Extended Progress Variable and Mixture Fraction Model for Turbulent Premixed Combustion

Transcription

1 Proceedngs of he World Congress on Engneerng 2010 Vol III, June 30 - July 2, 2010, ondon, U.K. Exended Progress Varable and Mxure racon Model for Turbulen Premxed Combuson Y.Y. Wu and C.K. Chan Absrac A approach combnng a progress varable and a mxure fracon ranspor equaon s used o descrbe a urbulen premxed flame wh non-unform equvalence rao, caused by laeral enranmen of ar. The equaons are solved usng a proecon-based fraconal sep mehod for low-march number flow n a large eddy smulaon (ES) framework. The numercal mehod wh a smply exended model s appled o smulae a slo Bunsen flame n wo dmensons. The compued mean flame fron s comparable o ha of expermen and 3D compuaon usng dealed chemcal knecs. The presen numercal smulaon can also well predc he flame hegh and he global urbulen flame speed. The compued flame surface densy profles mach wh hose of he expermen. Index Terms progress varable, mxure fracon, large eddy smulaon, non-unform equvalence rao. I. INTRODUCTION Amed o ncrease effcency and mnmze polluan emsson, urbulen premxed flames are praccally mporan. Many models have been developed for premxed urbulen combuson wh unform equvalence rao. However, hs knd of cases does no ofen happen n realy and n expermen. Accordng o Prasad e al. [12], spaal and emporal varaons n equvalence rao canno be avoded n praccal combuson, whch may be caused by hea losses, or mxng wh dluon flows of dfferen enhalpy. Many laboraory urbulen premxed flames are appled o furher undersand he neracon of urbulen and flame fron, such as urbulen V-flame [1]-[2], swrl-sablzed flame [10], and slo Bunsen flame [3], [15]. Coflow s ofen used o conrol he shear layers ha form around he fuel/ar mxure downsream he nflow. Then urbulen flames do no always burn a homogeneous premxed fuel/ar mxure because equvalence rao vares accordng o mxng beween he man fuel/ar nflow and he coflow. A sngle scalar of progress varable canno descrbe hs knd of problems compleely because he progress varable s based on unform equvalence rao. Addonally a mxure fracon equaon s employed o descrbe he varaon of fuel/ar mxure. Scalar dsspaon erm wll emerge n he Manuscrp receved March 17, Ths work was suppored by a sudenshp of The Hong Kong Polyechnc Unversy Y.Y. Wu s wh Deparmen of Appled Mahemacs, The Hong Kong Polyechnc Unversy, Hong Kong (e-mal: polyu.edu.hk). C.K. Chan s wh Deparmen of Appled Mahemacs, The Hong Kong Polyechnc Unversy, Hong Kong (phone: ; ax: ; e-mal: ck.chan@polyu.edu.hk). ISSN: (Prn); ISSN: (Onlne) progress varable equaon when a progress varable and a mxure fracon are appled ogeher o descrbe premxed flames wh non-consan equvalence rao. The erm wll dsappear only for homogenous premxed flames. In many smulaons of urbulen premxed flames consderng varyng equvalence rao, he erm s gnored [12]. In hs paper, a smply exended model combnng progress varable and mxure fracon s used o descrbe urbulen premxed flame wh varyng equvalence rao n mxng layer. And large eddy smulaon of a slo Bunsen flame [15] s used o valdae he numercal mehod wh he exended model. In ES mehod, large scale urbulence s solved drecly whereas small scale urbulence s modeled. The smples ES approach s o use no subgrd scales (SGS) [16]. In he smulaon, ES mehod wh a dynamc subgrd model s appled. The flame fron s followed by applyng a progress varable equaon. Addonally a mxure fracon equaon s employed o