Combust. Sci. andTech .,176 : 585-601, 2004Copyright (C5 Taylor & Francis Inc.

ISSN: 0010-2203 print/1563-521X online

DOI : 10.1080/0010220049027680 9

PDF MODELING OF CO AND NO FORMATION INLEAN PREMIXED METHANE FLAME S

WILLIAM VICENTE,* MARTÍN SALINAS ,AND ESTEBAN BARRIO S

Engineering Institute ,National Autonomous University of México ,México City, Méxic o

CÉSAR DOPAZO

LITEC and University of Zaragoza ,María de Luna,Zaragoza, Spai n

In this paper, CO and NO formation in a premixed turbulent methane flame i s

sirulated with a stochastic model of combustion . The model proposed is a

combination of both the computational fluid dynamics and the Monte Carl o

methods for the solution of the joint probability density function . Finite

chemical kinetics is represented by a GRI-derived reduced-chemistry model .

Tris resultant model is used to simulate a lean, premixed, bluff-body, stabi-

lized llame for which experimental data are available . Under this condition ,

the prediction of NO formation is a challenge because of its low concentra-

tions (typically a few parts per millian) and because every NO-formatio n

roule is relevant. The model used for the molecular mixing includes a variable

mixing time, covering the range from the Kolmogorov seale to the integra l

se-ale . A lookup tale is used to estirnate the thermochemical properties and i s

Received 14 May 2002 ; accepted 23 October 2003 .

The authors are very grateful to J .Y. Chen (University of California at Berkeley) for

his advice and stimulating discussions . Work in PDF modeling for turbulent flames by

the Fluid Mechanics Groups from the University of Zaragoza and UNAM is partially sup-

portecl by the European Union under project BE 95-1927 and at the Instituto de Ingenierí a

under project FI 3 .

*Address correspondence to wvr@vortex.iingen.unarn.mx

Taylor & Franci sTayíor Fx F rama Group

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found to be more adequate than direct integration . The results are comparedwith an experimental database .

Keywords : PDF, lean premixed combustion, CFD, Monte Garlo method ,NOx

INTRODUCTIO N

Lean premixed combustion (LPC) in low-NO X burners is becomin gan increasingly attractive option for practical combustion facilities .Compared with non-premixed flames, and with near-stoichiometri cpremixed combustion, LPC is the alternative technology with the lowes tNOx concentrations without a postflame treatment . Because of the lowtemperatures afforded by the excess percentage of air, the thel mal NO ,is far less active in LPC, though it is dominant in other combustio ntechnologies, and NO,, emissions below 10 ppm (at 15% excess oxygen )are often achieved .

However, LPC is not free of potential pitfalls ; among them, it is wort hmentioning the inconvenience of flame instability, as well as CO an dunburned hydrocarbon fotmation . All of these inconveniences (NO x, CO,UHC foinnation, and flame instability) are determined by the finite che-mical kinetics effects and by the interaction with the flow field . Designingmodem LPC systems such as gas turbines demands the use of numerica ltools to allow the study of this interaction within a turbulent flow .

One of these numerical tools is the solution of the joint probabilit ydensity function (PDF) of several scalars that determine the thermo-chemical state of the Huid in conjunction with the Huid dynamics equations .

Efficient methods for estimating this multidimensional PDF wereproposed in the early 1980s (Pope, 1981), but because of the CPU timecosts needed for these methods, this numerical approach is expensive . Thehigh cost of this method is a constraint, especiall .y because of the amountof independent variables to be solved . Given the available computingcapabilities at present, the PDF estimates must be limited to a rank from5 to 10 scalars, all of them depending on the space and time variables .

Nearly 300 steps and 50 species in methane—air combustion are used fo r

the detailed chemical kinetics system, with NO x fornation included .In this paper, a method based on joint PDF resolution for only fiv e

species is used to simulate the laboratory-scale methane—air flame o f

Nandula et al . (1996), as described in the next section. The stochastic

model is composed of a reduced chemical system, which considers NOS.

