Spectral radiative effects and turbulence/radiation interaction in a non-luminous turbulent jet diffusion flame

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1 Combuston and Flame 133 (2003) Spectral radatve effects and turbulence/radaton nteracton n a non-lumnous turbulent jet dffuson flame P.J. Coelho a, *, O.J. Teerlng a, D. Roekaerts b a Insttuto Superor Técnco, Mechancal Engneerng Department, Av. Rovsco Pas, Lsboa, Portugal b Delft Unversty of Technology, Thermal and Fluds Scences, Department of Mult-Scale Physcs, P.O. Box 5046, 2600 GA Delft, The Netherlands, Shell Research and Technology Centre, Amsterdam Receved 10 Aprl 2002; receved n revsed form 13 September 2002; accepted 12 November 2002 Abstract A non-lumnous turbulent jet dffuson flame s numercally smulated usng a Reynolds stress second-order closure, the steady lamnar flamelet model, and dfferent approaches for radatve transfer. The commonly used optcally thn approxmaton s compared wth the dscrete ordnates method. Calculatons usng the Planck mean absorpton coeffcent are compared wth computatons performed usng the spectral lne-based weghted-sumof-gray-gases model. The nteracton between turbulence and radaton s smulated, and ts nfluence on the predcted results s nvestgated. It s shown that the dscrete ordnates method and the optcally thn approxmaton yeld relatvely close results for the present flame f the medum s modelled as gray usng the Planck mean absorpton coeffcent. In both cases, the predcted fracton of radatve heat loss s sgnfcantly overestmated. However, f the spectral nature of gaseous radaton s accounted for, the computed radaton loss s closer to the expermental data. The fluctuatons of the speces have a mnor role n the nteracton between turbulence and radaton, whch s manly due to the temperature fluctuatons The Combuston Insttute. All rghts reserved. Keywords: Turbulent dffuson flames; Turbulence/radaton nteracton; Lamnar flamelet model; Radatve heat transfer 1. Introducton Flame radaton plays an mportant role n fres and n many combuston systems. Pollutant emssons are also nfluenced by flame radaton by means of ther dependence on the temperature feld. For example, the accurate predcton of NO n turbulent jet dffuson flames requres an accurate predcton of the flow/mxng feld and radaton losses, as well as a detaled mechansm of NO formaton and destructon. The present work s concerned wth the predcton of the radaton losses from non-lumnous turbulent dffuson flames. * Correspondng author. Tel.: ; fax: E-mal address: coelho@naver.st.utl.pt (P.J. Coelho). The calculaton of radaton from flames has often been based on the optcally thn approxmaton, e.g. [1], neglectng flame absorpton, partcularly n the case of non-lumnous flames. However, ths s a crude approxmaton n many applcatons, especally f an accurate predcton of the temperature feld s needed, as n the case where NO predctons are requred. More accurate radatve calculatons may be carred out usng, for example, the zone [2], Monte Carlo [3], sphercal harmoncs [4], dscrete transfer [5], dscrete ordnates [6], and boundary element [7] methods. However, the medum has been consdered as gray n most smulatons. Here, the word gray characterzes the radaton model, and not the gas radatve propertes model. Ths means that radatve calculatons are carred out usng total radatve propertes, no matter how these are calculated. An ad /03/$ see front matter 2003 The Combuston Insttute. All rghts reserved. do: /s (02)

2 76 P.J. Coelho et al. / Combuston and Flame 133 (2003) vanced non-gray model may be used to compute the gas radatve propertes, e.g., a narrow band model, but f t s used only to compute total radatve propertes, and the radaton model uses these total radatve propertes,.e., spectral band calculatons are not carred out, then we stll say that the medum s treated as gray. It s well known that gas radaton occurs n dscrete bands that comprse many thousands of spectral lnes. Lne-by-lne calculatons cannot be used n practcal problems because of ther computatonal costs, but several non-gray gas radatve property models are avalable. A survey of these models and ther couplng wth the radatve transfer equaton may be found n [8]. The statstcal narrow band model has been used to compute the radaton ntensty along lnes of sght [9], and as a post-processor (uncoupled calculatons) to calculate radatve wall fluxes and power n a turbulent flame nsde an axsymmetrcal furnace [10]. However, narrow band calculatons are too computatonally expensve for coupled flame structure and radaton smulatons. The exponental wde band model [11] may be used n flame calculatons, e.g. [12 14], but the correlaton between the spectral transmssvty and the radaton ntensty s generally gnored. These non-correlated calculatons may yeld large errors, as shown n [15]. The correlated formulaton s tme-consumng and dffcult to couple to dfferental soluton methods of the radatve transfer equaton (RTE), such as the dscrete ordnates method (DOM) [16]. The correlated k-dstrbuton method [17], ntally developed n the atmospherc radaton communty, has recently receved the attenton of the heat transfer communty [18,19]. It provdes the absorpton coeffcent as the basc radatve property, contrary to the narrow and wde band models, and therefore may be easly appled together wth any soluton method of the RTE. Unfortunately, t s stll qute expensve for most practcal applcatons. The computatonal requrements depend on the number of bands, number of quadrature ponts wthn a band and number of absorbng gases n the mxture. In the case of 44 bands and 7 quadrature ponts per band, the rato of the CPU tme for the correlated k-dstrbuton method to the CPU tme for the classcal weghtedsum-of-gray-gases (WSGG) model [20] wth 8 coeffcents s 28 for a H 2 O-N 2 gas layer and 11 for a CO 2 -N 2 gas layer [21]. In the case of a H 2 O-CO 2 -N 2 mxture, that rato generally exceeds two orders of magntude [22]. The WSGG model [20], ntally developed to use together wth the zone method, s probably the most popular method to compute the gas radatve propertes. However, t has generally been used to compute the total emssvty, treatng the gas as a gray medum. Only after the work of Modest [23] t has been mplemented as a non-gray gas model n the RTE, and coupled wth the dscrete ordnates and the dscrete transfer methods [10,24]. However, n the case of sgnfcant temperature gradents, the WSGG model may yeld mportant errors, hgher than 30%, as dscussed n [10]. The spectral lne-based weghted-sum-of-graygases (SLW) model developed by Denson and Webb [25] s an mproved verson of the WSGG model, whch has emerged n the last decade as a promsng alternatve model able to provde hgh accuracy at moderate computatonal cost, and compatblty wth any arbtrary soluton method of the RTE. A closely related model, referred to as absorpton dstrbuton functon model [26], dffers only n the calculaton of the weghts. The full-spectrum correlated-k dstrbuton [27] and the mult-scale correlated k-dstrbuton [28] methods, recently developed by Modest and co-workers, have many smlartes wth the SLW model. They extend the correlated-k dstrbuton to the entre spectrum by defnng a fractonal Planck functon, thus combnng all the advantages of the correlated k-dstrbuton method wth those of global models. In the case of turbulent flames, there s a lot of expermental and theoretcal evdence that the turbulence/radaton nteracton (TRI) has a sgnfcant nfluence due to the temperature and speces concentraton fluctuatons and the non-lnear relatonshp among temperature, radatve propertes and radaton ntensty [29]. Despte of ths, most works neglect such an nteracton. At present, the most accurate way to smulate the TRI seems to be the stochastc approach, frstly employed by Jeng et al. [9]. They assumed that the flow feld conssts of numerous statstcally ndependent turbulent eddes, each of them havng unform propertes, whch were assumed to be a sngle functon of mxture fracton. The nstantaneous mxture fracton was randomly selected accordng to a prescrbed cumulatve dstrbuton. Temporal correlatons were gnored n ther model. The model was modfed n [30] to account for temporal correlatons of mxture fracton, and n [31] to remove the restrcton of statstcally ndependent eddes, and some mnor mprovements were reported n [32,33]. Chan and Pan [34] have shown that cross temporal-spatal correlatons are also mportant. They further mproved the model by ncludng all together the mxture fracton correlatons n space and tme, ther cross correlatons, and postcorrelatons n space [35]. The above works were restrcted to the calculaton of spectral ntenstes along a lne of sght, and were not appled to the smulaton of radatve transfer n flames. Despte the promsng results acheved by the

