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Combustion and Flame 137 (2004) 444–457www.elsevier.com/locate/jnlabr/cn

Multidimensional mathematical modeling and numericinvestigation of co-flow partially premixed methane/ai

laminar flames

K. Claramunt,∗ R. Cònsul, C.D. Pérez-Segarra, and A. Oliva

Centre Tecnològic de Transferènciade Calor (CTTC), Universitat Politècnica de Catalunya (UPC), c/Colom 11,E-08222, Terrassa, Barcelona, Spain

Received 11 December 2002; received in revised form 23 January 2004; accepted 9 March 2004

Available online 15 April 2004

Abstract

The goal of this paper is to analyze, by means of detailed numerical simulations, the influence of the parmixing level and the adequacy of different mathematical submodels on the modeling of co-flow partially prmethane–air laminar flames. Five levels of premixing of the primary inlet are considered from an equivalence rof Φ = ∞ (non-premixed flame) toΦ = 2.464. Main flame properties are provided, giving special emphasis tanalysis of pollutant formation. Different mathematicalformulation aspects (several chemical mechanisms, ration effects, mass transport models, and inlet boundary conditions) are tested and validated against experimentdata available in the literature. Finite volume techniques over staggered grids are used to discretize the gequations. A parallel multiblock algorithm based on domain decomposition techniques running with loosely cou-pled computers has been used to obtain a competitive ratio between computational cost and resources.the quality of the numerical solutions presented in this article, a verification process based on the genRichardson extrapolation technique and on the grid convergence index (GCI) has been applied. 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords:Co-flow methane/air laminar flames; Combustion; Mathematical formulation analysis; Levels of premixing analysNumerical simulation; Verification; Validation

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1. Introduction

In recent decades, comprehensive studies hbeen reported in the literature concerning detailedperimental and numerical analysis of laminar flamin simple systems. Among these, and due to thwide application in household and industrial heatsystems, analysis of co-flow laminar flames has mvated special interest. These studies have usuallyfocused on non-premixed or premixed flames. M

* Corresponding author. Fax: 0034-93-739-81-01.E-mail address:[email protected]

(K. Claramunt).URL: http://www.cttc.upc.edu.

0010-2180/$ – see front matter 2004 The Combustion Institutdoi:10.1016/j.combustflame.2004.03.004

recently these studies have been extended to parpremixed situations [1,2] because of their significpractical and fundamental importance.

Partially premixed flames are formed when lethan the stoichiometric amount of oxidizer is mixeda fuel flow upstream of the reaction zone, in whichditional oxidizer is available to diffuse into the flamand provide complete combustion [3]. Due to thstability, partially premixed flames are used in Bunsburners, furnaces, gas-turbine combustor flames,fired domestic appliances, and other common cbustion devices. Recent studies suggest that antimum operating condition exists that minimizes tpollutant emissions and, thus, enhances the desigcleaner, practical burning combustors.

e. Published by Elsevier Inc. All rights reserved.

http://www.elsevier.com/locate/jnlabr/cnf

http://www.cttc.upc.edu

K. Claramunt et al. / Combustion and Flame 137 (2004) 444–457 445

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With the increase in computational power, the iprovement in numerical methods, and the use of maccurate experimental techniques, knowledge ofcombustion phenomenon taking place in these flahas been considerably increased.

Experimental studies have provided measuments of temperature, major species, radicals, ngen oxides, and soot. Mass spectrometry, RamanLIF techniques have been employed to study co-flflames under different geometrical configuratioequivalence ratios, and pressure conditions [1,4–6

With respect to numerical studies, from onethe first multidimensional simulations of co-flomethane–air laminar flames carried out by Mitchet al. [7], considerable improvement in the accuraof the mathematical models employed for the simutions has been achieved. Detailed numerical simtions with fully elliptic equations, complex transpoformulation, and detailed chemistry have beenported. Mainly C1 and C2 chemical mechanismsemployed and compared [8], molecular transpormodeled under different assumptions [9], soot formtion is sometimes modeled [5], and radiation transif considered, is usually evaluated with simplifimodels [10].

Despite the above-mentioned progress achiethere is a certain interest in the combustion commnity to continue working on the detailed analysisthese simple flames. Although combustion nearlyways takes place within turbulent flow fields whiincrease the mixing process, and thereby enhacombustion [11], a deep understanding of lamicombustion is a basic ingredient of the modelingturbulent flames, as well as of pollutant formation.

The aim of this work is to contribute to this interest by investigating, by means of detailed numcal simulations, the influence of the partial premixilevel and the adequacy of different mathematical smodels, in a co-flow partially premixed methane–flame.

The numerical study is performed by analyziboth main flame features and local data. The amoof pollutant produced is presented by means ofevaluation of emission indexesEIx , defined as theratio of grams per second of pollutant species to kgrams per second of methane burned.

