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Proceedings of the Combustion Institute 33 (2011) 2539–2546

www.elsevier.com/locate/proci

of the

CombustionInstitute

Effect of soot addition on extinction limitsof luminous laminar counterflow diffusion flames

Praveen Narayanan a, Howard R. Baum b, Arnaud Trouve a,b,⇑

a Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USAb Department of Fire Protection Engineering, University of Maryland, College Park, MD 20742, USA

Available online 16 September 2010

Abstract

The objective of the present study is to use large activation energy asymptotic (AEA) theory to bringbasic information on the extinction limits of non-premixed flames under sooting and radiating conditions.The AEA analysis assumes single-step global combustion chemistry, constant heat capacity and unityLewis numbers; it also includes a two-equation phenomenological model to describe soot formation,growth and oxidation processes, as well as a generalized treatment of thermal radiation that assumes spec-trally-averaged gray-medium properties and applies to flames with an arbitrary optical thickness. The focusof the present study is on the effect of external soot loading on flame extinction, and in particular on theslow-mixing/radiative-extinction limit that is believed to be the dominant mechanism that determines flameextinction in fires. External soot loading simulates non-local effects observed in multi-dimensional sootingflames in which soot mass may be produced at some flame locations and transported to others where it willincrease the flame luminosity and drive combustion conditions towards extinction. The AEA analysisshows that external soot loading results in a significant decrease of the size of the flammable domainand that the minimum value of flame stretch at the radiative-extinction limit is increased by more thanone order of magnitude compared to a non-soot-loaded-flame case. Multi-dimensional sooting flamesare therefore expected to be significantly more susceptible to radiative extinction than the one-dimensionalconfigurations that have been previously studied in microgravity combustion research.� 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Diffusion flame; Extinction; Soot; Radiation; Fire

1. Introduction

Laminar diffusion flames may be extinguishedby a number of different mechanisms. Forinstance, diffusion flames may be extinguished byaerodynamic quenching, a mechanism in which

1540-7489/$ - see front matter � 2010 The Combustion Institdoi:10.1016/j.proci.2010.07.003

⇑ Corresponding author at: Department of Fire Pro-tection Engineering, University of Maryland, CollegePark, MD 20742, USA. Fax: +1 301 405 9383.

E-mail address: atrouve@umd.edu (A. Trouve).

the flame is weakened by fast flow-induced pertur-bations and a critical decrease in the flame resi-dence time. Diffusion flames may also beextinguished by thermal quenching, a mechanismin which the flame is weakened by heat losses(e.g., convective cooling to cold wall surfaces,radiative cooling, or water evaporative coolingin fire suppression applications, etc) or by dilutionquenching, a mechanism in which the flame isweakened due to changes in the fuel or oxidizerstream composition (e.g., air vitiation in under-ventilated fires); in both thermal and dilution

ute. Published by Elsevier Inc. All rights reserved.

2540 P. Narayanan et al. / Proceedings of the Combustion Institute 33 (2011) 2539–2546

quenching, extinction occurs because of a criticalincrease in the flame chemical time. Laminarflame theory suggests that all these different phe-nomena may be explained by a single flame extinc-tion criterion known as a Damkohler numbercriterion [1–3]: the Damkohler number Da isdefined as the ratio of a characteristic fuel–airmixing time divided by a characteristic chemicaltime and extinction is predicted to occur for val-ues of Da that are critically low.

The existence of several flame extinction mech-anisms explains the different extinction limits thatare often observed in non-premixed combustionsystems. For instance, in the classical gaseous-fuellaminar counterflow diffusion flame configuration,the domain of flammability is limited by two fun-damental limits: a fast-mixing limit and a slow-mixing limit [4–10]. In the fast-mixing limit, Dais small because the mixing time is short. Thefast-mixing/aerodynamic-quenching limit is theclassical limit observed in the high Reynolds num-ber turbulent flames that are typical of many com-bustion engine applications. In contrast, in theslow-mixing limit, Da is small because the heatrelease rate is moderate and thermal radiationeffects are significant, the flame temperature is con-sequently low and the chemical time is long. Theslow-mixing/radiative-extinction limit is believedto be the dominant mechanism for extinction inthe low-to-moderate Reynolds number turbulentflames that are typical of fire applications.