3 P.J. Coelho et al. / Combuston and Flame 133 (2003) stochastc approach, t suffers from the tme-consumng need to take many trals for good statstcs, and may not be practcal n modellng complex combustor geometres, as recognzed by Hall and Vranos [36]. These authors have developed a smpler and faster sem-analytc approach, and appled ther method to a turbulent CH 4 -H 2 dffuson flame usng the wde band model. In another approach, whch avods a detaled knowledge of two-pont correlatons of temperature and concentraton fluctuatons, the ntegral form of the RTE s taken and the nstantaneous terms n the exponental expressons are replaced by tme averaged values [37]. Such an approxmaton s only vald f the optcal dmenson of the turbulent eddes s 0.3. Several authors have smplfed the tme averaged RTE by assumng that there s no correlaton between temperature and concentraton wthn each eddy. Ths s a consequence of assumng that the ndvdual eddes are homogeneous, optcally thn, and statstcally ndependent. Song and Vskanta [38] have used ths approach together wth the sphercal harmoncs P1 approxmaton for the smulaton of a natural gas fred furnace. Ths approach has also been used n [39] n the framework of a ray tracng method, n [40] together wth the DOM, and n [41] along wth the four-flux radaton model. The same approxmaton was recently used n [42] wth the P1 and wde band models for radatve calculatons and the veloctycomposton jont probablty densty functon (PDF) method [43] for flow and combuston modelng. These PDF methods for turbulent reactve flows can also be combned wth the boundary element method appled to radatve heat transfer and accountng for the TRI [44]. The motvaton for the present work stems from the questons rased at the Internatonal Workshop on Measurement and Computaton of Turbulent Non- Premxed Flames [45] regardng the approprateness of the optcally thn assumpton for the ploted flames. The large dscrepances between NO predctons made by dfferent research groups and nvestgators motvated the measurement of radatve heat loss from the flames. It was found that the measured radant fracton for the so-called flame D was 5.1% compared wth predcted values of 10.5% and 12.5% usng the PDF and condtonal moment closure (CMC) models along wth the optcally thn approxmaton [46]. However, ndependent calculatons based on a full jont PDF method and the same optcally thn approxmaton [47] gave a value of 2.85% for the total radant fracton accountng only for x/d 60. Ths suggests that the measured value mght be approached f the entre flame were consdered. Therefore, there s a need to a more accurate treatment of radatve transfer to address ths problem. In ths work, flame D s smulated usng a Reynolds stress model for turbulence modelng, the steady lamnar flamelet model for combuston, the DOM for radaton, the SLW model for the radatve propertes of the gas, and the approach of Song and Vskanta [38] to account for the TRI. For comparson purposes, calculatons have also been performed usng the optcally thn approxmaton. The man contrbuton of the present work s to present and apply a model for flame smulaton that smultaneously accounts for flame radaton usng advanced radaton and gas radatve propertes models, as well as TRI. Some researchers have used the DOM or the dscrete transfer method n flame smulatons. Others, but very few, have treated the medum as non-gray n the radaton model. Also very few have accounted for the TRI. Here, all these features are smultaneously accounted for, and ther role s nvestgated and compared wth smpler approaches. The nfluence of the spectral radaton effects and the TRI are nvestgated. The models are descrbed n the next secton, wth emphass on the radatve transfer calculatons. Then, some computatonal detals are provded, and the expermental confguraton s descrbed. Ths s followed by the presentaton and dscusson of the results. The man conclusons are drawn n the last secton. 2. Mathematcal modelng 2.1. Turbulence model Full second moment turbulence closure was used [48]. The transport equaton for the Reynolds stress was modeled after Launder, Reece, and Rod [49] (LRR-IP model), and the transport equaton for the Reynolds flux of a scalar was closed by a wdely used model descrbed by Jones [50]. Standard values were assgned to the constants of the model, wth one excepton: n the standard equaton for the dsspaton rate of turbulent knetc energy the constant C 1 s set to 1.60 to mprove the predcton of the spreadng rate of the jet. The computatonal flud dynamcs (CFD) code employed n the present work s an n-house code developed at the Delft Unversty of Technology. It solves the densty weghted averaged form of the governng equatons by usng the fnte volume method and a non-staggered grd varable arrangement Combuston model Combuston s modeled usng the conserved scalar approach wth a prescrbed probablty densty functon. Accordng to ths approach, n the case of