Special emphasis is placed on assessing the qity of the numerical solutions and reproducing eperimental conditions (i.e., verification and validatiprocesses). Numerical solutions are verified using thpostprocessing procedure described in [12] basethe generalized Richardson extrapolation technifor h-refinement studies and on the grid convergeindex (GCI) proposed by Roache [13]. Validationthe mathematical models is carried out considerthe experimental data provided in [2,6].

A parallel multiblock algorithm [14], specially developed to be used in loosely coupled parallel coputers (Beowulf clusters), was to perform the numical simulations, allowing exhaustive analysis of tflame with an excellent ratio between computatiotime and resources.

2. Problem definition

The flame considered is the co-flow partially pmixed methane–air laminar flame with the burncharacteristics defined in [2]. Experimental appatus and measurement techniques are extensivelyscribed in [5,6,15]. Hereafter, a brief descriptionthe test assay is presented. Fig. 1 is a schematicresentation of the burner.

Fuel mixed with primary air flows from an uncooledri = 5.55-mm-inner-radius brass tube withwall thickness ofwi = 0.8 mm. Air is injected fromthe annular region between this tube and a concer0 = 47.6-mm-inner-radius brass cylinder. The outube thickness isw0 = 3.4 mm. The fuel tube containa 110-mm length of glass beads to smooth the fland to ensure a fully developed velocity profile atexit. A perforated brass plate, glass beads, and fina 1.5-mm-cell-size ceramic honeycomb straightenairflow. The inner tube extends 4 mm above the heycomb surface to facilitate access to the lowest flaregions. The cylindrical brass chimney that confinthe flame and protects it from laboratory air movments has a diameter of 102 mm.

Different levels of premixing of the primary inleare considered from an equivalence ratio ofΦ = ∞(non-premixed flame) toΦ = 2.464. The lower limitwas just before flashback began to affect the flastructure [15]. Equivalence ratios and mass flowslisted in Table 1. Primary air is oxygen-enriched (25O2 by volume) and secondary air is “regular” (20.9%O2).

Due to the considerable complexity of the burnconfiguration below the bottom of the flame (preence of perforated brass plates, glass beads, an

Table 1Flame parameters

Φ mCH4 (g/min) mair (g/min)

Inner jet∞ 0.2165 0.000012.320 0.2165 0.24936.160 0.2165 0.49864.107 0.2165 0.74782.464 0.2165 1.2465

Outer jetAll 0 .0000 51.879

446 K. Claramunt et al. / Combustion and Flame 137 (2004) 444–457

thees

(a) (b)

Fig. 1. Confined co-flow methane–air laminar flame. (a) Burner idealized geometry and definition of the different zones fornon-equispaced cylindrical grid. (b) Definition of macrozones (I,II, and III) and their mesh node distribution (solid trianglindicate the direction of grid node intensification).

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ramic honeycomb structures), only the region abthe inner tube exit has been simulated (i.e.,z > 0).A computational height ofL = 200 mm was considered according to numerical results presented inSee Fig. 1 for details.

3. Mathematical model

The fluid flow and heat and mass transfer pnomena of the reactive gas in the co-flow partiapremixed methane–air laminar flame defined abovassumed to be described by the governing equatfor low-Mach-number flows (continuity, species, mmentum, energy and state equation) [14].

Modeling of the chemical mechanisms, moleclar transport fluxes, and radiation heat flux is requito close the problem. Furthermore, special attenhas to be paid to boundary conditions. Thermophcal properties are evaluated using the thermodynadata compiled in [16].

3.1. Mathematical submodels

The detailed chemical mechanism consideredGRI-Mech 3.0 [16], which involves 325 reactions a53 species. This mechanism is suitable for the

scription of pollutant formation because it includNOx reactions.

The shear stress tensor (τ ) is evaluated taking intoaccount Stokes’ law for Newtonian fluids consideria mixture viscosity. Diffusion heat flux (q) consid-ers Fourier’s conduction and the energy transporinterdiffusion. Mass fluxes of species are evaluaconsidering both an equivalent Fickian diffusion teand thermal diffusion term (Soret effect) [17],

(1)ji = −ρDim∇Yi − DTi ∇ lnT,

whereDim andDTi

are the multicomponent ordinaryand thermal diffusion coefficients, respectively.

Transport coefficients of the molecular fluxof momentumτ , heat q, and massji are evalu-ated considering a mixture-averaged formulatiPure-species transport properties are evaluateding CHEMKIN’s database [18]. For the mixtureaveraged viscosity and the thermal conductivity,semiempirical Wilke (1950) formula, modified bBird (1960), is used [17]. Mixture diffusion coefficientsDim, which appear in Eq. (1), are calculatfrom the Stefan–Maxwell equation and considertrace-speciesapproximation [17]. Assuming thatgiven species sees the rest moving with the saaverage velocity, and the mixture is composed


