determination of laminar flame speeds using stagnation and spherically expanding flames molecular...

Combustion and Flame xxx (2014) xxx–xxx

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Combustion and Flame

journal homepage: www.elsevier .com/locate /combustflame

Determination of laminar flame speeds using stagnation and sphericallyexpanding flames: Molecular transport and radiation effects

http://dx.doi.org/10.1016/j.combustflame.2014.03.0090010-2180/� 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

⇑ Corresponding author. Fax: +1 213 740 8071.E-mail address: [email protected] (F.N. Egolfopoulos).

Please cite this article in press as: J. Jayachandran et al., Combust. Flame (2014), http://dx.doi.org/10.1016/j.combustflame.2014.03.009

Jagannath Jayachandran, Runhua Zhao, Fokion N. Egolfopoulos ⇑Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 January 2014Received in revised form 28 February 2014Accepted 14 March 2014Available online xxxx

Keywords:Flame propagationLaminar flame speedExperimental uncertaintyFlame kinetics

The uncertainties associated with the extraction of laminar flame speeds through extrapolations fromdirectly measured experimental data were assessed using one-dimensional direct numerical simulationswith focus on the effects of molecular transport and thermal radiation loss. The simulations were carriedout for counterflow and spherically expanding flames given that both configurations are used extensivelyfor the determination of laminar flame speeds. The spherically expanding flames were modeled by per-forming high fidelity time integration of the mass, species, and energy conservation equations. The sim-ulation results were treated as ‘‘data’’ for stretch rate ranges that are encountered in experiments andwere used to perform extrapolations using formulas that have been derived based on asymptotic analy-ses. The extrapolation results were compared then against the known answers of the direct numericalsimulations. The fuel diffusivity was varied in order to evaluate the flame response to stretch and toaddress reactant differential diffusion effects that cannot be captured based on Lewis number consider-ations. It was found that for large molecular weight hydrocarbons at fuel-rich conditions, the flamebehavior is controlled by differential diffusion and that the extrapolation formulas can result in notableerrors. Analysis of the computed flame structures revealed that differential diffusion modifies the fluxesof fuel and oxygen inside the flame and thus affect the reactivity as stretch increases. Radiation loss wasfound to affect notably the extracted laminar flame speed from spherically expanding flame experimentsespecially for slower flames, in agreement with recent similar studies. The effect of radiation could beeliminated however, by determining the displacement speed relative to the unburned gas. This can beachieved in experiments using high-speed particle image velocimetry to determine the flow velocity fieldwithin the few milliseconds duration of the experiment. In general, extrapolations were found to be unre-liable under certain conditions, and it is proposed that the raw experimental data in either flame config-urations are compared against results of direct numerical simulations in order to avoid potentialfalsifications of rate constants upon validation.

� 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

The laminar flame speed, Sou, defined as the propagation speed of

a steady, laminar, one-dimensional, planar, adiabatic flame is afundamental property of any combustible mixture and it is a mea-sure of the mixture’s reactivity, diffusivity, and exothermicity. Theaccurate knowledge of So

u is essential towards validating kineticmodels (e.g., [1]) and constraining uncertainties of rate constants[2]. Furthermore, So

u along with the Markstein length, L, whichcharacterizes the response of laminar flame propagation to stretch,are inputs in turbulent flame models under conditions that theflamelet concept is applicable [3–5].

Measurement of Sou began as early as in the 1920s when Stevens

[6,7] studied flame propagation at constant pressure by trackingspherically expanding flames, SEF, in a soap bubble filled with aflammable mixture. Since then, significant progress has been madeboth in the experimental and numerical determination of So

u.However, notable scatter by as much as 25 cm/s was persistentin published So

u’s of methane flames [8] until the 1980s when theeffect of flame stretch [9] on flame propagation was accountedfor and subtracted from the measurements reducing thus theexperimental uncertainty notably [10–13]. Despite this progress,due to the relatively low sensitivity of So

u to chemical kinetics[14], there is need for experimental data with even lower uncer-tainty compared to what is reported currently so that they canbe used for kinetic model validation.

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Fig. 2. (a) Variation of Le with carbon number for n-alkane/air mixtures at p = 1 atmand Tu = 298 K, for / = 0.7 (dashed line) and / = 1.4 (solid line). (b) Variation of theratio of oxygen diffusivity to fuel diffusivity with carbon number for / = 1.4 n-alkane/air mixtures at p = 1 atm and Tu = 298 K.

2 J. Jayachandran et al. / Combustion and Flame xxx (2014) xxx–xxx

Among the various methods for measuring Sou, the counterflow

flame, CFF, and the SEF configurations are well established andwidely used, as they are considered to result in reliable data.Despite the fact that considerable effort has been devoted tounderstanding the intricacies and physics behind each approach,significant discrepancies persist in reported data, even when usingthe same method. Figure 1 depicts the relative deviation of exper-imental So

u with a normalized equivalence ratio U � //(1 + /) [15],where / is the equivalence ratio, of n-heptane/air mixturesreported in different studies from the data of Ji et al. [16] thatare used as the reference value. One can observe the increasing dis-crepancy between data obtained using the SEF [17] and CFF [18,19]configurations for off-stoichiometric / > 1 mixtures; corrections ofthe data reported in Refs. [18,19] to account for the differentunburned mixture temperatures, Tu, were made using the recom-mendation of Wu et al. [20]. It is evident that the disparity betweenthe So

u data sets increases with / and this trend persists for flamesof several high molecular weight, MW, fuels [16]. For / > 1 hydro-carbon/air mixtures, air is abundant on both mass and molar basiscompared to the fuel. Thus, the thermal diffusivity of the mixture isnearly that of nitrogen, and hence a Lewis number, Le, calculatedbased on oxygen, being the deficient reactant for / > 1 mixtures,will be close to unity as shown in Fig. 2a regardless of the fuelMW. Yet, a high sensitivity of L to / has been reported, for example,for rich n-butane/air mixtures [21].