Radiative-extinction has been studied in greatdetail over the past two decades in several laminardiffusion flame configurations, including solid fuelstagnation-point flames [11], spherical flamesaround liquid fuel droplets [12], and spherical orplanar gaseous fuel flames [4–10] (see [13] for areview). Note that these previous studies corre-spond to microgravity conditions and to one-dimensional flame configurations; they are alsocharacterized by extremely low values of the fuel–air mixing rates (i.e. low values of flame stretch).In microgravity configurations, radiative extinctionis achieved by gradually decreasing the flamestretch and thereby promoting sluggish combustionconditions that are vulnerable to radiative cooling.The values of flame stretch at the radiative-extinc-tion limit are significantly lower than those foundin earth-gravity laminar flames in which buoy-ancy-driven mixing will maintain a minimum valueof the fuel–air mixing intensity; therefore, the impli-cations of the results obtained in microgravity con-figurations to earth-gravity-flames in general, andto turbulent combustion applications in particularremain entirely open questions.

Note also, that in one-dimensional flame con-figurations, due to the very low flame tempera-tures observed at the radiative-extinction limit(as low as 1100–1300 K), laminar flames becomeblue-colored and soot-free prior to extinctionand that extinction is controlled by radiant

emissions from gaseous species (primarily CO2

and H2O). The implications of these results to fireapplications that emphasize the dominant role ofsoot and luminous radiation remain also unclear.The generic configuration used in the fire scienceliterature to understand flame extinction is a lam-inar co-flow jet diffusion flame configuration atsmoke point conditions [14–16]. In a laminar jetdiffusion flame configuration (and assuming asooty fuel), flame extinction is observed by gradu-ally increasing the fuel flow rate and therebylengthening the flame and promoting formationand growth of soot particles upstream of the flamesurface. The smoke point corresponds to the tran-sition from sooting flame conditions in which sootparticles are completely oxidized in the vicinity ofthe flame surface to smoking flame conditions inwhich a fraction of the soot mass leaks acrossthe reaction zone and is emitted downstream ofthe flame without oxidation. This transition isgenerally interpreted as a radiative-extinctionevent [17] but note that unlike the extinctionresults obtained in microgravity configurations,soot is the dominant factor that controls thesmoke point. One important feature of smokepoint configurations is that the flame is two-dimensional; multi-dimensional effects are likelyto play a major role in the flame dynamics: forinstance, the soot produced at low-elevationhigh-temperature flame locations is transportedinto the higher-elevation lower-temperature flametip region where it contributes to weaken the com-bustion. This non-local soot loading effect is notpresent in classical one-dimensional configura-tions, which suggests that these configurationsare not representative of multi-dimensional (lami-nar or turbulent) sooting flame conditions.

The objective of the present study is to evaluatethe effect of non-local soot loading on the extinc-tion limits of diffusion flames. The configurationcorresponds to steady, one-dimensional, planar,laminar, counterflow, diffusion flames; the fuel isethylene and the oxidizer is air; soot loading issimulated by adding a controlled amount of sootmass to the flow upstream of the flame. We usean extended large activation energy asymptotic(AEA) analysis to calculate the extinction limits;the AEA analysis is extended to include finite ratesoot chemistry and a generalized treatment ofthermal radiation that applies to participatingmedia ranging from optically-thin to optically-thick. The AEA formulation and modelingchoices are described in Section 2; results are thenpresented in Section 3.

2. Problem formulation

The present study is performed using largeactivation energy asymptotic theory. The AEAanalysis uses a classical two-layer decomposition

P. Narayanan et al. / Proceedings of the Combustion Institute 33 (2011) 2539–2546 2541

of the flame structure into an inner reaction zoneand an outer mixing zone. The inner reaction zoneis described using asymptotic expansions in termsof a small parameter related to an inverse Zeldo-vich number; the outer mixing zone is a convec-tion-diffusion chemically-inert zone modified bythermal radiation. The list of modeling simplifica-tions includes single-step global combustionchemistry, constant heat capacity, unity Lewisnumbers, spectrally-averaged gray-medium radia-tion properties, and a two-equation phenomeno-logical model to describe soot formation, growthand oxidation.