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one majority species, the equivalent Fickian difsion coefficient of one species into the mixture cbe formulated withDim = (1 − Yi)/X, whereX =∑N

j=1, j �=i Xj /Dij andDij represents the binary difusion coefficient [17]. Thermal diffusion coefficienDT

iare obtained relating calculatedDim values with

the thermal diffusion ratios (see [18]).Flame radiation is modeled using the assump

of optically thin transfer between the hot combution gases and the cold surroundings [19,20]. Tdefinition of an optically thin gas establishes thself-absorption is negligible compared with emissiA term for the absorption of radiation coming frothe surrounding background atTs temperature is included. WhenTs is low, this term can be neglecteThus, the radiative heat loss term per unit of voluis expressed as

(2)∇ · qR = 4σT 4N∑

i=1

(piκPi) − 4σT 4

s

N∑i=1

(piκIi ),

whereσ is the Stefan–Boltzmann constant;pi is thepartial pressure of speciesi; κPi

is the Planck-meanabsorption coefficient for speciesi; κIi is the incident-mean absorption coefficient for speciesi; andTs is thebackground temperature.

The radiating species considered in order ofportance are CO2, H2O, CH4, and CO. From runningRADCAL [21], Planck-mean and incident-mean asorption coefficients, which are fitted to polynomexpressions [19], are obtained at different tempetures.

3.2. Boundary conditions

Special attention has been paid to the inlet bouary conditions atz = 0 to relate their values to thknown values at the bottom of the burner (sectionin Fig. 1a). Species mass fractions are evaluateding species mass flow rates and considering thareaction occurs inside the tube:

(3)

(ρvzYi)B =[ρvzYi − ρDim

∂Yi

∂z− DT

im∂ lnT

∂z

]z=0

.

For the energy equation, in preceding works [2,1an ambient underpredicted (298 K) temperaturefixed at the entrance of the computational dom(z = 0). In this work, an enthalpy flux is evaluatedsection B and, assuming a temperature at sectionTB = 298 K, the temperature is estimated by solvin

(ρvzh)B =[ρvzh − λ

∂T

∂z−

N∑i=1

hi

(ρDim

∂Yi

∂z

(4)+ DTim

∂ lnT

∂z

)].

z=0

This boundary condition implies that no heat fluxtransferred through the inner tube between themary and secondary fluxes. Due to the configuraof the burner, the radial component of the velochas been neglected at the entrance of the comptional domain (z = 0) of both primary and secondaflows. To take into account the considerable gradieof temperature and mass fractions involved in thelet region, axial velocity is calculated using the locdensity value and assuming a mass flow rate proa parabolic mass flow rate for the primary inlet anplug-flow profile for the secondary one.

The chimney has been considered to be imperable and chemically inert (material without any cheical activity), being the total flux of species normalthe wall equal to zero. At the outlet of the burnerpressure outflow boundary condition is imposed [22]and a null gradient in the axial direction of tempeture and species is assumed.

4. Numerical methodology

The mathematical model is discretized usingfinite volume technique over cylindrical staggergrids. Central differences are employed for evaltion of the diffusion terms, while third-order boundschemes are used for evaluation of the convecones [23]. A time-marching SIMPLE-like algorithmis employed to couple velocity–pressure fields. Dcretized equations are solved in a segregated mner [24] using a multigrid solver [25]. The convegence of the time-marching iterative proceduretruncated once normalized residuals are below 10−8.

On the resolution of species equations, andovercome the stiffness of the governing equationpseudo-time splitting technique is used [14]. The eergy equation is considered in terms of enthalpy traport. Instead of solving this equation directly, a teperature convection–diffusion equation is consideand the full enthalpy transport equation is introducin the source term by means of a deferred correc(see [14] for details).

The computational domain has been discretiusing a cylindrical structured grid. Several zones wdifferent grid node distributions are defined (Fig. 1The number of grid nodes is increased at the ouof the inner tube where the gradients of methanehigher. As we move away from the bottom and frothe flame front, the grid node density is progressivdecreased by means of tanh-like functions. The nber of nodes corresponding to each zone is indicain terms of the grid parametern, and the direction othe intensified distribution is indicated by a solid tangle.


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The domain decomposition method is useda strategy to reduce the number of grid nodesfrom the flame fronts, and as a parallelization tenique. The whole domain is divided into several ovlappedsubdomainsjoined by the interpolation boundaries [14,26]. Each CPU solve one (or a group of) sdomain, and only one communication for each ouiteration is required. This is an important feature whBeowulfclusters are used. For further details see [26,27].

Due to the parabolic structure of the flow, the dmain is decomposed in eight subdomains in thzdirection. The computational behavior of the parlel multiblock algorithm confirms this strategy. Thuse of multiblock discretization allows the definitioof three macrozones, identified in the figure with Rman numbers, characterized by having the samenode distribution in ther direction.

All the numerical simulations have been peformed on aBeowulf clustercomposed of 48 standarPCs (AMD K7 CPU at 900 MHz and 512 Mbyte) wita conventional network.