4 78 P.J. Coelho et al. / Combuston and Flame 133 (2003) an adabatc flow the nstantaneous thermochemcal state of the gaseous mxture s a functon of a strctly conserved scalar varable, taken as the mxture fracton. Here, ths functon s obtaned usng the lamnar flamelet concept. The flamelet model s based on the dea that a turbulent flame may be regarded as an ensemble of flamelet structures attached to the nstantaneous poston of the flame surface, whch s corrugated by the turbulent flow feld [51]. The flamelet structure may be calculated from the soluton of the flamelet equatons. These equatons are derved from the transport equatons for speces mass fractons and energy by means of a coordnate transformaton. If the pressure s constant wth tme, and the Lews number for all the speces s equal to 1, then the flamelet equatons for non-premxed combuston may be wrtten as: T t 2 T 2 2 Z T 2c p Z c p Y T 2 c p Z c p Z Z 1 c p ẇ h q 0 (1) c p Y t 2 Y 2 Z 2 ẇ 0 (2) where T s the temperature, Y the mass fracton of speces, the densty, the scalar dsspaton rate, c p the specfc heat capacty, t the tme, Z the mxture fracton, ẇ the reacton rate, h the enthalpy, q the radatve source per unt volume, and the subscrpt refers to the th chemcal speces. Although prevous work [52] has shown that the unsteady lamnar flamelet model performs better than the steady model, the dfference between the H 2 O and CO 2 mass fractons predcted by the two models s relatvely small. Because these speces are responsble for most of the radaton loss, whch s of prmary concern n the present work, the steady flamelet model wll be used here. Therefore, only the steady soluton of Eqs. (1) and (2) s requred. However, the tme dependent terms are retaned n Eqs. (1) and (2), because these equatons are solved usng a marchng n tme soluton algorthm untl the steady state soluton s attaned. The temperature feld predcted by the steady lamnar flamelet model consderng radaton s unrealstc, as shown n [53]. Therefore, the radaton term was omtted n the soluton of the flamelet equatons, and the temperature computed from Eq. (1), correspondng to adabatc condtons, was used to calculate the densty feld n Eq. (2). The temperature and densty felds for non-adabatc condtons are obtaned from the enthalpy, as descrbed later. Accordng to ths approach, radatve transfer does not nfluence the flamelet relatonshps between nstantaneous values of chemcal composton and mxture fracton. Ths smplfcaton s expected to have a mnor nfluence for the present flame, because the total fracton of radatve heat loss s about 5%. The nfluence should be greater n strongly radatng flames. Ths expectaton s confrmed by prevous calculatons reported n [52], whch were carred out usng both the steady lamnar flamelet model, that neglects the nfluence of radatve transfer on the flamelet relatonshps, and the unsteady flamelet model, that accounts for t. It was found that the predctons of concentraton of major speces, ncludng CO 2 and H 2 O, were smlar for both models, showng that ths smplfcaton s acceptable. The scalar dsspaton rate, whch s a functon of the mxture fracton, s an mportant parameter n the flamelet equatons. It can be nterpreted as the nverse of a characterstc dffuson tme. Ths functon may be taken from counter-flow geometry [54] as Z a exp 2erfc1 2Z 2 (3) where erfc 1 s the nverse of the complementary error functon and a s the velocty gradent at the stagnaton pont. Applyng Eq. (3) to stochometrc condtons yelds Z st fz/fz st (4) where f(z) s the exponental term on the rght of Eq. (4) and the subscrpt st dentfes stochometrc condtons. Ths expresson for the scalar dsspaton rate s used n Eqs. (1) and (2). The scalar dsspaton rate for the stochometrc mxture, st, s taken as a flamelet parameter, that s, the soluton of the flamelet equatons s a functon of Z and st. For turbulent flows Eq. (4) must be averaged. Assumng that st and Z are statstcally ndependent [51] that equaton may be expressed as st0 1 fzp Z dz /fz st (5) where P (Z) s the pdf of mxture fracton. The averaged scalar dsspaton rate s the snk term of the transport equaton for the mxture fracton varance, Z 2, whch s modeled by C x k Z2 (6) C x s a constant set equal to 2.0 [55]. The condtonal mean scalar dsspaton rate at Z Z st, st, calculated from Eqs. (5) and (6) can be set equal to st n

5 P.J. Coelho et al. / Combuston and Flame 133 (2003) Eq. (4) because of the nertal range nvarance of scalar dsspaton rates [56]. In the CFD code transport equatons are solved for Z and Z 2. The mean and the varance of the mxture fracton completely defne the pdf, assumed to be a beta functon. The Favre mean values of the mass fractons are calculated by ntegratng the flamelet profles, parameterzed by the local value of st, over the mxture fracton range Ỹ 0 1 Y Z, st P Z dz (7) A transport equaton for the mean enthalpy, h, s also solved n the CFD code. It may be wrtten as ũ x j h j x j h u x j h j q (8) where ũ j s the Favre-averaged jth velocty component, x j s the coordnate along drecton j, s the vscosty, and s the Prandtl number. The tlde dentfes Favre-averaged quanttes. The mean temperature feld s computed from the enthalpy, as descrbed below. The enthalpy of the mxture s defned by h Y h Y h o T o T c p, dt (9) where Y s the mass fracton of speces, and h o s the enthalpy of formaton of the same speces at the standard reference temperature T o. The followng relaton holds for adabatc condtons h ad h ox 1 Z h fu Z (10) where h ox and h fu are the enthalpes of the oxdant and fuel, respectvely. The local fracton of radatve heat loss, X R,s defned as [57] X R h ad h (11) o h ad Y h Ths s a non-dmensonal quantty that s defned from local values of enthalpy and speces concentratons. It represents the rato of the local energy released by radaton to the energy that would have been released f the products were cooled down to the room temperature. Eq. (11) may be rearranged as follows, showng the ndependent varables for clarty o hz, st, X R X R Y Z, st h 1 X R h ad Z (12) The mass fractons of the chemcal speces are ndependent of the radatve heat loss fracton [39]. Insertng Eq. (10) nto Eq. (12) yelds hz, st, X R X R Y Z, st h o 1 X R h ox 1 Z h fu Z (13) The Favre-averaged enthalpy may then be computed as 1 o hz, st, X R P Z dz X R Ỹ h h 0 1 X R h ox 1 Z h fu Z (14) where Eq. (7) was used, and X R was assumed to be ndependent of Z. Ths equaton s used to compute X R from Z, h, and Ỹ, wth h gven from the soluton of ts transport equaton, and Ỹ gven from Eq. (7) for the local value of st. An mplct equaton for the temperature as a functon of Z, st, and X R s obtaned from Eqs. (9) and (13): 1 X R h ox1 Z h fu Z Y h o T T o c p (15) where Y T o T c p, dt c p T T o (16) Fnally, the mean temperature s calculated from T 0 or 1 TZ, st, X R P Z dz (17) T 1 TZ, st, X R Z, st, X R P Z dz (18) 0 where the overbar denotes the Reynolds average. The mean densty s computed dz from the deal gas law: 1 1 P Z Z, st, X R R o TZ, st, X R p 1 Y Z, st /W P Z dz (19)