descrbe he varaon of equvalence rao due o he mxng beween fuel/ar mxure nflow and coflow. A proecon-based fraconal sep mehod s employed o numercally solve he equaons. II. BASIC EQUATIONS Wh a smple global reacon rae, premxed combuson can be presened by a reacan mass fracon Y ( x, ), whch s descrbed by he followng equaon ( ρy ) + ( ρ uy ) = ( ρ D Y ) ɺ ω (1) where ρ, u, ɺ ω, D, are densy, velocy vecor, fuel reacon rae, molecular dffusvy, and me respecvely. or premxed combuson wh varyng equvalence rao, addonal scalar varable of mxure fracon Z( x, ) s needed. The ranspor equaon of mxure fracon s such as ( ρ Z) + ( ρuz) = ( ρ D Z) (2) The above wo equaons can be combned o appled n whchever premxed, parally premxed, or nonpremxed flames [14]. In he case of smple global reacon rae, progress varable c( x, ) s ofen used nsead of Y ( x, ) for convenence. In he hn premxed flame, progress varable changes from zero o uny. Wh progress varable and mxure fracon, lean reacan mass fracon can be defned by

2 Proceedngs of he World Congress on Engneerng 2010 Vol III, June 30 - July 2, 2010, ondon, U.K. Y ( x, ) = Y c( x, ), Z( x, ) [ ]. And for he premxed combuson wh coflow of ar or plo produc, followng equaon can be appled o express lean reacan mass fracon Y ( x, ) = YφZ ( x, ) [ 1 c( x, ) ] (3) In he unburn reacans Z = 1 and c = 0, whereas n he burn produc Z = 0 and c = 1. Y φ s he mass fracon of fuel n he man fuel/ar mxure nflow. When equaon (3) s used o replace mass fracon Y ( x, ) n (1), addonal erm relaed o Z( x, ) wll appear on he rgh hand of progress varable equaon. ( ρ c) 2ρ D + ( ρ uc) = ( ρ D c) + ɺ ω ( Yφ Z ) + Z c (4) Z or homogenous premxed combuson Z( x, ) = 1, and equaon (4) reduces o radonal progress varable equaon for premxed combuson ( ρ c) + ( ρuc) = ( ρ D c) + ɺ ω Yφ (5) where he wo erms on rgh hand sde can be combned o be expressed by a formulaon of flame surface densy ρus. ρ u s he densy of unburn reacan, S s he lamnar flame speed, and denoes he flame surface densy. or cases wh mxng layer beween man fuel/ar flow and coflow, he equvalence rao vares. In mxng layer, lamnar flame speed S and burn emperaure T should change accordng o he varaon of equvalence rao. The dependence of he scalar feld on mxure fracon s aken no accoun. The hrd erm on rgh hand sde of (4) s also gnored because s source of he dffuson erm n (1) s relavely small. The model s exended o consder varyng equvalen rao wh mxure fracon such ha ( ρ c) + ( ρ uc) = ρu Sd (6) φ local where Sd = Sd ( φ, Z) = S s he local flame speed, φr S s he lamnar flame speed for reacan wh equvalence rao φ R of man fuel/ar mxure, φ local s local equvalence rao, γ s a varable relaed o he local mxure fracon and s smlar o he funcon proposed by Prasad e al. [19]. Equvalence rao of φ mn = 0.5 corresponds o he lean exncon lm n mxng layer. Wh assumpon of low-mach number flow and flerng process of large eddy smulaon, non-dmensonal governng equaons can be obaned as ρ ( ρ uɶ ) + = 0 (7) x ( ρ u ) ( ρ uɶ uɶ ) p ' m + = + x x x ( ρ Z) ( ρ uɶ ) ( u z u Z) Z ρ ɶ + + = ( ρ D Z ) (9) x x ( ( ρ cɶ ) ( ρ uɶ ) u c u c) cɶ ρ ɶ ɶ + + = ρu Sd = ρu Sd (10) x x ISSN: (Prn); ISSN: (Onlne) γ b (8) where ρ, uɶ, p ', Zɶ, cɶ denoe average densy, flered velocy, pressure flucuaon, fler mxure fracon, and flered progress varable respecvely. m u u 2 uk µ + ɶ ɶ ɶ = + δ x x 3 x k and µ + = µ + µ / Re 2 1/ 2 wh urbulen vscosy µ = ρ C (2 Sɶ Sɶ ) and flered k l k l 1 uk ul sran rae ensor Sɶ ɶ ɶ k l = +. µ s dynamc 2 xl xk vscosy, Re s Reynolds number of he flow, and δ s Kronecker dela funcon. C s a model parameer obaned by he dynamc model of Germano e al. [18]. The unclosed ranspor fluxes n mxure fracon equaon and c-equaon are modeled wh smple graden expressons. A flame surface densy (SD) model s used o deermne he erm on he rgh hand sde of progress varable equaon. Based on curvaure of he flered progress varable, flame surface densy s compued usng a new model proposed by Knkker e al. [20]. The flame surface densy s decomposed no a resolved and an unresolved conrbuon such ha = cɶ + κ N M ( cɶ ) (11) where κ s a non-dmensonal model consan, N = cɶ cɶ s he un normal vecor normal o so-conours of he flered progress varable and M s a maskng funcon o avod undesred conrbuons n regons far away from he flame fron. III. NUMERICA METHODS or ncompressble flows, proecon-based fraconal sep mehod s an effcen dscrezaon sraegy. or low-mach number reacon flows, proecon-based fraconal sep mehods [11], [17], [21]-[23] and he generalzed proecon approach based on Helmholz-Hodge decomposng heory [4]-[5] boh have successful applcaons. Wh proecon-based fraconal sep mehod, momenum equaon s solved n wo pars. rsly, momenum equaons are negraed assumng consan pressure. The pressure flucuaon p ( x, ) s hen deermned before performng he second negraon sep. Veloces are correced subsequenly accordng o pressure flucuaon gradens. The correspondng equaons are ( ρ uɶ ) * uɶ m n ( ρ uɶ ) = ( ρ uɶ ) + + x x (12) n + 1 * p ' ( ρ uɶ ) = ( ρ uɶ ) +. x The pressure flucuaon s deermned by a Posson equaon for p '( x, ). Ths s obaned by akng dvergence of he second equaon of (12) and beng nroduced no connuy equaon o esmae ( ρ u ɶ ) n. The Posson equaon s such as 2 1 n + 1 * p ' = ( ρ ) + ( ρ u ɶ ) (13)

3 Proceedngs of he World Congress on Engneerng 2010 Vol III, June 30 - July 2, 2010, ondon, U.K. The densy varaon erm ( ) n 1 ρ + s ncluded above o elmnae dlaaon caused by hea release. or spacal dscrezaon, saggered grd s employed. A second order upwnd fne dfference mehod and a second order cenral fne dfference mehod are separaely used for he convecon and dffuson erms. or emporal dscrezaon, a hrd order Runge-Kua mehod s appled o mplemen me negraon. abou 4cm. The flame fron shape of smulaons and expermen are very smlar excep us near he burner ex, where he flame frons of expermen s more vercal. I may be caused by dversy of urbulen nflow. The downsream flame brush s much hcker han ha near he ex. And he downsream mean flame hckness of urbulence s much larger han ha of relaed lamnar flame. (a) (b) IV. RESUTS AND DISCUSSIONS The numercal mehod wh he smply exended model s appled o smulae he urbulen reacve flow feld of a slo Bunsen flame. The compuaon parameers are based on expermens of layev e al. [15]. The slo burner was composed of hree recangular burners: a cenral burner and wo sde burners. or all hree burners, sochomerc mehane-ar mxures were suppled. The cenral burner produced Bunsen flame of neres wh fla flames from sde burners, where ho producs flowed a a velocy o mach he velocy afer Bunsen flame [3]. In expermen, mean properes of he slo Bunsen flames are 2D. In presen work, one case of he expermenal flames (case 3b) s numercally smulaed n wo dmensons. or cenral fuel/ar mxure nflow, he dmenson s 25 mm and he nflow velocy s mean velocy of 3m/s supermposed wh 10% urbulen flucuaon. or sde burners, he nflow s ho combuson producs n unform velocy of 7m/s. Reacan