Second-order moment turbulence-chemistry models for simulating NOx

formation in gas combustionq

L.X. Zhoua,* , X.L. Chena, C.G. Zhengb, J. Yinb

aDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of ChinabNational Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China

Accepted 2 December 1999

Abstract

For simulating NOx formation in combustion presently used turbulence-chemistry models either cannot or are inconvenient to simulatefinite reaction rate, or need extremely large computation time and storage, difficult to be used in engineering complex flows. Hence, differentsecond-order moment (SOM) turbulence-chemistry models are proposed. The SOM models can well simulate CH4–O2 turbulent combustionand NOx formation, verified by experiments. The results indicate that the SOM models may be economic and reasonable for predicting NOx

formation in combustion.q 2000 Elsevier Science Ltd. All rights reserved.

Keywords: NOx formation; Gas combustion; Second-order moment model

1. Introduction

To reduce NOx formation in combustion, different tech-niques, such as air staging, reburning, flue-gas recirculation,are used in burners and furnaces. It is recognized that thenumerical modeling is a useful tool for optimizing theresults of these techniques. To find a reasonable andeconomical way for simulating NOx formation in engineer-ing complex turbulent flows, different turbulence-chemistrymodels were proposed. The PDF transport equation model[1] can well simulate detailed finite-rate kinetics. However,it needs the computation time and storage, two orders ofmagnitude larger than those for the presumed PDF model.Hence, it is used mainly for very simple flows, such as jetand channel flows, but is difficult to be used in complexengineering flows, such as 3-D recirculating and swirlingflows. Some investigators use the EBU–Arrhenius (EA)model [2] for simulating NOx formation, but the EAmodel actually cannot take the finite reaction rate intoaccount. Many investigators, including some commercialcodes adopt the presumed PDF-finite-reaction-rate model[3–7], in which the instantaneous reaction rate is taken asa function of two variables—temperature and oxygen

concentration or mixture fraction, and a product of two 1-D PDFs,p�T�p� f � or p�T�p�YO2

� is adopted as the presumedb -function PDF, instead of the real 2-D PDFp�T;YO2

�:Although the measurements of temperature fluctuationusing CARS demonstrated that theb -function PDF isappropriate in most of the flow regions [8], however, thepresumed 1-D PDF approach may cause some error.Actually, an elementary reaction rate is frequently deter-mined by the temperature and the two species concentration,so we have to use a product of the three 1-D PDFs, that maycause more serious error. Experiments point out that thismodel underpredicts NOx formation in some cases. In thepresumed PDF model with partial equilibrium [9] the reac-tivenessp , expressing the degree of deviation from theequilibrium state, is assumed to be independent of turbu-lence. This assumption is questionable, since turbulenceshould affect the reactiveness. Besides, in this model, it isnecessary not only to solve species equations and correl-ation transport equations, but also to do equilibrium calcu-lation. So the computation is time consuming. In this paper,we propose several second-order moment (SOM) turbu-lence-chemistry models tosimulate NOx formation incombus-tion. These models are rather simple, do not need muchcomputation time and can easily simulate finite-rate kinetics.

2. Second-order moment turbulence-chemistry models

The SOM models for simulating turbulence-chemistry

Fuel 79 (2000) 1289–1301

0016-2361/00/$ - see front matterq 2000 Elsevier Science Ltd. All rights reserved.PII: S0016-2361(99)00283-5

www.elsevier.com/locate/fuel

q Research results of the National Key Project of Fundamental Researchin 1999–2004, sponsored by the Ministry of Science and Technology,People’s Republic of China.* Corresponding author. Tel.:186-0-278-23141; fax:186-10-6278-

5569.E-mail address:zhoulx@mail.tsinghua.edu.cn (L.X. Zhou).

interactions are based on the Reynolds expansion of theexpression for the instantaneous reaction rate. A typicalinstantaneous reaction rate of Arrhenius type is

ws � Br2Y1Y2 exp�2E=RT�

Then, the time-averaged reaction rate is

�ws � Br2Y1Y2 exp�2E=RT� � Br2kY1Y2

� Br2 � �k 1 k 0�� �Y1 1 Y01�� �Y2 1 Y02� �1�

or

�ws � Br2� �Y1 1 Y 01�� �Y2 1 Y 02� exp 2E

R� �T 1 T 0�� �

�2�

where k � exp�2E=RT�: Different versions of SOMclosures are based either on Eq. (1), or on Eq. (2).