GO AND NO FORMATION IN LEAN FLAMES

587

formation through all the relevant routes, and the estimation of the jointPDF equation for the five scalars is based on a Monte Carlo method .This model, including the molecular mixing model developed by Dopaz o(1992), has been applied before for the lame experimental work b yFueyo et al . (2000) . Both numerical and experimental results have bee nsuccessfully compared in this work .

In the present work, the mixing time in the molecular mixing model isnot considered constant, and it is estimated from a local PDF (see Result sand Discussion) . The effect of the mixing time on the scalar is studied an dshown in the conclusions .

Finally, the chemical kinetic rates are not only estimated . from thelookup table (LUT) method developed by Chen et al . (1989), but alsofrom the direct integration of the ordinary differential equations (ODEs) .These results are commented on in the conclusions .

EXPERIMENTAL CASE CONSIDERE D

The experimental work simulated here is the methane–air turbulent flam emade by Nandula et al . (1996). This is a lean, premixed, confined, bluff-body-stabilized flame, A schematic of this setup is shown in Figure 1 . Theair and methane mixture is fed into the burner through an annular ring ,surrounding a circular obstarle that acts as a bluff body . A recirculationregion is produced downstream of the bluff body ín which the trappe dhot products serve to anchor the flame. Eddies (transient) issued from thebluff body are helpful to the reaction between the mixture and thecombustion products . The mixture in the flame is lean, with an equiva-lence ratio of 0 .586. The nondimensional Reynolds number in the flam eflow is 66,000 based on the inlet gas velocity and the disc diameter .

Detailed experimental data are available from this laboratory flam e(Nandula et al ., 1996 ; Pan et al ., 1991) . This data set consists mainly o f

radial distributions of velocity and temperature, as well as major an d

minor species, at several axial locations . The major species measured are

CO2, CH4, H2, 0 2, and H20, and the minor species are CO, OH, and NO .

MODEL1NG

Chemistry

A systematically reduced chemical system has been used for the con-

sideration of finite chemical rates. This chemical system is the one

588

W . VICENTE ET AL .

Figure 1 . Schematic view of the [)ame considered (dirnensions in mm) :

= 0.1, 0.3, 0 .6 ,

and LO are axial locatioris where experimental data are available .

reported by Mallampalli et al . (1998) and is based in the fuli Ga sResearch Institute (GRI) mechanism 2 .11 (Bowrnan et al ., 1995) .Automated reduction software reported by Chen (1988) was per-forrned ín the derivationsystem of five steps

of this reduced mechanism . The reducedfollowingand vine species is described by th e

reactions:

(1 )4OH

> 02 + 2H20

20H + 0 .33CH4 1 .67H 20 + 0 .33CO (2)

H2 + 0 .33C0 <

> 0 .33H20 + 0 .33CH4 (3)

79.00

44.45 --►1

metane + air

CO AND NO FORMATION IN LEAN FLAMES

589

H2 + 40H + 0,3300 + N2 <--->> 2 .33H20 + 0 .33CH 4 + 2N0

(4)

20H + CO 4-- ;> H2 O + CO2

(5)

It is well known (see Nicol et al ., 1999) that, for LPC combustion, al lthree NOX formation routes (therrnal or Zeldovich, prompt, and nitroge noxide route) may contribute significantly to the NO X levels . The forma-tion of NO through all of these routes is represented in the previou schemical system by reaction (4) .

After the end of this study a new version of the detailed GR Imechanism has been developed . This new GRI-3 .0 (Smith et al ., 1999 )reduces the possible errors in the NO formation observed in the 2 .1 1version. Many tests have been performed to determine the influence o fthis new GRI version on results .