These inconsistencies point to possible uncertainties in theexperimental determination of So

u, and could be associated withthe reactant flow rates, i.e. /, diagnostic equipment, the flow veloc-ity measuring approach, data analysis, and finally data interpreta-tion. In order to tackle uncertainties associated with eachexperimental approach, detailed understanding of the physics con-trolling the flame behavior and response to fluid mechanics andloss mechanisms is required.

At pressures less than 10 atm, Sou can be measured using the CFF

approach in which steady, laminar, and planar flames (e.g., [5,13])are established. Under such conditions, the only parameter thatcan be varied for a given set of thermodynamic conditions is theflame stretch, and this effect can be characterized readily usingavailable quasi-one dimensional codes (e.g., [22]).

Law and co-workers introduced the CFF approach to determineSo

u [5,13,23]. The method involves the determination of the axialvelocity profile along the system centerline and subsequently theidentification of two distinct observables. A reference flame speed,Su,ref, which is the minimum velocity just upstream of the flame,and a characteristic stretch, K, which is the maximum absolute

Fig. 1. Deviation of experimental Sou’s of n-heptane/air mixtures at p = 1 atm from

that of Ji et al. [16] (Tu = 353 K) represented by the solid line. Data represented bysymbols include: ( ) Kelley et al. [17] (Tu = 353 K), ( ) Smallbone et al. [18](Tu = 350 K) and ( ) Kumar et al. [19] (Tu = 360 K).

Please cite this article in press as: J. Jayachandran et al., Combust. Flame (201

value of the axial velocity gradient in the hydrodynamic zone.Thus, by varying Su,ref with K in the experiments, it was proposed[5,13,23] that So

u can be determined by performing a linear extrap-olation of the experimental data to zero stretch given that as K ? 0,Su,ref should degenerate to So

u. This approach was used in severalstudies (e.g., [24–26]) for H2 and C1–C2 hydrocarbon flames.

Subsequently, Tien and Matalon [27] demonstrated throughasymptotic analysis that the Su,ref vs. K response is non-linear asK ? 0, and that linear extrapolation of Su,ref to K = 0 results in theover-estimation of So

u; it should be noted that Su,ref is not thestretched flame speed, Su, as it is affected by thermal dilatationand flow divergence effect [13,27,28]. Tien and Matalon [27] pro-duced also a non-linear expression describing the variation Su,ref

with K, which subsequently was expressed by Davis and Law[29] in a more compact way as:

Su;ref ¼ Souf1� ðl� 1ÞKaþ Ka ln½ðr� 1Þ=Ka�g: ð1Þ

In Eq. (1), l is the Markstein number, Ka � aK=ðSouÞ

2 the Karlovitznumber, a the thermal diffusivity of the mixture, and r � (qu/qb)with qu and qb being the densities of the unburned and burnedstates at equilibrium respectively.

Chao et al. [30] used asymptotic analysis to show that the errorintroduced by linear extrapolations can be reduced for small Kaand large burner separation distance relative to the flame thick-ness. Vagelopoulos et al. [31] further showed computationallyand experimentally that in order for the linear extrapolation tobe accurate, Ka must be of the order of 0.1 for CH4/air, C3H8/air,and lean H2/air flames.

Recently, Egolfopoulos and co-workers [16,32,33] introduced acomputationally assisted approach in quantifying the non-linearvariation of Su,ref with K. Specifically, direct numerical simulations(DNS) of the experiments are carried out with detailed descriptionof molecular transport and chemical kinetics to avoid simplifyingassumptions used in asymptotic analysis. Thus, the variation ofSu,ref with K is computed and can be used to perform the non-linearextrapolations of the experimental data; indeed the DNS approachreproduces the non-linear behavior of Su,ref with K as predicted byTien and Matalon [27]. Given that the computed Su,ref vs. K curvemay lie over or below the data due to transport and kinetic modeluncertainties, it was shown that as long as the discrepanciesbetween data and predictions are not large, say within 30–40%,the shape of the Su,ref vs. K curve is minimally affected and couldbe translated to best fit the data and derive So

u at K = 0. This wasconfirmed through DNS in which the rates of main H + O2 ?OH + O branching or CO + OH ? CO2 + H oxidation reactions aswell as the diffusion coefficients of the reactants were modifiedintentionally by as much as 30–40%. It was shown that even undersuch notable but not excessive modifications of the overall reaction

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J. Jayachandran et al. / Combustion and Flame xxx (2014) xxx–xxx 3

rate, the shape of the computed Su,ref vs. K curves are nearly indis-tinguishable [32].

Ji et al. [16] showed that for the same sets of experimental dataof C5–C12 n-alkanes, linear extrapolation yield higher So

u’s for fuelrich mixtures, as compared to nonlinear extrapolation using thecomputationally assisted approach. Considering also the resultsshown in Fig. 1, it is reasonable to assume that the discrepanciesbetween reported So

u’s for / > 1 mixtures could be attributed par-tially to the extrapolations.

The SEF approach has been used extensively for measuring Sou

due to the wide pressure range of applicability, the relative sim-plicity of diagnostics, and the well defined stretch rate that makesthe determination of the burned gas Markstein length, Lb, straight-forward (e.g., [34–40]). The commonly used method involvestracking using Schlieren or shadowgraph the flame radius, Rf, ofthe expanding flame as a function of time, t, [36] during the initialphase of propagation during which the pressure rise is negligible.The flame speed with respect to the burned gas is defined as Sb -� dRf / dt, based on the assumption that the burned gas is station-ary, and the flame stretch is defined as K � (2/Rf)(dRf/dt) (e.g., [17]).During one experiment, the variation of Sb with K is monitored andthe through extrapolation to K = 0 the stretch free So

b value is deter-mined. Subsequently, So

u is determined through the density correc-tion as So

u � Sobðqb=quÞ.