2.1. SOM model-version 1

The first closure method is based on Eq. (2). Taking a seriesexpansion for the exponential function in Eq. (2), and assuming

E

R �T

T 0

�Tp 1

the closed form of time-averaged reaction rate is:

�ws � Br2 �Y1�Y2 exp

2

E

R �T

!

�"

1 1Y 01Y02�Y1

�Y21

E

R �T

T 0Y 01TY1

1T 0Y02TY2

!

1 12

E

R �T

!2 T 0

�T

!2#� Br2 �Y1

�Y2exp

2

E

R �T

!�1 1 F�

�3�All of the correlations of temperature fluctuation,concentration fluctuation and temperature–concentrationfluctuation are solved by their transport equations. Thegeneralized form of these transport equations is:

2

2t�r wc�1

2

2xj�rvjwc� � 2

2xj

me

sw

2wc

2xj

!1 mT

2w

2xj

2c

2xj

2 erwc=k (4)

wherew , c expressY1, Y2 or T, respectively.

2.2. SOM model-version 2

The advantage of version 1 is very simple, and its draw-back is the approximation made in series expansion of theexponential function. Since for most reactions, in particularfor reactions of the NOx formation, the activation energyEis in the range of 80 000–160 000 kJ kg21 mol21, then

L.X. Zhou et al. / Fuel 79 (2000) 1289–13011290

Nomenclature

x coordinateg mean square value of fluctuationk reaction rate coefficientm massp pressurew reaction rateA surface areaB pre-exponential factorE activation energyF function defined in Eq. (3)M molecular weightR universal gas constantT temperatureY mass fractionZ function defined in Eq. (8)

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

Experiment NF Model EA Model SOM1 Model SOM3 Model

k(m

2 /s2 )

x/Dr

Fig. 1. Turbulent kinetic energy�S� 0�:

E=R �T should be much larger than unity, for example, morethan 5. The assumption of

E

R �T

T 0

�Tp 1

will lead to underpredicting the time-averaged reaction rateof NO formation . Therefore, we should try to avoid usingthe approximation made in series expansion. The closuremethods in the following versions are based on Eq. (1).From Eq. (1), neglecting the third-order correlation, we havethe closed form of time-averaged reaction rate in version 2 as:

�ws � Br 2�� �Y1�Y2 1 Y01Y02� �k 1 �Y1k 0Y 02 1 �Y2k 0Y 01� �5�

where all of the correlations in Eq. (5) are solved using Eq. (4),assuming that the transport of correlations of concentrationfluctuation with the fluctuation of reaction rate coefficientk

likes the transport of other correlations. The time-averagedreaction coefficientk can be found by

�k �Z

exp�2E=RT�p�T� dT

wherep�T� is a presumed PDF for the temperature itself.

2.3. SOM model-version 3

In version 3, the correlations containing the fluctuation ofreaction rate coefficient are transformed into the integralsover PDFs. We have

k 0Y0 � kY 2 kY�ZZ

kYp�T�p�Y� dT dY 2 �YZ

kp�T� dT

So, we obtain the closed form of time-averaged reaction rate

L.X. Zhou et al. / Fuel 79 (2000) 1289–1301 1291

0 1 2 30

2

4

6

8

x/Df=1.50

Experiment NF Model EA Model SOM1 Model SOM3 Model

U(m

/s)

r/R0

0 1 2 30

2

4

6

8

x/Df=1.00

Fig. 3. Axial velocity profiles�S� 0�:

0 1 2 3

-0.5

0.0

0.5

1.0

x/Df=1.50

x/Df=1.00

Experiment (Baker, 1975) NF Model EA Model SOM1 Model SOM3 Model

U(m

/s)

r/R0

0 1 2 30.0

0.5

1.0

1.5

Fig. 4. Axial velocity profiles�S� 0:52�:

