International Journal of Applied Research & Studies ISSN

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1 Iteratioal Joural of Applied Research & Studies ISSN Research Article Effect of Mesoscale Wid o Primary ad Secodary Pollutats Emitted From a Area Source With Dry Depositio ad Chemical Reactio Authors 1Lakshmiarayaachari K, 2 Sudheer Pai K L*, 3Siddaliga Prasad M, 4 Paduragappa C Address for Correspodece: 1 Departmet of Mathematics, Sai Vidya Istitute of Techology, Bagalore - Idia 2 Pricipal, RNS First Grade College, Bagalore - Idia 3 Departmet of Mathematics, Siddagaga Istitute of Techology, Tumkur - Idia 4 Departmet of Mathematics, Raja Rajeswari College of Egieerig, Bagalore Idia ABSTRACT: A two dimesioal umerical model has bee developed to study the effect of mesoscale wid o primary ad secodary pollutats emitted from the groud based area source. The model takes ito accout the trasformatio process ad removal mechaisms through chemical reactio ad dry depositio. The umerical model is solved usig Crak-Nicolso fiite differece scheme uder the stability depedet meteorological parameters ivolved i wid velocities ad eddy diffusivity profiles. The mesoscale wid is cosidered to simulate the large scale wid produced by urba heat islad. The aalysis shows that the mesoscale wid produced by urba heat islad help the pollutats to circulate ad move i upwid directio, thus makig the problem of air pollutio more severe i urba areas. The aalysis of the proposed mathematical model leads to coclude that the cocetratio level of pollutats aggravates uder urba heat islad effect i both stable ad eutral atmospheric coditios. Key words: Mesoscale wid, Primary ad secodary pollutat, Crak-Nicolso method, Dry depositio, Chemical reactio. spkl@rediffmail.com *Correspodig author Id ijars/ Vol. II/ Issue I/Ja, 2013/292 1

2 Iteratioal Joural of Applied Research & Studies ISSN Itroductio Urbaizatio egatively impacts the eviromet maily by the productio of pollutio ad also by the modificatios i physical ad chemical properties of the atmosphere brought up due to the formatio of urba heat islads. Urba heat islad is referred as the pheomeo i which urba areas experiece warmer temperatures tha their rural surroudigs. The progressive replacemet of atural groud cover ad ladscape by built surfaces costitutes the mai cause of urba heat islad formatio. Waste heat from vehicles, idustries, air coditioers etc. further exacerbatig the urba heat islad effect. As far as pollutio i urba areas is cocered, the potetial hazards of air pollutats like carbo mooxide, sulphur dioxide, itroge oxides, ozoe, particulate matter etc. o huma life ad its eviromet has made it a major ad serious threat globally. The combustio of hydro carbo fuels i residetial area, vehicular exhausts due traffic flow ad several other major ad mior sources pollute a part or whole area of a urba eviromet. It is well kow that the pollutats are diffused ad advected dowwid; hece such area source pollutio ot oly affects people ad eviromet withi this area, but also the people stayig i adjacet rural opolluted area [1]. So, it is importat to determie the pollutio distributio i urba areas ad here, the wid flow over a particular area plays a very crucial role. The ature of wid flow varies accordigly with a regio; large urba areas due to formatio of urba heat islads ofte geerate their ow special type of wid, kow as mesoscale wid ad it plays a importat role i shapig the urba pollutio patter [2]. Mathematical models are of fudametal importace i describig ad uderstadig the dispersio (advectio-diffusio) of air pollutats i the atmosphere, sice all the parameters are expressed i mathematical forms ad therefore the ifluece of idividual parameters o pollutat cocetratio ca be easily examied. So i previous years, the effect of urba heat islads o pollutio patter through the use of mathematical models has bee ivestigated i some studies. Oe of the importat atmospheric pheomea is the coversio of air pollutats from gaseous to particulate form. The primary pollutats which are emitted directly ito the atmosphere are coverted ito secodary pollutats by meas of chemical reactio. The study of secodary pollutats is very importat as life period of secodary pollutats is loger tha primary pollutats ad it is more hazardous to huma life ad the surroudig eviromet. Several earlier studies have bee reported i which measuremets (usually airbore) have bee carried out i the dowwid of large urba complexes i order to obtai material balaces o gaseous ad particulate pollutats [3-9]. The umerical area-source model cosiders eddy diffusivity ad velocity profiles as fuctios of height, stability parameter ad frictioal velocity; see [10]. I this model the geostrophic wid, et heat flux, surface roughess, mixig height of the atmosphere ad emissio rate of the source is specified. Mass coversio equatios for a array of boxes (or odes) are solved simultaeously by a implicit fiite differece scheme for differet atmospheric coditios. Oe of the importat aspects of mathematical modelig of air pollutio which require immediate ijars/ Vol. II/ Issue I/Ja, 2013/292 2

3 Iteratioal Joural of Applied Research & Studies ISSN attetio is to study the diffusio of air pollutats i a urba area. The effect of mesoscale wid has to be take ito accout alog with large-scale wid to predict pollutat cocetratio. Large-scale wid is ot sufficiet for air pollutio forecasts i urba areas;see [11]. Mesoscale wid sharpes pollutio gradiets betwee urba ad rural areas [12,13]. Near the ceter of heat islad the vertical mixig would be ehaced by mesoscale wid; see [14]. Mathematical model with chemically reactive pollutats are studied; see [15-19]. These models do ot take ito the accout of mesoscale wid. The preset study takes ito cosideratio the effect of mesoscale wid o the distributio of pollutats i the atmosphere over a urba area. The limitatios of aalytical methods i obtaiig the solutios to developed mathematical model lead us to use umerical method. The well kow implicit Crak-Nicolso fiite differece umerical scheme, which has the advatage of beig ucoditioal stable, cosistet ad covergece, is employed to compute the cocetratio of pollutats i a give urba regio. We develop a umerical model for primary ad secodary pollutats i the atmosphere with more realistic large scale wid velocity, mesoscale wid velocity ad eddy diffusivity profiles. We study the effect of mesoscale wid velocity ad removal mechaisms such as dry depositio o primary ad secodary pollutats. I this model we have made geeral assumptio that the secodary pollutats are formed by meas of first order chemical coversio of primary pollutats. 2. Model developmet The physical problem cosists of a area source, which is spread out over the surface of a city with fiite dowwid distace ad ifiite cross wid dimesios. We assume that the pollutats are emitted at a costat rate from the area source ad spread withi the mixig layer adjacet to earth s surface where mixig takes place as a result of turbulece ad covective motio. This mixig layer exteds upwards from the surface to a height where all turbulet flux-divergeces resultig from surface actio have virtually falle to zero. The pollutats are trasported horizotally by large scale wid which is a fuctio of vertical height (z) ad horizotally as well as vertically by local wid caused by urba heat source, called mesoscale wid. We have cosidered the cetre of the heat islad at a distace x l/2 i.e. at the cetre of the city. We have cosidered the source regio withi the urba area which exteds to a distace l i the dowwid x directio (0 x l). I this model we have take the city legth l=6 km. Assumig the homogeeity of urba terrai, the mea cocetratio of pollutat is cosidered to be costat alog the crosswid directio i.e., pollutats cocetratio does ot vary i cross wid directio. Therefore, there is o y-depedece. The physical descriptio of the model is show schematically i figure 1. We ited to compute the cocetratio distributio i the urba area. We assume that the pollutats udergo the removal mechaisms, i.e., dry depositio ad chemical reactio. ijars/ Vol. II/ Issue I/Ja, 2013/292 3

4 Iteratioal Joural of Applied Research & Studies ISSN Figure 1: Physical layout of the model 2.1 Primary pollutat The basic goverig equatio of primary pollutat ca be writte as C p t + U x, z C p x + W z C p = K z z C p kc p (1) where C p = C p x, z, t is the ambiet mea cocetratio of pollutat species, U is the mea wid speed i x-directio, W is the mea wid speed i z- directio, K z is the turbulet eddy diffusivity i z-directio ad k is the first order chemical reactio rate coefficiet of primary pollutat C p. Equatio (1) is derived uder the followig assumptios: The lateral flux of pollutats alog crosswid directio is assumed to be small i.e., V C p y ad y K y C p y 0 where V is the velocity i the y - directio ad K y is the eddydiffusivity coefficiet i the y directio. Horizotal advectio is greater tha horizotal diffusio for ot too small values of wid velocity, i.e., meteorological coditios are far from stagatio. The horizotal advectio by the wid domiates over horizotal diffusio, i.e., U C p C K p x x, where U ad K x x are the horizotal wid velocity ad horizotal eddy diffusivity alog x directio respectively. x We assume that the regio of iterest is free from pollutio at the begiig of the emissio. Thus, the iitial coditio is ijars/ Vol. II/ Issue I/Ja, 2013/292 4

5 Iteratioal Joural of Applied Research & Studies ISSN C p = 0 at t = 0, 0 x l ad 0 z H (2) Where l is the source legth i the dowwid directio ad H is the mixig height. We assume that there is o backgroud pollutio of cocetratio eterig at x = 0 ito the domai of iterest. Thus C p = 0 at x = 0, 0 z H ad t > 0 (3) We assume that the chemically reactive air pollutats are beig emitted at a steady rate from the groud level. They are removed from the atmosphere by groud adsorptio. Hece the correspodig boudary coditio takes the form K z C p = p C p Q at z = 0, 0 < x l ad t > 0 (4) Where Q is the emissio rate of primary pollutat species ad p is the dry depositio velocity. The pollutats are cofied withi the mixig height ad there is o leakage across the top boudary of the mixig layer. Thus K z C p = 0 at z = H, x > 0, 0 < x l ad t > 0 (5) The term kc p i equatio (1) represets coversio of gaseous pollutats to particulate material as log as the process ca be represeted approximately by first-order chemical reactio. We assume that the gaseous species is coverted ito particulate matter. Sulphate represets a example of gas-to-particle coversio. 2.2 Secodary pollutat The goverig equatio for the secodary pollutat C s is C s t + U x, z C s + W z C p x = K z z C s + V g kc p (6) Where, V g is the mass ratio of the secodary particulate species to the primary gaseous species, which is beig coverted. I derivig equatio (2.6) we have made similar assumptios as i the case of primary pollutat. The appropriate iitial ad boudary coditios o Cs are: C s = 0 at t = 0, for 0 x X 0 ad 0 z H (7) C s = 0 at x = 0, for 0 z H ad t > 0 (8) Sice there is o direct source for secodary pollutats, we have K z C s = s C s at z = 0, 0 x X 0 ad t > 0 (9) K z C s = 0 at z = H ad t > 0 (10) Where s is the dry depositio velocity of the secodary pollutat C s. ijars/ Vol. II/ Issue I/Ja, 2013/292 5