The majority of the Sou’s reported in the literature (e.g.,

[36,37,39,40]) were extracted using linear extrapolations to deter-mine So

b [36]. Kelley and Law [21] identified that for mixtures withLe – 1.0 and Ka relevant to experiments, Sb varies non-linearlywith K and proposed a quasi-steady non-linear extrapolation equa-tion, which was derived originally by Ronney and Sivashinksy [41]for flames of mixtures sufficiently far from stoichiometry, constanttransport properties, and one-step reaction:

Sb=Sob

� �2 ln Sb=Sob

� �2 ¼ �2LbK=Sob ð2Þ

Subsequently, Kelley et al. [42] relaxed the quasi-steadyassumption and proposed an improved non-linear extrapolationformula; a review of all extrapolation methodologies can be foundin Refs. [42,43].

Uncertainties associated with prolonged effects of ignition dur-ing the initial stages of flame propagation [38,44] and fluiddynamic effects induced by using a cylindrical rather than a spher-ical chamber [45] have been studied and accounted for.

Mclean et al. [46] carried out a numerical study and determinedthat the radiation losses in SEF’s can cause a systematic underesti-mation of So

u. It has been established also from previous studiesthat radiation affects flame propagation notably only for mixturesat near limit conditions [47]. Chen [48] performed DNS of near-limit CH4/air mixtures to investigate the effects of radiation lossand reabsorption. It was shown that radiation heat loss from theburned gas could cause an inward flow and that accounting forreabsorption moderates the overall heat loss from the burnedgas. Santner et al. [49] showed that So

b of slow flames could beaffected significantly by the burned gas cooling. In the same studya correction methodology to obtain accurate So

b values was pro-posed based on an analytical model for fluid flow coupled withan optically thin limit (OTL) model for radiation. However, this cor-rection method has to be used with caution, especially for high-pressure flames for which reabsorption can be important (e.g.,[50,51]).

Lecordier and coworkers [52,53] performed for the first timedirect measurements of the flow velocities in SEF experiments byseeding the flow with silicon oil droplets and using kHz-level par-ticle image velocimetry (PIV). Thus, the displacement velocity withrespect to the unburned gas, Un, was determined as Un � Ug � Sb

[34], where Ug is the maximum velocity upstream of the flame.


In these experiments Sb and K are determined readily once thetemporal variation of Rf is known; Rf is defined as the location atwhich the droplets have been vaporized completely. So

u is thenobtained by extrapolating Un to K = 0. This approach providesdirect measurements and does not require the use of the densitycorrection to obtain So

u, which introduces the questionable assump-tion of equilibrium in the burned gas that in reality is affected byradiation and its density varies both in time and space. Renouand coworkers (e.g., [54]) extended this technique to flames ofliquid fuels and demonstrated that discrepancies exist betweenSo

u values measured using PIV and those determined using theSchlieren or shadowgraph approach.

Based on the aforementioned considerations, the main goal ofthis investigation was to perform DNS of CFF and SEF configura-tions and assess uncertainties in determining So

u under realisticexperimental conditions. The emphasis was on transport and radi-ation effects. While Le effects have been studied extensively in pastpertinent studies, this is not the case for reactant differential diffu-sion that can be rather important for large MW fuels as evidentfrom Fig. 2b in which the ratio of oxygen to fuel diffusivities isshown to increase with the fuel carbon number for / = 1.4 n-alkane/air mixtures. The effect of radiation was assessed also forSEF’s in the context of the various approaches available for derivingthe raw experimental data.

2. Numerical approach

2.1. Freely propagating and counterflow flames

In order to assess the validity of current practices in determin-ing So

u using the CFF and SEF approaches, DNS of both types ofexperiments were performed using a variety of codes and detaileddescription of chemical kinetics and molecular transport. The DNSresults were treated as ‘‘data’’ for the range of K’s that are typicallyused in both types of experiments, and subsequently Eqs. (1) and(2) were used to perform extrapolations. The advantage of thisapproach is that both the response of flame propagation to K fromhigh to near-zero values and So

u are known so that the merits andshortcomings of Eqs. (1) and (2) can be assessed.

Furthermore, the DNS approach allows for the rigorous assess-ment of reactant differential diffusion effects. A parametric studywas performed on the effect of the fuel diffusivity on the responseof CH4/air flames to K given the relatively small size of the kineticmodel, essential for unsteady DNS of SEF’s, and the fact that diffu-sivities of CH4 and O2 do not differ substantially. The variation ofthe CH4 diffusivity was implemented through modification of itsLennard-Jones (L-J) parameters. The unperturbed case is referredto as OD (original diffusivity). ID (increased diffusivity) and DD(decreased diffusivity) refer to the cases in which the L-J parame-ters of CH4 were replaced with those of H2 and n-C12H26 respec-tively. This approach ensures that the chemistry is consistent inall computations, and also circumvents the complexities associatedwith fuel cracking which high MW fuels are susceptible to. The val-ues of Le and ratio of fuel to oxygen diffusivities in the mixture areshown in Table 1 for / = 0.7, 1.0 and 1.4. DNS were performed alsofor steady n-C12H26/air CFF’s in order to verify the results obtainedfrom CH4/air flames.

Sou’s and variation of Su,ref with K for CFF’s were computed

respectively, using the PREMIX code [55,56] and an opposed-jetflow code [57] that was originally developed by Kee and co-work-ers [22]. Both codes were integrated with the CHEMKIN [58] andthe Sandia transport [59,60] subroutine libraries. The H and H2 dif-fusion coefficients of several key pairs are based on the recentlyupdated set [61]. Both codes have been modified to account forthermal radiation (OTL) of CH4, CO, CO2, and H2O [57,62].



Table 1Lewis number, Le and ratio of fuel to O2 diffusivities for the mixtures used in thepresent study.

