A General Lagrangian Approach to Simulate Pollutant Dispersion in Atmosphere for Low-wind Condition

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1 Amercan Journal of Envronmenal Engneerng 213, 3(1: 8-12 DOI: /j.ajee A General Lagrangan Approach o Smulae Polluan Dsperson n Amosphere for Low-wnd Condon Jonas C. Carvalho 1,*, Marco T. M. B. Vlhena 2, Gerváso A. Degraza 3, Marel Salle 1 1 Unversdade Federal de Peloas, Pós-Graduação em Meeorologa, Peloas, RS, Brazl 2 Unversdade Federal do Ro Gre do Sul, Pós-Graduação em Eng. Mecânca, Poro Alegre, RS, Brazl 3 Unversdade Federal de Sana Mara, Pós-Graduação em Meeorologa, Sana Mara, RS, Brazl Absrac In hs work we presen a sem-analycal Lagrangan parcle model o smulae he polluan dsperson durng low wnd speed condons. The model s based on a mehodology, whch solves he Langevn equaon hrough he assumpon ha coeffcen of he negrang facor s a complex funcon. The mehod leads o a non-lnear sochasc negral equaon, whch s solved by he Mehod of Successve Approxmaons or Pcard s Ierave Mehod. Takng no accoun he somorphsm beween he complex real plane by wrng down he low wnd formulaon n polar form, he procedure allow o deermne a formula for he low wnd drecon. Furhermore, an expresson analogous o he Euleran auocorrelaon funcon suggesed by Frenkel[1] appears n he real componen soluon. The model resuls presen an mprovemen n relaon o he oher models are shown o agree very well wh he feld racer daa colleced durng sable condons a Idaho Naonal Engneerng Laboraory (INEL. Keywords Low Wnd Speed Condon, Ar Polluon Modellng, Lagrangan Parcle Model, Pcard Ierave Mehod, Auocorrelaon Funcon, Model Evaluaon 1. Inroducon Recen years have been seen he flowerng of he work of searchng analycal soluon for he Langevn equaon wh he man purpose of smulang polluan dsperson n he amosphere. The meanng of analycy reles on he fac ha no approxmaon s made n he dervaves or doman dscrezaon along he soluon dervaon. In hs drecon, appeared n he leraure he works of Carvalho e al.[2-3], whch solves he Langevn equaon by he followng seps: lnearzaon of he Langevn equaon soluon of he resulan sochasc negral equaon by he Pcard erave scheme. Ths procedure leads o an analycal soluon n each erave sep. Carvalho Vlhena[4] solved by hs mehodology he Langevn equaon for low wnd speed condon. In order o model he polluan dsperson durng meerng effec n he soluon, he auhors made he assumpon ha he coeffcen of he negrang facor of he frs order lnear dfferenal equaon s a complex funcon, he magnary componen models he low wnd condon. Furhermore, he auhors consdered only he real componen of he negrang facor. A hs pon, s relevan o menon ha * Correspondng auhor: jonas.carvalho@ufpel.edu.br (Jonas C. Carvalho Publshed onlne a hp://journal.sapub.org/ajee Copyrgh 213 Scenfc & Academc Publshng. All Rghs Reserved by hs procedure, he Frenkel[1] auocorrelaon funcon naurally appears n he soluon. In hs work we oban a more general model, unlke he work of Carvalho Vlhena[4], consderng he real magnary pars of he complex funcon before performng he mu lp lcaon of he negrang facor, expressed by he Euler formula, nsde ousde of he negral soluon. Takng no accoun he somorphsm beween he complex real plane by wrng down he low wnd formulaon n polar form, he procedure allow o deermne a formula for he low wnd drecon. Furhermore, an expresson analogous o he Euleran auocorrelaon funcon suggesed by Frenkel[1] appears n he real componen soluon. Fnally, s necessary o menon ha when he non-dmensonal quany ha conrols he meerng oscllaon frequency goes o zero hs soluon reduces o he soluons encounered by Carvalho e al.[2-3] for wndy condon. The low wnd speed daa colleced durng sable condons a Idaho Naonal Engneerng Laboraory (INEL[5] has been used o evaluae he new model. The paper s oulned as follows: n secon wo he model s presened, n secon hree he modellng resuls are dscussed n secon four he conclusons. 2. The Low Wnd Model The approach consss n he lnearzaon of he Langevn equaon as sochasc dfferenal equaon:

