Center for T - Stanford University

Center for Turbulence Research

Proceedings of the Summer Program ��

��

E�ect of chemistry and turbulence on NOformation in oxygen�natural gas �ames

By J��M� Samaniego�� F� N� Egolfopoulos� AND C� T� Bowman�

The e�ects of chemistry and turbulence on NO formation in oxygen�natural tur�bulent di�usion �ames gas �ames have been investigated� The chemistry of nitricoxides has been studied numerically in the counter�ow con�guration� Systematiccalculations with the GRI �� mechanism for combustion of methane and NOchemistry were conducted to provide a base case� It was shown that the �simple�Zeldovich mechanism accounts for more than � of N� consumption in the �amein a range of strain�rates varying between �� and �� s�� The main shortcom�ings of this mechanism are �� overestimation �� of the NO production rateat low strain�rates because it does not capture the reburn due to the hydrocarbonchemistry� and �� underestimation �� of the NO production rate at high strain�rates because it ignores NO production through the prompt mechanism� Reburnthrough the Zeldovich mechanism alone proves to be signi�cant at low strain�rates�A one�step model based on the Zeldovich mechanism and including reburn has beendeveloped� It shows good agreement with the GRI mechanism at low strain�ratesbut underestimates signi�cantly N� consumption �about �� at high strain�rates�The role of turbulence has been assessed by using an existing ��D DNS data base ofa di�usion �ame in decaying turbulence� Two PDF closure models used in practicalindustrial codes for turbulent NO formation have been tested� A simpler versionof the global one�step chemical scheme for NO compared to that developed in thisstudy was used to test the closure assumptions of the PDF models� because the database could not provide all the necessary ingredients� Despite this simpli�cation� itwas possible to demonstrate that the current PDF models for NO overestimate sig�ni�cantly the NO production rate due to the fact that they neglect the correlationsbetween the �uctuations in oxygen concentration and temperature� A single scalarPDF model for temperature that accounts for such correlations based on laminar�ame considerations has been developed and showed excellent agreement with thevalues given by the DNS�

�� Introduction

This study is an investigation of the e�ects of chemistry and turbulence on nitricoxide formation in oxygen�natural gas �ames� The choice of oxygen as the oxidizeris related to current interest in use of oxygen for high temperature combustion in

� Air Liquide� Centre de Recherche Claude�Delorme� France

� University of Southern California� Los Angeles� CA

� Stanford University� Stanford� CA

�� J��M� Samaniego� F� N� Egolfopoulos � C� T� Bowman

industry� In terms of NO emissions� oxygen is advantageous compared to preheatedair in high�temperature processes such as those encountered in the glass and steelmaking industry due to the low nitrogen content� However� molecular nitrogen stillis present in certain amounts in the natural gas �� for Algerian gas to �� forGroningue gas� and in the oxygen stream� In the latter case� the nitrogen contentdepends on the production method from virtually �� for cryogenic oxygen to �for vacuum swing absorption� Other sources of nitrogen for oxygen�natural gascombustion include air leaks into the furnace� Despite the relatively low levels ofNO emissions from oxy�combustion� ever more stringent emission standards requirea better understanding of NO formation in oxy��ames�

The chemical mechanisms controlling NO formation are well known �Fenimore�� De Soete �� Miller � Bowman �� Drake � Blint �� Bozelli et al�� Bowman �� Nitric oxide is formed from two sources molecular nitro�gen� N�� and fuel�bound nitrogen� In the case of oxygen�natural gas �ames� theonly source is molecular nitrogen since natural gas� whatever its origin� does notcontain nitrogen�bound species� In this case� the three main pathways for NO for�mation are the Zeldovich� prompt� and N�O mechanisms� and the main pathwayfor NO destruction is the reburn mechanism� The Zeldovich mechanism is basedon O�atom attack of N� through O � N� � NO � N � and is active both in the�ame zone and the post�ame zone� It is strongly dependent on temperature dueto the high activation energy� EZeldovich� of the N��consuming step �EZeldovich �� kcal�mole�� A particular case of the Zeldovich mechanism is the equilibriumZeldovich mechanism� where O�atoms are in partial equilibrium with molecular O��In this case� a global one�step reaction for NO formation can be derived with anoverall activation energy� Eequilibrium � �� kcal�mole� The prompt mechanismis active only within the �ame zone since it requires the presence of CHx radicalsfor consumption of molecular nitrogen� mainly through CH � N� � HCN � N �Subsequent elementary reactions lead to the formation of NO� This mechanism isweakly dependent on temperature due to the low activation energy of the main N��consuming step �Eprompt