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

Experiment NF Model EA Model SOM1 Model SOM3 Model

k(m

2 /s2 )

x/Dr

Fig. 2. Turbulent kinetic energy�S� 0:52�:

in version 3 as

�ws � Br2�� �Y1�Y2 1 Y01Y02�

Zkp�T� dT

1 �Y1�ZZ

kY2p�T�p�Y2� dT dY2 2 �Y2

Zkp�T� dT�

1 �Y2�ZZ

kY1p�T�p�Y1� dT dY1 2 �Y1

Zkp�T� dT�� �6�

where p�Y1�; p�Y2� are presumed PDFs of concentrationfluctuation. In the simplest case, if the PDFs of temperatureand concentration fluctuation are given as the two-deltadistribution (not necessarily this form of PDF), we have

�ws � Br2 �Y1�Y2 exp�2E=R �T�Z �7�

where the functionZ reflects the effect of turbulent fluctua-tion on the reaction rate

Z � chE

R �T

g1=2T�T

!

� 1 1Y01Y02�Y1

�Y21

g1=2Y1�Y1

1g1=2

Y2�Y2

!th

E

R �T

g1=2T�T

!" #�8�

where

gT � T 02; gY � Y02 ch�x� � �ex 1 e2x�=2;

th�x� � sh�x�ch�x� sh�x� � �ex 2 e2x�=2

The correlationsgT; gY are determined by Eq. (4).

3. Simulation of methane–air turbulent combustion

In order to verify the models, the SOM models are firstused to simulate methane–air diffusion combustion. Forcomparison, beside the SOM models, other models suchas the EA model and the model taking no account of turbu-lence fluctuations (NF) are also used. For the NF model, inwhich the effect of temperature and concentration fluctua-tions is not taken into account, the time-averaged reactionrate is simply expressed by time-averaged variables, that is

ws � Br2Y1Y2 exp 2E

RT

� ��9�

For the EA model [2], the time-averaged reaction rate is

L.X. Zhou et al. / Fuel 79 (2000) 1289–13011292

3.4E2

4.8E26.2E2

7.6E2

9E2

1E3 1.2E3

1.3E3

1.5E3

1.5E3

0.0 0.5 1.0 1.5 2.0 2.5 3.0-3

-2

-1

0

1

2

3

r/R0

x/Df

Fig. 5. Isotherms (NF model,S� 0; a � 1:05�:

4.2E2

4.2E2

5.4E2

5.4E2

6.6E2

6.6E2

7.8E2

7.8E2

9E2

9E2

9E2

9E2

1E3

1E3

1E3

1E3

1.1E3

1.1E3

1.3E3

1.3E3

1.4E3

1.4E3

0.0 0.5 1.0 1.5 2.0 2.5 3.0-3

-2

-1

0

1

2

3

r/R0

x/Df

Fig. 6. Isotherms (NF model,S� 0; a � 0:72�:

determined by

ws � min�wsT;wsA�;wsT�ceburge=k;

wsA � Br2Y1Y2 exp

2

ERT

!�10�

wherewsT is the reaction rate controlled by turbulent fluc-tuation,cebu is an empirical constant with a value of 0.35–0.4, obtained by many numerical tests, andg can be deter-mined by solving its transport equation or algebraic expres-sion. wsA is the reaction rate controlled by the laminarreaction mechanism, that is Eq. (9).

In predicting methane–air turbulent combustion, thek–emodel [2] is adopted to simulate gas turbulence, and the fluxmethod [9]is adopted for simulating radiative heat transfer.The transport equations governing the net flux in axial andradial directions are expressed by [9]:

ddx

1a 1 s

dRx

dx

� �� a�Rx 2 sT4�1

s2�Rx 2 Rr �

1r

ddr

rra 1 rs 1 1

dRr

dr

� �� a�Rr 2 sT4�1

s2�Rr 2 Rx�

�11�

wherea is the absorption factor of gas ands the reflectivefactor of gas.