2.1. Adiabatic AEA flame structure

We start from a classical asymptotic treatmentapplied to the centerline of a counterflow diffusionflame [18] (see [19] for a general discussion ofAEA). We choose to describe the outer mixingzone in physical space and the inner reaction zonein mixture-fraction space. We use the Howarth-transformed coordinate to indicate spatial loca-tion, n ¼

R x0ðq=q2Þdx, where x is distance along

the flame normal direction and q is mass density,where also x = 0 denotes the stagnation-pointflame location and the subscript 2 indicates condi-tions in the oxidizer stream. The mixture-fractionequation becomes:

�udZdn¼ D2

d2Z

dn2ð1Þ

where �u is the mass-density-weighted flow velocityin the flame normal direction, �u ¼ ðqu=q2Þ, wherealso we assume steady state and use the approxi-mation: q2D ¼ q2

2D2. Mass conservation showsthat the flow field is simply related to the strainrate a (measured on the air side of the flame):�u ¼ �an. The classical solution of Eq. (1) is:

Z ¼ 1

2� 1

2erf

nffiffiffiffiffiffiffiffiffiffiffiffiffi2D2=a

p !

ð2Þ

The flame location is then given by nf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2D2=a

perf�1ð1� 2ZstÞ, where Zst is the stoichi-

omteric value of mixture fraction. And the scalardissipation rate at the flame location isvst ¼ ða=pÞ expð�n2

f =ðD2=aÞÞ.In the case of adiabatic combustion ( i.e.,

without thermal radiation transport), the flametemperature may be estimated from its Burke–Schumann expression, T st ¼ ZstT 1 þ ð1� ZstÞT 2þZstY F ;1ðDH F =cpÞ, where Y F is the fuel mass frac-tion, DH F the heat of combustion (per unit massof fuel), cp the heat capacity (assumed constant),and where the subscript 1 indicates conditions inthe fuel stream. In the following, we usecp ¼ 1006 J=kg-K, and DH F ¼ 32:7 MJ=kg, whichgives an adiabatic flame temperature equal to2368 K.

T st and vst are the two quantities from the solu-tion of the outer mixing zone problem that arerequired in the treatment of the inner reactionzone. We now discuss this treatment and startfrom the classical flamelet equations:

� vst

2

d2T

dZ2¼ DH F

cp

� �_xF

qst

� vst

2

d2Y F

dZ2¼ � _xF

qst

� vst

2

d2Y O2

dZ2¼ �rs

_xF

qst

ð3Þ

where T is the fluid temperature (K), Y k the spe-cies k mass fraction, _xF the fuel mass reaction rateðkg=s=m3Þ, qst the mass density at the flame loca-tion (qst is simply related to T st via the ideal gaslaw) and rs the stoichiometric oxygen-to-fuel massratio. The fuel mass reaction rate follows anexpression proposed in Ref. [20] for ethylene–aircombustion:

_xF ¼A

ð106Þpþq�1�MF

qstY F

MF

� �p qstY O2

MO2

� �q

� expð�T a=T Þ ð4Þ

where A is a model coefficient, A ¼ 2:0 �1012ðmol=m3Þð1�p-qÞs�1, p and q the model fueland oxygen concentration exponents, p ¼ 0:1;q ¼ 1:65;M k the species k molecular weight(kg/mol), and T a a model activation temperature,T a ¼ 15107K. Following standard practice[18,19], inner variables are now expanded in termsof a small parameter e ¼ ð1=ZeÞ ¼ ðT 2

st=T aÞðcp=DH F =Y F ;1Þ, where Ze is a Zeldovich number.We write:

Z ¼ Zst þ ef

T ¼ T st � ehDH F Y F ;1

cp

� �Y F ¼ e½ð1þ aÞfþ h�Y F ;1

Y O2¼ e ða� bÞfþ h½ �ðrsY F ;1Þ

ð5Þ

where a ¼ ðT 1 � T 2Þðcp=DH F Y F ;1Þ, b ¼ Zst=ð1� ZstÞ, and where h is a function of the innervariable f. The system of equations in Eq. (3) re-duces to a single ordinary differential equation:

d2h

df2¼ d½ð1þ aÞfþ h�p½ða� bÞfþ h�q expð�hÞ

ð6Þwhere d is a Damkohler number, d ¼2qðpþq�1Þ

st Y ðpþq�1ÞF ;1

rqs A expð�T a=T stÞ

ð106Þpþq�1M ðp�1ÞF Mq

O2vstZeðpþqþ1Þ . The boundary conditions

associated with Eq. (6) are obtained by matchingthe inner and outer solutions: ðdh=dfÞf!�1 ¼�ð1þ aÞ and ðdh=dfÞf!þ1¼ðb�aÞ. A final change

of variable is introduced: g¼ 12ððdh=dfÞf!þ1

2542 P. Narayanan et al. / Proceedings of the Combustion Institute 33 (2011) 2539–2546

�ðdh=dfÞf!�1Þ�1;c¼ðdh=dfÞf!þ1þðdh=dfÞf!�1ðdh=dfÞf!þ1�ðdh=dfÞf!�1

;~h¼h�cg; and Eq. (6) becomes:

d2~hdg2¼ d�½~hþ g�p½~h� g�q expð�~h� cgÞ ð7Þ

where d� is a modified Damkohler number,

d� ¼ dð4=ð1þ bÞ2Þ. The boundary conditions are:

ðd~h=dgÞg!�1 ¼ �1 and ðd~h=dgÞg!þ1 ¼ þ1. Inthe present study, we solve Eq. (7) using a two-point boundary value problem solver availablein MATLAB.

2.2. AEA flame structure with thermal radiation

Radiative cooling effects are described using atemperature deficit equation that calculates devia-tions from the reference Burke–Schumannsolution.

anddnðDT Þ ¼ �D2

d2

dn2ðDT Þ � _qrad

qcpð8Þ

where DT ¼ ðT adBS � T Þ, with T the radiation-mod-

ified temperature and T adBS the (known mixture-

fraction-based) adiabatic Burke–Schumann tem-perature, and where _qrad is the radiation coolingrate (W=m3 – see below). We solve Eq. (8) usinga Green’s function approach:

DT ðnÞ ¼Z þ1

�1

_qrad

qcpðn�ÞGrðn; n�Þdn� ð9Þ

where

Grðn; n�Þ ¼1�erf n�=

ffiffiffiffiffiffiffiffiffi2D2=ap� �

2D2exp n�2

2D2=a

ffiffiffiffiffiffipD2

a

q1þ erf n=

ffiffiffiffiffiffiffiffiffiffiffiffiffi2D2=a

p ; �1 < f < n�

1þerf n�=ffiffiffiffiffiffiffiffiffi2D2=ap� �

2D2exp n�2

2D2=a

ffiffiffiffiffiffipD2

a

q1� erf n=

ffiffiffiffiffiffiffiffiffiffiffiffiffi2D2=a

p ; n� < f < þ1

8><>: ð10Þ

We now turn to the treatment of the inner reac-tion zone. We assume that the thickness of thereaction zone is much smaller than the thicknessof the layer where radiation is active (i.e., theflame optical depth). Under such conditions, theinner layer problem can be de-coupled fromthe radiation problem and Eqs. (6) and (7) remainunchanged. Note that: the boundary conditions ofEq. (6) are slightly modified by thermal radiation:ðdh=dfÞf!�1 ¼ �ð1þ aÞ þ ðdðDT Þ=dZÞZ¼Zst

andðdh=dfÞf!þ1 ¼ ðb� aÞ þ ðdðDT Þ=dZÞZ¼Zst

; theexpressions for the Damkohler numbers d andd� are unchanged provided that qst and T st areevaluated using T st ¼ ðT ad

BS � DT ÞZ¼Zst.

Next, we consider the model description of theradiation cooling rate _qrad that appears in Eq. (8).We start from the general spectrally-averagedgray-medium expression [21,22]:

_qrad ¼ jð4rT 4 � GÞ ð11Þwhere j is the mean absorption coefficient of themedium ðm�1Þ; r the Stefan–Boltzmann constant,and G the integrated incident radiation ðW=m2Þ.The mean absorption coefficient in Eq. (11) is ex-pressed in terms of contributions due toCO2;H2O and soot:

j ¼ pðxCO2aCO2

þ xH2OaH2OÞ þ Csoot � fvT ð12Þwhere p is pressure (atm), xk the species k molefraction, ak the Planck mean absorption coeffi-cient of species k ðm�1 atm�1Þ; Csoot a model coef-ficient, and fv the soot volume fraction. ak isdescribed using temperature-dependent curve-fitexpressions given in Ref. [23]. The soot volumefraction is related to the soot mass fraction Y soot:fv ¼ ðqY soot=qsootÞ with qsoot the soot mass density,qsoot ¼ 1800 kg m�3 [21,22]; the calculation of Y soot

is discussed in Section 2.3. Csoot is determined bythe value of the soot refractive index [21,22] andbased on recent measurements [24], we use:Csoot ¼ 1817 m�1 K�1. The CO2 and H2O molefractions in Eq. (12) are obtained from theBurke–Schumann solution; temperature in Eqs.(11) and (12) is obtained from the radiation-mod-ified Burke–Schumann solution, T ¼ ðT ad