6 Iteratioal Joural of Applied Research & Studies ISSN Meteorological parameters The treatmet of equatios (1) ad (6) maily depeds o the proper estimatio of diffusivity coefficiet ad velocity profile of the wid ear the groud/or the lower layers of the atmosphere. The meteorological parameters ifluecig eddy diffusivity ad velocity profile are depedet o the itesity of turbulece, which is iflueced by atmospheric stability. Stability ear the groud is depedet primarily upo the et heat flux. I terms of boudary layer otatio, the atmospheric stability is characterized by the parameter L; see [20], which is also a fuctio of et heat flux amog several other meteorological parameters. It is defied by L = u 3 ρc p T κgh f (11) Where u * is the frictio velocity, H f the et heat flux, the ambiet air desity, C p the specific heat at costat pressure, T the ambiet temperature ear the surface, g the gravitatioal acceleratio ad κ the Karma s costat 0.4. H f < 0 ad cosequetly L > 0 represets stable atmosphere, H f > 0 ad L < 0 represet ustable atmosphere ad H f = 0 ad L represet eutral coditio of the atmosphere. The frictio velocity u is defied i terms of geostrophic drag coefficiet c g ad geostrophic wid u g such that u = c g u g (12) where c g is a fuctio of the surface Rossby umber R 0 = u fz 0, where f is the Coriolis parameter due to earth s rotatio ad z 0 is the surface roughess legth. The value of c g, the drag coefficiet for a eutral atmosphere i the form: see [21]. c g = 0.16 log 10 R (13) The effect of thermal stratificatio o the drag coefficiet ca be accouted through the relatios: c gus = 1.2 c g for ustable flow, (14) c gs = 0.8 c g for slightly stable flow ad (15) c gs = 0.6 c g for stable flow. (16) I order to evaluate the drag coefficiet, the surface roughess legth z 0 may be computed accordig to the relatioship i.e., z 0 = (H a) (2A ); see [22], where H is the effective height of roughess elemets, a is the frotal area see by the wid ad A is the lot area (i.e., the total area of the regio divided by the umber of elemets). ijars/ Vol. II/ Issue I/Ja, 2013/292 6

7 Iteratioal Joural of Applied Research & Studies ISSN Fially, i order to coect the stability legth L to the Pasquill stability categories, it is ecessary to quatify the et radiatio idex. Followig values of H (Table 1) for urba area; see [10]. Table 1: Net heat flux H f (lagley mi 1 ) Net radiatig idex : Net heat flux H f : f 3.1 Eddy diffusivity profiles Followig gradiet trasfer hypothesis ad dimesioal aalysis, the eddy viscosity K M is defied as K M = u 2 (17) U Usig similarity theory, the velocity gradiet may be writte as; see [20] U = u φ M κz (18) Substitutig this i the equatio (17), we have K M = κu z φ M (19) The fuctio φ M depeds o, z L where L is Moi-Obukhov stability legth parameter. It is assumed that the surface layer termiates at z = 0.1κ u for eutral stability. For stable f coditios, surface layer exteds to z = 6L. For eutral stability with z < 0.1κ u (withi surface layer) f φ M = 1 ad K M = κu z (20) For stable flow with 0 < z/l < 1, φ M = 1 + α z ad (21) L K M = κu z 1+ α L z (22) For stable flow with 1 < z/l < 6, φ M = 1 + α ad K M = κu z 1+α (23) ijars/ Vol. II/ Issue I/Ja, 2013/292 7

8 Iteratioal Joural of Applied Research & Studies ISSN It has bee show that = 5.2 ; see [23]. I the PBL (plaetary boudary layer), where z/l is greater tha the limits cosidered above ad z > 0.1κ u f for K M. For eutral stability with z > 0.1κ u f,, we have, the followig expressios K M = 0.1κ 2 u 2 f (24) For stable flow with z > 6L, up to H, the mixig height, K M = 6κu L 1+α. (25) Equatios (19) to (25) give the eddy viscosity for the coditios eeded for the model. However, the model deals with the trasport of mass rather tha the trasport of mometum, as implied by the use of viscosity. Sice both the mass ad the mometum are trasported by turbulet eddies, it is physically reasoable to assume that the turbulet viscosity coefficiet K M is umerically equivalet to the eddy diffusivity coefficiet K z. Also, there is some experimetal evidece that the ratio K z K M remais costat ad equal to uity, at least i the surface layer; see [23]. The commo characteristic of K z is that it has a liear variatio ear the groud, a costat value at mid mixig depth ad a decreasig tred as the top of the mixig layer is approached. The theoretical aalysis of eutral boudary layer gives the expressio i the form; see [24], K z = 0.4u ze 4z H (26) where H is the mixig height. For stable coditio, we use the followig form of eddy-diffusivity; see [25], K z = κu z z L exp bη, (27) b = 0.91, η = z L μ, μ = u fl The above form of K z was derived from a higher order turbulece closure model which was tested with stable boudary layer data of Kasas ad Miesota experimets. Eddy-diffusivity profiles give by equatios (26) ad (27) have bee used i this model developed for eutral ad stable atmospheric coditios. ijars/ Vol. II/ Issue I/Ja, 2013/292 8