s mehane-ar mxure (CH4/ar) wh equvalence rao of 1.0. Densy rao of sochomerc reacan o produc s gven by ρ ρ = 7.47 and lamnar burnng speed s aken as S u b 1 = The compuaonal doman s m s [ 37.5, 37.5] mm [0, 100] mm and sascal resuls are obaned by averagng he nsananeous values over 10,000 me seps. (c) g.2 Mean flame fron descrbed by progress varable conours of (a) Expermen (b) Smulaon resul of Bell e al. (c) Presen smulaon g. 3 conrasvely shows he mean axal velocy of presen smulaon and expermen. The dfference may be also nduced by urbulen nflow dscrepancy. g.1 Insananeous compued slo Bunsen flame g.1 shows an nsananeous compued slo Bunsen flame. The flame fron s descrbed by progress varable. In fg.2, he mean flame fron s presened by progress varable conour of (a) s from expermen of layev e al. [15] and (b) s from 3D smulaon of Bell e al. [3] consderng dealed chemcal knecs. (c) presens 2D smulaon resul. All he average flame heghs from expermen and smulaons are g.3 Mean axal velocy along he burner cenerlne ISSN: (Prn); ISSN: (Onlne)

4 Proceedngs of he World Congress on Engneerng 2010 Vol III, June 30 - July 2, 2010, ondon, U.K. The mass of reacans mxure enerng he cener burner per second s mɺ R = ρrdvɶ 0h, and D s he lengh of cener burner, h s he lengh along spread drecon, V ɶ 0 s he mean nflow axal velocy. Accordng o expermen, here s no fuel escape from he burner, he reacans mxure consumed per second can be obaned by mass conservaon mɺ R = ρrst 0.5h. 0.5 s he lengh of flame fron measured from mean progress varable conour c ɶ = 0.5. So he average global urbulen flame speed s S = mɺ ( ρ ) = Vɶ / 2.3S. The relaed T R R 0.5 h D quany of expermen s S 2.55S. T (a) (b) g.4 Mean flame surface densy a (a) y=2.8cm and (b) y=5cm g.4 depcs mean flame surface densy a y=2.8cm and y=5cm. ne shows he compued resuls and do presens he expermen resuls. The fgure does no show symmery. In expermen, s due o nonunform flow n he burner nernal flow feld [15]. In numercal smulaon, may be generaed by random nflow velocy flucuaons. However, he resuls of smulaon and expermen agree well. V. CONCUSION In praccal and expermenal combusons, spaal and emporal varaons n equvalence rao ofen happen. I may be caused by hea losses, or mxng wh dluon flows of dfferen enhalpy. In hs paper, combned progress varable and mxure fracon wh a smply exended model s used o ISSN: (Prn); ISSN: (Onlne) descrbe urbulen premxed flames wh varyng equvalence rao n mxng layer. arge eddy smulaon of a slo Bunsen flame s performed. Compared wh expermen and 3D compuaon wh dealed chemcal knecs, presen numercal mehod wh exended model well predcs mean flame fron, flame hegh, and global urbulen flame speed. The compued flame surface densy profles also mach wh hose of he expermen. The resuls ndcae ha presen numercal mehod can reasonably smulae urbulen premxed flame wh varyng equvalence rao, whch has wo dmensonal mean properes. In fuure work, we wll ry o explore he exended model n hree dmensons. ACKNOWEDGMENT The auhors wsh o hank Dr. Karl-Johan Nogenmyr for hs useful dscusson and consrucvely suggeson. REERENCES [1] R.K. Cheng, Condonal samplng of urbulence nenses and Reynolds sress n premxed urbulen flames, Combuson Scence and Technology. 