BS � DT Þ.We now turn to the calculation of the incident

radiation G. Following classical solutions developedfor a one-dimensional non-scattering medium, wecalculate G from the following expression [21,22]:

G¼ 2p Ib;1E2ðsÞþ Ib;2E2ðsR� sÞþZ s

0

Ibðs0ÞE1ðs� s0Þds0�

þZ sR

sIbðs0ÞE1ðs0 � sÞds0

�ð13Þ

where q is the local optical depth, s ¼R x

xLjdx, and

qR the flame optical depth, sR ¼R xR

xLjdx, with

xLðxRÞ the x-location of the left (right) boundaryof the computational domain, where Ib is the localblack body radiation intensity, Ib ¼ ðrT 4=pÞ;Ib;1ðIb;2Þ the value of Ib in the fuel (oxidizer) stream,and E1 and E2 are exponential integrals, E1ðxÞ ¼R1

x ðexpð�uÞ=uÞdu and E2ðxÞ ¼ expð�xÞ � xE1ðxÞ.

2.3. AEA flame structure with soot

Soot formation is described using a phenome-nological modeling strategy previously developedby Moss et al. [25,26] and Lindstedt et al.[27,28]. The strategy consists in solving two trans-port equations for soot mass fraction Y soot andsoot number density nsoot:

Fig. 1. Peak flame temperature versus strain rate a (log-linear plot). Top dashed line: adiabatic flame. Threelower solid lines: sooting and radiating flames withY soot;R ¼ 0 (top), 1% (middle), 5% (bottom). The endpoints of each solid line mark the lower and upper limitsof the flammable domain.

P. Narayanan et al. / Proceedings of the Combustion Institute 33 (2011) 2539–2546 2543

ð�uþ �V tÞdQdnþ d �V t

dnQ ¼ _xQ

qð14Þ

where Q is Y soot or ðnsoot=q=NAÞ, with NA the Avo-gadro number, N A ¼ 6:022� 1026 particles=kmol,and where �V t is the mass-density-weighted ther-mophoretic velocity in the flame normal direction,�V t ¼ �0:54m2 @=@nðlnðT ÞÞ, with m2 the kinematicviscosity in the oxidizer stream. The source termon the RHS of Eq. (14) incorporates semi-empir-ical descriptions of important physical and chem-ical soot processes, e.g. particle inception, surfacegrowth, oxidation, and coagulation [25–28]. Wewrite:

_xY soot ¼ ðq2T 1=2xF Þ � ðcdca exp � T a

T

� �

þ cc exp � T c

T

� �nsootÞ � cO

xO2

T 1=2

� �

� exp � T O

T

� �ðqY sootÞ2=3n1=3

soot

_xðnsoot=q=NAÞ ¼ caðq2T 1=2xF Þ exp � T a

T

� �

� cbT 1=2 nsoot

NA

� �2

ð15Þ

where ca; cb; cc; cd; cq; T a; T c; T q are model coeffi-cients; values for these coefficients are taken fromRef. [25]. The expressions in Eq. (15) are based ona number of simplifying assumptions, for instancethe model ignores the role of soot precursors andassumes a mono-dispersed soot particle size distri-bution. In addition, because it is combined withsingle-step combustion chemistry, the modeladopts ethylene as the controlling species for sootinception and O2 as that for soot oxidation. Thevalues of the fuel and oxygen mole fractions, xF

and xO2, and of the temperature T required inEqs. (14) and (15) are obtained from a recon-structed flame solution that combines the innerand outer solutions discussed in Section 2.2: theinner solution is used for ðZst � 0:01Þ 6Z 6 ðZst þ 0:02Þ; the outer solution is used outsidethat range. Eq. (14) is solved using a second-orderfinite difference method and is coupled to theequations presented in Sections 2.1 and 2.2 viaan iterative algorithm.

Soot mass is added to the flow on the air sideof the flame. While adding soot on the fuel sidemay appear as a better representation of themulti-dimensional sooting flame configurationsthat motivate the present study, it is importantto realize that since the counterflow flame islocated on the air side of the stagnation plane,soot particles added to the fuel will essentially fol-low flow streamlines and will never reach theflame because of the adverse effects of convectionand thermophoresis; thus, to evaluate their impacton flame structure, soot particles must be added atlocations where they can be convected into the

reaction zone. This observation illustrates someof the challenges and limitations found in usingcounterflow flame configurations as a representa-tive model for sooting flames.