9 Iteratioal Joural of Applied Research & Studies ISSN Mesoscale wid velocity profiles It is kow that i a urba city the heat geeratio causes the vertical flow of air with maximum velocity (risig of air) at the cetre of the city. Hece the city ca be called as heat islad. This risig air forms a air circulatio ad this circulatio is completed at larger heights. This is called mesoscale circulatio. I order to icorporate more realistic form of velocity profile i the models, we itegrate equatio U = u φ M from z κz o to z + z o for stable ad eutral coditios which depeds o Coriolis force, surface frictio, geostrophic wid, stability characteristic parameter L ad vertical height z. But large urba areas geerate a additioal circulatio called Mesoscale circulatio. Therefore, to take ito accout the mesoscale wid over the urba areas, for realistic form of velocity profiles, it is ecessary to modify the wid velocity profiles. So we obtai the followig expressios for large ad mesoscale wid velocities; see[2,7]. I case of eutral stability with z < 0.1κ u f, we get U = u κ l z+z 0 z 0 (28) It is assumed that the horizotal mesoscale wid varies i the same vertical maer as u. The vertical mesoscale wid W e ca the be foud by itegratig the cotiuity equatio ad we obtai i the form U e = a x x o l z+z 0 z 0 (29) Where a is proportioality costat. Thus we have U x, z = u + u e = u κ a x x o l z+z 0 z 0 (30) W z = W e = a z l z+z 0 z 0 z + z 0 l z + z 0 (31) I case of stable flow with 0 < z L < 1, we get U = u κ l z+z 0 + α z (32) z 0 L U e = a x x o l z+z 0 + α z (33) z 0 L U x, z = u + u e = u a x x κ o l z+z 0 + α z (34) z 0 L W z = W e = a z l z+z 0 z 0 z + z 0 l z + z 0 + α 2L z2 (35) ijars/ Vol. II/ Issue I/Ja, 2013/292 9

10 Iteratioal Joural of Applied Research & Studies ISSN I case of stable flow with 1 < z L < 6, we get U = u κ l z+z 0 z (36) U e = a x x o l z+z 0 z (37) U x, z = u + u e = u κ a x x o l z+z 0 z (38) W z = W e = a z l z+z 0 z 0 + z 0 l z + z z (39) I the plaetary boudary layer, above the surface layer, power law scheme has bee employed. U = u g u sl z z sl H z sl p + usl (40) U e = a x x o z z sl H z sl p + usl (41) U x, z = u + u e = u g u sl a x x o z z sl H z sl p + 1 a x xo u sl (42) W z = W e = a z z sl p+1 z z sl H z sl p + zusl (43) Where, u g is the geostrophic wid, u sl the wid at z sl, z sl the top of the surface layer, H the mixig height ad p is a expoet which depeds upo the atmospheric stability. Suggested values for the expoet p, obtaied from the measuremets made from urba wid profiles, as follows; see [26]. p = 0.20 for eutral coditio 0.35 for slightly stable flow 0.50 for stable flow Wid velocity profiles give by equatios (34), (35), (38), (39), (42) ad (43) are used i this model; see [2]. 4. Numerical Method We ote that it is difficult to obtai the aalytical solutio for equatios (1) ad (6) because of the complicated form of wid speed ad eddy diffusivity profiles cosidered i this model. Hece, we have used umerical method based o Crak-Nicolso fiite differece scheme to obtai the solutio. The depedet variable C p is a fuctio of the idepedet variables x, z ad t, i.e., C p = C p (x, z, t). First, the cotiuum regio of iterest is overlaid with or subdivided ito a set of equal rectagles of sides x ad z, by equally spaced grid lies, parallel to z axis, ijars/ Vol. II/ Issue I/Ja, 2013/292 10

11 Iteratioal Joural of Applied Research & Studies ISSN defied by x i = i 1 x, i = 1,2,3, ad equally spaced grid lies parallel to x axis, defied by z j = (j 1) z, j = 1,2,3, respectively. Time is idexed such that t = t, = 0, 1, 2, 3, where is the time step. At the itersectio of grid lies, i.e. grid poits, the fiite differece solutio of the variable C p is defied. The depedet variable C p (x, z, t) is deoted by, C p = C p (x i, z j, t ), where (x i, z j ) ad t idicate the (x, z) value at a ode poit (i, j) ad t value at time level respectively. We employ the implicit Crak-Nicolso scheme to discretize the equatio (1). The derivatives are replaced by the arithmetic average of its fiite differece approximatios at the t ad ( + 1) t time steps. The equatio (1) at the grid poits (i, j) ad time step + 1/2 ca be writte as C p 1 2 t U z C p C 2 x + U z p x K z z C p + K z z C p W z C p 2 2 k C p + W z C p +1 + C p +1 = for i = 1,2, j = 1,2, (44) This aalog is actually the same as the first order correct aalog used for the forward differece equatio, but is ow secod order-correct, sice it is used to approximate the derivative at the poit (x i, z j, t +1 2 ). We use the backward differeces for advective term i the primary pollutat equatio. i.e C p t + 12 = C +1 p Cp The simplest way to model the trasport properties is to use upwid differecig where backward differeces are used whe the velocities are positive ad forward differeces are used whe the velocities are egative. U x, z C p x U C p = Cpi 1j x U C pi +1j C p x for U > 0 for U < 0 (45) We ote that i the preset problem the velocity is always positive ad hece we always use U x, z C p x = C p U Cpi 1j x (46) U x, z C p x +1 = C p Cpi 1j U x (47) W z C p = C p Wj Cp 1 z (48) ijars/ Vol. II/ Issue I/Ja, 2013/292 11

