1) at the grid points (i, j ) and time step n + 1 2 can be written as 1 n+ n n + 1 n n + 1 ∂Cp 2+ 1 U ( z) ∂Cp + U ( z) ∂Cp C C = 1 ∂ K ( z) ∂ p + ∂ K ( z) ∂ p ∂t ij 2 ∂x ij ∂x ij 2 ∂z z ∂z z ∂z ij ∂ z ij 1 − 2 ( k + kwp ) (Cpij + Cpij+1 ) , i = 1,2,....., n n j = 1,2,..... (17) On simplifying, equation (17) can be written as n+ n+ n+ Aj C pi+−1ij + B j C pij11 + D j C pij1 + E j C pij1 1 = Fj C pi −ij + G j C pij −1 + M j C pij + N j C pij +1 , n − + n n n n (18) for each i = 2,3,4,….. i max l …… i max X 0 , j=2,3,4,……jmax-1 and n=0,1,2,3,……, . and i maxl and imaxX0 are the i values at x = l and X0 respectively and jmax is the value of j at z = H. The finite difference equations for the secondary pollutant Cs obtained from the partial differential equation (6) can be written as n+1 n+1 n+1 n+1 n n n ACs i−1j +BCs ij−1 +DCs ij +ECs ij+1 = FCsni−1j +GCs ij−1 +MjCs ij +NjCs i j+1 +VgkCsnij j j j j j j , (19) The above system of equations has a tridiagonal structure and is solved by Thomas Algorithm [12]. 273 International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 5. RESULTS AND DISCUSSIONS A numerical model to study the effects of primary and secondary pollutant concentration horizontally and vertically is developed. The results of this model have been illustrated graphically in figures 2 - 7 to analyse the dispersion of air pollutants in the urban area downwind and vertical direction for stable and neutral conditions of atmosphere. In figure 2 the ground level concentration at various heights for primary and secondary pollutants of stable case is studied. For removal mechanisms Ws=0 and Vd=0 for stable and neutral cases are analysed. The primary pollutant concentration attains maximum value of 180. As height increases i.e. and the primary pollutant concentration decreases up to 7kms and there is no substantial decrease with height thereafter. This is because we have considered source region up to 6kms and no source beyond 6 to 12 kms. The same effect is observed for secondary pollutants but the concentration of secondary pollutants is maximum at the end of source region and outside source region. The ground level concentration of secondary pollutants increases up to 6 kms and remains constant thereafter in source free region. As height increases the concentration of secondary pollutants decreases. In figure 3 the concentration at various distances for primary and secondary pollutants of stable case is studied. As distance increases the primary pollutant concentration increases in the source region and is maximum at the end of source region and is rapidly decreases thereafter. The concentration is zero around 55mts height for stable case because we have considered the sources at the ground level. The secondary pollutant concentration increases with downwind distance and is maximum in the source free region. The concentration of secondary pollutants is zero about 70mts height. In figure 4 the Ground level concentration at various heights of primary and secondary pollutants for neutral case is obtained. It is observed that the pollutant concentration is 55 at the end of source region. The concentration is low in neutral atmosphere when compared to stable case. This shows that neutral atmosphere enhances vertical diffusion. In figure 5 the same effect is observed but the concentration of primary and secondary pollutants are zero at the heights 350 and 400 respectively. This shows that neutral condition enhances vertical diffusion and the concentration is low in downwind distance when compared to stable condition. 200 0.06 Secondary pollutants Primary pollutants ws=0 Ws=0 180 C(X,2) Vd=0 Vd=0 0.05 160 C(X,2) 140 C(X,4) 0.04 120 Concentration Concentration C(X,6) C(X,4) 100 0.03 C(X,6) 80 0.02 60 40 0.01 20 0 0.00 0 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 8000 10000 12000 Distance X Distance X Fig. 2. Ground level concentrations at various heights for primary and secondary Pollutants (stable case) 274 International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME In figure 6 the ground level concentration at various heights for primary and secondary pollutants with removal mechanisms Ws=0.001 and Vd=0.01 for stable case is analysed. From the graph we notice that the concentration of primary and secondary pollutants decreases as height increases along downwind distance. 