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A Numerical Investigation of Thermal Diffusion Influence on Soot Formation in Ethylene/Air Diffusion Flames

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A Numerical Investigation of Thermal Diffusion Influence on Soot Formation in Ethylene/Air Diffusion Flames
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  A Numerical Investigation of Thermal DiffusionInfluence on Soot Formation in Ethylene/AirDiffusion Flames HONGSHENG GUO a, *, FENGSHAN LIU a , GREGORY J. SMALLWOOD a and O¨MER L. GU¨LDER b a Combustion Research Group, Institute for Chemical Process and Environmental Technology, National Research Council of Canada,1200 Montreal Road, Building M-9, Ottawa, Ont., Canada K1A 0R6;  b  Institute for Aerospace Studies, University of Toronto,4925 Dufferin Street, Toronto, Ont., Canada M3H 5T6  Thermal diffusion, caused by temperature gradients, tends to draw lighter molecules towarmer regions and to drive heavier molecules to cooler regions of a mixture. The influenceof thermal diffusion on soot formation in coflow laminar ethylene/air diffusion flames isnumerically investigated in this paper. Detailed reaction mechanisms and complex thermal andtransport properties are employed. The fully elliptic governing equations are solved.Radiation heat transfer from the flames is calculated by the discrete-ordinates method coupledto an SNBCK-based wide band model. A simplified two-equation soot model is used.The interactions between soot and gas-phase chemistry are taken into account. The thermaldiffusion velocities are calculated according to the thermal diffusion coefficients evaluated basedon multicomponent properties.The results show that thermal diffusion does affect soot formation in ethylene/air diffusionflames. Although the effect on soot formation in pure ethylene/air flame is not significant, theinfluence is enhanced if lighter species, such as helium, are added to the fuel or the air stream.The peak integrated soot volume fraction doubles if thermal diffusion is not taken into account inthe simulation of the flame with helium addition to the air stream. Keywords : Combustion; Thermal diffusion; Laminar flame; Soot; Simulation; Radiation INTRODUCTION Heat and mass transport processes have been shown to beof great importance for soot formation (McLintock, 1968;Schug  et al. , 1980; Axelbaum  et al. , 1988; Axelbaum &Law,1990;Gu¨lder etal. ,1996;Glassman,1998;Guo etal. ,2002a). Thermal diffusion (Soret effect), caused bytemperature gradients in a mixture, gives an additionalterm in the diffusion velocity of a chemical species.It tends to draw lighter molecules to hot regions and todrive heavier molecules to cold regions of the mixture. In a pioneering work, Dixon-Lewis (1968) observedthat the thermal diffusion flux of hydrogen could be of the same order of magnitude as the ordinary diffusionflux, caused by concentration gradients, in a hydrogen/airflame. The same observation was made byGreenberg (1980) in the study of one-dimensionalhydrogen/air flames using a one-step chemistry modeland phenomenological expressions for the thermaldiffusion coefficients. Later it was found by Warnatz(1982) that the laminar flame speeds of both lean and richhydrogen/air flames were lower when thermal diffusionwas taken into account, although only the thermaldiffusion fluxes of atomic and molecular hydrogen wereconsidered in the simulation. In the study of vortex-flameinteractions in hydrogen jet diffusion flames, Hancock  et al.  (1996) showed that the thermal diffusion effectcouldn’t be neglected in the numerical simulation. Recentstudies of Ern and Giovangigli (1998, 1999) furtherindicated the importance of thermal diffusion in variousreactive flows. In the paper of Ern and Giovangigli (1998),it was shown that thermaldiffusion was important notonlyfor the prediction of structures of hydrogen/air andmethane/air Bunsen flames, but also for the prediction of NO in a counterflow methane/air flame. Being differentfrom the result of Warnatz (1982), the study of Ern and ISSN 1061-8562 print/ISSN 1029-0257 online q 2004 Taylor & Francis LtdDOI: 10.1080/10618560310001634203 *Corresponding author. Tel.:  þ 1-613-991-0869. Fax:  þ 1-613-957-7869. E-mail: hongsheng.guo@nrc-cnrc.gc.ca  International Journal of Computational Fluid Dynamics , February 2004  Vol.  18 (2), pp. 139–151  Giovangigli (1999) indicated that for hydrogen/airflames, while the speeds of lean flames were lower,those of rich flames were higher when thermal diffusionof all species was considered. The influence of thermaldiffusion on the speeds of methane/air flameswas negligible. More recently, Williams (2001) revealedthat thermal diffusion caused an increase in the predictedextinction strain rate of methane/air counterflow flames. In spite of the importance of thermal diffusion,little attention has been paid to the influence of thermal diffusion on soot formation processes.It was totally neglected in some studies (McEnally et al. , 1998; Smooke  et al. , 1999; Bennett  et al. , 2001),while only the thermal diffusion of light species(such as H 2  and H) was taken into account inother studies, such as our two recent papers (Guo  et al. (2002a,b) and Kennedy  et al.  (1996)). To ourknowledge, the relative influence of thermal diffusionon soot formation has not been reported previously inthe literature. In the present paper, soot formation processes in apure ethylene/air and four helium or argon dilutedethylene/air coflow laminar diffusion flames aresimulated. The objective is to investigate the relativeinfluence of thermal diffusion on soot formation. Weemploy the primitive variable method in which the fullyelliptic governing equations are solved with detailedgas-phase chemistry and complex thermal and transportproperties. The effects of soot inception, growth andoxidation on gas-phase chemistry are taken intoaccount. For the soot kinetics process, a simplifiedtwo-equation soot model is used. Radiation heat transferfrom CO 2 , CO, H 2 O and soot is calculated using thediscrete-ordinates method coupled to an SNBCK-basedwide band model. NUMERICAL MODEL The flame configuration studied is a coflowaxisymmetric laminar diffusion flame. The fuelstream flows from the centre pipe, and the oxidantstream flows from the annular concentric space. Exceptfor the pure ethylene/air diffusion flame, flames withhelium or argon addition to the fuel or air stream arealso studied. Gas-phase Governing Equations The numerical model solves the fully elliptic governingequations for the conservation of mass, momentum,energy, gas species mass fractions, soot mass fractionand soot number density. In cylindrical coordinates( r  ,  z ), the governing equations for the gas-phase are(Kuo, 1986): Continuity: ›› r  ð r  r  v Þ þ ››  z ð r  r  u Þ ¼  0 :  ð 1 Þ Axial momentum: r  v › u › r  þ r  u › u ›  z ¼ 2 ›  p ›  z  þ 1 r  ›› r  r  m › u › r     þ  2  ››  z  m › u ›  z   2 23 ››  z m r  ›› r   ð rv Þ   2 23 ››  z  m › u ›  z   þ 1 r  ›› r  r  m › v ›  z    þ r  g  z : ð 2 Þ Radial momentum: r  v › v › r   þ r  u › v ›  z ¼ 2 ›  p › r  þ ››  z m › v ›  z    þ 2 r  ›› r r  m › v › r    2 231 r  ›› r  m ›› r  ð rv Þ   2 231 r  ›› r r  m › u ›  z   þ ››  z m › u › r    2 2 m vr  2  þ 23 m r  2 ›› r  ð rv Þ þ 23 m r  › u ›  z :  ð 3 Þ Energy: c  p  r  v › T  › r   þ r  u › T  ›  z   ¼ 1 r  ›› r  r  l › T  › r     þ ››  z  l › T  ›  z   2  X KK  þ 1 k  ¼ 1 r  c  pk  Y  k   V  kr  › T  › r  þ  V  kz › T  ›  z    2  X KK  þ 1 k  ¼ 1 h k  W  k  v  k   þ  q r  : ð 4 Þ Gas species: r  v › Y  k  › r  þ r  u › Y  k  ›  z ¼ 2 1 r  ›› r  ð r  r  Y  k  V  kr  Þ 2  ››  z ð r  Y  k  V  kz Þþ  W  k  v  k  ;  k   ¼  1 ; 2 ;  . . . ; KK  ;  ð 5 Þ where  u  and  v  are the velocities in axial (  z ) and radial ( r  )directions, respectively;  T   the temperature of the mixture; r   the density of the mixture (soot and gas);  W  k   themolecular weight of the  k  th gas species;  l  the mixturethermal conductivity;  c  p  the specific heat of the mixtureunder constant pressure;  c  pk   the specific heat of the  k  th gas H. GUO  et al. 140  species under constant pressure;  v  k   the mole productionrate of the  k  th gas species per unit volume;  p  the pressure.It should be pointed out that the production rates of gasspecies include the contribution due to the soot inception,surface growth and oxidation (see the next section).Quantity  h k   denotes the specific enthalpy of the k  th gas species;  g  z  the gravitational acceleration in the  z  direction;  m  the viscosity of the mixture;  Y  k   the massfraction of the  k  th gas species;  V  kr   and  V  kz  the diffusionvelocities of the  k  th gas species in  r   and  z  directions;and  KK   the total gas-phase species number. The quantitieswith subscript  KK   þ  1 correspond to those of soot. As anapproximation, the thermal properties, obtained fromJANAF thermochemical tables (Chase  et al. , 1985), of graphite are used to represent those of soot. The last term  q r   on the right-hand side of Eq. (4) is thesourcetermduetoradiationheattransfer.Itisobtainedbythediscrete-ordinates method coupled to a statistical narrow-band correlated-K (SNBCK)-based wide bandmodel fortheproperties of CO, CO 2 , H 2 O and soot (Liu  et al. , 1999).The spectral absorption coefficient of soot is obtained byRayleigh’stheoryforsmallparticlesandtherefractiveindexof soot due to Dalzell & Sarofim (1969) is  5 : 5  f  v = l , with  f  v being the soot volume fraction and l the wavelength. The diffusion velocity consists of three terms:ordinary diffusion, thermal diffusion and correctiondiffusion velocities. Therefore: V  kx i  ¼  V  okx i  þ  V  Tkx i  þ  V  cx i ; k   ¼  1 ; 2 ;  . . . ; KK  ;  x i  ¼  r  ;  z : ð 6 Þ Both the ordinary and thermal diffusion velocities canbe obtained by the detailed multicomponent method, orthe approximate mixture-average method. The former isaccurate, but computationally expensive. The latter iscorrect asymptotically in some special cases, such as in abinary mixture, in diffusion of trace amounts of speciesinto a nearly pure species, or systems in which all speciesexcept one move with nearly the same diffusion velocity.Since the current study concentrates on the relativeinfluence of thermal diffusion, the ordinary diffusionvelocity  V  okx i , caused by concentration gradient, isobtained by the approximate mixture-average formulation(Kee  et al. , 1986), i.e. V  okx i  ¼ 2  1 Y  k   D k  › Y  k  ›  x i k   ¼  1 ; 2 ;  . . . ; KK  ;  x i  ¼  r  ;  z :  ð 7 Þ Quantity  V  Tkx i is the thermal diffusion velocity,whose influence will be investigated in the presentpaper, in  x i  ( r   or  z ) direction for the  k  th gas species.It is obtained by the detailed multicomponent formulation(Kee  et al. , 1986): V  Tkx i  ¼ 2  D T k  r  Y  k  1 T  › T  ›  x i k   ¼  1 ; 2 ;  . . . ; KK  ;  x i  ¼  r  ;  z  ð 8 Þ where  D T k   is the thermal diffusion coefficient obtained bythe method given by Kee  et al.  (1986). The correction diffusion velocity  V  cx i is used to ensurethat the net diffusive flux of all gas species and soot is zero(Kee  et al. , 1986). Quantity  D k   in Eq. (7) is related to the binary diffusioncoefficients through the expression:  D k   ¼ 1 2  X  k  X KK  j – k   X   j  D  jk  ;  k   ¼  1 ; 2 ;  . . . ; KK   ð 9 Þ where  X  k   is the mole fraction of the  k  th species, and  D  jk   is the binary diffusion coefficient. Soot Model Although some detailed kinetic models of soot inception,growth and oxidation have been derived, such as those byFrenklach  et al.  (1984) and Frenklach and Wang (1990,1994), they are too complex and computationallyexpensive to be implemented in simulations of multi-dimensional combustion systems. Conversely, the appli-cability of purely empirical soot models is questionableunder conditions different from those under which theywere originally formulated. Based on some semi-empirical assumptions, McEnally  et al.  (1998) andSmooke  et al.  (1999) used the sectional model to simulatethe soot formation processes. In addition to themomentum, energy and gas species conservationequations, several soot section equations (usually morethan 10) need to be solved. The model developed byLeung  et al.  (1991) and Fairwhether  et al.  (1992)has been successfully used in our previous studies(Guo  et al. , 2002a,b) for the simulations of ethylene/airdiffusion flames. It has been shown that this modelcan capture the features of the effects of inertspecies dilution on soot formation in ethylene/air diffusionflames. As only two additional equations need to be solvedfor soot processes in this model, it is used again in thepresent paper. Two transport equations are solved for soot massfraction and number density. They are r  v › Y  s › r   þ r  u › Y  s ›  z ¼ 2 1 r  ›› r  ð r  r  V  T  ; r  Y  s Þ 2  ››  z ð r  V  T  ;  z Y  s Þ þ  S  m ð 10 Þ r  v ›  N  › r  þ r  u ›  N  ›  z ¼ 2 1 r  ›› r  ð r  r  V  T  ; r   N  Þ 2  ››  z ð r  V  T  ;  z  N  Þ þ  S   N   ð 11 Þ where  Y  s  is the soot mass fraction,  N   is the sootnumber density defined as the particle number per unitmass of mixture. Quantities  V  T ; r   and  V  T ;  z  are theparticle thermophoretic velocities. They are obtained bythe expression for a free molecular aerosol NUMERICAL INVESTIGATION OF THERMAL DIFFUSION 141  (Talbot  et al. , 1980): V  T  ;  x i  ¼ 2 0 : 55  mr  T  › T  ›  x i  x i  ¼  r  ;  z :  ð 12 Þ Although the particle thermophoretic motion is alsoa kind of thermal diffusion, its effect was not studiedin this paper, since the emphasis in this paper is thethermal diffusion of gas species (Eq. 8).The source term  S  m  in Eq. (10) accounts for thecontributions of soot nucleation ( v  n ), surface growth( v  g ) and oxidation ( v  O ). Therefore, S  m  ¼  v  n  þ v  g 2 v  O :  ð 13 Þ The model developed by Leung  et al.  (1991) andFairwhether  et al.  (1992) is used to obtain the threeterms on the right-hand side of Eq. (13). The modelassumes the chemical reactions for nucleation andsurface growth, respectively, as: C 2 H 2  ! 2C ð S Þ þ  H 2  ð R1 Þ C 2 H 2  þ  n C ð S Þ ! ð n  þ  2 Þ C ð S Þ þ  H 2  ð R2 Þ with the reaction rates given by the expressions: r  1  ¼  k  1 ð T  Þ½ C 2 H 2  ð 14 Þ r  2  ¼  k  2 ð T  Þ  f  ð  A s Þ½ C 2 H 2  ð 15 Þ where  f  (  A s ) denotes the functional dependence on sootsurface area per unit volume. Similar to our previousstudies (Guo  et al. , 2002a,b), a simple linear functionaldependence is used, i.e.  f  (  A s )  ¼  A s . Neoh etal. (1981)investigatedthesootoxidationprocessin flames, and found that the oxidation due to both O 2 and OH is