12 Iteratioal Joural of Applied Research & Studies ISSN W z C p +1 = C p Cp 1 Wj z (49) Also, for the diffusio term, we use the secod order cetral differece scheme Hece, K z z C p K z z C p Similarly, = K z z C p = 1 = 1 z + 1 K z z C p z K j +1 +K j 2 C p +1 C p z 2 z 2 K j +1 + K j C p +1 1 z K j +K j 1 2 C p Cp 1 z C p K j + K j 1 C p C p 1 (50) K z z C p +1 = z 2 K j +1 + K j C p C p K j + K j 1 C p C p 1 (51) Substitutig equatios (46) to (51) i equatio (44) ad rearragig the terms we get the fiite differece equatios for the primary pollutat C p i the form A j C +1 pi 1j + B j C +1 p 1 + D j C p + E j C p +1 = F j C pi 1j + G j C p 1 + M j C p + N j C p +1 (52) for each i = 2,3,4, imaxl imaxx 0, for each j = 2,3,4, jmax 1 ad = 0,1,2,3, where, A j = U j 2 x F j = U j 2 x B j = 4 z 2 K j + K j 1 + W j 2 z G j = 4 z 2 K j + K j 1 + W j 2 z E j = K 4 z 2 j + K j +1 N j = K 4 z 2 j + K j +1 D j = 1 + U j M j = 1 U j 2 x + W j 2 x W j 2 z + 2 z K 4 z 2 j K j + K j 1 + k 2 K 4 z 2 j K j + K j 1 k 2 i maxl ad imaxx 0 are the i values at x = l ad X 0 respectively ad jmax is the value of j at z = H. Equatio (52) is true for iterior grid poits. At the boudary grid poits we have to use the boudary coditios (2) to (5). The iitial coditio (2) is 0 C p = 0 for j = 1,2, j max, i = 1,2, imaxl imaxx 0 The coditio (3) becomes +1 = 0 for i = 1 ad j = 1,2, jmax, = 0,1,2, (53) C p The boudary coditio (4) ca be writte as ijars/ Vol. II/ Issue I/Ja, 2013/292 12

13 Iteratioal Joural of Applied Research & Studies ISSN z 1 + p C K p C p +1 = Q z (54) j K j for j = 1, i = 2,3,4,. imaxl ad = 0,1,2,3 The boudary coditio (5) ca be writte as C pmax 1 C pmax = 0 (55) for j = jmax, i = 2,3,4, imaxl imaxx 0 The above system of equatios (52) to (55) has a tridiagoal structure ad is solved by Thomas Algorithm. The ambiet air cocetratio of primary pollutats (gaseous) is obtaied for various atmospheric coditios ad the values of dry depositio, wet depositio ad chemical reactio rate are costat. Similarly the fiite differece equatios for the secodary pollutat Cs ca be writte as A j C +1 si 1j + B j C +1 s 1 + D j C +1 s + E j C +1 s +1 = F j C si 1j + G j C s 1 + M j C s + N j C s v gkc p (56) for i = 2,3,4 imaxl, imaxx 0, j = 2,3,4, jmax 1 The iitial ad boudary coditios o secodary pollutat C s C are 0 = 0 for j = 1,2, jmax, i = 1,2, imaxl imaxx 0 (57) C +1 s = 0 for i = 1, j = 1,2, jmax, = 0,1,2, for i = 2,3,4 imaxl, imaxx 0, j = 2,3,4, jmax 1 s 1 + s z K j C +1 s C +1 s +1 = 0 for j = 1, i = 2,3, imaxl imaxx 0 (58) C s 1 - C s = 0 for j = jmax, i = 2,3,4, imaxl, imaxx 0 (59) where, A j = U j 2 x F j = U j 2 x B j = 4 z 2 K j + K j 1 + W j 2 z G j = 4 z 2 K j + K j 1 + W j 2 z E j = K 4 z 2 j + K j +1 N j = K 4 z 2 j + K j +1 D j = 1 + U j M j = 1 U j 2 x + W j 2 x W j 2 z + 2 z 4 z 2 K j K j + K j 1 4 z 2 K j K j + K j 1 V g is the mass ratio of the secodary particulate species to the primary gaseous species which is beig coverted. ijars/ Vol. II/ Issue I/Ja, 2013/292 13