/ Le Dfuel DO2

�Original L-J Parameters (OD)0.7 1.0 1.141.0 N/A 1.161.4 1.1 1.17

n-C12H26 L-J Parameters (DD)0.7 2.3 0.451.0 N/A 0.481.4 1.0 0.51

H2 L-J Parameters (ID)0.7 0.7 1.671.0 N/A 1.671.4 1.2 1.68


Sou’s and the variation of Su,ref with K in CFF’s were computed

using the USC-Mech II [63] and JetSurF 1.0 [64] kinetic modelsfor CH4/air and n-C12H26/air flames respectively. The CFF simula-tions were performed for the twin flame configuration and for alarge burner separation distance (10 cm) to avoid conductive heatloss to the burner at very low K’s.

2.2. Spherically expanding flames

In order to perform one-dimensional DNS of SEF’s, a transientone-dimensional reacting flow code (TORC) was developed usingthe PREMIX code [55,56] as the framework. The conservation equa-tions for mass, species, and energy are solved numerically as afunction of time in spherical coordinates [38].

The method of lines approach was adopted to solve the stiffnon-linear system of equations, which involves the replacementof spatial derivatives with discrete difference approximations, rely-ing on an ordinary differential equation (ODE) solver to performthe time integration. This simplifies the method of solution asthe ODE solver takes the burden of time step selection to maintainstability and local error control of the evolving solution. Finite dif-ference approximations were used for spatial discretization. A firstorder windward scheme was used for the convective terms and asecond order central difference scheme for the diffusion terms,similar to the PREMIX code [55,56]. The set of discretized equa-tions form a differential algebraic equation (DAE) system of index1 [65], with velocity being the only algebraic variable among theprimitive variables that are temperature, T, velocity, u, and speciesmass fractions, Y. Time integration of this DAE system was per-formed using the DASPK [66–69] solver, which implements a back-ward-difference formula with adaptive time step and ordercontrol.

An adaptive grid methodology was developed as computationalefficiency is severely compromised if a static mesh is used for amoving flame problem. The spatial domain was divided into fiveregions of different grid point densities (L1–L5), each having uni-form grid spacing. The algorithm ensures that the flame is alwayslocated within the region of highest mesh point density (L1) andthat the other regions are distributed around the L1 such thatthe furthest region (L5) will have the least mesh density. Gridrestructuring was performed every time the flame separation fromthe L1 boundary was within a user defined tolerance. In order toovercome the computational overhead of restarting DASPK everytime a re-gridding process was completed, a ‘‘flying/warm restart’’was facilitated by interpolating the solutions at previous time stepson to the new grid [70–72]. Thus, the solver can continue integrat-ing using the higher order multistep method and/or using a largertime step. The restructuring of the mesh was done in such a way


that the grid points located in the flame remain intact to avoidinterpolation errors. In order to ensure zero gradients upstreamof the flame, R2u where R is the spatial coordinate was chosen asthe dependent variable instead of u. This facilitates further reduc-tion of interpolation errors. A monotone cubic Hermite interpola-tion [73] technique as suggested by Hyman et al. [72] was foundto work the best.

DASPK requires the initial condition to satisfy the governingDAE system of equations [69]. To obtain the initial condition, thetransient terms were discretized using the backward Euler methodand the system of equations were integrated in time over a coupleof small time steps (depending on the time scales of the problem)using a modified Newton method implemented in the TWOPNT[74] solver. In order to validate TORC, simulations of planar flameswere performed and the results were compared against thoseobtained using PREMIX [55,56].

The kinetic, thermodynamic, transport and radiation calcula-tions were performed similarly to PREMIX [55,56]. In order toreduce the computational cost, DNS of SEF’s were performed onlyfor CH4/air flames and employed two models obtained from USC-Mech II using the DRG reduction strategy [75]. The reductionwas done using an array of PREMIX solutions for lean and rich mix-tures separately. For / = 0.7 and 1.0 a model consisting of 17 spe-cies and 78 reactions was used, while for / = 1.4 the reduced modelincluded 24 species and 137 reactions.

A domain of radius Re = 25 cm was used in the simulations. Att = 0, a stagnant pocket of hot burned gases of radius 2.5 mm sur-rounded by the unburned combustible gas mixture was used toachieve ignition. It should be noted that while Re = 25 cm is of norelevance to the actual experimental conditions it is large enoughradius at which near-zero stretch are reached and at which theextrapolations are carried out.

At t > 0 the boundary conditions are:

dTdR

� �R¼0¼ 0; uR¼0 ¼ 0;

dYK

dR

� �R¼0¼ 0;

dTdR

� �R¼Re

¼ 0;dYK

dR

� �R¼Re

¼ 0;

where YK is the mass fraction of species K. Note that only oneboundary condition for u is allowed and is specified at the centerof the domain. The maximum heat release rate was used as a mar-ker to track Rf as a function of t. Sb is obtained by differentiating a3rd order polynomial used to fit locally the variation of Rf with t,and as mentioned earlier, K � (2/Rf)(dRf/dt).

3. Results and discussion

3.1. Differential diffusion effects for CFF’s and SEF’s

Figure 3 depicts the variation of Sou with / for CH4/air mixtures

for various CH4 diffusivities, DCH4 , while in Fig. 4 the logarith-mic sensitivity coefficients of So

u to the CH4–N2 and O2–N2 binarydiffusion coefficients are shown. Results indicate that the modifica-tion of DCH4 has an opposite effect on So

u for / < 1.0 and / P 1.0.Details of the flame structure are shown in Fig. 5, and it can be

seen that a change in DCH4 results in a corresponding change in itsdiffusion length relative to O2. The diffusion length of CH4 for a /= 1.4 CH4/air flame computed with DD (Fig. 5b) is reduced com-pared to the OD case (Fig. 5a). Thus, YCH4 and the local equivalenceratio, /local, increase at the location at which the CH4 consumptioninitiates as shown in Fig. 6, which results in the reduction of reac-tivity, shown also in Fig. 6. Similar analysis can be used to explainthe dependence of So

u on DCH4 for all mixtures shown in Table 1.Furthermore, it is of interest to note that the dependence of So

u



Fig. 3. Computed Sou’s of CH4/air flames at p = 1 atm and Tu = 298 K using USC-Mech

II. ( ) OD; ( ) DD; ( ) ID.

Fig. 4. Logarithmic sensitivity coefficients of Sou to the CH4–N2 (black) and O2–N2

(grey) binary diffusion coefficients for CH4/air flames at p = 1 atm, Tu = 298 K, andvarious /’s.