6 80 P.J. Coelho et al. / Combuston and Flame 133 (2003) where R o s the unversal gas constant and W s the molar weght of speces Gas radaton propertes model The radatve propertes of the gas mxture are computed usng the SLW model. Ths model s summarzed below for the sake of completeness. In a mxture wth two partcpatng gases, namely CO 2 and H 2 O, the total emssvty, g, may be wrtten as N g g k0 N g a jk1 exp jkl (20) j0 where ndces j and k dentfy the jth gray gas component for H 2 O and the kth gray gas component for CO 2, N g s the number of gray gases and L s the path length. The values j 0 and k 0 account for the spectral wndows where H 2 O and CO 2 are transparent to radaton, respectvely. It s assumed here that the contrbuton from other radatng speces, such as CO and CH 4, s neglgble. The contrbuton from CO n the combuston gases s neglgble, as long as ts concentraton does not exceed relatvely hgh values of the order of 5% [58], whle the contrbuton from CH 4 s even lower than that of CO. The weght a jk n Eq. (20) s defned as the fracton of blackbody energy n the spectrum where the effectve absorpton cross-secton of H 2 OsC wj and where the effectve absorpton cross-secton of CO 2 s C c,k. The spectral regons where the effectve absorpton cross-secton s C s, j are those where the absorpton cross-secton s between C s, j and C s, j1, wth subscrpt s standng for the chemcal speces (w or c). The supplemental absorpton cross-sectons, denoted wth a tlde, are used to calculate the weghts a jk, but they do not appear drectly n the correspondng gray gas absorpton coeffcents jk. The absorpton cross-secton doman s dvded nto N g ntervals, equally spaced n a logarthmc scale, provded that N g s large enough, typcally 10 or 20. The lmts of each nterval defne the values of the supplemental absorpton cross-sectons C s, j. The absorpton crosssectons C s, j are computed as follows C s, j exp ln C s, j ln C s, j1 /2 (21) The weghts a jk, whch add up to unty, are calculated accordng to the double ntegraton approach [59], yeldng a jk F w C w, j1 F w C w, j F c C c,k1 F c C c,k (22) The absorpton-lne blackbody dstrbuton functon, F s,sdefned as the fracton of the blackbody energy n the portons of the spectrum where the hgh-resoluton spectral absorpton cross-secton of the gas, C s,, s less than a prescrbed value C s : F s C s, T b, T g, P T, X s 1 4 T b E b, T b d (23) C s,t g,p T,X s where s the Stefan-Boltzmann constant and E b s the spectral emssve power of a blackbody evaluated at wave number and blackbody (source) temperature T b. The subscrpt refers to the th spectral segment where C s, C s, whch depends on C s, gas temperature, T g, total pressure, P T, and molar fracton of speces s, X s. The correlatons for F w and F c gven n [60] and [61], respectvely, were employed n all the calculatons reported below. The absorpton coeffcents jk n Eq. (20) are calculated as jk N w C w, j N c C c,k (24) where N w and N c are the molar denstes of H 2 O and CO 2, respectvely. In the general case of a nonsothermal and/or nonhomogeneous medum, the absorpton cross-sectons defned by Eq. (21) are assocated wth a reference state defned by a temperature, T ref, a total pressure, P T,ref, and molar fractons X s,ref, and are denoted by C s,ref. That reference state s determned as the spatal average of the temperature, total pressure, and chemcal composton felds. The absorpton-lne blackbody dstrbuton functons n Eq. (22) are calculated for T g T ref, P T P T,ref, X s X s,ref, and T b T loc or T b T wall dependng on the source of radaton beng the gas or the wall. Here, T loc s the local gas temperature and T wall s the wall temperature. The local molar denstes and the absorpton cross-sectons n Eq. (24) are both evaluated from the local propertes, dentfed wth subscrpt loc,.e., T g T loc, P T P T,loc and X s X s,loc. The local values of the absorpton cross-secton, C s, n that equaton are calculated from the followng mplct equaton [62]: F s C s, T b T ref, T g T loc, X s X s,loc, P T P T,loc F s C s,ref, T b T ref, T g T ref, X s X s,ref, P T P T,ref (25) where the reference absorpton cross-sectons, C s,ref, are determned from Eq. (21), as stated above. Equaton (25) s solved at each spatal locaton usng an teratve method. The total number of gases, N g, ranges between 10 and 20. However, t s possble to acheve good accuracy wth only N g 3 by means of an optmzaton procedure based on the mnmzaton of the