2 Amercan Journal of Envronmenal Engneerng 213, 3(1: du + f ( U = g(, (1 d whch has he well known soluon n erms of he negrang facor: U = f ( d 1 ( g e d. f ( d (2 e In order o embody he low wnd speed condon n he Langevn equaon, s assumed ha U f ( are complex funcons wren as: U = u + v (3 f = p +, (4 ( q where u v are he real magnary pars of U, respecvely, p q are he real magnary pars of f (, respecvely. Therefore, he exponenals appearng n Equaon (2 reads lke: f ( d. (5 p q e = e + Applyng he Euler formula, Equaon (2 becomes: p U = e cos( q + sn( q u + v + g [ ][ ( ( ] p e [ cos( q sn( q ] 1 ( d p. (6 e [cos( q sn( q ] Mulplyng he Equaon (6 by he complex conjugae performng he mulplcaons, we can oban p p ( ( [ cos( sn( ] ( ( U = e q + q u + v + { cos ( sn ( } g e q q d Consderng = τ, we can wre he Equaon (7 as p U = e [ cos( q + sn( q ] u ( + v( +. (8 pτ g ( e cos ( qτ sn ( qτ d In order o deermne he wnd drecon we cas equaon (8 no: U = e e p p cos( q u sn( q v [ ] pτ ( + g( e cos( qτ [ ] d pτ ( g( e sn( qτ d + (7, (9 where by comparson wh Equaon (3 we have p [ ] d pτ ( + g( e cos( qτ u = e cos( q u (1a p [ ] d pτ ( g( e sn( qτ v = e sn( q v. (1b Bearng n mnd he somorphsm beween he complex real planes, he low wnd expresson gven by Equaon (9 s descrbed n he complex plane. Ths procedure allows as deermnng he low wnd drecon, usng polar form. For hs end, we rewre Equaon (9 lke 2 θ = u 2 + v e, (11 U where θ s he low wnd drecon relave he x-axs v θ = arcan (12 u Noe ha he real componen of he Equaon (9, pτ e cos( qτ s analogous o he Euleran auocorrelaon funcon suggesed by Frenkel[1] (p. 8 wren n a dfferen way by Murgaroyd[6]. Therefore, p q are gven by 1 m p = q = ( m T ( m T where T s he me scale for a fully developed urbulence m s a non-dmensonal quany ha conrols he meerng oscllaon frequency. A hs pon, s mporan o menon ha when m goes o zero he Equaon (9 reduces o he soluon for wndy condons, whch s wren n erms of he exponenal form of he auocorrelaon funcon. The Equaon (9 s a non-lnear sochasc negral equaon, whch mus be solved eravely. The mehod appled o solve he Equaon (9 s he Mehod of Successve Approxmaons or Pcard s Ieraon Mehod[7], assumng ha he nal guess for he erave approxmaon s deermned from a Gaussan dsrbuon. For applcaons, he values for he parameers m T have been calculaed accordng o Carvalho e al.[8]: * + * 16π T T T m = (13 4πT T =, (14 u* where h s he sable PBL hegh, u * s he frcon velocy T* 2s. As he urbulence s consdered Gaussan n he horzonal drecon, he funcon g ( can be gven by:.64 h

3 1 Jonas C. Carvalho e al.: A General Lagrangan Approach o Smulae Polluan Dsperson n Amosphere for Low-wnd Condon σ u σ 2σ g( = + + ξ ( 2 2 x j 2σ x, (15 j τ L 2 where σ s he urbulen velocy varance, τ L s he Lagrangan me scale ξ s a normally dsrbued (average varance d rom ncremen. For he vercal componen, we solve he Langevn equaon by he ILS approach as suggesed by Carvalho e al.[4]. 3. Modellng Resuls The daa ulzed o evaluae he model performance are composed by a seres of feld expermens conduced under sable condons n low wnds over fla erran. The racer daa were colleced a Idaho Naonal Engneerng Laboraory (INEL he resuls are publshed n a U.S. Naonal Oceanc Amospherc Admnsraon (NOAA repor[5]. For he smulaons, he urbulen flow s assumed nhomogeneous only n he vercal he ranspor s realzed by he longudnal componen of he mean wnd velocy. The horzonal doman was deermned accordng o sampler dsances he vercal doman was se equal o he observed PBL hegh. The me sep was mananed consan was obaned accordng o he value of he Lagrangan decorrelaon me scale ( = τ L / c, where τ L mus be he smaller value beween τ L (wh = 1,2, 3 c s an emprcal coeffcen se equal o 1. The values of σ τ L were parameerzed accordng o scheme developed by Degraza e al.[9]. The negraon mehod used o solve he negrals appearng n Equaon (1 was he Romberg echnque. Because of wnd drecon varably durng he INEL expermen, a full 36 o samplng grd was mplemened. Arcs were lad ou a rad of 1, 2 4 m from he emsson pon. Samplers were placed a nervals of 6 o on each arc for a oal of 18 samplng posons. The recepor hegh was.76 m. The racer SF 6 was released a a hegh of 1.5 m. The 1 h average concenraons were deermned by means of an elecron capure gas chromaography. Wnd measuremens were provded by lghwegh cup anemomeers bvanes a he 2, 4, 8, 16, m levels of he 61-m ower locaed on he 2 m arc. Table 1 shows he meeorologcal daa ulzed for he valdaon of he proposed model. Observed wnd speeds were used o calculae he coeffcens for he exponenal wnd profles. The Monn-Obukhov lengh L he frcon velocy u * were approxmaed hrough he numercal bes f beween he observed wnd speeds calculaed wnd profle suggesed by Busnger e al.[1]. To calculae h (he sable PBL hegh, he relaon h=.4( ul 1/2 * fc was used[11], where f c s he Corols parameer. The model performance s shown n Tables 1 2 Fgures 1. Table 1 shows he comparson beween observed predced ground-level cenerlne concenraons. Table 2 presens he resul of he sascal analyss made wh observed predced values of ground-level cenerlne concenraon followng Hanna s[12] sascal ndces. Gvng a look a he resuls we observe ha he model smulaes que well he expermenal daa n sable condon, ndeed, presens resuls comparable or even beer han ones obaned by Carvalho Vlhena[4] Carvalho e al.[8]. The sascal analyss reveals ha all ndces are whn accepable ranges, wh NMSE, FB FS values are relavely near o zero R FA2 are relavely near o 1. Table 1. Observed predced ground-level cenerlne concen ra ons run dsance (m observed (µ predced (µ gm -3 gm Table 2. S a s cal evalua on Model NMSE R FA2 FB FS Proposed Model Carvalho Vlhena[4] Carvalho e al.[8]