5. Verification and validation processes

5.1. Verification procedure and results

A postprocessing procedure [12], based ongeneralized Richardson extrapolation forh-refine-ment studies and on theGCI proposed by Roache [13was used to establish a criterion on the sensitivitythe simulation to the computational model paramethat account for the discretization: the mesh spacand the order of accuracy. This tool estimates theder of accuracy of the numerical solution (observorder of accuracyp), and the error band where thgrid independent solution is expected to be contai(uncertainty due to discretizationGCI), also provid-ing criteria on the credibility of these estimation

See [12,14] for further details about this postproceing procedure.

Local estimators of theGCI andp are calculatedat the grid nodes where monotone convergencobserved. These grid nodes are named RicharnodesRn. Global values ofGCI andp are calculatedby means of volumetric and arithmetic averaging,spectively. It is considered that an estimation is crible when the global observed order of accuracyp

approaches the theoretical value, and when the nber of Richardson nodes is high enough. See [12]details.

Theh-refinement study is performed with five leels of refinement (n = 1, 2, 4, 8, and 16). For examplfor the finest discretizationn = 16, 141.856 CVs areemployed. Estimations are given for a zone limited0 � r � 1.59 cm and 0� z � 20 cm, which is in factthe space region that encloses the flame.

In Table 2, the main flame features, together wthe uncertainty estimates for temperature and mfractions of CO, NO, and NO2, are given for the lasthree highest levels of refinement (n = 4, 8, and 16)and both extreme cases of premixing (i.e.,Φ = ∞ andΦ = 2.464). Asymptotic convergence behavior of tmaximum temperature is observed at the centerlthe height of the flame, and the emission indexesCO, NO, and NO2. GCI values for the third level orefinement (n = 4) already reach an adequate levelaccuracy. For instance, nondimensional temperahas an average uncertainty of±0.34% (i.e., approx-imately±1 K for its dimensional value). These esmates agree with the asymptotical behavior of mflame properties.

The percentage of Richardson nodes (Rn) of thenumerical solutions presented in this article has bfound to be sufficiently large for all the variables (i.in general, higher than 75%). When the grid is refinthe number of Richardson nodes increases, andcertainty estimates reduce their value approximaby 4, as expected. Furthermore, the global orde

Table 2Verification of the numerical solutions: main flame features and postprocessing resultsa

Gridn

Tmax,C Hf EICO EINO EINO2 GCI∗ (%)

T ∗ = T /298 YCO YNO YNO2

Φ = ∞4 1908.37 5.860 0.331 3.181 0.450 3.4× 10−1 4.4× 10−4 9.2× 10−5 2.6× 10−6

8 1908.56 5.852 0.323 3.224 0.452 6.0× 10−2 1.0× 10−4 1.6× 10−5 1.2× 10−6

16 1908.51 5.838 0.321 3.248 0.452 9.4× 10−3 3.9× 10−5 6.0× 10−6 5.2× 10−7

Φ = 2.4644 2032.55 4.012 0.225 2.692 0.372 1.6× 10−1 4.7× 10−4 6.2× 10−6 1.4× 10−6

8 2033.59 3.985 0.217 2.699 0.366 3.1× 10−2 2.6× 10−4 1.7× 10−6 2.4× 10−6

16 2033.65 3.982 0.216 2.700 0.365 7.7× 10−3 2.8× 10−5 2.7× 10−7 2.9× 10−7

a Φ, equivalence ratio;n, grid parameter;Tmax,C, maximum temperature at the symmetry axis;Hf , flame height;EIx ,emission index. (For table description see Section 5.)


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Computational costs, in terms of CPU secondsouter iteration, for the three above-mentioned levof refinements are 11.12, 38.57, and 146.1, respec-tively [14]. Approximately 3000 outer iterations aneeded to reach a highly converged solution (nmalized residuals below 10−8). According to thesecomputational costs and the uncertainty estimatestained, the third level of refinement (n = 4) has beenconsidered the most appropriate to perform themerical studies presented hereafter, in terms of baccuracy (numerical credibility) and computationtime.

5.2. Validation

The numerical simulation results are compawith experimental data [2]. The conclusions on tagreement to experimental data should be carefconsidered due to the measurement uncertaintiesscribed in [2].

Table 3 lists the main flame characteristicsall levels of partial premixing. The flame height dcreases while the partial premixing level increasThe amount of oxygen introduced through the pmary inlet reduces the axial distance that it hasovercome to diffuse into the flame and to create schiometric conditions. Flame height predictions aregood agreement with experimental results (discrepcies are always lower than 7%). In all cases an oprediction is obtained.

Computational temperatures also show goagreement with experimental data. Maximum cterline temperatures disagree less than 3.4% fopremixing cases. General trends observed in eximental studies are appropriately reproduced innumerical results. Fig. 2 shows the temperature pfiles along the centerline for the different equivalenratios. As can be seen, as equivalence ratio decretemperature increases and profiles present shagradients.

,

Well-known temperature maps are obtained fornon-premixed situation, whereas for the flame wequivalence ratioΦ = 2.464, the structures of nonpremixed and premixed flames are mixed, formwhat is known as adouble flame. The outer flameregion clearly has a non-premixed structure, wherthe inner region defines a classic premixed shapethe non-premixed situation, a larger amount of freair (from the secondary inlet) enters the flame, decreasing temperature maps.