For the numerical procedure, the differential equationsare integrated in the control volume to obtain finite differ-ence equations using a hybrid scheme. The FDEs are solvedusing the SIMPLE algorithm, i.e.p–v corrections with line-by-line iterations and under-relaxation. The temperature iscalculated from the enthalpy and species concentration, andthe density of the gas mixture is calculated according to theideal gas state equation. For the boundary conditions,uniform distribution condition, no-slip condition, fullydeveloped flow condition and axi-symmetrical conditionare taken at the inlet, wall, exit and axis, respectively.The criterion of convergence is the summation ofresidual mass sources less than 1023. The laminar reac-tion rate for methane–air combustion is taken from Ref.[10] as:

wfu � 1 × 1010r2YfuYox exp�1:84× 104=T� �12�

The predictions are made for two different cases—sudden-expansion flows (swirl numberss� 0 and s�0:52� and jet flows with turbulent diffusioncombustion.

L.X. Zhou et al. / Fuel 79 (2000) 1289–1301 1293

4.2E2 5.4E2

6.6E27.8E2

9E2

1E3

1.1E3

1.1E3

1.3E3

1.3E3

1.4E3

1.4E3

0.0 0.5 1.0 1.5 2.0 2.5 3.0-3

-2

-1

0

1

2

3

r/R0

x/Df

Fig. 7. Isotherms (EA model,S� 0; a � 1:05�:

4.1E2

5.2E26.3E27.4E2

8.5E2 9.6E2

1.1E3

1.1E31.2E3

1.2E3

1.3E3

1.3E3

0.0 0.5 1.0 1.5 2.0 2.5 3.0-3

-2

-1

0

1

2

3

r/R

0

x/Df

Fig. 8. Isotherms (EA model,S� 0; a � 0:72�:

3.1. The sudden-expansion combusting flows

An axi-symmetrical sudden-expansion combustor withcentral fuel injection and annular air supply was experimen-tally investigated by Baker et al. [11]. Measurements ofturbulent kinetic energy and mean velocity were reported.Predicted turbulent kinetic energy change along the axis inthe case of non-swirling flows using four models and itscomparison with measurements are shown in Fig. 1. Inthis figure and the following figures, NF (model 1), EA(model 2), SOM-1 (model 3), SOM-3 (model 4) denotethe no-fluctuation model given by Eq. (9), in which thereaction rate is a quasi-laminar one and the effect oftemperature and concentration fluctuations are neglected,the EBU–Arrhenius model given by Eq. (10), SOMmodel-version 1 and SOM model-version 3, respectively.All four models can predict the peak value near the inlet.The location of the peak value, predicted by models 2,3 and4 is near to that measured, but the predicted peak values arehigher than those measured. On the other hand, the locationof the peak value predicted by model 1 is not in agreementwith that measured, but the predicted peak value is near tothat measured. Fig. 2 shows the predicted turbulent kineticenergy change along the axis in the case of swirling flows

using four models and its comparisons with measurements.The results give similar trends as those in Fig. 1, but thepeak values of turbulent kinetic energy are about twice thevalues in Fig. 1.

Figs. 3 and 4 give the comparison between the predictedaxial velocities using four different models and thosemeasured. Fig. 3 corresponds to the case of non-swirlingflows. Although four models give the same trends but theresults given by the SOM-3 model are in best agreementwith the experiment. Fig. 4 corresponds to the case ofswirling flows. It can be seen that the differences amongthe four model predictions are obvious. The SOM-3model gives better results than the other models. Themeasured central recirculation zone appears at the locationof x=Df � 1:00: All NF, EA, and SOM-3 models can predictthis recirculation zone with some quantitative differences,while the recirculation zone predicted by the SOM-1 modelis much smaller and shorter than the measured one anddisappears quickly near the inlet. This implies that thetemperature predicted by the SOM-1 model is much higherthan that measured. No temperature information wasreported in reference [11]. Therefore, the effect ofcoefficient of excessive aira on the flame length is studied,in order to validate the models’ capability of simulating

