CONDITIONAL MOMENT CLOSURE MODELLING OF SOOT FORMATION IN TURBULENT NON-PREMIXED, ETHYLENE-AIR FLAMES

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1 CONDITIONAL MOMENT CLOSURE MODELLING OF SOOT FORMATION IN TURBULENT NON-PREMIXED, ETHYLENE-AIR FLAMES Yunard, Robert M. Woolley and Mchael Farweather Energy and Reource Reearch Inttute, School of Proce, Envronmental and Materal Engneerng, Unverty of Leed, Leed, Wet Yorkhre, LS 9JT, UK ABSTRACT Preented are reult obtaned from the ncorporaton of a em-emprcal oot model nto a frt-order Condtonal Moment Cloure (CMC) approach to modellng turbulent, non-premxed, ethylene-ar flame. Soot formaton determned va the oluton of two tranport equaton for the oot ma fracton and the partcle number denty, wth acetylene and benzene employed a the ncpent pece reponble for oot nucleaton. The concentraton of thee pece are calculated ung a detaled ga-phae knetc cheme nvolvng 463 reacton and 70 pece. The tudy focue on the nfluence of dfferental dffuon of oot partcle on oot volume fracton predcton, and the reult of calculaton are addtonally compared wth expermental data for the flow feld and temperature of the three ethylene-ar flame nvetgated. Good agreement wth oot data found provded dfferental dffuon effect are ncluded. INTRODUCTION Depte dwndlng reource, fol fuel combuton tll play a major role n the world economy and wdely ued for the producton of energy. The formaton and emon of partculate pollutant uch a oot, a a conequence of hydrocarbon combuton, fat becomng a major concern n both developed and more o, developng countre. Wthn mot combuton ytem, oot formaton and oxdaton uually occur n hghly turbulent zone whch nvolve the nteracton of complex chemcal and phycal phenomena. In order to accurately predct the producton and detructon of oot n uch ytem, an ntegrated model repreentng turbulence, the nteracton between turbulence and ga-phae chemtry, oot partcle producton and removal, and radaton heat loe requred. The ablty to apply uch an ntegrated model of demontrable accuracy to mnme oot producton and emon n relaton to afety and envronmental conderaton, would repreent a major tep-forward n our ablty to degn and manage combuton procee. Soot partcle are of gnfcantly greater ma than gaphae pece, and therefore they dffue conderably more lowly than the latter. However, the majorty of model ued n predctng oot n non-premxed turbulent flame aume equal dffuvty of oot partcle and ga-phae pece (Wen et al., 003; Ma et al., 005; Mau et al., 006). Kronenburg et al. (000) were amongt the frt to pont out the mportance of accountng for the dfferental dffuon of oot partcle when predctng methane ootng flame. Ptch et al. (000) alo obtaned mproved predcton of oot volume fracton n the context of ethylene nonpremxed flame when the effect of dfferental dffuon were taken nto conderaton. When predctng turbulent combuton, dffculty encountered n modellng the chemcal ource term that appear n the pece contnuty equaton. The hghly nonlnear dependence of th term on pece concentraton and temperature, whch fluctuate rapdly n turbulent flow, mpede any attempt at a lnear frt-order cloure n term of the averaged local temperature and concentraton. The CMC method addree th problem by utlng moment condtoned on a value of a conerved calar, namely the mxture fracton. Undertakng calculaton n conerved calar pace allow the removal of much of the non-lnearty of the chemcal ource term by aumng fluctuaton due to turbulence to be neglgble (Klmenko and Blger, 1999). Frt-order CMC ha been appled n the form of a parabolc equaton et to a wde range of combuton problem and ha performed uccefully for attached nonpremxed jet flame (Farweather and Woolley, 003; 004) and lfted turbulent jet dffuon flame (Km and Matorako, 005). In addton to thee uccee n