Under the hypothesis of an equal mass diffusion coefficient for all o fthe species, and in the absence of any hect loss, the therrnochernical stateof this system is determined by five scalars in the present work ; they areassurned to be the specific mole numbers (or moles per unit mass) fo rH2O, CH.4, CO, OH, and NO . Heat losses by radiation are neglecte din this work because their effect on the formation of minor species is

negligible (see Fueyo et al ., 2000) .A direct integration during the PDF calculation of the system com-

posed of the ODEs [In Eq . (6), Sa dq /dt] for a reduced mechanism ina turbulent-flarne calculation is impractical because it requires a CP Utime overhead . An alternate solution to alleviate this problem includesthe precalculation of the related thermochemistry and its storage i nthe forro of a LUT (Chen et al ., 1989), to be recalled during the maincalculation. However, the LUT method performed in this work couldpresent problerns with the grid size and with the limits for the spac e

composition required in the calculation of the PDF .

Aerodynamics

The flame aerodynamics is rnodeled via Favre-averaged continuity an d

rnomentum equations . The turbulent-convective terrn, resulting from th eaverage of the convective terms in the conservation equations, wa smodeled with the Renormalization Group (RNG) version of k–s turbu-lent model (Yakhot and Orszag, 1986) .

59 0

W. VICENTE ET AL.

The llame aerodynamic equations are solved using a finite-volum emethod, with a SIMPLE-type algorithm (Patankar and Spalding, 1972) t osolve the pressure—velocity coupling . For more details see Vicente (2000) .

Composition PDF

The joint PDF for the scalars that determine the thermochemical state ,namely ,

P( nH,o, n CH 4 , nco, nOH ) nNO )

shall be estimated from the solution of its transport equation in a Favre-average form (Dopazo, 1992) :

0(PP)+v . pvP + D0t (P"pc =0)P)

(6)((D Ja lc = 0)P) — 12. (SÉ)

a- f -ea

a= * 30,

where the overbar denotes Reynolds averages, the tilde is used fo rFavre averages, V is the mean Huid velocity, P" is its Favre fluctuation ,the subindex a denotes the ath scalar, e is the scalar vector, O.; is a vectorof values of the scalar, O is a component of the preceding vector, Ja is thediffusion flux of the ath scalar, and Sa is the sourceterm due to chemica lreaction of the ath scalar .

The joint multiscalar PDF [Eq . (6)] is solved using a cell- based ,Monte Carlo particle method (Pope, 1981) in which the PDF is repre-sented locally (i .e ., at each computational cell) by an ensemble of sto-chastic partirles .

The first term in Eq . (6) represents the local temporal change in th ePDF, the second term is the convection of the PDF by the mean flow, thethird is the turbulent convection, the fourth represents the effects o fmixing at the molecular level, and the fifth is the PDF change because o fchemical reactions . Among them, the temporal, mean convection, an dchemical reaction tenns are closed but the turbulence convection an dmixing terrns are not .

Turbulent convection is closed in the present work using an eddy-diffusivity model, namely,

CO AND NO FORMATION IN LEAN FLAMES

59 1

0 (P" c = 19 5) _ — v . (rr, O)5)

( 7 )

where F t is an eddy-diffusivity coefficient and F t = ,ut/0 .7 .The molecular mixing term inhibits scalar fluctuations, bringing th e

probability of the scalar toward the scalar mean . Closure of this termrequires in principie the knowledge of the joint statistics of the scalarsand their gradients (Dopazo, 1992) . The need for such knowledge isusually solved by resorting to the so-called molecular mixing models . Atthis point, the linear mean square estimation (LMSE ; Dopazo, 1973) i sperformed . This model postulates a proportionality between a char-acteristic mixing time and k/e . In the literature (Fueyo et al ., 2000), thi srelationship has been assumed to be

T mix = k/CD i

( 8 )

where the CD term is equal to 4.0 in the baseline case . However, the valuefor this constant may change. In this work, different values of CD areused to evaluate its effect on the predicted values ; CD Cakes the additiona lvalues of 3 .0 and 8 .0 . (For CD = 2.0, the reaction on the flame loes no texist .)