L.X. Zhou et al. / Fuel 79 (2000) 1289–13011294

3.4E2 4.8E2

6.2E2

7.6E2 9E2

1E3

1.2E3

1.3E3

1.5E3

1.5E3

0.0 0.5 1.0 1.5 2.0 2.5 3.0-3

-2

-1

0

1

2

3

r/R0

x/Df

Fig. 9. Isotherms (SOM-3 model,S� 0; a � 1:05�:

4E2

5E2

6E2

7E28E2 9E2

1E3

1.1E3

1.2E3

1.2E3

0.0 0.5 1.0 1.5 2.0 2.5 3.0-3

-2

-1

0

1

2

3

r/R

0

x/Df

Fig. 10. Isotherms (SOM-3 model,S� 0; a � 0:72�:

finite chemical kinetics. Figs. 5–10 give the predictedisotherms, from which the flame length can be determined.Figs. 5 and 6 are isotherms predicted by the NF model. Theflame length is the shortest whena is about unity andincreases rapidly whena increases or decreases fromunity. It can be seen that the predicted flame length exceeds

the computation domain whena is 0.72, which seemsunreasonable. Figs. 7 and 8 are isotherms predicted by theEA model. The flame length and the temperature distri-bution across the flow field almost remain unchangedwhena changes from 1 to 0.72. This implies that althoughin principle the EA model can consider the interactionbetween the turbulence and the chemical reaction, actually,during numerical simulation the reaction rate is controlledonly by turbulence in most regions of the flow field. There-fore, the EA model cannot simulate finite reaction rate effec-tively and hence cannot give reasonable results. Figs. 9 and10 are the prediction results given by the SOM-3 model.Whena is near unity, the flame length reaches minimum.As a decreases from unity, the flame length increasesobviously and reasonably. The SOM-1 model gives similarresults except that it gives much shorter flame length than

L.X. Zhou et al. / Fuel 79 (2000) 1289–1301 1295

Fig. 11. Jet burner configuration.

0 2 4 6 8 10

500

1000

1500

2000

x/D=45

x/D=30

x/D=15

x/D=2

x/D=1

x/D=75

Experiment NF Model

SOM1 Model

SOM3 Model

EA Model

T(k

)

r/R

0 2 4 6 8 10

500

1000

1500

2000

0 2 4 6 8 10

500

1000

1500

2000 0 2 4 6 8 10

500

1000

1500

2000 0 2 4 6 8 10

500

1000

1500

2000 0 2 4 6 8 10

500

1000

1500

2000

Fig. 12. Temperature profiles.

L.X. Zhou et al. / Fuel 79 (2000) 1289–13011296

0 2 4 6 8 100.0

0.1

0.2 Experiment NF Model SOM1 Model SOM3 Model EA Model

x/D=7.5

x/D=15

x/D=30

x/D=45

x/D=75

x/D=1

Ych

4

r/R

0 2 4 6 8 100.0

0.1

0.20 2 4 6 8 10

0.0

0.1

0.20 2 4 6 8 10

0.0

0.1

0.20 2 4 6 8 10

0.0

0.1

0.20 2 4 6 8 10

0.0

0.1

0.2

Fig. 13. CH4 concentration profiles.

Table 1Geometrical sizes and flow parameters

Main jet inner diameterDJ

(mm)7.2 Ucofl(m s21), Tcofl (K) 0.9, 291

Pilot annulus inner diameterDP,I (mm)

7.7 Main-jet speciesconcentration(volume fraction)

25% CH4

75% dry airPilot annulus outer diameterDP,O (mm)

18.2 UJ(m s21), TJ (K) 49.6, 294

Burner outer wall diameterDB,O

(mm)18.9 UP(m s21), TP (K) 11.4, 1880

the SOM-3 model. As shown in the above statements, SOM-1 and SOM-3 models can simulate interaction between theturbulence and the chemical kinetics, while the NF modeland EA model can consider the influence of either onlychemical kinetics or only turbulence.