14 Iteratioal Joural of Applied Research & Studies ISSN The system of equatios (52) to (55) has tri diagoal structure but is coupled with equatios (56) to (59). First, the system of equatios (52) to (55) is solved for C p, which is idepedet of the system (56) to (59) at every time step. This result at every time step is used i equatios (56) to (59). The the system of equatios (56) to (59)is solved for C s at the same time step. Both the systems of equatios are solved usig Thomas algorithm for tri-diagoal equatios (52) to (55) ad (56) to (59). Thus, the solutios for primary ad secodary pollutat cocetratios are obtaied. 5. Results ad discussio A umerical model is developed to study the effect of mesoscale wid o the cocetratio of primary ad secodary pollutats. The pollutats are emitted at a costat rate from a uiformly distributed area source. We have cosidered the source regio withi the urba city 0 x l which exteds up to 6000 meters. The primary pollutat is cosidered to be chemically reactive to form secodary pollutat by meas of first order chemical reactio. The results of this model have bee preseted graphically from figure 2 to figure 7. I figure 2, the effect of mesoscale wid o the groud level cocetratio of primary pollutat with the chemical reactio rate coefficiet with respect to distace for stable ad eutral atmospheric coditios are aalyzed. The cocetratio of primary pollutat decreases rapidly as the value of chemical reactio rate icreases. The cocetratio of pollutat is less i upwid side of the cetre of heat islad ad more i the dowwid side of the cetre of heat islad i the presece of mesoscale wid () whe compared to that of without mesoscale wid (a=0). This is because the horizotal compoet of mesoscale wid is alog the large scale wid o the left ad agaist it o the right. Thus i the presece of mesoscale wid, the advectio is more o the left ad less o the right. Therefore the cocetratio is less o the left ad more o the right i the presece of mesoscale wid. I geeral, the cocetratio of primary pollutat icreases i the dowwid directio. Comparig figures (a) ad (b), we fid that the cocetratio of pollutat at give distace is much smaller i the eutral atmospheric coditio tha that i the stable atmospheric case. The maximum cocetratio of pollutat is aroud 245 i stable case ad is ear to 65 i eutral atmosphere at x=6000 meters for k c = ijars/ Vol. II/ Issue I/Ja, 2013/292 14

15 Cocetratio Cocetratio Cocetratio Cocetratio Iteratioal Joural of Applied Research & Studies ISSN a=0 Primary pollutat (a) Distace k c = k c = a=0 Primary pollutat (b) Distace k c = k c = Figure 2: Effect of chemical reactio rate coefficiet o groud level cocetratio primary pollutat with respect to distace for (a) stable ad (b) eutral atmospheric coditios a=0 Primary pollutat = a=0 Primary pollutat = = = (a) Distace (b) Distace Figure 3: Effect of dry depositio velocity o groud level cocetratio primary pollutat with respect to distace for (a) stable ad (b) eutral atmospheric coditios. I figure 3, the effect of mesoscale wid o the groud level cocetratio of primary pollutat for differet values of dry depositio velocity with respect to distace for stable ad eutral atmospheric coditios are studied. As dry depositio velocity icreases the cocetratio of pollutat decreases. The magitude of cocetratio of pollutat is higher i stable case ad lower i eutral case. This is because the eutral case ehaces vertical diffusio to the greater heights ad thus the cocetratio is less. The cocetratio decreases rapidly i the stable case ad decreases slowly i the eutral case because the pollutat s cocetratio is high i stable case comparig to that i eutral case. ijars/ Vol. II/ Issue I/Ja, 2013/292 15

16 Iteratioal Joural of Applied Research & Studies ISSN I figure 4, the effect of mesoscale wid o the groud level cocetratio of secodary pollutat for differet values of dry depositio velocity with respect to distace for stable ad eutral atmospheric coditios are studied. As dry depositio velocity icreases, the cocetratio of secodary pollutat decreases. The magitude of the secodary pollutat is higher i the stable case ad lower i the eutral case. The cocetratio of secodary pollutat icreases as distace icreases. We observe that the cocetratio of secodary pollutat is more at the ed of the city regio ear the groud level (z = 2 meter). I figure 5, the effect of mesoscale wid o the groud level cocetratio of primary pollutat with the chemical reactio rate coefficiet with respect to height for stable ad eutral atmospheric coditios with ad without mesoscale wid is aalyzed. The cocetratio of pollutat decreases as the chemical reactio rate icreases. The magitude of cocetratio of pollutat is higher i stable case ad lower i eutral case. The cocetratio of pollutat decreases as height icreases. I the stable case, the cocetratio is zero aroud 22 meters height ad i eutral case the cocetratio reaches zero at 100 meters height from the groud level. This is because the eutral case ehaces vertical diffusio to the greater heights ad thus the cocetratio is less. I figure 6, the effect of mesoscale wid o the groud level cocetratio of primary pollutat for differet values of dry depositio velocity with respect to height for stable ad eutral atmospheric coditios is studied. As dry depositio velocity icreases, the cocetratio of primary pollutat decreases with respect to height. I stable case the magitude of cocetratio reaches zero aroud 25 meters height ad i eutral case the cocetratio is zero at 110 meters height from the groud level surface. Near the groud level the cocetratio of primary pollutat is aroud 140 i stable case whereas it is ear to 55 i eutral atmospheric coditio. I figure 7, the effect of mesoscale wid o the groud level cocetratio of secodary pollutat for differet values of dry depositio velocity with respect to height for stable ad eutral atmospheric coditios is studied. As dry depositio velocity icreases, the cocetratio of secodary pollutat decreases with respect to height. I the stable case the groud level cocetratio of secodary pollutat is more comparig to that i the eutral case. The cocetratio reaches zero aroud 25 meters height i stable case. But i eutral atmospheric coditio the cocetratio is zero at 110 meters height. ijars/ Vol. II/ Issue I/Ja, 2013/292 16