Fig. 6. Variation of /local with temperature in a / = 1.4 freely propagating CH4/airflame at p = 1 atm, and Tu = 298 K computed using USC-Mech II with OD ( ) and DD( ), and variation of CH4 consumption rate with temperature with OD ( ) and DD( ).


on DCH4 is not captured by the following equation that is based onLe considerations [76]:

SouðLe – 1Þ ¼ So

uðLe ¼ 1ÞffiffiffiffiffiLep

ð3Þ

The effect of reactant differential diffusion on the propagation ofstretched flames was assessed in the CFF configuration. Figures 7–9 depict the variation of Su;ref=So

u with Ka for / = 0.7, 1.0, and 1.4mixtures respectively. These figures include also the extrapolationcurves using Eq. (1) that fit the DNS results for a range of Ka that

Fig. 5. (a) Normalized mass fraction profiles of CH4 ( ) and O2 ( ), and CH4 consumpp = 1 atm, computed using USC-Mech II and OD. (b) Normalized mass fraction profiles offlame at Tu = 298 K, p = 1 atm, and / = 1.4, computed using USC-Mech II and DD.


are representative of those used in experiments (e.g., [27,29]). UsingOD, Eq. (1) predicts closely the DNS results. As DCH4 starts deviatingfrom the oxygen diffusivity, DO2 , for the ID and DD cases, a discrep-ancy is observed between the extrapolated So

u from its known valueby as much as 5% for / = 0.7 with ID and 30% for / = 1.4 with DD. Asummary of extrapolation errors resulting from linear and asymp-totically derived nonlinear methods is reported in Table S1 in thesupplementary material.

From Figs. 7–9 it is apparent also that there is a significantchange in slope of the Su;ref=So

u vs. Ka curve when DCH4 is modifiedfor the / = 0.7 and 1.4 mixtures. In CFF’s, it is not possible to mon-itor the modification of the burning intensity with K by simplytracking the variation of Su,ref with K, as Su,ref is affected also bythermal dilatation and flow divergence [13,27,28]. On the otherhand, the burning intensity is best described by the total heatrelease rate per unit area, HRRtot, obtained by integrating the heatrelease rate over the entire flame. Figures 10–12 depict the varia-tion of HRRtot with Ka for / = 0.7, 1.0 and 1.4 mixtures respectively.The results for the / = 0.7 mixture shown in Fig. 10 can beexplained based on Le – 1.0 effects caused by the imbalance ofenergy loss from and energy gain by the reaction zone [76]. Forthe / = 1.4 mixture however, even though Le � 1.0 for all hydrocar-bons, a substantial increase in HRRtot with Ka is seen for the DD

tion rate profile (—) for a / = 1.4 freely propagating CH4/air flame at Tu = 298 K andCH4 ( ) and O2 ( ), and CH4 consumption rate profile (—) for a freely propagating



Fig. 7. Variation of Su;ref=Sou with Ka of a / = 0.7 CH4/air CFF at p = 1 atm and

Tu = 298 K computed using USC-Mech II with ID ( ), OD ( ), and DD ( ). ID ( ),OD ( ), and DD ( ) correspond to fitting using Eq. (1). The full range DNS resultsare shown in hollow symbols, while the DNS results used for fitting Eq. (1) areshown in solid symbols.





Fig. 10. Variation of HRRtot with Ka of a / = 0.7 CH4/air CFF at p = 1 atm andTu = 298 K computed using USC-Mech II with ID ( ), OD ( ), and DD ( ).




case for which there is a notable difference between DCH4 and DO2 .Thus, the diffusion rate of O2 towards the reaction zone increasescompared to CH4 with increasing K, making thus the mixture morestoichiometric and increasing the overall reactivity [76]. For the /


= 1.0 mixture the slope of HRRtot with Ka does not change for thedifferent DCH4 values. This is due to the fact that for near-stoichi-ometric mixtures there is a minor sensitivity of the overall reactiv-ity to modifications in / as it reaches a maximum value.

Figures 13 and 14 depict the variations of /local and the con-sumption rate of CH4 for a / = 1.4 flame at K = 30 and 200 s�1

respectively and computed with OD and DD. The results confirmthat as K increases, /local decreases at the locations at which theCH4 consumption begins. As a result, there is a notable increase



Fig. 13. Variation of /local with temperature in a / = 1.4 CH4/air CFF at p = 1 atm,Tu = 298 K, and K = 30 s�1 computed using USC-Mech II with OD ( ) and DD ( ),and variation of CH4 consumption rate with temperature with OD ( ) and DD( ).

Fig. 14. Variation of /local with temperature in a / = 1.4 CH4/air CFF at p = 1 atm,Tu = 298 K, and K = 200 s�1 computed using USC-Mech II with OD ( ) and DD ( ),and variation of CH4 consumption rate with temperature with OD ( ) and DD( ).

Fig. 15. Variation of Su;ref=Sou with Ka of a / = 0.7 n-C12H26/air CFF at p = 1 atm and

Tu = 443 K computed using JetSurF 1.0 with ID ( ) and OD ( ). ID ( ) and OD ( )correspond to fitting using Eq. (1). The full range DNS results are shown in hollowsymbols, while the DNS results used for fitting Eq. (1) are shown in solid symbols.