3.2. The jet combusting flows

The Experimental results of piloted jet flame werereported by Barlow et al. Detailed information is providedin the web pages [12]. The geometrical configuration of theburner for predicted methane–air turbulent combustion isthat reported in Ref. [12] (Fig. 11). The computationaldomain is a half of the burner and 35× 25 grid nodes aretaken inside the domain. The geometrical sizes and flow

parameters of the burner are given in Table 1. The recom-mended boundary conditions by [12] are taken at the inlet,wall, exit and axis, respectively. Running a case in aPentium-133 PC takes about 40 min.

The prediction results are shown in Figs. 12–17. It can beseen that although compared with the experimental results,the predicted temperature profiles (Fig. 12) using the fourmodels have the same right trend, however, quantitativelythe SOM-3 gives the best agreement with experimentalresults than all the other models. The location of tempera-ture peaks predicted by the SOM-1 model deviates mostobviously from the experimental one. The NF model clearlyunderpredicts the temperature everywhere because of notconsidering the turbulent fluctuation, particularly at theaxial location ofx=D � 75; where the predicted temperature

L.X. Zhou et al. / Fuel 79 (2000) 1289–1301 1297

0 2 4 6 8 100.0

0.1

0.2

x/D=45

x/D=30

x/D=15

x/D=7.5

x/D=1

x/D=75

Yo 2

r/R

0 2 4 6 8 100.0

0.1

0.2

Experiment NF Model SOM1 Model SOM3 Model EA Model

0 2 4 6 8 100.0

0.1

0.20 2 4 6 8 100.0

0.1

0.20 2 4 6 8 100.0

0.1

0.20 2 4 6 8 10

0.00.10.2

Fig. 14. O2 concentration profiles.

is much lower than the experimental one. The EA modelunderpredicts the temperature in some radial locations andoverpredicts it in other locations. Figs. 13 and 14 show CH4

and O2 concentration profiles. Once again, the SOM-3model gives the best results than other models. Thepredicted CH4 concentration (Fig. 13) using the NF modelis much larger than that measured, since the fuel is muchless consumed under much lower reaction rate predicted bythe NF model. The CH4 concentration using other models isalso underpredicted. Similar results are obtained for oxygenconcentration profiles (Fig. 14). The superiority of theSOM-3 model over other models is made evident by Fig.15, showing the predicted and measured RMS values oftemperature fluctuations. Two peaks can be seen in most

of the regions. Both the two SOM models can predict thetwo-peak distribution. Nevertheless, the predicted locationsof these peaks using the SOM-1 model obviously deviatefrom the measured ones, while the SOM-3 predictions are inmuch better agreement with the measured results. The othermodels cannot predict the two-peak distribution. Figs. 16and 17 show the changes of temperature and temperaturefluctuation along the axial direction. All EA, SOM-1, andSOM-3 models well predict the peak temperature distri-bution (Fig. 16) and the predicted peaks are all near theposition of the experimental one:x=Df � 47: However, thetemperature predicted by the SOM-1 model has an apparentdeviation from the experimental one. The NF model isincapable of predicting the peak and the predicted

L.X. Zhou et al. / Fuel 79 (2000) 1289–13011298

0 2 4 6 8 100

200

400 Experiment SOM1 Model SOM3 Model

Trm

s(k)

r/R

0 2 4 6 8 100

200

400x/D=2

x/D=1

0 2 4 6 8 100

200

400x/D=15

0 2 4 6 8 100

200

400 x/D=30

0 2 4 6 8 10

0200400

x/D=45

0 2 4 6 8 100

200

400 x/D=70

Fig. 15. RMS value of temperature fluctuation.

temperature is much lower than that measured. Thetemperature predicted using the EA model is lower thanthat measured all along the centerline. The temperaturedistributions predicted by the two SOM models are in betteragreement with experiments after the peak than before thepeak. It can be seen from Fig. 17 that the distributions oftemperature fluctuation predicted by the two SOM modelsare qualitatively different. The SOM-3 model can wellpredict the two-peak distribution of temperature fluctuationalong the centerline, which is close to experiments, whilethe SOM-1 model is incapable of predicting the two-peakdistribution. The temperature fluctuation has great effect onthe formation of NOx and the SOM-3 model can reasonablysimulate the temperature fluctuation, therefore, the SOM-3model is expected to be used for simulating NOx formationin practical combustors and furnaces.