modellng ga-phae combuton, CMC ha alo hown prome n the calculaton of oot formaton n nonpremxed flame (Kronenburg et al., 000). In th paper, the reult of an applcaton of a frt-order CMC approach (Klmenko and Blger, 1999) to the calculaton of turbulent non-premxed, ethylene-ar flame and oot formaton are preented. The oot model ued n the calculaton baed on that preented by Leung et al. (1991) and Lndtedt (1994), wth tranport equaton for oot ma fracton and partcle number denty ncorporated nto the CMC approach. The nfluence of dfferental dffuon of oot partcle, prevouly nvetgated by Kronenburg et al. (000), further aeed wthn the computaton of thee turbulent ethylene flame. MATHEMATICAL MODELLING Expermental Condton and Flow Calculaton The three non-premxed ethylene-ar flame condered n the preent tudy have been expermentally reported by 1119

2 Kent and Honnery (1987), Coppalle and Joyeux (1994), For the dervaton of the condtonal ga-phae pece and Young and Mo (1995), denoted from now on a ma fracton equaton, both reactve and conerved calar flame KH, CJ and YM. The mportant charactertc of are aumed to dffue equally, whch mple D = D ξ. each of thee flame are preented n Table 1 and further Wth th aumpton, the econd and lat term on the rght detal of the flame geometry, method of data collecton, hand-de of Eq. 1, repreentng the ource term that and proceng can be found n the relevant reference. generate dfferental and patal dffuon, repectvely, are cancelled. The condtonal calar dpaton ( χ η ) wa Table 1: Operatng condton for ethylene-ar flame modelled ung the approach of Grmaj (1991), whle the remanng non-lnear chemcal ource term ω η wa Ethylene-ar flame KH CJ YM Abolute preure/atm modelled a for mple frt-order cloure, aumng the Fuel ext velocty/m condtonal fluctuaton of the reactve calar around the Fuel ext temperature/k mean to be neglgble. Mean value were obtaned ung the Nozzle dameter/mm CHEMKIN package (Kee et al., 1996), together wth a full Ext Reynold number chemcal knetc cheme contng of 70 pece and 463 r Co-flow ar velocty/m eacton (Qn et al., 000). In addton to the CMC pece tranport equaton, the oot model employed n the preent tudy, a decrbed n Leung et al. (1991) and Lndtedt (1994), requre the oluton of two addtonal tranport equaton for the oot The calculaton of the flow and mxng feld wa acheved by olvng the Favre-averaged form of the partal dfferental equaton whch decrbe conervaton of ma, momentum and the tranport of mxture fracton and t varance. A tandard k-ε turbulence model wa ued to cloe the above equaton et. Standard turbulence modellng contant approprate to axymmetrc flow were employed, although to enure the accurate predcton of the preadng rate of the jet, an adjutment wa made to the value of C ε from 1.9 to 1.84 to affect an ncreae n the dpaton rate of turbulence knetc energy. A modfed veron of the GENMIX (Spaldng, 1977) code wa mplemented n the oluton of the two-dmenonal, axymmetrc form of the tranport equaton. The code apple an mplct formula n the tream-we drecton and a hybrd-dfferencng cheme n the cro-tream drecton for t marchng ntegraton procedure. Frt-Order CMC-Baed Soot Model A general frt-order, parabolc CMC equaton can be obtaned by averagng the ntantaneou equaton governng pece ma fracton, Y n tattcally tatonary, turbulent reactng flow, on the condton that the ntantaneou mxture fracton ξ equal an arbtrary value η. Aumng a neglgble varaton n condtonal tattc acro the wdth of the flow, alo allow the mplementaton of a onedmenonal form. However, n order to account for any mall varaton that may be preent, the CMC equaton can be radally averaged by ntegratng acro the flow (Barlow et al., 1999). When the conerved calar and reactve calar have dfferng dffuon coeffcent, that D D ξ, an uncloed form of the CMC equaton can be