Major species centerline profiles are also plotteFig. 2. Although significant disagreements have bobtained between numerical and experimental dglobal trends of the influence of the premixing levare well predicted.

Results for the non-premixed flame show twell-known diffusion flame structure. Maximum CH4mole fractions are held at the axis, while O2 and N2surround the flame. Combustion products are formnear the stoichiometric surface. Fuel decompoforming H2, CO, and H2O on the rich side of theflame front. CO oxidizes, forming CO2 mainly atthe top of the flame. As the level of premixing icreases, the reactants (CH4 and O2) need further timebefore they start to react. The higher injection velities produce relatively flatprofiles at the inner tubexit. Sharper gradients are also predicted.

The double flame structure effects can also beserved in the H2O mole fraction profiles presentein Fig. 2 for Φ = 2.464. The presence of a markpeak followed by a second smaller one denotesmain zones of H2O production. At the first flamefront, with premixing features, the mole fraction peincreases with the level of premixing. H2O measure-ments present a more aleatory behavior (experimecalibration difficulties for H2O mole fractions due tothe low vapor pressure conditions are reported in [However, both computational and experimentalsults coincide with the fact that the greatest amoof H2O is produced at the inner flame front, and wthe observation of an inner peak for the higher pmixing conditions.

en-

Table 3Main flame characteristics for the different levels of premixinga

Φ Tmax,C(K)

Hf(cm)

Tmax(K)

(r, z)

(mm)RHL(W)

HR(W)

EICO(g/kg)

EINO(g/kg)

EINO2(g/kg)

∞ 1908(1960) 5.9 (5.7) 2005 0.66, 0.66 35.11 181.05 0.33 3.2 0.4512.32 1933(2000) 5.6 (5.2) 2009 0.65, 0.72 34.83 180.99 0.30 3.2 0.466.160 1956(2020) 5.2 (4.9) 2017 0.63, 0.97 34.75 180.91 0.28 3.1 0.434.107 1980(2040) 4.8 (4.5) 2031 0.61, 1.41 34.79 180.74 0.26 3.0 0.412.464 2033(2090) 4.0 (3.8) 2083 0.57, 1.97 35.04 180.35 0.22 2.7 0.37

a Φ, equivalence ratio;Tmax,C, maximum temperature at the symmetry axis;Hf , flame height;Tmax, (r, z), maximum flametemperature and location;RHL, radiant heat loss;HR, heat release;EIx , emission index. Experimental results [2] are in partheses.


Fig. 2. Comparison of numerical results (lines) and experimental data (dots) by [2] along the symmetry axis.


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OH mole fractions are also given in Fig. 2. Genetrends coincide for both computational and expemental profiles. Once again, the double flame strture for the highest levels of premixing is predictedthe presence of an inner peak. In this case the amof OH formed in the inner flame is considerably lowthan that produced in the outer one.

6. Discussion

6.1. Pollutant formation

One of the most relevant aspects in the anasis of co-flow methane–air flames are the studiepollutant formation and the influence of equivalenratio in partially premixed flames. Global trendsNOx emission predicted by GRI-Mech 3.0 are givin Table 3. As can be seen, similar amounts of Nand NO2 are predicted for equivalence ratios btweenΦ = 4.107 andΦ = ∞. When the premixinglevel increases, the higher production of thermalis balanced by a reduction of prompt NO [28]. Athough the residence time decreases when the levpartial premixing increases, the higher temperatuachieved between inner and outer flames increaseamount of NOx produced. For lower values ofΦ, areduction in NOx is predicted. Similar trends werobserved by Gore and Zhan [28] in their experimenmeasurements. They suggested that these reduccould be associated with the reduction of prompt Ncaused by intermediate hydrocarbon chemistrycontributions from the reverse prompt mechanismscribed by Takagi and Xu [29]. Isopleths for NO aNO2 are plotted in Fig. 3. Higher NOx emissionsare observed qualitatively for the non-premixed fla(left figures). A larger amount of prompt NO formemainly at the stoichiometric surface is predictedthe non-premixed flame. NO2 contours show the double flame structure forΦ = 2.464.

Although CO mole fraction experimental mesurements are not provided in [2], computationalsults are also given to show the premixing dependeof its production (Fig. 2). As can be seen, for tnon-premixed flame, CO production does not presa remarkable peak. It is progressively producedthe inner flame region. At the top of the flame itpartially consumed to produce CO2. As the level ofpremixing increases, the CO peak becomes slenO2 injected through the primary inlet does not reduCO concentration, as CO is oxidized mainly byactions involving OH radical. At the inner premixeflame front, a considerable amount of CO is producAt the inner diffusion flame, carbon monoxide is prgressively consumed.