3.3. NO formation during methane–air combustion

For predicting the NOx formation during methane–aircombustion within sudden-expansion flows, the velocityvectors and temperature map predicted using the SOM-3

model are shown in Figs.18and 19. The predicted NOconcen-tration map, accounting for the Zeldovich mechanism ofthermal NO and the DeSoete mechanism of fuel NO dueto the formation of HCN [13], using the SOM-3 model, isshown in Fig. 20.

Figs. 21–23 are NO concentration profiles predicted atdifferent axial distances using four different turbulence-chemistry NO models. The solid lines express the resultsobtained by the SOM-3 model. The dots denote the experi-mental results taken from Ref. [3]. The dotted lines, dashedlines and dash-dotted lines are results obtained by threepresumed-PDF-finite-rate models. In the last three modelsthe time-averaged reaction rate of NO formation is obtainedusing the following expression:

�WNO �Z1

0

Z1

0WNO� �r =r�P�u�P�YO2

� dudYO2�13�

where

u � �T 2 Tmin�=�Tmax 2 Tmin�and the PDF is taken in the form of ab function:

P�f� � fa21�1 2 f�b21Z1

0fa21�1 2 f�b21 df

where

a� �f � �f �1 2 �f �=f 02� b� a�1 2 �f �= �ff denotes temperature, oxygen concentration or mixturefraction. The RM value of its fluctuation is determined bythree different models:

Model 1: empirical expression

f 02 � s�1 2 �f �= �f �14�wheres� 0:6

L.X. Zhou et al. / Fuel 79 (2000) 1289–1301 1299

1 2X

0 .05

0.1

0 .15

0.2

0 .25

0.3

Y

Fig. 18. Velocity vectors (SOM-3 model).

0 20 40 60 80

500

1000

1500

2000

Experiment NF EA SOM1 SOM3

Te

mp

(k)

x/D

Fig. 16. Axial temperature distribution.

0 20 40 60 800

100

200

300

400

500

Experiment SOM1 SOM3

Trm

s(k)

x/D

Fig. 17. Axial distribution of temperature fluctuation.

Model 2: algebraic expression of second-order moment

f 02 � C1mT2 �f

2xj

!2

=�C2re=k� �15�

Model 3: SOM transport equation

2

2xj�rujf

02� � 2

2xj

mT

sf

2f 02

2xj

!1 C1mT� 2

�f

2xj�2

2 C2re f 02=k 1 2f 0Sf 0 �16�where the last term on the right-hand side is the term dueto radiation and combustion. It can be seen that in generalthe SOM-3 model predictions are near to that obtainedusing the PDF model 3, and are better than those obtainedby other models. Although the SOM-3 model and presumed

PDF model with SOM transport equations (PDF model 3) givesimilar results, the last model needs much more computationaltime.

4. Conclusions

1. Different versions of SOM turbulence-chemistry modelsfor NOx formation in turbulent flows were proposed. TheSOM models can simulate detailed chemistry with finite-rate kinetics and computationally is acceptable for simu-lating engineering complex flows.

L.X. Zhou et al. / Fuel 79 (2000) 1289–13011300

1 2

X

0 .05

0.1

0 .15

0.2

0 .25

0.3

Y

36.6857

22.9501

29.9

368

22.9 50 1

3 .22507

29.9368

49.8387

41.2984

19.3

406

7 .29081

13.4283

Fig. 20. NO concentration map (SOM-3 model).

1 2X

0 .05

0.1

0 .15

0.2

0 .25

0.3

Y

2155.58

2049.16

1975.43

1834.33 1724.4

2484

.96

2309.252178.72

2131.03

2227.14

Fig. 19. Temperature map (SOM-3 model).