wrtten for a condtonal tranported calar ( ) a: ξ Q Q 1 D Q u η = χ η x D η D ξ Q 1 ρdξ η ω η Dξ x x η * e y, (1) ma fracton, Y, and the oot partcle number denty,. In the cae of dfferental dffuon beng neglected, N the tranport equaton for Y and N are obtaned n a mlar way a for the ga-phae pece, ettng D = D = D ξ for = Y, N n Eq. 1, to gve: Y () N QY x Q x 1 QY u η = χη ω η Y η 1 QN u η = χη ω η N η N where the upercrpt refer to a calar of equal dffuvty. When the dfferental dffuon of oot partcle taken nto account, fxng the molecular coeffcent of oot partcle and nucle equal to zero,.e. D = D = 0 for = Y, N, Eq. 1 mplfe by neglectng the dpaton term, th beng the frt term on the rght-hand de. However, the e y, patal dffuon term een n Eq. 1 now become gnfcant. The current modellng approach follow the work of Kronenburg and Blger (001a, 001b) n the repreentaton of e y,, and the tranport equaton conderng the effect of dfferental dffuon are modelled a Eq. 4 and 5, where τ K the Kolmogorov tme cale. ( QY Q ) Y Y QY ξ Q Y uη = ρdξ η ωy η x x x η τ ( η) K QN ξ Q N uη = ρdξ η ωn η x x x η ( QN Q ) N τ ( η) K * * N (3) (4) (5) 110

3 The ource term ω η n Eq. and 4 account for Y the effect of oot nucleaton, urface growth and oxdaton. Acetylene and benzene were elected a the ncpent pece reponble for oot nucleaton (Lndtedt, 1994). However, the former pece wa condered a the only chemcal contrbutor to the ncreae n oot ma va urface growth. The oot nucleaton proceed va the reacton C H C H and C 6 H 6 6 C 3H and t aumed that urface growth contnue va an acetylene reacton mlar to that of oot nucleaton. Aumng oot oxdaton to proceed through C 0.5 O CO and C OH CO H, the ource term for the oot ma fracton equaton can now be wrtten a: ω η = k ( Q ) Q M 6 k ( Q ) Q M Y 1 T CH T C6H6 k3( QT) AQ C H M k4( QT) AQ O M (6) k ( Q ) AQ M 5 T OH where M the oot molecular weght. The ource term ω η n Eq. 3 and 5 repreent the N producton and reducton of oot partcle number denty due to nucleaton and agglomeraton, repectvely. Wth N A repreentng the Avogadro number, C an agglomeraton a contant, σ B the Stefan-Boltzmann contant, and C the mn mnmum carbon atom number, they can be expreed a: NA ω η N = k1( Q ) C H 6 ( ) T Q k QT Q C6H6 Cmn 1/ 6 1/ (7) 6 Q Y 6σ BQT Ca ( ρηq ) πρ N ηqn ρ Reacton rate contant for nucleaton, urface growth and oxdaton that appear n Eq. 6 and 7 are preented n Table (Lndtedt, 1994; Kronenburg et al., 000). dfferencng approxmaton. The computatonal grd conted of 68 radal node n mxture fracton pace. Table : Reacton rate contant for oot formaton and oxdaton, n the form of the Arrhenu expreon k j = AT b exp (-T a /T) (unt K, kmol, m, ). k j A b T a k x ,000 k 0.75 x ,000 k x ,100 k x ,680 k x RESULTS AND DISCUSSION Predcted and meaured value of temperature along the centrelne, and radally at three dfferent heght above the burner, n flame KH are hown n Fg. 1. The temperature predcton dplay quanttatvely good reult n comparon to expermental data. It hould be noted that the flat calculated axal temperature profle cloe to the nozzle ndcate a regon pror to CMC calculaton commencng. It een that the progre of the computed axal temperature n lne wth the meaurement up to around 500 mm above the burner. Beyond th heght, however, the temperature over-predcted by up to 00 K n the furthermot downtream poton. Th content wth the calculaton of Ma et al. (005) whch ued a flamelet approach wth a k-ε turbulence cloure, and Lndtedt and Louloud (005) who ued a tranported PDF approach. Predcted radal temperature at x = 138 and 41.5 mm downtream of the nozzle are n excellent agreement wth data throughout the flame. Further downtream, at x = 345 mm, although the core and radal peak temperature are well captured, the predcton n the fuel-lean regon devate omewhat from expermental