CO emission indexes are also given in TableCO emissions are small in comparison with tho

Fig. 3. Pollutant formation. Mole fraction isopleths for: ni-trogen oxide, NO (top); nitrogen dioxide, NO2 (bottom).Two premixing levels:Φ = ∞ (left); Φ = 2.464 (right).

of NO. An increase of about 10% is predicted froΦ = 12.32 toΦ = ∞. These differences increase ntably when the highest level of premixing is consered, reducing CO emission index to about 33%. Ttendency confirms the experimental results presein [28].

6.2. Mathematical submodel analysis

The adequacy of different mathematical submels is presented here. For comparison purposesmathematical model described in Section 2 is refeto as thereference model.

A summary of these studies is presented inble 4 for the extreme premixing situations (Φ = ∞andΦ = 2.464). Maximum temperature at the centline, flame height, maximum flame temperature, aits location and emission indexes for CO, NO, aNO2 are listed in this table for all modeling testeThe comparison is performed in such a way thatreference modelcriteria are always applied except t


Table 4Mathematical submodel analysis: main flame characteristicsa

Approach Modelalternatives

Tmax,C(K)

Hf(cm)

Tmax(K)

(r, z)

(cm)EICO(g/kg)

EINO(g/kg)

EINO2(g/kg)

Φ = ∞Experimental [2] 1960 5.7 – – – – –Ref. model 1908 5.9 2005 0.66, 0.66 0.33 3.2 0.45Boundary T fixed 1885 6.1 1968 0.61, 1.19 0.37 2.7 0.43Chemistry GRI-Mech 2.11 1904 5.7 2000 0.65, 0.78 0.35 1.6 0.21

GRI-Mech 1.2 1905 5.7 2002 0.66, 0.59 0.37 – –42-Step-Mech 1900 5.6 2025 0.66,0.66 0.66 – –Flame-Sheet 1971 6.6 2244 0.64, 0.00 – – –

Radiation No-Radiation 2106 5.7 2106 0.00, 5.69 0.19 4.4 0.42Transport Lefixed 1891 5.9 2001 0.65, 0.78 0.33 3.2 0.46

Leunity 1997 6.1 2042 0.67, 0.48 0.44 4.0 0.45No-Soret 1910 5.9 2013 0.66, 0.66 0.32 3.2 0.41

Φ = 2.464Experimental [2] 2090 3.8 – – – – –Ref. model 2033 4.0 2083 0.57, 1.97 0.22 2.7 0.37Boundary T fixed 2005 4.2 2047 0.54, 2.42 0.26 2.1 0.34Chemistry GRI-Mech 2.11 2029 3.9 2068 0.54, 2.12 0.24 2.0 0.23

GRI-Mech 1.2 2030 3.9 2068 0.54, 2.12 0.26 – –42-Step-Mech 2024 3.9 2085 0.59, 0.84 0.46 – –

Radiation No-Radiation 2212 4.0 2211 0.00,3.90 0.14 4.1 0.39Transport Lefixed 2016 4.0 2078 0.63, 1.34 0.22 2.7 0.36

Leunity 2120 4.3 2139 0.63, 0.84 0.29 3.7 0.38No-Soret 2033 4.0 2084 0.63, 1.26 0.22 2.8 0.35

a Φ, equivalence ratio;Tmax,C, maximum temperature at the symmetry axis;Hf , flame height;Tmax, (r, z), maximum flametemperature and location;EIx , emission index.

ned

i-re

tsted,m

ofile

nt

toumsionaf-lly

or

chevi-

ch-1].ap-c-

et fortheom-,.5%erRI-ts ofter-the

ac-ionlar. InchI-con-

model alternative underanalysis. The results obtaiare given below.

6.2.1. Energyboundary conditionThe energy boundary condition used in therefer-

ence model(Eq. (4)) is compared with a simple condtion, which consists of fixing an ambient temperatuof 298 K atz = 0. The influence of both treatmencan be observed in Table 4 and Fig. 4. As expecwhen the inlet temperature is fixed, lower maximutemperatures are predicted. The temperature pralong the symmetry axis for both flames (Φ = ∞andΦ = 2.464) is plotted. There is better agreemewith experimental data when thereference modelisapplied. Temperature profiles for both flames tendmove toward the right, also increasing the maximtemperature at the centerline. Furthermore, emisindexes of pollutant (see Table 4) are significantlyfected for the energy boundary condition, especiaNO formation, which is underpredicted at 18% fΦ = ∞ and at 28% forΦ = 2.464.

6.2.2. Chemical modelsNumerical results obtained employing GRI-Me

3.0 are compared with those obtained using its pr

ous releases (1.2, 2.11) [30] and with a 42-step meanism (hereinafter referred to as 42-Step-Mech) [3The simple irreversible single-step flame-sheetproach [7] is also included, giving an idea of its acuracy.