7 P.J. Coelho et al. / Combuston and Flame 133 (2003) squared relatve error n emssvty over the range of path lengths typcal of the flame under nvestgaton, and for the temperature and speces concentratons at the reference state [25]. In such a case, Eq. (21) s not used, and the absorpton cross-sectons as well as the supplemental absorpton cross sectons at the reference state are determned from the optmzaton procedure. Eq. (25) s stll used to calculate the absorpton cross-sectons at the local condtons. Ths strategy was used n the present work Radaton model The code for radatve transfer calculatons s based on the DOM. The radatve transfer equaton for the jth gray gas component of H 2 O and the kth gray gas component of CO 2 may be wrtten as follows [23]: di jk ds jki jk a jk jk I b (26) where I jk s the radaton ntensty for those gray gas components, s s the drecton of propagaton of radaton, and I b s the blackbody radaton ntensty. Scatterng has been neglected, snce t s not present n the case of a gaseous non-lumnous flame. Ths equaton s tme-averaged to account for the TRI, yeldng di jk ds jki jk a jk jk I b (27) The dffculty wth ths equaton s the correlaton between the radaton ntensty and the absorpton coeffcent n the frst term on the rght sde. Followng Song and Vskanta [38], t wll be assumed that the ndvdual eddes are homogeneous, optcally thn, and statstcally ndependent. The accuracy of ths method was recently nvestgated by means of radatve transfer calculatons along lnes of sght of the same flame consdered here. The results are reported n [63] and show that the predctons obtaned usng the DOM together wth the SLW model and the optcally thn eddy approxmaton of Song and Vskanta [38] for the TRI are wthn 4% of the results obtaned usng the stochastc method by Chan and Pan [35], whch were taken as benchmark. Ths shows that the method of Song and Vskanta s reasonably accurate for the present flame. Accordng to ths method, the correlaton on the rght of Eq. (27) may be approxmated by jk I jk jk I jk (28) and so Eq. (27) may be smplfed as di jk ds jki jk a jk jk I b (29) The boundary condton for a gray dffuse surface may be wrtten as I w, jk w a jk I bw 1 w ns0 n si jk s d (30) where w s the surface emssvty, n s the unt vector normal to the surface, and s s the drecton of propagaton of the ncdent radaton assocated wth the sold angle d. In the present case w 1, because the boundary of the computatonal doman s treated as a blackbody at room temperature. The tme-averaged form of the boundary condton s wrtten as I w, jk w a jk I bw 1 w ns0 n si jk s d (31) Applyng the DOM, the tme-averaged equaton for the jth component of H 2 O, the kth component of CO 2, and the mth drecton can be wrtten n the form: m di jk ds jki m jk a jk jk I b (32) The total radaton ntensty for drecton m s gven by N I m g N g I m jk (33) k0 j0 The dscretzaton of Eq. (32) follows standard practces, as descrbed n detal n [64,65]. That equaton may be re-wrtten n cylndrcal coordnates as I m jk m m x r ri m jk 1 r r m I jk m jk I jk m a jk jk I b (34) where m, m, and m are the drecton cosnes of the axal, radal and tangental drectons, respectvely, gven by m sn m cos m, m sn m sn m, m cos m (35) Here, s the azmuthal drecton angle, measured from the local radal drecton. The 3rd term on the left of Eq. (34) s dscretzed as follows, for fxed m :

8 82 P.J. Coelho et al. / Combuston and Flame 133 (2003) m I jk m m1/ 2 I jk m1/ 2 m1/ 2 I jk m1/ 2 /w m (36) The drectons m 1/2 defne the edges of angle assocated wth drecton m, and w m s the quadrature weght for that drecton. The geometrcal coeffcents satsfy the followng relaton, drawn on the bass of sotropc radaton m1/ 2 m1/ 2 w m m (37) The coeffcent 1/2 s equal to zero [64]. The coeffcents m1/2 for the other drectons are determned recursvely from Eq. (37). The ntegraton of Eq. (34) over a control volume yelds m m A x I jk, x,out m I jk, x,n m m A r,out I jk,r,out m A r,n I jk,r,n A r,out A r,n m1/ m1/ 2 I 2 m1/ 2 jk,p m1/ 2 I jk,p w m jk I m jk,p a jk jk I b,p V (38) where A are the areas of the cell faces and V s the volume. The ndces n (out) denote a cell face where radaton flows nto (out from) the control volume, and the subscrpts r and x stand for the radal and axal drectons, respectvely. The subscrpt P dentfes the control volume under consderaton. The radaton ntenstes at the cell faces have been calculated accordng to the STEP dscretzaton scheme: m I jk, x,out m m1/ I jk,r,out I 2 m jk,p I jk,p (39) Insertng Eqs. (39) nto Eq. (38), an explct relaton for the radaton ntensty at pont P s obtaned. The soluton algorthm requres the soluton of a set of M (1 m M) dfferental equatons (Eq. (34)) for every j and k gray gas component of H 2 O and CO 2, respectvely. These equatons are decoupled, because there s no scatterng, and may be effcently solved as descrbed n [64,65]. The source term of the transport equaton for enthalpy s gven by N q g k0 N g j0 M 4a jk jki b jk m1 w m I jk m (40) The above equatons show that the mean value of the absorpton coeffcent jk, as well as the mean value of the product a jk jk I b, are requred to account for the TRI. The absorpton coeffcent depends on the molar fractons of H 2 O and CO 2, the weght a jk s a functon of the same molar fractons and temperature, and I b s a functon of the temperature. Furthermore, the molar fractons of H 2 O and CO 2 and the temperature depend on the mxture fracton and scalar dsspaton rate, and n addton the temperature depends on the radatve heat loss fracton. Therefore, the requred mean values were computed by means of ntegraton over the mxture fracton range, takng nto account all the dependences referred above: 1 jk Z, st jk Z, st, X R P Z dz (41) 0 a jk jk I b 0 1 a jk Z, st, X R jk Z, st I b Z, st, X R Z, st, X R 3. Computatonal detals P Z dz (42) The computatonal doman extends from x/d 0 to x/d 150, and from r/d 0tor/d 50, where d s the dameter of the fuel nozzle. A non-unform grd wth grd nodes was used, wth the grd nodes concentrated close to the centrelne, n such a way that 16 grd nodes n the radal drecton are placed nsde the fuel jet, and 22 grd nodes are nsde the plot fuel flame. The radatve transfer calculatons were performed usng the same spatal grd, and the T 7 quadrature [66]. The SLW model requres the defnton of a reference state. To defne ths reference state, only the subdoman where the local mxture fracton exceeds 5% of the stochometrc mxture fracton was consdered. The subdoman where the local mxture fracton s below that value corresponds to ponts whch are suffcently far from the flame regon, so that they should not nfluence a reference state that s representatve of average condtons n the flame regon. It was checked that ths threshold of 5% has no nfluence on the predctons. The boundary condtons at the nlet sectons are prescrbed accordng to the recommendatons reported n [45]. The flamelet profles for the speces concentratons were generated usng a computer code and a detaled chemcal mechansm developed at ITM- RWTH, as descrbed n [52]. A flamelet lbrary was generated for values of the scalar dsspaton rate rangng from a low value close to chemcal equlbrum to a large value close to extncton (0.1 s 1 and 200 s 1, respectvely, for the present flame). The mean values of speces mass fractons are computed