reult. At the ame axal poton, Ptch et al. (000) found that the calculated temperature over-predcted the radal data due to an overetmaton of the jet preadng rate. Soluton of the CMC Equaton Flow and mxng feld nformaton from the turbulent flow calculaton employng a reactng-flow denty were paed to the CMC model, where the et of pece ma fracton, oot ma fracton, partcle number denty and enthalpy equaton were olved n mxture fracton pace. The flow and mxng feld are related to the reactve calar feld through the mean denty, and comparon between dente obtaned from the CMC oluton and precrbed equlbrum value howed lttle varaton at the locaton examned n the flame condered. Couplng of the flow feld and CMC calculaton wa therefore deemed unneceary for the calculaton reported. Soluton of the CMC equaton n real pace wa acheved ung a fractonal tep method, mplemented ung the tff ODE olver VODE (Brown et al., 1989), whch apple a backward dfferentaton formula approach to the oluton of the non-lnear equaton et. Second-order dfferental ample pace term were determned ung a central Temperature /K Axal dtance /mm 500 x= 41.5 mm Radal dtance /mm x= 138 mm x= 345 mm Radal dtance /mm Fg. 1: Axal and radal temperature profle for flame KH (ymbol meaured, lne predcted). 111

4 The accuracy of the turbulent flow feld predcton, n addton to the ncluon of the effect of radaton heat tranfer nto a combuton model, of prme mportance n order to correctly predct the temperature of a combuton ytem. A mple radaton model employed heren, where emon from pece CO, H O, CH 4, CO and oot were ncluded, ha been found to yeld reaonable accuracy n many non-ootng combuton applcaton (Farweather and Woolley, 003; 004). The prece predcton of peak temperature obtaned n both the centrelne and radal drecton, a hown n Fg. 1, ndcate the method of accountng for oot radaton atfactorly mplemented n th flame. Predcton of axal and radal dtrbuton of oot volume fracton are preented alongde expermental value n Fg.. The old lne repreent the mulaton reultng from conderng the effect of dfferental dffuon, and the dahed lne the mulaton whch neglected dfferental dffuon n the CMC calculaton. When the aumpton of equal dffuvty appled, t clearly een that the centerlne oot volume fracton profle gnfcantly under-predcted n the oot formaton zone. The predcted oot formaton tart at a lower rate than that meaured, leadng to a hft of peak oot volume fracton nto the oxdaton zone. The meaured peak oot volume fracton of 1.57 x 10-6 under-predcted by two order of magntude. A key ue that the dcrepancy between the computed and meaured oot volume fracton n the formaton zone drectly related to the modellng of the urface growth rate n the oot model. When the effect of dfferental dffuon neglected, the approxmaton that the urface growth rate proportonal to the local oot urface area, f( A) = A, reult n gnfcantly low oot volume fracton, a oberved by Lndtedt (1994) and Ma et al. (005) n ethylene flame, and Kronenburg et al. (000) n methane flame. However, when the dfferental dffuon between oot partcle and ga-phae pece taken nto account, the oot volume fracton predcton brought n lne wth meaurement. The magntude of the peak oot volume fracton lghtly below the meaured value, but the hape of the meaured profle very well captured n both the formaton and oxdaton zone. Inpecton of the radal profle n Fg. at the downtream locaton of x = 41.5 and 345 mm ndcate that poor agreement between predcton and meaurement obtaned wth the aumpton of equal dffuvty. However, at axal poton x = 138 and x = 483 mm better agreement obtaned. Reult baed on the dfferental dffuon model are very encouragng. Overall, the hape of the radal oot volume fracton profle precely predcted. At the locaton cloet to the nozzle, x = 138 mm, the predcted oot volume