250 0.060 Primary pollutants Ws=0 Secondary pollutants 0.055 Ws=0 Vd=0 Vd=0 0.050 200 0.045 C(6000,Y) 0.040 150 Concentration Concentration C(3000,Y) 0.035 0.030 C(9000,Y) 100 0.025 C(6000,Y) 0.020 0.015 50 C(3000,Y) 0.010 C(9000,Y) 0.005 0 0.000 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Height Y Height Y Fig. 3. Concentrations at various distances for primary and secondary Pollutants (stable case) 60 Primary pollutants Secondary pollutants Ws=0 0.010 Ws=0 C(X,2) Vd=0 Vd=0 50 C(X,4) 0.008 40 C(X,6) C(X,2) C(X,4) Concentration Concentration 0.006 C(X,6) 30 0.004 20 0.002 10 0.000 0 0 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 8000 10000 12000 Distance X Distance X Fig. 4. Ground level concentrations at various heights for primary and secondary Pollutants (neutral case) In figure 7 the concentration at various distances of primary and secondary pollutants with removal mechanisms Ws=0.0001 and Vd=0.01 for neutral case is studied. It is observed that the concentration of primary and secondary pollutants is high in the lower heights and as move upwards the concentration decreases and is zero around 350 mts in neutral case. 275 International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 70 Primary pollutants 0.010 Secondary pollutants W s=0 W s=0 60 C(6000,Y) 0.009 Vd=0 Vd=0 C(6000,Y) 0.008 50 0.007 Concentration Concentration 40 0.006 0.005 30 C(3000,Y) C(3000,Y) 0.004 20 0.003 C(9000,Y) 0.002 10 C(9000,Y) 0.001 0.000 0 0 50 100 150 200 250 300 350 0 100 200 300 400 500 Height Y Height Y Fig.5. Concentrations at various distances for primary and secondary pollutants (neutral case) 60 Prim ary pollutant S econdary pollutants W s=0.001 0.005 W s=0.001 Vd=0.01 C (X ,2) 50 C(X,2) V d=0.01 C (X ,4) 0.004 40 C(X,4) C (X ,6) Concentration Concentration C(X,6) 0.003 30 0.002 20 10 0.001 0 0.000 0 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 8000 10000 12000 Distance X D istance X Fig. 6. Ground level concentration at various heights for primary and secondary pollutants (stable case) with removal mechanisms Ws=0.001 and Vd=0.01 Primary pollutants secondary pollutants 35 0.0035 Ws=0.001 W s=0.001 Vd=0.01 C (6000,Y) Vd=0.01 30 0.0030 25 0.0025 Concentration Concentration 20 0.0020 C (9000,Y) 15 0.0015 C(80,Y) C (3000,Y) 10 0.0010 C(40,Y) 5 C(120,Y) 0.0005 0 0.0000 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 400 450 Height Y H eight Y Fig.7. Concentrations at various distances for primary and secondary pollutants with removal mechanisms Ws=0.001 and Vd=0.01 (neutral case) 276 International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 6. CONCLUSIONS A numerical model for the computation of the ambient air concentration emitted from an urban area source undergoing various removal mechanisms and transformation process is presented. The concentration of primary pollutants and secondary pollutants attains peak value at the downwind end of the source region. The concentration of primary and secondary pollutants is less in magnitude for neutral atmosphere when compared to stable condition. The neutral atmospheric condition enhances vertical diffusion carrying the pollutant concentration to greater heights and thus the concentration is less at the surface region. Hence neutral case is favorable condition in air pollution point of view. REFERENCES [1]. Pope I C A, Dockery D W, Schwartz J, Routlegde, Review of Epidemiological Evidence of Health Effects of Particulate Air Pollution, Inhalation Toxicology 7(1995) 1-18. [2]. Brook R D, Franklin B, Cascio W, Hong Y, Howard G, Lipsett M, Luepker R, Mittleman M, Samet J, Smith Jr S C, Tager I, Air Pollution and Cardiovascular Disease: A Statement for Healthcare Professionals from the Expert Panel on Population and Prevention, Sci. Amer. Heart Assoc 109 (2004) 2655- 2671. [3]. Riediker M, Cascio W E, Griggs T R, Herbst M C, Bromberg P A, Neas L, Williams R W, Devlin R B, Particulate Matter Exposure in Cars Is Associated with Cardiovascular Effects in Healthy Young Men, Am. J. Respir. Crit. Care Med 169 (2004) 934-940. [4]. Arif A A, Shah S M, Association between Personal Exposure to Volatile Organic compounds and Asthma among Us Adult Population, Int. Arch. Occup. Environ. Health 80 (2007) 711-719. 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Dhiren Dave, Dr. Sanjay Nalbalwar and Dr. Ashok Ghatol, “Location Aware Management and Control for Pollution Prevention”, International Journal of Electronics and Communication Engineering & Technology (IJECET), Volume 4, Issue 2, 2013, pp. 338 - 347, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472. 277

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