4 Amercan Journal of Envronmenal Engneerng 213, 3(1: F gure 1. Sca er dagram be ween observed (C o predced (C p ground-level cenerlne concenraon. Dashed lnes ndcae facor of 2, doed lnes ndcaed facor of 3 sold lne ndcaes unbased predcon 4. Conclusons In hs paper was presened a more general model o smulae he polluan dsperson n meerng low wnd condons. The model was obaned by solvng he Langevn equaon hrough he negrang facor wh s coeffcen beng a complex funcon. The mehod leads o a sochasc negral equaon whose soluon was obaned hrough he Mehod of Successve Approxmaons or Pcard s Ieraon Mehod. Takng no accoun he somorphsm beween he complex real plane by wrng down he low wnd formulaon n polar form, was possble o deermne a formula for he low wnd drecon. An expresson analogous o he Euleran auocorrelaon funcon for meerng condons appeared n he real componen soluon. The proposed mehod can be used o smulae de conamnan dsperson n meerng or non-meerng suaons. The model was evaluaed hrough he comparson wh expermenal daa. The resuls obaned by he new model agree very well wh he expermenal daa, ndcang ha represens he dsperson process correcly n low wnd speed condons. ACKNOWLEDGEMENTS The auhors acknowledge he fnancal suppor provded by CNPq. (Conselho Naconal de Desenvolvmeno Cenífco e Tecnológco CAPES (Coordenação de Aperfeçoameno de Pessoal de Nível Superor. [1] Franços N. Frenkel, "Turbulen dffuson: Mean concenraon dsrbuon n a flow feld of homogeneous urbulence", Advances n Appled Mechancs, vol.3, pp.61-17, [2] Jonas C. Carvalho, Ézo R. Nchmura, Marco T. M. B. Vlhena, Davdson M. Morera, Gerváso A. Degraza, "An erave langevn soluon for conamnan dsperson smulaon usng he Gram-Charler PDF", Envronmenal Modellng Sofware, vol.2, no 3, pp , 25. [3] Jonas C. Carvalho, Marco T. M. B. Vlhena Davdson M. Morera, "An alernave numercal approach o solve he Langevn equaon appled o ar polluon dsperson", Waer, Ar Sol Polluon, vol.163, no 1-4, pp , 25. [4] Jonas C. Carvalho, Marco T. M. B. Vlhena, "Polluan dsperson smulaon for low wnd speed condon by he ILS mehod", Amospherc Envronmen, vol.39, pp , 25. [5] Jerrold F. Sagendorf, C. Ray Dckson, "Dffuson under low wnd-speed, nverson condons", U.S. Naonal Oceanc Amospherc Admnsraon Tech. Memorum ERL ARL-52, [6] R. J. Murgaroyd, Esmaons from geosrophc rajecores of horzonal dffusvy n he md-laude roposphere lower sraosphere, Quarerly Journal of he Royal M eeorologcal Socey, vol.95, pp.4-62, [7] W. Boyce, R. DPrma, 21. Elemenary dfferenal equaons boundary value problems, Wley, New York, 749 pp, 21. [8] Jonas C. Carvalho, Gerváso A. Degraza, Sérgo G. Magalhães, "Parameerzaon of meerng phenomenon n a sable amospherc boundary layer", Physca A, vol.368, pp , 26. REFERENCES [9] Gerváso A. Degraza, Osvaldo L. L. Moraes, Amaur P. Olvera, "An Analycal mehod o evaluae mxng lengh scales for he planeary boundary layer", Journal of Appled

5 12 Jonas C. Carvalho e al.: A General Lagrangan Approach o Smulae Polluan Dsperson n Amosphere for Low-wnd Condon Meeorology, vol.35, pp , [1] J. A. Busnger, Turbulen ransfer n he amospherc surface layer. In D.A. Haugen (Ed., Workshop n Mcromeeorology, Amercan Meeorologcal Socey, Boson, pp.67-1, 1973 [11] Serge S. Zlnkevh, On he deermnaon of he hegh of he Ekman boundary layer. Boundary-Layer Meeorology vol.3, pp , [12] Seven R. Hanna, "Confdence lm for ar qualy models as esmaed by boosrap jacknfe resamplng mehods", Amospherc Envronmen, vol.23, pp , 1989.