41(1984) [2] J.B. Bell, M.S. Day, I.G. Shepherd, M. Johnson, R.K. Cheng, J.. Grcar, V.E. Becker and M.J. ewsk, Numercal smulaon of a laboraory-scale urbulen V-flame. awrence Berkeley Naonal aboraory repor BN Journal. [3] J.B. Bell, M.S. Day, J.. Grcar, M.J. ewsk, J.. Drscoll, and S.A. layev, Numercal smulaon of a laboraory-scale urbulen slo flame, Proc. Combus Ins. 31 (2007) [4] J.B. Bell, M.S. Day, J.. Grcar, M.J. ewsk, M. Johnson, R.K. Cheng, I.G. Shepherd, Numercal smulaon of a premxed urbulen V-flame. In. Coll. Dyn. Explo. Reac. Sys., Japan, [5] J.B. Bell, M.S. Day, J.. Grcar, M.J. ewsk, A compuaonal sudy of equvalence rao effecs n urbulen, premxed Mehane-Ar flame. Proc. ECCOMAS-CD, he Neherlands, [6] C.K. Chan, K.S. au, B.. Zhang, Smulaon of a premxed urbulen flame wh he dscree vorex mehod. In. J. Numer. Meh. Engng. 48 (2000) [7] C.K. Chan, H.Y. Wang, H.Y. Tang, Effec of nense urbulence on urbulen premxed V-flame. In. J. Eng. Sc. 41 (2003) [8] C.K. Chan, B. Sewar, C.W. eung, Numercal Smulaon of Premxed V-lame, Proc. In. Conf. Mech. Eng. 2007, [9] T. Ponso, D. Veynane, Theorecal and Numercal Combuson [10] K.-J. Nogenmyr, P. Peersson, X.S.Ba, A. Nauer, J. Olofsson, C. Brachman, H. Seyfren, J. Zeerberg, Z.S., M. Rcher, A. Drezler, M. nne and M. Alden, arge eddy smulaon and expermens of srafed lean premxed mehane/ar urbulen flames, Proc. Combus Ins. 31(2007) [11] Y. u, K.S. au, C.K. Chan, Y.C. Guo, W.Y. n, Srucures of scalar ranspor n 2D ransonal e dffuson flames by ES, In. J. Hea Mass Tran. 46(2003) [12] R. O. S. Prasad, R. N. Paul, Y. R. Svahanu, and J. P. Gore, An evaluaon of combned flame surface densy and mxure fracon models for nonsenhalpc premxed urbulen flames, Combus. lame. 117(1999) [13] P. Bgo, M. Champon and D. Garréon-Bruguères, Modelng a urbulen reacve flow wh varable equvalence rao: Applcaon o a flame sablzed by a wo-dmensonal sudden expanson, Combus. Sc. and Tech. 158(2000) [14] K. Bray, P. Domngo and. Vervsch, Role of he progress varable n models for parally premxed urbulen combuson, Combus. lame. 141(2005) [15] S.A. layev, J.. Drscoll, C.D. Carer, and J.M. Donbar, Measured properes of urbulen premxed flames for model assessmen, ncludng burnng veloces, srech raes, and surface denses, Combus. lame. 141(2005)1 21. [16] J. Jancka, A. Sadk, arge eddy smulaon of urbulen combuson sysems. Proc. Combus Ins. 30 (2005) [17] J. de Charenenay, D. Thévennn, B. Zamuner, Comparson of drec numercal smulaon of urbulen flames usng compressble or low-mach number formulaons, In. J. Numer. Meh. l. 39 (2002)

5 Proceedngs of he World Congress on Engneerng 2010 Vol III, June 30 - July 2, 2010, ondon, U.K. [18] M. Germano, U. Ponmell, P. Mon, W.H. Cabo, A dynamc subgrd-scale eddy vscosy model. Phys. luds, 3(1991) [19] R. O. Prasad, R. N. Paul, Y. R. Svahanu, J. P. Gore, An evaluaon of combuson flame surface densy and mxure fracon models for nonsenhalpc premxed urbulen flames, Combus. lame. 117 (1999) [20] R. Knkker, D. Veynane, J.C. Rolon, C. Meneveau, Planar laser-nduced fluorescence n a urbulen premxed flame o analyze large eddy smulaon models, Proc. In. Symp. Turb., Hea and Mass Tran., sbon, [21] B. essan, M.V. Papalexandrs, Tme-accurae calculaon of varable densy flows wh srong emperaure gradens and combuson. J. Compu. Phys. 212 (2006) [22] H.N. Nam, P.S. Wyckoff, O.M. Kno, A sem-mplc numercal scheme for reacng flow: I. Sff Chemsry. J. Compu. Phys. 143 (1998) [23]. Ncoud, Conservave hgh-order fne-dfference schemes for low-mach number flows. J. Compu. Phys. 158 (2000) [24] [24] H. ahaly, M, Champon, D. Karmed and P. Bruel, Inroducon of Dluon n he BM Model: Applcaon o a sagnang urbulen flame, Combus. Sc. and Tech. 135(1998) ISSN: (Prn); ISSN: (Onlne)