Fig. 16. Variation of Su;ref=Sou with Ka of a / = 1.4 n-C12H26/air CFF at p = 1 atm and

Tu = 443 K computed using JetSurF 1.0 with ID ( ) and OD ( ). ID ( ) and OD ( )correspond to fitting using Eq. (1). The full range DNS results are shown in hollowsymbols, while the DNS results used for fitting Eq. (1) are shown in solid symbols.

Fig. 17. Variation of (/local)peak ( ) and HRRtot ( ) with Ka of a / = 1.4 n-C12H26/airCFF at p = 1 atm and Tu = 443 K computed using JetSurF 1.0 with OD.


of the CH4 consumption rate as K increases for the DD case com-pared to OD. More specifically, the maximum CH4 consumptionrate is about 40% higher for the DD case for K = 30 s�1, and by a fac-tor of 3.5 for K = 200 s�1. These results reveal the basis physics thatcontrol the dependence of the overall flame reactivity with stretchfor rich mixtures of high MW fuels and which need to be accountedfor when raw experimental data are interpreted to determine non-directly measured properties such as So

u.n-C12H26/air CFF’s were computed also in order to verify the

findings for CH4/air flames. The diffusivity of n-C12H26/air wasmodified also by using the L-J parameters of CH4 and this case isreferred to as ID given that n-C12H26 becomes more diffusive. Fig-ures 15 and 16 depict the variation of Su;ref=So

u with Ka for / = 0.7and 1.4 respectively and the behavior is consistent with thatobserved for CH4/air flames. The / = 0.7 flame computed with ODmixture exhibits lower Su,ref values and the flame extinguishes atlower K compared to the ID case as shown in Fig. 15. Furthermore,the use of Eq. (1) in both OD and ID cases results in So

u’s that areclose to its known value. On the other hand, for the / = 1.4 flame,using Eq. (1) results in the over-prediction of the known So

u valueby 9% and 4% for the OD and ID cases respectively. The variationsof the peak local equivalence ratio, of (/local)peak and HRRtot areshown in Fig. 17 for a / = 1.4 n-C12H26/air flame computed with


OD. Similarly to CH4/air flames computed with DD, (/local)peak

decreases and HRRtot increases as Ka increases given that the flamebecomes more stoichiometric.

CH4/air SEF’s were computed for a wide range of Rf’s and thecomputed Sb values for 1 cm < Rf < 3 cm were used for performingextrapolation using Eq. (2). This Rf range is used typically in exper-iments (e.g., [39,54]) so that the data are not affected by the



Fig. 18. Variation of Sb=Sob with Ka of a / = 0.7 CH4/air SEF (ADB) at p = 1 atm and

Tu = 298 K computed using a reduced USC Mech II with ID ( ), OD ( ), and DD ( ).ID ( ), OD ( ), and DD ( ) correspond to fitting using Eq. (2). The full range DNSresults are shown in hollow symbols, while the DNS results used for fitting Eq. (2)are shown in solid symbols.





Fig. 21. Variation of HRRtot with Ka of a / = 1.4 CH4/air SEF at p = 1 atm andTu = 298 K computed using a reduced USC-Mech II with ID ( ), OD ( ), and DD( ).

Fig. 22. Variation of Sb=Sob with Ka of CH4/air SEF’s (ADB) at p = 1 atm and Tu = 298 K

computed using a reduced USC Mech II with DD for / = 0.7 ( ), / = 1.0 ( ), and /= 1.4 ( ).


ignition energy and pressure rise [38,45]. The simulations wereperformed for adiabatic flames (ADB), i.e. without considering radi-ation, with ID, OD, and DD similarly to the CFF’s, and the variationof Sb=So

b with Ka is shown in Figs. 18–20 for / = 0.7, / = 1.0, and /= 1.4 respectively. With DD, the extrapolation error in So

b is approx-imately 6% for the lean and rich flames. Extrapolations of the /= 1.4 results obtained with OD and ID result in errors of 4.5% and10% respectively. It is also of interest to note that for flames withpositive Lb, the variation of Sb with Ka is nearly linear despite thehigh sensitivity of Sb to Ka, and that Eq. (2) always generates ahighly non-linear curve for such flames especially when the Sb vs. Kslope is steep. In general, mixtures with Le – 1.0 and/or notabledifferences between the fuel and O2 diffusivities, the magnitudeof the extrapolation error increases. Other than the aforemen-tioned conditions, the extrapolation error was determined to bein general below 3%.

For SEF’s, Sb represents the flame propagation speed withrespect to the stationary burned gas at adiabatic conditions. Thevariation of HRRtot with Ka is shown in Fig. 21 for / = 1.4 and itcan be seen that the behavior computed using ID, OD, and DD issimilar to that of Sb=So

b with Ka shown in Fig. 20 indicating thatSb is a good indicator of the overall burning intensity, similar tothe results obtained for CFF’s.

Figure 22 depicts the variation of Sb=Sob with Ka for / = 0.7, /

= 1.0, and / = 1.4 computed with DD that is representative ofhigh MW hydrocarbons, and the results are consistent with


experimental results of n-butane/air mixtures [21]. More specifi-cally, the sign of Lb for heavy hydrocarbons changes from positivefor / < 1.0 to negative for / > 1.0. For / < 1.0 mixtures of high MWhydrocarbons the flame response is controlled by Le effects,whereas for / > 1.0 differential diffusion is the controlling factor.

Kelley et al. [42] used asymptotic analysis to account fordifferential diffusion effects, and they showed that there is anon-monotonic variation of the flame speed with stretch for nearstoichiometric mixtures with contrasting fuel and oxygen diffusiv-ities. Using the formulation obtained in Ref. [42] for extrapolationsto obtain accurate flame speeds was not feasible due to the large



Fig. 23. Temperature profiles for a / = 1.0 CH4/air SEF at p = 5 atm, Tu = 298 K, andRf = 8 cm computed using a reduced USC Mech II with ADB ( ), OTL ( ), and HOTL( ).