In a general way, on the prediction of main flamfeatures (see Table 4), there is certain agreemenall mechanisms employed (without consideringflame-sheet approach). The most unfavorable cparison with thereference modelis for 42-Step-Mechwhich predicts a temperature at the centerline 0lower (Φ = 2.464) and also a flame height 5.4% low(Φ = ∞). These differences decrease when GMechs 2.11 and 1.2 are employed. Disagreemenabout 0.2 and 3.5% for temperature at the cenline and flame height, respectively, are obtained inmost unfavorable situations compared with therefer-ence model.

GRI-Mechs 2.11 and 1.2 contain the same retions except the reactions involved in the predictof NOx , which are added in 2.11. Thus, very simiresults for main flame characteristics are obtainedcenterline and radial profiles (Figs. 5–9), GRI-Me1.2 is not plotted due to its visual fitting with GRMech 2.11 results. This agreement suggests the


pro-ary

op:

ng

xesCOs0%.spe-

-

dis-ses.re-b-nes

pro-

ualeen

m-

ter-

dic-t atsee

ainlo-

benceeg-

cal

Fig. 4. Mathematical submodel analysis: temperaturefiles along the symmetry axis. Energy equation boundcondition comparison:reference model(solid line) versusfixed temperature boundary condition (dashed line). TΦ = ∞; bottom:Φ = 2.464. Experimental results in [2].

sideration of NOx mechanisms in a postprocessiprocedure (see [32]).

Greater differences occur when emission indeare evaluated. 42-Step-Mech clearly overpredictsformation (approximately 100%), while GRI-Mech2.11 and 1.2 present disagreements of about 1Relative to NOx formation, GRI-Mech 3.0 predicthigher production of these species. This fact is escially important in the non-premixed condition (approximately 50%).

Radial profiles of NO and NO2 mole fractions(Figs. 8 and 9) emphasize the above-mentionedagreements between GRI-Mech 3.0 and 2.11 releaAs previously commented, Version 2.11 underpdicts nitrogen oxide production. The differences otained for these species are the most important oin the comparison of mathematical approaches.

Fig. 5. Mathematical submodel analysis: temperaturefiles along the symmetry axis. Top:Φ = ∞; bottom:Φ = 2.464. Experimental results in [2]. There are no visdifferences between GRI-Mechs 2.11 and 1.2, and betwthereference modeland the No-Soret model.

6.2.3. RadiationWhen the radiation model is not considered, i

portant differences as to thereference modelare ob-served. For the maximum temperatures at the cenline, differences of nearly 200 K for theΦ = ∞ flameand approximately 180 K for theΦ = 2.464 flame areobtained (see Table 4). These temperature overpretions are maintained along the centerline, excepthe inner tube exit, where both profiles concur (Fig. 5).

The distributions of CO2 and H2O mole fractionshave a similar trend (Figs. 6 and 7). In fact, the mdifferences between both modelings appear whencal radiative heat loss is important. At the inner tuexit and for low enough temperatures, the influeof radiant heat source on the energy equation is nligible. When this term increases in importance, lo


er

gni-

r-theO

ced-re

ms.thareon

eg

er

re-

rm-

es,

isg.,-ted

edhe

Fig. 6. Mathematical submodel analysis: CO2 mole frac-tion profiles along the symmetry axis. Top:Φ = ∞; bottom:Φ = 2.464. Experimental results in [2]. See Fig. 5 for furthexplanation.

temperature is clearly affected, decreasing its matude notably.

Referring to nitrogen oxides, important diffeences are also predicted. In general, neglectingradiative heat loss implies an overprediction of Nformation by a factor of 2, which is in concordanwith the results of Barlow et al. [19]. In fact, anbeing thatthermal NOx is one of the major contributions to NOx formation, an increase in temperatuconsequently supposes an increase in NOx due to thegreat temperature dependence of these mechanis

For NO2 predictions, the levels of the peak wiand without consideration of the radiation modelsimilar (Table 4), but a delay on this peak formatiis revealed (Fig. 9). Most NO2 is formed at the flamefront and it is basically attributed to aprompt pro-duction. NO2 radial profile at the base of the flam(z = 10 mm) for the maximum level of premixin

Fig. 7. Mathematical submodel analysis: H2O mole frac-tion profiles along the symmetry axis. Top:Φ = ∞; bottom:Φ = 2.464. Experimental results in [2]. See Fig. 5 for furthexplanation.

considered in this work (i.e.,Φ = 2.464) shows thedouble flame structure characteristic of partially pmixed flames.

6.2.4. Mass transport coefficientsThe reference modelcalculation of mixture dif-

fusion coefficientsDim is compared with two othepossibilities based on the definition of the Lewis nuberLei = λ/(ρDimcpi ):

• Assuming a fixed Lewis number for each specifor instance, LeCH4 = 0.97, LeO2 = 1.11,LeH2 = 0.3. For major species, these fixed Lewnumbers are provided in the literature (e.see [33]). In this work, when fixed Lewis numbers are not provided, they have been evaluaby averaging the local Lewis values obtainfrom the numerical results performed using t


ion

e

x-s aes.xi-en-ed

mewist isix-

ed.is

-en-oth

ons

a--

r5

ms

owf-

Fig. 8. Mathematical submodel analysis: NO mole fractradial profiles at the axial position of 10 mm. Top:Φ = ∞;bottom:Φ = 2.464. See Fig. 5 for further explanation.

reference model(e.g., LeNO = 1.09, LeNO2 =1.25, LeN2O = 1.39, LeCN = 1.09, LeCH2CO =1.56,LeCH2OH = 1.35).