The use of a single scale mixing time (from the author's poínt of view)is unrealistic. Therefore, a PDF is used both to create a variable mixin gtime (`zmix) and to study its effects on the num.erical results . The PDF isassumed here to be a double delta density function . One delta is in theKolrnogorov scale (-Kol ) and the other is in an integral scale ('TI) . Then thePDF for local mixing time results in the foliowing :

P('Vmix-PDF) = a3(Zmix — TKol) + bb(Tmix — TI)

(9 )

where a + b = 1 and 'rmix = k/ CD 1 .Equation (9) evaluates the local mixing time from a randorn selectio n

between values of the Kolmogorov time and the integral time . This

selection is a function of the ratio (zmix — tI)/('Kol — TI) .

Numerical Details

The simulation is a two-dimensional domain on the bluff body . Just a

half of the total diameter of the bluff body is considered in the radial

direction and the axial direction extends 4.6 diameters . A 40 x 45

59 2

W. VICENTE ET AL .

(radial x axial) mesh is used with a refinement near the wall and in th eshear layer. A grid-refinement test was carried out using a (cheaper) edd ybreak up model (Spalding, 1971) ; the results obtained in this mesh prove dto be grid independent .

Typically, 100 stochastic particles/cell were used for the Monte Carl omethod . Steady accumulation of particles was implemented to avoi dstochastic fluctuations when calculations of mean values were performed .The number of particles was doubled lo 200/cell in the course of test run swithout noticeable change in the average values .

CPU running time was around 28 h on a Pentium II PC at 300 MHzrunning Linux for the LUT method . Six days were necessary in a 64 + 2Pentium III CPU's cluster for the direct integration of the ODE system .The CPU time for this simulation in a sequential computation is up to200 times the time spent in the LUT cases .

RES ULTS AND DISCUSSIO N

Figure 2 shows the radial evolution of the axial velocity at two axia llocations . Three different CD values are considered, and results arecompared with the experimental data of Nandula et al . (1996), Theresults are satisfactory in general, both in the shear layer and in therecirculation region . Differences are due to the turbulence model used,which does not consider the anisotropies of the flow (Vicente, 2000) . Theeffect from the CD value on the axial velocity is negligible .

Figure 3 shows the radial evolution of the concentration of twomajor species, CH4 and H2O, and of temperature at two axial loca-tions, Major species are in good agreement, both in the shear layer an din the recirculation zone . Temperature is over-estimated by around 10 0K in the recirculation region . Because CH 4 levels are accurately pre-dicted and near equilibrium values, the neglect of radiative effects i nthe heat transfer process is the main cause of this overprediction (se eVicente, 2000) . Major species and temperature are affected . by the CDvalue in the shear layer . Here, the high temperature produces thereaction of the air–methane mixture in the flue gases . The scalar gra-dient is increased when values for CD are enhanced, because the mixingmodel approaches the probability of the scalar toward its mean value

with a bigger rase . As a consequence, the chemical rates of pro-duction/destruction of the major species become fast enough to relaxthe concentration into equilibrium when they are competing with

CO AND NO FORMATION IN LEAN FLAMES

593

z/lio=1 .0

CD=3 .0CD=4 . 0CD=8 . 0

Exp

z/Do=0 . 3

0.3

0 . 6r/Do

0.3

0 .6rlDo

Figure 2. Radial profiles of axial velocity at axial stations a/D0 = 0.3, 1 .0 with three Cj va-

lues of local rnixing time and comparisons with experimental data .

mixing. The preceding effects produce a reaction zone with a narrowe r

thickness as CD is increased .

Figure 4 shows the radial evolution of concentrations for some ke y

minor species for which experimental data are available. These mino r

species are CO, OH, and NO. CO is assumed to be completely burned i n

the recirculation region, which is in disagreement with experimental data .