Fig. 25. Variation of Sb with K of a / = 1.0 CH4/air SEF at p = 1 atm and Tu = 298 Kcomputed using a reduced USC Mech II with ADB ( ), OTL ( ), and HOTL ( ). ADB( ), OTL ( ), and HOTL ( ) correspond to fitting using Eq. (2). The full rangeDNS results are shown in hollow symbols, while the DNS results used for fitting Eq.(2) are shown in solid symbols.


number of parameters that have to be determined by fitting theextrapolation equation to the experimental data. Hence all extrap-olation equations were obtained by invoking the assumption thatthe reactant differential diffusion effect is negligible (for off-stoi-chiometric mixtures) and that Le solely governs the flame dynam-ics. In the present study it was shown that this assumption is notvalid for rich mixtures of large MW hydrocarbons and thus usingsuch an equation to extrapolate experimental data can lead tonotable errors.

3.2. Radiation effects for SEF’s

Figure 23 depicts the spatial temperature profiles for a / = 1.0CH4/air SEF at p = 5 atm that has propagated to Rf = 8 cm. Recall,that the adiabatic and optically thin limit cases are representedby ADB and OTL respectively. In order to account for potentialre-absorption, additional simulations were carried out by usinghalf of the Planck’s mean absorption coefficient represented byHOTL, which is in a way equivalent to reabsorption of half theenergy emitted by the burned gas. It is apparent that the presenceof radiation results in a notable reduction of the temperature of theburned gases and the assumption of equilibrium does not hold.

The spatial variation of the gas velocity for the conditions ofFig. 23 is shown in Fig. 24. The radial inward flow, observed as neg-ative velocities, is a result of density change in the burned gas dueto radiative heat loss. It is seen that the extent of this inward flow

Fig. 24. Velocity profiles for a / = 1.0 CH4/air SEF at p = 5 atm, Tu = 298 K, andRf = 8 cm computed using a reduced USC Mech II with ADB ( ), OTL ( ), and HOTL( ).


is reduced with reabsorption, e.g. HOTL, as the total heat loss isreduced, which is consistent with the findings of Chen [48].

Figure 25 illustrates the variation of Sb with K, and the discrep-ancies that can be induced due to extrapolations to K = 0 are appar-ent. As expected, in the presence of radiation Sb is reduced and as aresult the effect of inward flow is augmented at large radii, whichis evident also by the equation derived by Santner et al. [49]. How-ever, the reported results for OTL and HOTL may not be entirelyphysical, given that at very large radii some of the lost energymay be reabsorbed due to large optical thickness. Nevertheless,caution is recommended when SEF raw data are interpreted asSantner et al. [49] has suggested also.

Table 2 illustrates the errors induced when the Sb data derivedusing OTL are used to extract So

b for CH4/air mixtures at variousconditions. The percentage difference in So

u calculated using thePREMIX code [55,56] between the OTL and ADB conditions repre-sents the difference that should be obtained after extrapolatingSb at the corresponding conditions to obtain So

b, as they are linearlyrelated through the continuity equation. Results show that theerror due to cooling of the burned gas increases for off-stoichiom-etric mixtures and at high pressures, in agreement with the find-ings of Santner et al. [49] in that the slower flames are affectedmost by the radiative heat loss given that more time is availablefor the burned gases to cool during the duration of the experiment.

The alternative approach of Lecordier and coworkers [52,53]was evaluated also using the computed SEF structures from whichin addition to Sb the values of Ug were extracted as well so that thedisplacement velocity Un � Ug � Sb [34] can be evaluated. Figure 26compares the variations of Sb and Un with K, for the ADB, OTL, andHOTL cases. It is of interest to note that while radiation has a majoreffect on Sb, its effect on Un is minor as its values computed in allthree cases collapse in a single curve. Mathematically, radiation

Table 2Differences in So

u’s between the ADB and OTL cases as obtained from freelypropagating flame simulations, and in So

b’s obtained by extrapolating the SEF resultsfor the ADB and OTL cases.

/ Pressure(atm)

% Difference between OTL &ADB So

u’s from freelypropagating flame simulations

% Difference betweenOTL & ADB extrapolatedSo

b’s from SEF data

0.7 1 �0.4 �4.01.0 1 �0.2 �2.01.4 1 �2.4 �6.20.8 5 �0.1 �6.01.0 5 �0.1 �4.6



Fig. 26. Variation of Sb (solid) and Un (hollow) with K of a / = 1.0 CH4/air SEF atp = 5 atm and Tu = 298 K computed using a reduced USC Mech II with ADB ( ), OTL( ), and HOTL ( ).

Fig. 27. Variation of Un=Sou with Ka for a / = 1.4 CH4/air SEF at p = 1 atm and

Tu = 298 K computed using a reduced USC Mech II with ID ( ), OD ( ), and DD ( ).ID ( ), OD ( ), and DD ( ) correspond to fitting using the equation derived byKelley and Law [21]. The full range DNS results are shown in hollow symbols, whilethe DNS results used for fitting the extrapolation equation are shown in solidsymbols.

Fig. 28. Variation of temperature at location of maximum velocity (Ug) with stretchfor a / = 1.4 CH4/air SEF at p = 1 atm and Tu = 298 K.


affects similarly Sb and Ug through the inward flow, so that itseffect is subtracted out. From a physical point of view this resultis reasonable, as Un is a measure of the stretched flame speed withrespect to the unburned gas, which is not affected from radiationfor mixtures that are not close to the flammability limits [47].

Radiation calculations involving spectrally resolved emissionand absorption are computationally expensive and prone to uncer-tainty due to simplifying assumptions required to make thenumerical simulation feasible [77]. Therefore, by measuring Un

the complications associated with accurate reabsorption calcula-tions needed to model the experimental data are circumvented,and enables using the OTL model to simulate SEF’s to reasonableaccuracy for mixtures which have limited overlap of spectral bandswith the burned products and are far from the flammability limit.Any approach in SEF’s is not applicable for near-limits for which So

u

can be of the order of 10 cm/s or less [47], given that buoyancy willdominate the flame behavior and will cause severe distortion ofthe flame surface in experiments.