• Assuming a unity Lewis number for all thspecies involved in the chemical model (Lei =1.0, i = 1,2, . . . ,N ).

Both approximations are commonly used, for eample, for the flamelet approach [34,35], which itechnique usually applied to model turbulent flamThus, knowledge of the accuracy of these appromations is necessary for both the CPU savingscountered and for proper application of the mentionflamelet approach.

Excellent agreement for global and detailed flaproperties is achieved (see Table 4) when fixed Lenumbers are used, as the computational efforless intensive in comparison with the complete mture averaged formulation employed in thereferencemodel. CPU time savings of about 25% are obtainOn the other hand, when a unity Lewis number

Fig. 9. Mathematical submodel analysis: NO2 mole fractionradial profiles at the axial position of 10 mm. Top:Φ = ∞;bottom:Φ = 2.464. See Fig. 5 for further explanation.

considered (second approach), considerable disagreements occur. The maximum temperature at the cterline and the flame height are overpredicted for bextreme levels of premixing (see Table 4). CO2 andH2O mass fraction profiles present relevant deviatito experimental measurements (Figs. 6 and 7).

From observation of the emission indexes,EICOis overpredicted with respect to thereference model.Differences of about 33% forΦ = ∞ and 30% forΦ = 2.464 were observed. EvenEINO2 does not varysignificantly; EINO is also overpredicted with 25%for Φ = ∞ and 39% forΦ = 2.464. Radial profilesof NO show an overprediction of this pollutant formtion (Fig. 8), while profiles of NO2 are not so dissimilar (Fig. 9).

Thermal diffusion ratios are important only fochemical species with mass weights lower thang/mol, so for the methane combustion mechanisconsidered in this work, only H and H2 are taken intoaccount. Main flame features listed in Table 4 shthat the contribution of thermal diffusion (Soret e


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eth-five

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fect) is not very important in these flames. The mimportant contributions in species discretized eqtions are due to chemical reactions and ordinaryfusion fluxes, and thermal diffusion has a minor cotribution.

7. Conclusions

The influence of the level of premixing on thmain features of a partially premixed co-flow meane–air laminar flame has been investigated forlevels of premixing fromΦ = ∞ (non-premixedflame) toΦ = 2.464. Special emphasis has been givto pollutant formation (CO and NOx ). In addition, theefficacy/suitability of various existing mathematicsubmodels for the solution of these kind of flamesbeen analyzed for both extreme levels of premixin

Computations were submitted to a verificatiprocedure to estimate the accuracy of the numersolutions. This analysis gives an error band wheregrid independent solution is expected to be contai(GCI), and also gives a criterion for mesh selection

A validation process of the numerical solution fall levels of premixing selected was carried out oserving that the better agreement with experimedata provided in the literature is achieved whenproposed energy boundary condition is employed.

The influence of the level of premixing on polutant formation has been studied, and concordawith results published by other authors was noted.low Φ, a reduction in NOx formation is predictedFor the non-premixed flame, CO production doespresent a remarkable peak. As the level of preming increases, the CO peak becomes slender. Theemission index increases about 10% fromΦ = 12.32to Φ = ∞.

On the prediction of main flame features, theis a certain agreement for all mechanisms employHowever, significant disagreements have beentained for the estimation of emission indices. WhGRI-Mech 2.11 is used, a lower prediction of NOx

is found compared with GRI-Mech 3.0, especiallynon-premixed conditions. GRI-Mech 3.0 doubles theNOx emission with respect to GRI-Mech 2.11 for tnon-premixed flame, while forΦ = 2.464 it is 25%higher. While GRI-Mechs predict similar amountsCO formation, 42-Step-Mech overpredicts considably.

With respect to the transport modeling approaanalysis, the formulation considering nonunity Lewnumbers was shown to be suitable in terms of bagreement with complex transport and computatiosavings. The employment of fixed Lewis numbemeans a 25% CPU time reduction. On the other haconsideration of a unity Lewis number notably ov

predicts the maximum temperature at the centerand the flame height. It is also shown that thermalfusion (Soret effect) has a minor contribution in thekind of flames.

The consideration of radiant heat exchangebeen shown to affect flame temperature considerand, therefore, pollutant formation. Thus, a betreatment of this phenomenon should to be invegated.

Acknowledgments

This work has been financially supported by tComisión Interministerial de Ciencia y TecnologSpain (Project TIC1999-0770), and by the Comsionat per Universitats i Recerca de la GeneralitaCatalunya. The authors also acknowledge the ussuggestions given by Professor Raymond Viskant

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