However, the peak values in the shear layer are in good agreement. OHprofiles show great agreement on the centerline, where the OH con-

centration is close to its equilibrium value of 200 ppm . The trend towar d

superequilibrium values is well predicted in the shear layer, although th e

calculated peak values exceed the experimental value . The overall shap e

of the NO profile is also well captured, although values in the early par t

of the recirculation regían are underpredicted by 5 ppm . Nandula et al .

(1996) point out that NO measurements are likely to be 2 or 3 ppm i n

excess due to interferente of 02 in the NO channel .

594

W. VICENTE ET AL .

0.18

0.16 -

0.14--

0 .12

0.10 -

0 .08 -

0 .06 -

0.04

0.02 -

z/Do=0 .3 z/Do=O, 3z/Do=0 . 3

z/Do=1 .0

z/Do=1 .0

z/Do=1 .00 .1 8

0 .16

0.14

0 .12

0 .1 0

0.08

0.06

0.04

0.02

0 .00

0 0.3

0 . 6r/D o

o

Figure 3. Radial evolution of majar species (molar fractions) and temperature at two axia llocations (top: z/D = 0 .3 ; bottom : z/D = 1 .0) . Comparison with three values of CD : 3 .0, 4 .0(baseline case), 8.0, and comparison with experimental data .

The effect of the CD value on CO, OH, and NO predictions i srepresented in Figure 4 . Differences can be remarkable in the shear layer ,and particularly for CO and OH, for which an increase in the CD value isrelated to an enhancement in the peak values . This effect is caused by

CO AND NO FORMATION IN LEAN FLAMES

595

z/Do=1 .Q

z/Do=L0

zJDo=1 .0

z/Do=0.31

CD=3 .0

Co-s .0 - --• -Exp ♦

z/Do=0.3 z/Do=0 . 3

V 0

0 .3

0 .6

0

0 .3

0 . 60 .3

0, 6rlL7o

r/Do

r/Do

Figure 4 . Radial evolution of minor species at two axial locations (top : z/D = 0 .3 ; bottom:

z/D = 1 .0) ; comparison with experimental data .

reasons similar to those observed in the major species case. Changes

of CD for the NO concentrations are very small . This is probably because

the chemical rates for the NO-related reactions work too

m mixingo warto

relax the NO concentration to the equilibrium, wherea sl

faster .

596

W . VICENTE ET AL .

The effect of a local variable mixing time can be ascertained fro mFigure 5, which shows the radial profiles of temperature and CO and N Oconcentrations at z/D = 0 .3 and 1 .0 . Both cases are similar and, satis-factorily, in respect of minor species and temperature . Differences can b e

z/Do=1 .0

z/Do=0 . 37000

600 0

500 0

1000 -

0 .3

0 . 6

z/Do=1 .0

0.3

0 . 6r/Do

z/Do= 1 . 0

Si ngléPDF(t-mix )

Exp • -

Figure 5. Radial evolution oF temperature, CO and NO concentrations at z/D = 0.3 (CO)

and 2/D = 1 .0 (T, CO, and NO) ; comparison between a single value and a variable loca l

mixing time (PDF) .

CO AND NO FORMATION IN LEAN FLAMES

597

remarkable in the shear layer, for which the variable mixing time predict shigher peak values . The effects of the variable mixing time on tempera-ture and species are similar to when CD values are increased . The NOconcentrations are not affected by this modification to the characteristi cmixing time .

Figure 6 is used to compare the radial profiles of temperature, CO .,and NO concentrations with both the LUT method and the direct inte-gration of the ODE system . Very small differences can be noticed at thelast axial position (z/D = 1 .0) for T, CO, and NO . The LUT metho dshows very small errors, even for the minor species, and when its CP Utime is compared with the corresponding CPU time spent for the direc tintegration, it is about 200 times smaller .

Figure 7 shows the differences between the two versions of the ful lGRI mechanism . Note differences are observed in the main species and i n

the temperature . Differences can be spotted in the CO profiles, for whic h

GRI-3 .0 predicts a peak value 5% higher . The principal modification in

the new GRI version is in the CH kinetics to prompt NO . However, thi s

mechanisrn has shown a very small (<5%) contribution to the NO for-

mation for LPC (Vicente, 2000 ; }die et al ., 1996) . We can conclude tha t

modification in the new GRI version does not affect the results of ou r

study .