9 P.J. Coelho et al. / Combuston and Flame 133 (2003) a pror, that s, before executng the BIGMIX code, for a dscrete representatve set of values of mxture fracton, mxture fracton varance, and scalar dsspaton rate, and the results are stored n three-dmensonal tables. Smlarly, the mean values of temperature and densty are calculated a pror for the same dscrete set of parameters plus one addtonal parameter, the fracton of radatve heat loss. They are based on the flamelet profles for speces and temperature n adabatc condtons, and n Eq. (15) for non-adabatc condtons. The results are stored n four-dmensonal tables. The mean values gven by Eqs. (41) and (42) are also calculated a pror, and stored n four-dmensonal tables. Durng the flame smulaton, the mean values of these quanttes are obtaned from nterpolaton of the stored data, rather than by explct evaluaton of the ntegrals. The numercal accuracy was checked by comparng the predcted results calculated usng the grd mentoned above wth those obtaned usng a coarser grd wth 80 grd nodes n the axal drecton. It was found that the two sets of results were very close to each other, and therefore may be regarded as grd ndependent. The calculatons were performed usng a workstaton wth an AMD Athlon Duron 750 MHz processor. The radatve transfer calculatons are performed only after a reasonably converged soluton has been acheved, and only every 20 teratons of the CFD code. Each CFD teraton requres 12.5 s of CPU tme. The computatonal requrements for radatve transfer depend on the selected modelng approach. The CPU tme for the optcally thn approxmaton s neglgble, whle the DOM requres about 4.2 s per call to the radaton solver n the case of a gray medum, and 67 s n the case of a non-gray medum dealt wth the SLW model. The rato between these two values s 16, whch corresponds to the number of tmes that the radatve transfer equaton s solved to account for all the gray gas components n the SLW model (N g 3 n Eq. (21)). Ths does not nclude the tme requred to compute the radatve propertes of the medum. The calculaton of these propertes requres less than 2 s for a gray medum f the TRI s neglected or accounted for only va I b, and 27 s f the TRI f fully accounted for. If the SLW model s used, then 21 s and 148 s are needed, dependng on whether the TRI s fully taken nto account or not. No efforts were made to optmze these tmes, for example, t s lkely that a lower order quadrature can be employed n the DOM wthout sgnfcant nfluence on the accuracy. 4. Expermental confguraton The ploted methane/ar jet flame nvestgated n ths study was expermentally studed n [67], the so-called flame D, and the expermental data are avalable n [45]. The ploted burner has a man jet dameter of 7.2 mm and a plot wth an nner dameter of 7.7 mm and an outer dameter of 18.2 mm. The fuel jet mole composton s 25% CH 4 and 75% ar at 294 K. The annular plot burns a mxture of C 2 H 2, H 2, ar, CO, and N 2, wth the same enthalpy and equlbrum composton as methane/ar at 0.77 equvalence rato (Z 0.27), and at 1880 K. The fuel jet Reynolds number s The mean veloctes of the fuel and plot jets are 49.6 and 11.4 m/s, respectvely. In addton, there s a coflow of ar wth a velocty of 0.9 m/s. At these condtons, the flame burns as a dffuson flame, and no evdence of premxed reacton n the fuel-rch methane/ar mxture was found. 5. Results and dscusson The calculatons were carred out for sx dfferent stuatons: 1. Adabatc flame. 2. Optcally thn flame. In ths case the radatve transfer equaton s not solved, and the source term of the enthalpy equaton s calculated from q 4T 4 T 4 (43) where s the absorpton coeffcent and T s the background temperature. The absorpton coeffcent s calculated as recommended n [45],.e., usng the curve fts for the Planck mean absorpton coeffcents of H 2 O, CO 2, CO, and CH 4. The TRI s partally taken nto account by usng T 4 rather than T 4 n the evaluaton of the source term. However, the TRI s not fully accounted for, because the correlaton kt 4 n the emsson term s gnored and approxmated as T 4, and s calculated usng the local mean temperatures and mean speces concentratons. 3. The radatve transfer equaton s solved usng the DOM, and the Planck-mean absorpton coeffcent, hereafter denoted by P, s calculated as n Case 2. The TRI s only accounted for n the evaluaton of I b,.e., T 4 rather than T 4 s used to calculate I b. However, the turbulent fluctuatons are gnored n the calculaton of the mean absorpton coeffcent, and the cor-

10 84 P.J. Coelho et al. / Combuston and Flame 133 (2003) Fg. 1. Predcted and measured axal profles of temperature, mxture fracton, H 2 O and CO 2 mass fractons. relaton between the absorpton coeffcent and the blackbody emssve power s neglected. Ths s the most common approach n flame radaton calculatons whenever the radatve transfer equaton s solved. 4. The radatve transfer equaton s solved usng the DOM, P s calculated as n Case 2, and the TRI s fully accountng for,.e., and I b are calculated from equatons smlar to Eqs. (41) and (42). 5. The radatve transfer equaton s solved usng the DOM, the gas radatve propertes are calculated usng the SLW model, and the TRI s only taken nto account n the evaluaton of I b. 6. The radatve transfer equaton s solved usng the DOM, the gas radatve propertes are calculated usng the SLW model, and the TRI s fully accounted for as descrbed before. The predcted temperature, mxture fracton, H 2 O, and CO 2 mass fracton profles along the centrelne are plotted n Fg. 1 along wth the expermental data. The temperature s accurately predcted up to the measured stochometrc length, L stoch 47d. Upto ths dstance, radaton s of margnal mportance, snce n ths part of the flame the temperature s determned by the rate of turbulent combuston. The maxmum predcted temperature occurs slghtly downstream of the maxmum measured temperature. Further downstream, the temperature s overpredcted, regardless of the radaton model. Ths s consstent wth the overpredcton of mxture fracton and combuston products (H 2 O and CO 2 ) downstream of L stoch. Radaton plays a more mportant role n ths regon, as expected [68]. The radal temperature profles at statons x/d 30, 45, 60, and 75, whch are gven n Fgs. 2 to 5, respectvely, suggest that the spreadng rate of the fuel jet s overestmated, even though the C l constant of the turbulence model has been decreased to mprove ths. In the vcnty of the centrelne the mass fractons of H 2 O and CO 2, plotted n the same fgures, are also overpredcted, except the CO 2 mass fracton at x/d 30 and 45. The overpredcton of temperature, H 2 O and CO 2 mass fractons contrbute to overestmate the radatve heat loss. The varatons n the temperature predctons usng dfferent radaton models are of the same order of magntude as the systematc uncertanty of the temperature measurements (3%, as reported n [67]). Moreover, the dfference between the temperatures calculated wth radaton and those computed for adabatc condtons do not exceed 150 K, no matter