fracton reache an off-ax peak, th fndng beng content wth that of Ma et al. (005) and Ptch et al. (000) who ued a flamelet approach. The expermental data do not clearly dplay uch a peak value, although Kent and Honnery (1987) do tate that cloe to the nozzle the meaured oot volume fracton exhbt off-ax peak whch approach the ax at x = 345 mm. Turnng to the econd flame nvetgated, Fg. 3 preent axal profle of temperature and oot volume fracton, a well a radal profle of oot volume fracton at Soot volume fracton 9.0x x x10-7 Soot volume fracton.5x10-6.0x Axal dtance /mm 1.x10-6 x= 138 mm x= 41.5 mm.5x10-6 x= 345 mm x= 483 mm.0x Radal dtance /mm Fg. : Axal and radal oot volume fracton profle for flame KH (ymbol meaured, old lne predcted wth dfferental dffuon, dahed lne predcted neglectng dfferental dffuon). Soot volume fracton Temperature /K x10-6 x = 8 mm.0x Soot volume fracton.5x10-6.0x10-6 Axal dtance /mm Radal dtance /mm x = 338 mm Fg. 3: Axal temperature and oot volume fracton, and radal oot volume fracton profle for flame CJ. Notaton a n Fg.. two dfferent downtream locaton, n flame CJ. Agan, the reult how that the predcted temperature n excellent agreement wth data, wth temperature captured along the centrelne of the flame, and wth good agreement n term of the magntude and locaton of the peak temperature. Thee reult are n lne wth thoe acheved ung the tranported PDF method (Lndtedt and Louloud, 005), although t hould be noted that no adjutment wa made to the enthalpy ource term n the preent work n order to obtan 11

5 th level of agreement. Unfortunately, the expermental data lack radal temperature profle o comparon n th regard are not poble. Fgure 3 alo how the centrelne profle of oot volume fracton, ndcatng mlar trend to thoe found n flame KH where neglectng the effect of dfferental dffuon yeld lower predcton at all axal locaton. Accurate predcton of oot volume fracton along the centrelne up to x = 400 mm, wthn the oot formaton regon, are acheved when the dfferental dffuon effect condered. Beyond th heght, however, the calculated oot volume fracton n the oxdaton zone evolve at a lower rate than oberved n the experment, leadng to lghtly hgher predcted oot concentraton. Th could be due to a lower oot oxdaton rate produced n the CMCbaed oot model than oberved n the experment, although no uch under-predcton of oxdaton rate evdent n the reult for flame KH, nor ndeed n flame YM examned later n th tudy. Th dcrepancy could be explaned by comparng meaured oot concentraton dtrbuton on the centrelne between flame KH and flame CJ. Inpecton of meaured oot dtrbuton on the centrelne of flame KH, hown n Fg., tend to confrm that the rate of oot formaton comparable wth the rate of oot oxdaton. The CMC-baed oot model atfactorly predct the oot dtrbuton n th flame n both of thee zone. In contrat, the meaured oot dtrbuton on the centrelne of flame CJ, preented n Fg. 3, ndcate a rate of oot oxdaton hgher than that of oot formaton, a reflected by a harp fall-off of oot concentraton after reachng t peak value. Th n accord wth the expermental obervaton, reported by Coppalle and Joyeux (1994), whch compared three ethylene flame of dfferent Reynold number and found that the flame condered n the preent tudy ha the hghet oxdaton rate among the three flame nvetgated. Comparon between predcted and meaured radal profle of oot volume fracton at two dfferent axal locaton for flame CJ are alo preented n Fg. 3. An mprovement n oot volume fracton predcton agan acheved when the CMC-baed oot model account for the nfluence of dfferental dffuon. The radal poton of the maxmum oot volume fracton at x = 8 mm, a well a the overall hape of the oot profle at both axal poton, een to agree well wth the data. Flame YM repreent the fnal flame nvetgated n th tudy. Retrcton of pace