17 Cocetratio Cocetratio Cocetratio Cocetratio Iteratioal Joural of Applied Research & Studies ISSN a=0 Secodary pollutat = a=0 Secodary pollutat = = = (a) Distace (b) Distace Figure 4. Effect of dry depositio velocity o groud level cocetratio of secodary pollutat with respect to distace for (a) stable ad (b) eutral atmospheric coditios Primary pollutat k c = a= Primary pollutat a=0 80 k c = k c = k c = (a) 0 (b) Height Height Figure 5. Effect of chemical reactio rate coefficiet o groud level cocetratio primary pollutat with respect to height for (a) stable ad (b) eutral atmospheric coditios. ijars/ Vol. II/ Issue I/Ja, 2013/292 17

18 Cocetratio Cocetratio Cocetratio Cocetratio Iteratioal Joural of Applied Research & Studies ISSN Primary pollutat = a= Primary pollutat a= = = = (a) Height 0-5 (b) Height Figure 6: Effect of dry depositio velocity o groud level cocetratio primary pollutat with respect to height for (a) stable ad (b) eutral atmospheric coditios Secodary pollutat a= Secodary pollutat a= = = = = (a) (b) Height Height Figure 7: Effect of dry depositio velocity o groud level cocetratio of secodary pollutat with respect to height for (a) stable ad (b) eutral atmospheric coditios ijars/ Vol. II/ Issue I/Ja, 2013/292 18

19 Iteratioal Joural of Applied Research & Studies ISSN Coclusio The urba heat islad effect geerates their ow mesoscale wids ad cosequetly prevets the dispersal of pollutats which will result i a icrease i pollutio cocetratio i the atmosphere. The urba heat islad adds to the developmet of hazehood of cotamiated pollutats ad also helps these pollutats to circulate i upward directio, thus makig the pollutio problem more severe. It should be uderstood that the reasos for the trasformatio of big cities ito urba heat islads is attributed to huma factors, hece, collective efforts should be made i the process of reducig the urba heat islad ad for the creatio of cooler ad healthy city. The effect of mesoscale wid o a two dimesioal air pollutio due to area source is preseted usig a mathematical model to simulate the dispersio processes of primary ad secodary pollutats i a urba area with dry depositio ad chemical reactio. This model takes ito accout more realistic form of large scale wid, mesoscale wid ad eddy diffusivity profiles. I order to clearly visualize the role of mesoscale wid ( ad hece of urba heat islad) i shapig urba pollutio patter, the whole aalysis has also bee doe i the absece of mesoscale wid ad their comparative study shows substatial chages i pollutio distributio. It has bee foud that the mesoscale wid aggravates the groud level cocetratio of air pollutats i stable ad eutral atmospheric coditios. The results also demostrate the icrease i cocetratio level up to a cosiderable height uder the mesoscale wid, thus it helps i circulatig ad movig the pollutats i vertical directio. It ca be cocluded that the presece of mesoscale wid ehaces the cocetratio level of pollutats i urba areas for all vertical ad dowwid distaces uder all atmospheric coditios. The aalysis reveals that the groud level cocetratio of primary ad secodary pollutats attais peak value at the dowwid ed of the urba area. I the case of stable atmospheric coditio, the cocetratio of primary ad secodary pollutats is high at the surface regio. I the case eutral atmospheric coditio the pollutats reaches to more heights. This idicates that eutral case ehaces vertical diffusio of pollutats. Also, the results obtaied from the preset work shows that i the presece of mesoscale wid the advectio are more o the left ad less o the right of cetre of heat islad. The cocetratio of primary ad secodary pollutats is decreased o the left ad icreased o the right of cetre of heat islad i the presece of mesoscale wid compared to that i the absece of mesoscale wid. Though it is true, that owadays the air pollutio problems are ot hadled i the way described i the preset study, there are various air pollutio situatios that require the use of complex mesoscale models to adequately describe the processes ad dyamics as well as icorporate chemistry ad emissios i a adequate maer. Complex modelig studies such as CMAQ (Commuity Multiscale Air Quality Modelig) has bee desiged to approach air quality as a whole by icludig state of the sciece capabilities for modelig multiple air quality issues. ijars/ Vol. II/ Issue I/Ja, 2013/292 19