3. Results

The flame structure was calculated using theAEA formulation presented in Section 2 and fordifferent strain rate conditions while systemati-cally changing these conditions until the extinc-tion limits were reached (the limits are simplyidentified by the absence of numerical conver-gence). The calculations were also performed fordifferent soot mass loading levels; the soot loadinglevel is characterized by the value of the soot massfraction upstream of the flame, noted Y soot;R; thecorresponding soot number density is not impor-tant and is arbitrarily chosen as a very small num-ber. The selected values for Y soot;R are up to 5%,0 6 Y soot;R 6 0:05; these values are believed to berepresentative of non-local soot loading effects inmulti-dimensional flames: for instance, the caseY soot;R ¼ 0:05 leads to values of the soot mass frac-tion in the flame region that are comparable tovalues previously observed in direct numericalsimulations of turbulent flames [10].

Figure 1 presents the variations of peak flametemperature with strain rate a for three differentsoot loading levels. Similar variations for the cor-responding adiabatic flame case are also includedfor comparison. The figure shows that the flam-mable domain of laminar counterflow diffusionflames is limited by upper and lower limits at largeand low values of a. As discussed in Section 1, theupper limit corresponds to the classical flame

Fig. 3. Stoichiometric value of the soot volume fractionfv;st versus strain rate a (log-log plot). Three solid lines:sooting and radiating flames with Y soot;R ¼ 0 (bottom),1% (middle), 5% (top).

2544 P. Narayanan et al. / Proceedings of the Combustion Institute 33 (2011) 2539–2546

response to increasing mixing rates, i.e. to anintensification of combustion at moderate-to-highvalues of a (or equivalently vstÞ, followed by aero-dynamic quenching once a P aULðvst P vUL

st Þ. Thisupper limit is the only extinction limit observedunder adiabatic combustion conditions. The lowerlimit corresponds to the flame response todecreasing mixing rates, i.e. to a progressive weak-ening of combustion at moderate-to-low values ofa, followed by radiative-extinction oncea 6 aLLðvst 6 vLL

st Þ. In the absence of soot loadingðY soot;R ¼ 0Þ, we find: aLL � 0:8 s�1ðvLL

st �0:025 s�1Þ and aUL � 1955 s�1ðvUL

st � 61 s�1Þ. Inthe case with Y soot;R ¼ 0:05, we find:aLL � 13 s�1ðvLL

st � 0:4 s�1Þ and aUL � 1360 s�1

ðvULst � 42 s�1Þ. Thus, we find that soot mass load-

ing results in a significant decrease of the size ofthe flammable domain, and that for the range ofconditions considered in the present study, themaximum value of flame stretch at the upperextinction limit may be decreased by as much as30% while the minimum value of flame stretch atthe lower limit may be increased by more thanan order of magnitude.

These modifications of the flammable domainin the presence of soot loading are further illus-trated in Fig. 2. Figure 2 presents the variationsof the Damkohler number d� with a and showsthat consistent with classical laminar flame the-ory, flame extinction occurs when d� takes criti-cally low values. An important result in Fig. 2 isthat the critical values of d� at the lower limitare approximately equal to those at the upperlimit: in the case with Y soot;R ¼ 0, d�LL � 1:1 andd�UL � 1:2, whereas in the case withY soot;R ¼ 0:05; d�LL � 0:9 and d�UL � 1:2, withd�LLðd�ULÞ the value of d� at the lower (upper)

Fig. 2. Damkohler number d� versus strain rate a (log-log plot). Top dashed line: adiabatic flame. Three lowersolid lines: sooting and radiating flames with Y soot;R ¼ 0(top), 1% (middle), 5% (bottom). The critical values of d�

at the extinction limits are close to 1.

extinction limit. This result lends support to theunifying concept of a flame Damkohler numberas the basis to predict all flame extinction limits:an approximate but reasonably accurate expres-sion of the extinction criterion is d� 6 1.

Figure 3 presents the variations of soot volumefraction fv;st with strain rate a; fv;st is the soot vol-ume fraction measured at the stoichiometric loca-tion where Z ¼ Zst. In the absence of soot loading,significant amounts of soot may be produced onthe fuel side of the flame but this soot mass doesnot accumulate in the high-temperature regionof the flame because of adverse effects of both oxi-dation chemistry and convective and thermopho-retic transport: fv;st remains below 0.01 ppm. Incontrast, in the presence of soot loading, thereare significant amounts of soot mass present inthe high-temperature region of the flame: fv;st ison the order of 1 ppm in the case withY soot;R ¼ 0:01, and on the order of 5 ppm in thecase with Y soot;R ¼ 0:05. This high-temperaturesoot is responsible for the increase in the flameluminosity and for the associated changes in theflame dynamics described in Figs. 1 and 2.