11 P.J. Coelho et al. / Combuston and Flame 133 (2003) Fg. 2. Predcted and measured radal profles of temperature, H 2 O and CO 2 mass fractons at x/d 30. the radaton model employed, and t s somewhat dffcult to dstngush the dfferent curves n Fgs. 1 to 5. Therefore, enlarged vews of the temperature profles are provded n those fgures to hghlght the dfference between the modelng approaches. The temperature profles for Cases 3 and 5 are not shown because they are almost concdent wth the profles for Cases 4 and 6, respectvely. The mxture fracton and major speces mass fracton profles are almost concdent for all cases, and therefore they have only been plotted for Case 6. However, t s worth notng that although the radaton loss from the flame s relatvely small, ts nfluence on the temperature dstrbuton may have an mportant mpact on the NO emsson, as dscussed n [67]. It s clear from the temperature profles that f P Fg. 3. Predcted and measured radal profles of temperature, H 2 O and CO 2 mass fractons at x/d 45.

12 86 P.J. Coelho et al. / Combuston and Flame 133 (2003) Fg. 4. Predcted and measured radal profles of temperature, H 2 O and CO 2 mass fractons at x/d 60. s employed, then the DOM (Cases 3 and 4), whch accounts for both gas emsson and absorpton, and the optcally thn approxmaton (Case 2), whch only accounts for emsson, gve relatvely close results. However, the radatve heat loss for the optcally thn approxmaton s larger, and the temperatures lower, than for the DOM, because absorpton s neglected n the former case. If the medum s modeled as nongray usng the SLW approach (Cases 5 and 6), then the predcted temperatures are lower than those n adabatc condtons (Case 1), but hgher than the temperatures calculated usng the DOM/gray gas assumpton (Cases 3 and 4) or the optcally thn approxmaton (Case 1). Fgs. 1 to 5 show that, for all the radaton models, the mean temperature and the mean mass fractons of H 2 O and CO 2 are overpredcted compared wth the experments over a large part of the flame. Bascally, Fg. 5. Predcted and measured radal profles of temperature, H 2 O and CO 2 mass fractons at x/d 75.

13 P.J. Coelho et al. / Combuston and Flame 133 (2003) Fg. 6. Predcted and measured non-dmensonal radant power along the axal drecton. ths has nothng to do wth the choce of the radaton model, but shows a lmtaton of the other models, namely the turbulence and combuston models. Such dscrepances are common to all the predctons that have been reported for ths flame, as descrbed n the Proceedngs of the recent Workshops on Measurement and Computaton of Turbulent Nonpremxed Flames avalable n [45]. Overall, our predctons are not far from the others. On the other hand, t was found that the radant fracton strongly depends on the radaton model, as revealed n Fg. 6. Fg. 6 shows the measured and predcted nondmensonal radant power, C*, along the axal drecton, as defned n [46,69]: C* 4R2 q R (44) S rad,exp where q R s the radatve heat flux, whch s a functon of the axal poston, and R s the radus of the computatonal doman. C* s non-dmensonalzed by the measured total radant power, S rad,exp. The heat fluxes on the boundary cannot be computed usng the optcally thn approxmaton, and therefore, C* s not gven for Case 2. Fg. 6 shows that f the spectral nature of gaseous radaton s taken nto account usng the SLW model (Cases 5 and 6), the radaton loss s sgnfcantly lower than treatng the medum as gray usng P (Cases 3 and 4), and a very satsfactory agreement wth the expermental data s found. The TRI enhances the radaton loss, but t plays a secondary role n the present flame. However, the good agreement found n Cases 5 and 6 may be fortutous, because the temperature and the H 2 O and CO 2 mass fractons are generally overestmated, and therefore C* should also be overestmated. To further nvestgate ths ssue, we have performed addtonal calculatons of radatve transfer based upon the measured temperature, H 2 O and CO 2 mass fracton profles. These calculatons are decoupled from CFD. Therefore, errors arsng from turbulence and combuston models do not nfluence these radatve calculatons, except n the calculaton of mean values not drectly avalable from the expermental data, such as I b and those n Eqs. (41) and (42). The calculaton of these mean values assumes a clpped Gaussan pdf shape, defned from the measured mean and varance of mxture fracton. The relatonshps between nstantaneous values of temperature/speces and mxture fracton are also taken from expermental data. However, the expermental data s lmted to the centrelne profle up to x/d 80 and to a few radal profles up to x/d 75. Therefore, the expermental data has been nterpolated for x/d 75, and extrapolated further downstream. Ths extrapolaton s subject to uncertantes, whch certanly nfluence the predctons to some extent. The results obtaned are plotted n Fg. 7. It shows that the values of C* are consstently lower than those n Fg. 6, as expected. However, C* s agan strongly overpredcted for Cases 2 and 3, whle t s underpredcted for Cases 5 and 6. The predcted total radaton heat loss for all the dfferent methods s gven n Table 1 along wth the measured value. Ths corresponds to the ntegral of q over the computatonal doman. The rato of ths value to the power released n combuston s the total fracton of radatve loss, R, whch s also gven n Table 1. The optcally thn approach overpredcts the radaton loss by a factor of 2.6 and 1.9, for coupled CFD/radaton and uncoupled radatve calculatons, respectvely,.e., t estmates a radatve heat loss fracton close to that obtaned n the calculatons reported n [46], usng also the optcally thn approxmaton. As mentoned before, n that work radant fractons of 10.5% and 12.5% were computed usng the PDF and the CMC combuston models, respectvely. In the former case the calculatons were performed up to a downstream locaton of x/d 90,