0.00 0.05 0.10 0.15 0.20 0.25 0.300

5

10

15

20

25

30

35

40

45 x=0.39 m NSO M Exp PDF-v1 PDF-v2 PDF-v3

NO

(p

pm

)

r (m )

Fig. 21. Predicted NO concentration�x� 0:39 m�:

0.00 0.05 0.10 0.15 0.20 0.25 0.300

10

20

30

40

50

60

x=0.82 m

NSOM Exp PDF-v1 PDF-v2 PDF-v3

NO

(p

pm

)

r(m)

Fig. 22. Predicted NO concentration�x� 0:82 m�:

0.00 0.05 0.10 0.15 0.20 0.25 0.300

5

10

15

20

25

30

35

40

45

50

x=1.505 m

NSOM Exp PDF-v1 PDF-v2 PDF-v3

NO

(ppm

)

r(m)

Fig. 23. Predicted NO concentration�x� 1:505 m�:

2. The SOM models, verified by experiments, can wellsimulate turbulent combustion.

3. The SOM-3 model in predicting NOx formation givesresults at least as good as those obtained using thepresumed PDF model with transport equations ofSOMs (PDF-3), however, the SOM-3 model needsmuch less computation time than the PDF-3 model.

4. The development of SOM-3 model needs further experi-mental verification.

Acknowledgements

This study is sponsored by the National Key Project ofFundamental Research in 1999–2004, Ministry of Scienceand Technology, People’s Republic of China and PhDThesis Foundation of Tsinghua University.

References

[1] Pope SB. PDF methods for turbulent reactive flows. Prog EnergyCombust Sci 1985;11:119.

[2] Zhou LX. Theory and numerical modeling of turbulent gas-particleflows and combustion. Beijing: Science Press and CRC, 1993.

[3] Beretta A, Mancini N, Podenzani F, Vigevano L. The influence of thetemperature fluctuation variance on NO predictions for a gas flame.Proceedings of the Third International Conference on CombustionTechnology for a Clean Environment, Paper 18.4, 1995.

[4] DalSecco S, Mechitoua N, Simonin O. Three-dimensional modelingof coal combustion and NO formation in a 250MW coal-fired furnace,

Proceedings of the Third International Conference on CombustionTechnology For a Clean Environment, Paper 31.2, 1995.

[5] Faltsi-Saravelon O, Wild PN. Prediction of NO emissions from aswirling coal flame. Proceedings of the Third International Con-ference on Combustion Technology For a Clean Environment,Paper 20.2, 1995.

[6] Okasanen A, Maki-Mantila E. Use of PDF in modeling of NO forma-tion in methane combustion. Proceedings of the Third InternationalConference on Combustion Technology For a Clean Environment,Paper 18.3, 1995.

[7] Schlatter M, Flury M. Modeling of NO formation in turbulent H2

flames. Proceedings of the Third International Conference onCombustion Technology For a Clean Environment, Paper 18.1, 1995.

[8] Mantzaras J, van der Meer. A study of temperature fluctuation intwo-shear flow, natural gas-fueled piloted jet diffusion flames,Proceedings of the Second International Symposium on Turbulence,Heat and Mass Transfer, Delft, 1997. p. 665–74.

[9] Smoot LD, Smith PJ. Coal combustion and gasification. New York:Plenum Press, 1985.

[10] Spalding DB. Mixing and reaction in steady confined turbulent flame.Proceedings of the 13th International Symposium on Combustion,1970. p. 649–57.

[11] Baker RJ, Hutchingson P, Khalil EE, Whitelaw JH. Measurements ofthree orthogonal velocity components in confined coaxial jet flowswith and without combustion, Proceedings of the 15th InternationalSymposium on Combustion, 1975.

[12] Barlow R. Sandia piloted CH4/air jet flames: preliminary data release.3rd International Workshop on Measurement and Computation ofTurbulent Nonpremixed Flames.http://www.ca.sandia.gov/work-shop/workshop.html.

[13] DeSoete GG. Overall reaction rate of NO and N2 formation from fuelnitrogen. Proceedings of the 15th International Symposium onCombustion, 1975. p. 1093–1102.

L.X. Zhou et al. / Fuel 79 (2000) 1289–1301 1301