Figs. 4–6 present two representative flamestructures, as obtained for a ¼ 20 s�1, with andwithout external soot loading. Under these lowstrain rate conditions, the effects of radiation cool-ing are pronounced and are responsible for a dra-matic decrease in peak flame temperature equal toapproximately 330 K when Y soot;R ¼ 0 and 500 Kwhen Y soot;R ¼ 0:02 (Fig. 4).

Figure 5 presents the corresponding spatialvariations of soot volume fraction across theflame. In the case with Y soot;R ¼ 0:02, the sootmass is released upstream of the flame atx � 17mm. As they approach the flame zone(i.e., moving from right to left in Fig. 5), a

Fig. 4. Temperature versus normal distance to theflame, a ¼ 20 s�1. Top line: flame with Y soot;R ¼ 0;bottom line: soot-loaded-flame with Y soot;R ¼ 2%.x 6 �5mmðx P 12mmÞ corresponds to the fuel (air)side of the flame.

Fig. 5. Soot volume fraction versus normal distance tothe flame, a ¼ 20 s�1. Bottom line: flame with Y soot;R ¼ 0;top line: soot-loaded flame with Y soot;R ¼ 2%. Sootaddition occurs at x � 17 mm.

Fig. 6. Mean radiation absorption coefficient versusnormal distance to the flame, a ¼ 20 s�1. Bottom lineswith square symbols: flame with Y soot;R ¼ 0; top lineswithout symbol: soot-loaded flame with Y soot;R ¼ 2%.For each flame case, the plot shows the total absorptioncoefficient j (upper solid curve) and its soot contribu-tion, Csoot � fvT (lower dashed curve); the differencebetween the two curves is the contribution of CO2 andH2O (see Eq. (12)).

P. Narayanan et al. / Proceedings of the Combustion Institute 33 (2011) 2539–2546 2545

significant fraction of the soot particles is firstconsumed by oxidation (a consequence of addingsoot on the air side of the flame); the rest is trans-ported across the reaction zone where asmentioned above, the particles contribute toincreasing the flame luminosity and to weakeningthe flame.

Figure 6 presents the corresponding spatialvariations of the mean absorption coefficient j.In the absence of soot loading, the flame opticaldepth sR takes low values, sR � 0:035, and theflame remains in the optically-thin regime, i.e., aradiation regime dominated by emission; in con-trast, in the case Y soot;R ¼ 0:02; sR � 0:12 and theflame belongs to a mixed radiation regime in

which absorption becomes important. Also, whilein the case with Y soot;R ¼ 0, the contributions ofsoot and gas radiation to the mean absorptioncoefficient j have comparable weights, in the casewith Y soot;R ¼ 0:02, the soot contribution is clearlydominant.

4. Conclusion

The effect of external soot loading on theextinction limits of laminar counterflow ethyl-ene–air diffusion flames is studied using largeactivation energy asymptotic theory. The AEAanalysis is extended to include a phenomenologi-cal soot model that accounts for particles incep-tion, growth and oxidation, and a generalizedtreatment of thermal radiation that accounts forboth emission and absorption phenomena andapplies to participating media ranging from opti-cally-thin to optically-thick. Soot loading is simu-lated by adding a controlled amount of soot massto the flow upstream of the flame.

Note that in the present study, soot is added onthe air side of the flame. In a counterflow flameconfiguration in which soot is added to the fuelside, convection and thermophoresis will preventsoot particles from being transported into theflame. This observation illustrates some of thechallenges and limitations found in using counter-flow flame configurations as a representativemodel for multi-dimensional sooting flames.

The AEA analysis shows that soot loadingresults in a significant decrease of the size of the

2546 P. Narayanan et al. / Proceedings of the Combustion Institute 33 (2011) 2539–2546

flammable domain: the minimum value of flamestretch at the radiative-extinction limit is increasedby more than an order of magnitude. These resultssupport the idea that in multi-dimensional sootingflames, soot first produced at locations character-ized by fast fuel–air mixing and vigorous combus-tion conditions and then transported intolocations characterized by slow-mixing and slug-gish combustion conditions will play a dominantrole by increasing the local flame luminosity anddriving the flame towards radiative extinction.This non-local multi-dimensional effect is believedto be a dominant mechanism to explain flameextinction in fires; this effect is not present in clas-sical (i.e., non-soot-loaded) one-dimensionalflame configurations, which suggests that theseconfigurations are not representative of multi-dimensional sooting flame conditions.