20 Iteratioal Joural of Applied Research & Studies ISSN However, i such complex models a umber of processes viz. sea breeze circulatios, urba heat islads, lee waves etc. goig o iside them, so they appear as black boxes ad oe caot easily uderstad the effects of idividual processes o the air quality. Apart from this, for may policy ad scietific applicatios o air quality modelig, it is desirable ot oly to kow the ambiet pollutat cocetratios that would result from a certai situatio, but also the extet to which those cocetratios would chage uder various perturbatios. Thus, the model proposed here helps i uderstadig oe of these processes that is, urba heat islad effect, by allowig cotrol over evirometal parameters. Hece, it would be easy to determie the steerig factor for such pheomeo ad also to test its sesitivity agaist chages i atmospheric coditios. Thus, the results of the proposed model ca be used to icrease the cofidece i complex model predictios ad idetify variables like wid field, atmospheric stability, etc. which should be ivestigated more closely i such complex modelig studies. Refereces [1] M. Vekatachalappa, S.K. Kha, K.A.G. Kakamari, Time depedet mathematical model of air pollutio due to area source with variable wid velocity ad Eddy diffusivity ad chemical reactio, Proc. Idia Nat. Sci. Acad. 69, pp , [2] J.F. Dilley, K.T. Ye, Effect of a mesoscale type wid o the pollutat distributio from a lie source, Atmospheric Eviromet, 6, pp , [3] P.LHaagese, A.L.Morris, Forecastig the behavior of the St. Louis, Missouri pollutat plume, Joural of Applied Meteorology 13, pp , [4] J.F.Stampfer Jr., J.A.Aderso, Locatig the St. Louis urba plume at 80 ad 120 km ad some of its characteristics, Atmospheric Eviromet 6, pp , [5] R.J.Breedig, P.L.Haagese, J.A.Aderso, J.P.Lodge Jr., J.F.Stampfer Jr., The urba plume as see at 80 ad 120 km by five differet sesors, Joural of Applied Meteorology 14, pp , [6] R.J.Breedig, H.B.Klois, J.P.Lodge Jr., J.B.Pate, D.C.Sheesley, T.R.Eglert, D.R.Sears, Measuremet of atmospheric pollutats i the St. Louis area, Atmospheric Eviromet 10, , [7] K.L.Calder, Atmospheric diffusio of particulate material cosidered as a boudary value problem, Joural of Meteorological society 18, pp , [8] T.J.Chadler, Discussio of the paper by MARSH ad FPSTER, The bearig of the urba temperature field upo urba pollutio patters, Atmospheric Eviromet 2, pp , [9] D.L.Ermak, A aalytical model of air pollutat trasport ad dispersio from a poit source, Atmospheric Eviromet 11, pp , [10] K.W.Raglad, Multiple box model for dispersio of air pollutats from area sources, Atmospheric Eviromet 7, pp , [11] J.F. Griffiths, Problems i urba air pollutio, AIAA 8th Aerospace Sciece Meetig, AIAA, Paper No , New York, ijars/ Vol. II/ Issue I/Ja, 2013/292 20

21 Iteratioal Joural of Applied Research & Studies ISSN [12] Maju Agarwal, Abhiav Tado, Modelig of the urba heat islad i the form of mesoscale wid ad its effect o air pollutio dispersal. Applied Mathematical modelig 34, pp , [13] B.Davidso, A summary of the New York urba air pollutio dyamics program, Joural of air pollutio cotrol 17, pp , [14] B.A.Eaga, Mahoey JR. Numerical modelig of advectio ad diffusio of urba area source pollutats, Joural of Applied Meteorology 11, pp , [15] K.Lakshmiarayaachari, C.Paduragappa, M.Vekatachalappa, Mathematical model of air pollutat emitted from a time defedat area source of primary ad secodary pollutats with chemical reactio, Iteratioal Joural of Computer Applicatios i Egieerig, Techology ad Scieces 4, pp , [16] S.K.Kha, Time depedet mathematical model of secodary air pollutat with istataeous ad delayed removal, Associatio for the advacemet of modelig ad simulatio techiques i etreprises 61, pp 1-14, [17] C.Paduragappa, K.Lakshmiarayaachari, M.Vekatachalappa, Effect of mesoscale wid o the pollutat emitted from a time depedet area source of primary ad secodary pollutats with chemical reactio, Iteratioal Joural of Computer Applicatios i Egieerig, Techology ad Scieces 4, pp , [18] N.Rudraiah, M.Vekatachalappa, Sujit Kumar Kha, Atmospheric diffusio model of secodary pollutats, Iteratioal Joural of Evirometal Studies 52, pp , [19] D.Pal, D.K.Siha, A area source umerical model o atmospheric dispersio with chemical reactio ad depositio, Iteratioal Joural of Evirometal Studies 29, pp , [20] A.S.Moi, A.M.Obukhov, Basic laws of turbulet mixig i the groud layer of the atmosphere, Doklady Akademii SSSR 151,pp , [21] H.H.Lettau, Wid surface stress ad Geostrophic Drag Coefficiets i the Atmospheric Surface Layer i Advaces i Geophysics, Atmospheric Diffusio ad Air pollutio Academic Press New York 6, pp , [22] H.H.Lettau, Physical ad Meteorological Basis for Mathematical Models of Urba diffusio processes, Proceedigs of Symposium o Multiple Source Diffusio Models USEPA publicatio AP-86, [23] E.K.Webb, Profile relatioships: the log liear rage, ad extesio ito strog stability, Quarterly Joural of Royal Meteorological Society 96, pp 67-90, [24] C.C.Shir, A prelimiary umerical study of a atmospheric turbulet flows i the idealized plaetary boudary layer, Joural of Atmospheric Scieces 30, pp , [25] J.Y.Ku, S.T. Rao, K.S.Rao, Numerical simulatio of air pollutio i urba areas; model developmet, Atmospheric Eviromet, 21(1) pp , [26] P.M.Joes, M.A.Larriaga, C.B.Wilso, The urba wid velocity profile. Atmospheric Eviromet 5, pp , ijars/ Vol. II/ Issue I/Ja, 2013/292 21