In order to extrapolate the experimental Un data to obtain Sou,

Renou and coworkers [54] used the formula of Kelley and Law[21] derived for the unburned flame propagation speed, definedas the flow velocity relative to the flame upstream of the preheatzone where the temperature rise is negligible. Figure 27 depictsUn as a function of Ka for / = 1.4 CH4/air mixtures computed withID, OD and DD, the cases examined in Fig. 20 along with the extrap-olation curves obtained using the formula of Kelley and Law [21]. Itis evident from Figs. 20 and 27 that the trends of variation of Sb andUn with Ka are different and that Un unlike Sb and HRRtot (Fig. 21) isnot a proper indicator of the burning intensity.

Extracting the unburned flame propagation speed from anunsteady flame in a non-planar flow geometry in not trivial.Dixon-Lewis and Islam [28] simulated a planar steady flame in aquasi-1D diverging flow geometry and showed that the flamespeed can be obtained by density correction of the flow velocityat the location of peak reaction rate. It became evident also fromthe simulations in the current study that this point of maximumvelocity (Ug), used to compute Un, is influenced by thermal dilata-tion as shown in Fig. 28 which is a plot depicting the variation oftemperature at the location of maximum velocity with stretch.These observations are indicative of the fact that Un, obtained fromUg, is not the actual unburned stretched flame speed, but a refer-ence flame speed similarly to CFF’s. Using the equation of Kelleyand Law [21], which does not account for thermal dilatation andgeometric effects, to extrapolate Un could result in substantial errorin the extrapolated value of So

u. This increased uncertainty is evi-dent from the difference in extrapolation errors of Un and Sb


depicted by Figs. 27 and 20 respectively and the consistent overprediction of the known So

u values in all cases.

3.3. Experimental uncertainties

Compared to the DNS results presented in this study, experi-mental data exhibit larger uncertainty and/or scatter due to severalfactors.

Clearly, in steady state experiments like those of CFF’s thedirectly measured Su,ref’s can be optimized so that the uncertaintiesare minimized. In carefully performed CFF experiments, the uncer-tainty based on 2r, where r is the standard deviation, can be aslow as 5% [16]. However, the uncertainty of CFF experiments canbe 10% or higher if issues related to the quality of the flow, reactantconcentrations especially for / > 1.0 flames of liquid fuels, flow tra-cer seeding density, the implementation of particle image veloci-metry (PIV) or laser Doppler velocimetry (LDV) to measure flowvelocities, and interpretation of the raw data are not addressedcarefully and rigorously.

Performing SEF experiments is by far more challenging. First, instatic experiments the equivalence ratio uncertainties can begreater compared to steady state experiments. Second, the experi-ments last only few milliseconds making thus the implementationof any type of diagnostics with satisfactory temporal resolutionchallenging. Uncertainties in SEF experiments are expected to begreater for very slow (buoyancy) or very fast (time resolution)




flames as well as for higher pressures in which flames tend to beunstable.

It should be noted that uncertainties in the reported Sou of the

order of 10% or higher are not desirable as such data cannot be usedeffectively for the validation of kinetic models given the relativelylow sensitivity of So

u to kinetics. An alternative viable approach invalidating kinetic models is to compare the raw experimental datafrom either CFF or SEF experiments against corresponding DNSresults, so that the uncertainties associated with extrapolationsare removed.

4. Concluding remarks

Direct numerical simulations of counterflow and sphericallyexpanding flames were carried out in order to assess uncertaintiesstemming from current practices that are used to interpret exper-imental data and derive the laminar flame speed. The analysisfocused on the effects of molecular transport and thermal radia-tion. The counterflow and freely propagating flames were simu-lated using established codes. A new code was developed inorder to simulate spherically expanding flames by integratingaccurately in time the one-dimensional transient conservationequations in spherical coordinates. The results of the simulationswere treated as data in the range of stretch rates that are encoun-tered in experiments, and were used to perform extrapolations tozero stretch using formulas that have been derived from asymp-totic analyses. The validity of these practices was tested upon com-paring the results against the known answers of the directnumerical simulations.

The effect of molecular transport was studied by varying thefuel diffusivity. It was concluded that for fuel lean hydrocarbon/air mixtures, the preferential diffusion of heat or mass as mani-fested by the Lewis number dominates the flame response tostretch. For fuel rich mixtures, the controlling factor was deter-mined to be the differential diffusion of the reactants into the reac-tion zone for heavy hydrocarbons. It was found also that usingextrapolation equations derived based on asymptotics analysisand simplifying assumptions to obtain the laminar flame speeds,could result in significant errors for rich flames of heavyhydrocarbons.

Numerical simulations of spherically expanding flames withradiative heat loss revealed that the standard approach of measur-ing the flame propagation speed relative to the burned gas usingthe shadowgraph/Schlieren techniques, could result in a system-atic under-prediction of the true laminar flame speed due to aninward flow induced by the density change in the burned gas. Fur-thermore, it was shown that by simultaneously measuring themaximum velocity upstream of the flame and the burned flamespeed and evaluating thus the displacement speed relative to thefresh gases, this error could be avoided. It was determined thatthere is a negligible effect of the density change in the burnedgas due to radiation on the displacement speed relative to the freshgas.

Acknowledgments

This material is based upon work supported as part of theCEFRC, an Energy Frontier Research Center funded by the U.S.Department of Energy, Office of Science, and Office of Basic EnergySciences under Award Number DE-SC0001198. The authors wouldlike to thank Dr. Linda Petzold and Dr. Shengtai Li for providingsubroutines to implement the ‘‘warm restart’’ for the DASPK solver.


Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.combustflame.2014.03.009.

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