The present AEA results indicate that in multi-dimensional laminar or turbulent sooting flames,stretched flame elements with values of the strainrate above approximately 13 s�1 (or stoichiome-tric values of the scalar dissipation rate above0.4 s�1) will be susceptible to radiative extinction.The AEA results also support the concept of asingle critical value of the Damkohler number topredict all flame extinction limits. Future workwill consider an application of the present AEAanalysis to the construction of flammability mapsrelevant to fire problems.

Acknowledgments

This research is supported by the US NationalScience Foundation, Division of Chemical, Bioen-gineering, Environmental, and Transport Systems(Grant No. CTS-0553508).

References

[1] A. Linan, Acta Astronaut. 1 (1974) 1007–1039.[2] F.A. Williams, Combustion Theory, 2nd ed., The

Benjamin/Cummings Publishing Company, 1985.[3] N. Peters, Turbulent Combustion, Cambridge Uni-

versity Press, 2000.

[4] T. Daguse, T. Croonenbroek, J.C. Rolon, N.Darabiha, A. Soufiani, Combust. Flame 106 (1996)271–287.

[5] S.H. Chan, J.Q. Yin, B.J. Shi, Combust. Flame 112(1998) 445–456.

[6] K. Maruta, M. Yoshida, H. Guo, Y. Ju, T. Niioka,Combust. Flame 112 (1998) 181–187.

[7] X.S. Bai, L. Fuchs, F. Mauss, Combust. Flame 120(2000) 285–300.

[8] F. Liu, G.J. Smallwood, O.L. Gulder, Combust.Flame 121 (2000) 275–287.

[9] H.Y. Wang, W.H. Chen, C.K. Law, Combust.Flame 148 (2007) 100–116.

[10] P. Narayanan, A. Trouve, Proc. Combust. Inst. 32(2009) 1481–1489.

[11] J.L. Rhatigan, H. Bedir, J.S. T’ien, Combust. Flame112 (1998) 231–241.

[12] B.H. Chao, C.K. Law, J.S. T’ien, Proc. Combust.Inst. 23 (1990) 523–531.

[13] J.S. T’ien, H. Bedir, Proc. First Asia–Pacific Conf.Combust., Osaka, Japan, 1997.

[14] G.H. Markstein, Proc. Combust. Inst. 20 (1984)1055–1061.

[15] G.H. Markstein, J. de Ris, Proc. Combust. Inst. 20(1984) 1637–1646.

[16] L. Orloff, J. de Ris, M.A. Delichatsios, Combust.Sci. and Tech. 84 (1992) 177–186.

[17] P. Joulain, Proc. Combust. Inst. 27 (1998) 2691–2706.[18] G.F. Carrier, F.E. Fendell, F.E. Marble, SIAM J.

Appl. Math. 28 (1975) 463–500.[19] C.K. Law, Combustion Physics, Cambridge Univer-

sity Press, 2006.[20] C.K. Westbrook, F.L. Dryer, Combust. Sci. Tech-

nol. 27 (1981) 31–43.[21] R. Siegel, J.R. Howell, Thermal Radiation Heat

Transfer, 4th ed., Taylor and Francis, 2001.[22] M.F. Modest, Radiative Heat Transfer, 2nd ed.,

Academic Press, 2003.[23] International Workshop on Measurement and

Computation of Turbulent Nonpremixed Flames,Sandia National Laboratories, http://public.ca.san-dia.gov/TNF/radiation.html.

[24] T.C. Williams, C.R. Shaddix, K.A. Jensen, J.M.Suo-Anttila, Intl. J. Heat Mass Transfer 50 (2007)1616–1630.

[25] J.B. Moss, C.D. Stewart, K.J. Young, Combust.Flame 101 (1995) 491–500.

[26] S.J. Brookes, J.B. Moss, Combust. Flame 116 (1999)486–503.

[27] K.M. Leung, R.P. Lindstedt, W.P. Jones, Combust.Flame 87 (1991) 289–305.

[28] M. Fairweather, W.P. Jones, H.S. Ledin, R.P.Lindstedt, Proc. Combust. Inst. 24 (1992) 1067–1074.