preclude preentaton of graph depctng flow and temperature feld for th flame. The reult howed that although the calculated centrelne mean mxture fracton n very good agreement wth the meaurement, the predcted axal temperature le conformng. In the lower part of the flame, partcularly n the range between 150 and 350 mm above the nozzle, the temperature over-predcted by up to 00 K. However, outde th range the temperature n good agreement wth data. A mlar dcrepancy alo noted by Ma et al. (005), and Mau et al. (006) when modellng th flame ung a flamelet approach. Fgure 4 preent axal and radal profle of predcted and meaured oot volume fracton. Inpecton of the axal profle agan how that mproved oot predcton can be acheved by ncorporatng the dfferental dffuon effect of oot partcle nto the CMC model. The hape of the oot volume fracton dtrbuton dentcal to the meaurement, although a lower predcton obtaned n the oot oxdaton regon. In comparon, Ba et al. (1998) lghtly over-predcted oot concentraton n the oxdaton regon, wth mproved agreement wth data obtaned when the rate of partcle ncepton wa decreaed by 0 percent. It hould be noted that no adjutment wa made n the preent tudy ether to the oot ncepton or oxdaton rate. Wthout uch an adjutment, the reult demontrate a mlar level agreement wth data to thoe of Ma et al. (005) and Mau Soot volume fracton.5x10-6.0x10-6 et al. (006) Axal dtance /mm.0x10-6 x= 00 mm x= 50 mm 1.6x x x x10-7.5x10-6 x= 350 mm.0x x= 160 mm 8.0x x x10-7.0x Radal dtance /mm Radal dtance /mm x= 450 mm Fg. 4: Axal and radal oot volume fracton profle for flame YM. Notaton a n Fg.. Radal oot predcton, gven n the ame fgure, how that cloe to the nozzle, at x = 160 mm, the oot level omewhat over-predcted, but the hape and peak locaton are n agreement wth data. Soot predcton at the heght of x = 00 and 50 mm are n excellent agreement wth the data. CMC predcton alo demontrate greater accord wth the data when compared to reult obtaned at the ame heght of x = 50 mm n other tude (Ma et al., 005; Mau et al., 006). Further downtream the predcted radal oot profle tll follow the pattern of the meaurement, although oot volume fracton are lghtly under-predcted at all radal locaton. A wth the prevou flame condered, predcton that do not account for dfferental dffuon effect do not capture the expermental data CONCLUSIONS A frt-order CMC-baed oot model ha been appled to the calculaton of oot level n three turbulent nonpremxed ethylene flame, wth one am beng to nvetgate the nfluence of dfferental dffuon of oot partcle on predcton. The reult demontrate that predcton are n 113

6 better accord wth data when uch effect are accounted for, wth predcton n the oot formaton and oxdaton zone of the flame n good agreement wth obervaton, apart from n the trongly oot oxdng flame CJ whch how dfference between predcted and meaured oot value n the oxdaton zone. Predcton that gnore dfferental dffuon gnfcantly under-predct oot level. Reult therefore upport the mportance of accountng for the dfferental dffuon of oot partcle n predctng ootng flame, a prevouly noted by Kronenburg et al. (000). In general, predcton of mean mxture fracton, temperature and oot volume fracton n the flame tuded how good qualtatve and quanttatve agreement wth data, and compare favorably wth the reult of earler nvetgaton of thee flame that employed flamelet and tranported PDF approache. Wth repect to axal temperature predcton, an over-predcton occurred n the far-feld regon of flame KH, and wthn the lower part of flame YM, although an excellent repreentaton of temperature obtaned along the core of flame CJ, ndcatng that the radaton model employed atfactory. The ue of Reynold tre turbulence cloure may be requred n CMC calculaton to mprove the centerlne temperature predcton n ome flame. REFERENCES Ba, X.S., Balthaar, M., Mau, F., and Fuch, L., 1998, Detaled Soot Modellng n Turbulent Jet Dffuon Flame, Proceedng of the Combuton Inttute, Vol. 7, pp Barlow, R.S., Smth, N.S.A., Chen, J.