Discrepancy between numerical and experimental data for mino r

species are found for both GRI versions, principally in the case of CO i n

zlDo=1 .o

zJDo=1 .0 z/Do=1 .03o

LUT —DI

-Exp

25

20

o- 1 5aoz 1 0

o

0

0 .3

0, 6r/Do

Figure 6. Radial evolution of temperature and CO and NO concentrations at z fD = 1 .0 ;

comparison of the LUT method and the direct integration of the ODE system .

598

W. VICENTE ET AL .

2500

2000

1-

zIDo=0.38000

GRl-2 .11 -GRI-3 .0 - -

Exp +

6000

5000

7000

400 0

° 3000

zlDa=0 .3

zlDo=0. 330

GRI-2 .11 - GRI-2,11 -GRI-3 .0 -- - GRI-3.0 - -_

Exp + 25 Exp +

_

2 0

1 5

. .++ + + + +10 -

1000200 0

1000-500

0

0.3r)Do

zJDa=1 . 02500

800 0

7000

600 0

0.6

20

1 5

10 -

000 +

0 .3

0,6

0.3

0.6rlDo

0.3

0 .6r1Do

GRI-2 .11-GRI-3 .0 –-

Exp +

3 0

2 5

Figure 7 . Radial evolution of temperature and CO and NO concentrations at z/D = 0.3 and1 .0 ; comparison between GRI versions 2.11 and 3 .0 .

the flame recirculation zone . Although the numerical approach results i n

values close to the equilibrium one for such equivalence ratio (which i s

zero), the experiments report finite values . The same results have been

found from different chemistry models, and they also seem to be (largely )

uninfuenced by the consideration of the radiative heat losses . The same is

observed when different turbulence and molecular mixing models ar e

used (Vicente, 2000) .

CO AND NO FORMATION IN LEAN FLAMES

599

Vicente (2000) used two molecular mixing models : one of them is theLMSE rnodel (Dopazo model, 1973) and the other is the modified Cur lmodel (Janicka et al., 1979) . Chemical models in the literature inchidetwo global systems, one of four steps (Iones and Linsdtedt, 1988) and th eother of five steps Nicol et al . (1999) : another is the systematicall yreduced chemical system of Mallampalli et al . (1998) . Turbulent model sinclude the standard k—c model (Launder and Spalding, 1972), the RNGversion (Yakhot and Orszag, 1986), and the Reynolds stress transpor tmodel of Launder et al . (1975) .

CONCLUSION S

This papel- has described a stochastic model for the prediction of NOformation and combustion in lean premixed turbulent llames, and itsvalidation is supported with experimental data. A GRI-derived reduced-

chemistry model allows the use of finite chemical rates and helps t o

represent all the NO formation routes .

The local mixing time 'Emixx is calculated with a local PDF, and the CDvalue of this characteristic time has been varied arbitrarily to study it s

effects primarily on the predictions of CO and NO . When this constant is

changed, or if is achieved frote a PDF, the CO is lightly affected i n

the shear layer. The peak values in this zone are increased approximatel y

25% when CD is enhanced from 3 .0 to 8 .0 . I-owever, the axial velocity

and concentrations for NO are not affected by these parameters. It was

shown that the local mixing time significantly affects the minor species i n

the shear layer, and alternate forms for the characteristic mixing time

should be investigated .

The LUT method is found to be a good alternative to predict th e

thermochemical state in the calculation of PDF . The LUT method has an

error rate lower than 5%, and it is conveniently less expensive in CP U

time than the direct integration method.

The two GRI versions used show weak differences in results . The CH

kinetics to pronzpt NO modification in GRI 3 .0, the principal change in

the new GRI version, is not important in NO formation for LPC .

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