14 88 P.J. Coelho et al. / Combuston and Flame 133 (2003) Fg. 7. Predcted and measured non-dmensonal radant power along the axal drecton. The predcted values are based on temperature and speces concentraton felds nterpolated and extrapolated from the expermental data. and n the second case up to x/d 100. However, the regon downstream of x/d 100 (x/l stoch 2.13) stll contrbutes to the radaton loss, although that contrbuton s relatvely small, as shown n Fgs. 6 and 7. Therefore, the total fracton of radaton loss for the whole flame would exceed 12.5% accordng to those results. The calculatons reported n [47], usng agan the optcally thn approxmaton, predct a value of 2.85% for the radant fracton up to x/d 60. Therefore, they approach the measured value of R 5.1% for the whole flame, because only 50 60% of the total radaton loss occurs up to that axal staton. Nevertheless, they are n contradcton wth our calculatons and wth those descrbed n [46], whch both predct a much hgher radatve loss when the optcally thn approxmaton s employed. More accurate results are expected f radatve transfer s calculated usng the DOM, whch s based on the numercal soluton of the RTE. However, f P s used, then R s stll too large compared wth the expermental data. In fact, Table 1 shows that R 11.4% n Case 3, and R 12.1% n Case 4, for coupled CFD/radatve calculatons. The correspondng values for uncoupled calculatons are lower, but stll too hgh f compared wth the expermental data. The most accurate predctons are obtaned usng the SLW model, whch accounts for the spectral nature of gaseous radaton. In the case of coupled CFD/radatve calculatons, R s slghtly underestmated f the TRI s only accounted for va I b (Case 5), and margnally overpredcted f the TRI s fully taken nto account (Case 6). However, ths good agreement reles upon computed temperature and speces concentraton felds that tend to overestmate the expermental data, as dscussed above. If uncoupled radatve calculatons are carred out, then R s underestmated by about 25% for both Cases 5 and 6. From the results obtaned, t can be concluded that the DOM along wth P yelds only a margnal mprovement of the radatve heat loss over the optcally thn approxmaton. In the former case, and accountng for the TRI only va I b (Case 3), the radatve heat source s gven by q 4I b G (45) Table 1 Predcted and measured radatve heat loss and fracton of radatve heat loss Test Case Coupled CFD/Radaton calculatons Radatve calculatons based on expermental data Radatve heat loss (kw) Fracton of radatve heat loss (%) Radatve heat loss (kw) Fracton of radatve heat loss (%) Expermental

15 P.J. Coelho et al. / Combuston and Flame 133 (2003) Table 2 Predcted total radaton ntensty (W/m 2 st) at the end of the radal lne of sght at x/d 45 Emsson and absorpton Emsson only RADCAL (narrow band model) Analytcal ntegraton of the radatve transfer equaton usng P where G s the ncdent radaton, whle n the second case t s gven by Eq. (43). Hence, emsson s calculated n the same way by the two models, and ths mples that the dfference between them s due to absorpton. A comparson between the results computed for the two terms nto parenthess n Eq. (45) has shown that the frst term s at least one order of magntude larger than the ncdent radaton, except n the coldest regons of the flame. Ths explans why the DOM results usng P (Cases 3 and 4) yeld only a small mprovement of the predcted fracton of radatve heat loss compared wth the optcally thn results. The RADCAL code [70], whch s based on the Goody narrow band model [71], was used n [46] to calculate the spectral radaton ntensty along the radal cross secton at x/d 45. The measured speces and temperature profles were used as nput data. Accordng to these calculatons, whch have been confrmed by us, f only emsson s accounted for, then the total radated power s 39% hgher than that of the emsson-absorpton computaton. The values obtaned are gven n Table 2, along wth those obtaned from the analytcal soluton of the radatve transfer equaton usng P. It can be seen that the emsson-only calculatons performed usng P dffer by less than 10% from the narrow band results. However, the emsson-absorpton results dffer by more than 40%. The gray gas calculatons are margnally affected by absorpton, contrary to the spectral calculatons, and predct a sgnfcantly hgher radaton ntensty. Ths s entrely consstent wth a radatve heat loss sgnfcantly hgher when P s used (Cases 3 and 4) than n the SLW computatons (Cases 5 and 6). The reason for ths behavor les n the error resultant from usng P to calculate absorpton, as dscussed below. Accountng for the TRI only va I b, the exact radatve source term s gven by q 0 4I b 4 4 P I b 0 4 I d d I d d (46) The emsson term s correctly represented usng P, but not the absorpton one [70]. The use of P n the absorpton term nvolves the followng approxmaton 0 4 I d d P0 4 I d d P G (47) However, ths may be a rather crude approxmaton n the present case. In fact, we have computed both terms of ths equaton usng the SLW method, and t was found that the frst term s about one order of magntude larger than the second one over a large part of the flame. Therefore, t can be concluded that radatve calculatons usng P obtaned from RAD- CAL are expected to overestmate the radatve heat loss, n agreement wth what we, as well as other researchers [46], have found. The SLW model provdes more accurate results, but the uncoupled radatve calculatons based upon the expermental data underpredct the fracton of radatve heat loss by about 25%. There are several reasons that may explan ths dscrepancy. Among these are the nterpolaton of expermental data, and specally the extrapolaton downstream of x/d 75, the modelng assumptons nherent to the SLW model, the expermental errors of the spectroscopc data base from whch the absorpton-lne blackbody dstrbuton functon s obtaned, and the role of CO and CH 4, whch has been neglected. Another source of error that may be mportant s the expermental error n temperature. In fact, an error of 3% n the temperature yelds an error of about 12% n I b. Addtonal coupled CFD/radatve calculatons were performed usng T 4 rather than I b T 4 n the calculaton of the emsson term of the radatve transfer equaton,.e., fully gnorng the TRI. The predcted radatve heat loss was kw for the DOM/gray gas calculatons, and kw for the DOM/SLW calculatons. Ths means that the predcted radaton heat loss ncreases by about 30% for the present flame when the TRI s fully smulated, for both gray and non-gray smulatons. The radaton loss from the present flame s relatvely low. Therefore, the ncrease n accuracy resultant from the full smulaton of the TRI may not