-Y., and Blger, R.W., 1999, Ntrc Oxde Formaton n Dlute Hydrogen Jet Flame: Iolaton of the Effect of Radaton and Turbulence-Chemtry Submodel, Combuton and Flame, Vol. 117, pp Brown, P.N., Byrne, G.D., and Hndmarh, A.C., 1989, VODE, a Varable-Coeffcent ODE Solver, SIAM Journal on Scentfc and Stattcal Computng, Vol. 10, pp Coppalle, A., and Joyeux, D., 1994, Temperature and Soot Volume Fracton n Turbulent Dffuon Flame: Meaurement of Mean and Fluctuatng Value, Combuton and Flame, Vol. 96, pp Farweather, M., and Woolley, R.M., 003, Frt-Order Condtonal Moment Cloure Modellng of Turbulent, Nonpremxed Hydrogen Flame, Combuton and Flame, Vol. 133, pp Farweather, M., and Woolley, R.M., 004, Frt-Order Condtonal Moment Cloure Modellng of Turbulent, Nonpremxed Methane Flame, Combuton and Flame, Vol. 138, pp Grmaj, S.S., 1991, Aumed β-pdf Model for Turbulent Mxng: Valdaton and Extenon to Multple Scalar Mxng, Combuton Scence and Technology, Vol. 78, pp Kee, R.J., Rupley, F.M., and Mller, J.A., 1996, CHEMKIN II: A FORTRAN Chemcal Knetc Package for the Analy of Ga-Phae Chemcal Knetc, Report No. SAND B, Sanda Natonal Laboratore. Kent, J.H., and Honnery, D., 1987, Soot and Mxture Fracton n Turbulent Dffuon Flame, Combuton Scence and Technology, Vol. 54, pp Km, I.S., and Matorako, E., 005, Smulaton of Turbulent Lfted Jet Flame wth Two-Dmenonal Condtonal Moment Cloure, Proceedng of the Combuton Inttute, Vol. 30, pp Klmenko, A.Yu., and Blger, R.W., 1999, Condtonal Moment Cloure for Turbulent Combuton, Progre n Energy and Combuton Scence, Vol. 5, pp Kronenburg, A., and Blger, R.W., 001a, Modellng Dfferental Dffuon n Nonpremxed Reactng Turbulent Flow: Applcaton to Turbulent Jet Flame, Combuton Scence and Technology, Vol. 166, pp Kronenburg, A., and Blger, R.W., 001b, Modellng Dfferental Dffuon n Nonpremxed Reactng Turbulent Flow: Model Development, Combuton Scence and Technology, Vol. 166, pp Kronenburg, A., Blger, R.W., and Kent, J.H., 000, Modelng Soot Formaton n Turbulent Methane-Ar Jet Dffuon Flame, Combuton and Flame, Vol. 11, pp Leung, K. M., Lndtedt, R. P., and Jone, W. P., 1991, A Smplfed Reacton Mechanm for Soot Formaton n Nonpremxed Flame, Combuton and Flame, Vol. 87, pp Lndtedt, R.P., 1994, Smplfed Soot Nucleaton and Surface Growth Step for Nonpremxed Flame, In Soot Formaton n Combuton, Bockhorn, H., ed., Sprnger- Verlag, Berln, pp Lndtedt, R.P., and Louloud, S.A., 005, Jont-Scalar Tranported PDF Modelng of Soot Formaton and Oxdaton, Proceedng of the Combuton Inttute, Vol. 30, pp Ma, G., Wen, J.Z., Lghttone, M.F., and Thompon, M.J., 005, Optmzaton of Soot Modellng n Turbulent Nonpremxed Ethylene/Ar Jet Flame, Combuton Scence and Technology, Vol. 177, pp Mau, F., Netzell, K., and Lehtnem, H., 006, Apect of Modellng Soot Formaton n Turbulent Dffuon Flame, Combuton Scence and Technology, Vol. 178, pp Ptch, H., Remeer, E., and Peter, N., 000, Unteady Flamelet Modelng of Soot Formaton n Turbulent Dffuon Flame, Combuton Scence and Technology, Vol. 158, pp Qn, Z., Lank, V.V., Yang, H., Gardner, W.C., Dav, S.G., and Wang, H., 000, Combuton Chemtry of Propane: A Cae Study of Detaled Reacton Mechanm Optmzaton, Proceedng of the Combuton Inttute, Vol. 8, pp Spaldng, D.B., 1977, GENMIX: A General Computer Program for Two-dmenonal Parabolc Phenomena, Pergammon Pre, Oxford. Wen, Z., Yun, S., Thomon, M.J., and Lghttone, M.F., 003, Modelng oot formaton n turbulent keroene/ar jet dffuon flame, Combuton and Flame, Vol. 135, pp Young, K.J., and Mo, J.B., 1995, Modellng Sootng Turbulent Jet Flame ung an Extended Flamelet Technque, Combuton Scence Technology, Vol. 105, pp