important, depending on the local equivalenceratio. The radical O also contributes to soot oxidationin some regions. Therefore the soot oxidation by O 2 , OHand O are accounted for by the following reactions: 0 : 5O 2  þ  C ð S Þ ! CO  ð R3 Þ OH  þ  C ð S Þ ! CO  þ  H  ð R4 Þ O  þ  C ð S Þ ! CO :  ð R5 Þ The reaction rates for these three reactions wereobtained by: r  3  ¼  k  3 ð T  Þ T  1 = 2  A s ½ O 2  ð 16 Þ r  4  ¼  w  OH k  4 ð T  Þ T  2 1 = 2  A s  X  OH  ð 17 Þ r  5  ¼  w  O k  5 ð T  Þ T  2 1 = 2  A s  X  O  ð 18 Þ where  X  OH  and  X  O  denote the mole fractions of OHand O, and  w  OH  and  w  O  are the collision efficiencies forOH and O attack on soot particles, respectively.The collision efficiency of OH is treated as thatdescribed by Kennedy  et al.  (1996), who accounted forthe variation of the collision efficiency of OH with timeby assuming a linear relation between the collisionefficiency and a dimensionless distance from thefuel nozzle exit. A collision efficiency of 0.5 forradical O attack on the particles is used in this study(Bradley  et al. , 1984). All the reaction rate constants,  k  i  ð i  ¼  1 ;  . . . ; 5 Þ ;  aresummarized in Table I. The source term  S   N   in Eq. (11) accounts for the sootnucleation and agglomeration, and is calculated as: S   N   ¼ 2 C  min  N  A r  1 2 2 C  a 6  M  C  ð S  Þ pr  C  ð S  Þ   1 = 6 £ 6 k  T  r  C  ð S  Þ   1 = 2 C  ð s Þ½  1 = 6 r   N    11 = 6 ð 19 Þ where  N  A  is Avogadro’s number ( 6 : 022 £ 10 26 particles = kmol Þ ;  C  min  is the number of carbon atoms inthe incipient carbon particle  ð 9 £ 10 4 Þ  (Fairwhether  et al. ,1992),  k   is the Boltzman constant  ð 1 : 38 £ 10 2 23 J = K  Þ ; r  C  ( S  )  is the soot density (1800kg/m 3 ),  C  (s) is the moleconcentration of soot (kmol/m 3 ),  M  C  (s)  is the molar massof soot (12.011kg/kmol), and  C  a  is the agglomeration rateconstant for which a value of 3.0 (Fairwhether  et al. , 1992)is used. Numerical Scheme The flames modeled in this study are generated with aburner (Gu¨lder  et al. , 1996) in which the fuel streamflows from a 10.9mm inner diameter vertical tube, andthe oxidant stream flows from the annular regionbetween the fuel tube and a 100mm inner diameterconcentric tube. The wall thickness of the fuel tube is0.95mm. The computational domain covers an area from0 to 3.0cm in the radial direction and 0 to 11.0cm inthe axial direction. The inflow boundary  ð  z  ¼  0cm Þ corresponds to the region immediately above thefuel nozzle exit. This computational domain hasbeen shown to be large enough by a sensitivity calculation.Totally, 104 £ 71 non-uniform grids are used. Finergrids are placed in the reaction zone and near the fuelnozzle exit region by a grid adaptive refinement method. TABLE I Rate constants, as  k  i  ¼  A exp ð 2 E  =  RT  Þ  (units are kg, m, s,kcal, kmol and K) k  i  A E Reference k  1  1.35E  þ  06 41 Fairwhether  et al.  (1992) k  2  5.00E  þ  02 24 Fairwhether  et al.  (1992) k  3  1.78E  þ  04 39 Fairwhether  et al.  (1992) k  4  1.06E  þ  02 0 Neoh  et al.  (1981) k  5  5.54E  þ  01 0 Bradley  et al.  (1984) H. GUO  et al. 142  FIGURE 1 Comparison of predicted and measured flame temperature and soot volume fraction.FIGURE 2 Comparison of soot volume fraction for pure ethylene/airflame. a. Integrated soot volume fraction; b. Radial soot volume fractionat axial height of   z  ¼  1 : 0cm :  TD represents  Thermal Diffusion . FIGURE 3 Thermal diffusion factors of some main species and radicalsat the axial height of   z  ¼  1 : 0cm  for pure ethylene/air flame.NUMERICAL INVESTIGATION OF THERMAL DIFFUSION 143
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