�� kcal�mole�� The N�O mechanism is due to reactionof O�atoms with N� in a three�body reaction� i�e� O �N� �M � N�O �M � withthe subsequent reaction of N�� to form mainly through O�N�O � NO�NO� Thereburn mechanism is responsible for consumption of NO within the �ame zone toproduce N�� and it is controlled by CH� CH�� and CH� radicals�

These various mechanisms are present in air �ames and must be accounted forto predict accurately the level of nitric oxide emissions� The usual picture for NOproduction in air �ames can be split in two parts �� production in the �ame zonethrough a balance between the prompt� N�O� and reburn mechanisms � in thiszone� the Zeldovich mechanism can be neglected� �� production of NO in the post��ame zone by the Zeldovich mechanism alone� The picture is di�erent in oxygen�ames since in this case� the destruction rate of N� is controlled by the Zeldovichmechanism� with a small contribution of the prompt mechanism �Samaniego et al�� The main reason is that the higher temperatures of the oxygen �ame tendto increase the destruction rates of N�� and this acceleration is more pronounced

NO formation in oxy��ames ��

for the Zeldovich mechanism due to its higher activation energy� As a result� theZeldovich mechanism becomes faster than the prompt NO mechanism in the �amezone of an oxygen �ame� while it is slower than the prompt NO mechanism in the�ame zone of an unpreheated air �ame� However� it is not clear whether the NOproduction rates can be derived from the Zeldovich mechanism alone since reburnmay be taking place as indicated by the high HCN concentration levels observedin the numerical study of Samaniego et al� �� Furthermore� it is necessaryto clarify the role of non�equilibrium atomic oxygen since it is unclear whether thepartial�equilibrium assumption holds �Samaniego et al� ��

The e�ect of turbulence on NO emissions in jet di�usion �ames has been studiedextensively �Peters � Donnerhack �� Turns � Myrh �� Chen � Kollman ��Driscoll et al� �� The turbulent mixing process results in temporal �uctuationsin temperature and species composition which in�uence the NO formation rates�Since the relationships between NO formation rate� temperature� and species arehighly non�linear� NO emissions cannot be predicted from mean temperature andspecies concentrations alone� Therefore� accurate predictions of NO formation ratesrequire the knowledge of temperature and composition �uctuations� Modelers havederived PDF formulations that account for the e�ect of these �uctuations on theturbulent NO production rate �Janicka � Kollmann �� Pope � Correa ��Correa � Pope �� The closure of the turbulent source term for NO is madepossible through the use of a joint PDF which� in the case of Zeldovich NO withthe partial equilibrium assumption for atomic oxygen� can be expressed as

d�NO�

dt�

Zk�O��

��N�� exp��E�RT �P ��O�� N�� T � d�O�� d�N�� dT ��

where P ��O�� N�� T � is a joint PDF of oxygen concentration� �O�� nitrogen concen�tration� �N�� and temperature� T � Usually� the nitrogen concentration is consideredconstant and �N�� is taken out of the PDF� Furthermore� in practice� �O�� and Tare assumed to be independent variables and the joint PDF� P � can be expressedas the product of two single�variable PDF�s� PO�

and PT � such as

P ��O�� T � � PO��O��PT �T � ��

In such a case� the turbulent NO production term becomes

d�NO�

dt� k�N��

�Z�O��

��PO��O��d�O��

��Zexp��E�RT �PT �T �dT

��a�

In some cases� the same term is calculated from a single PDF by assuming that�O�� uctuations can be neglected

d�NO�

dt� k�N��O��

��

�Zexp��E�RT �PT �T �dT

��b�


In both cases� this �nal assumption has a limited domain of validity since itpresumes that the oxygen concentration �uctuations are not correlated with thetemperature �uctuations� This may be valid in the post��ame zone where thecombustion process is completed� but it is highly questionable in the �ame zonewhere the temperature and oxygen concentration levels are obviously correlated�Since the amount of NO produced in the �ame zone of oxygen �ames can be verysigni�cant� it is expected that such PDF modeling approaches are inapplicable�The goals of this study are to derive a simple chemical model for NO formation in

oxygen di�usion �ames and to propose a model for the closure of the turbulent NOproduction term based on a PDF approach� For this purpose� two complementarynumerical approaches are used� one addressing the e�ects of chemistry� and anotheraddressing the e�ect of turbulence�

�� Eects of chemistry

�� Numerical approach

Concerning chemistry� NO formation mechanisms are investigated using thecounter�ow �ame problem� The counter�ow con�guration is often used to addresschemistry�turbulence interactions with detailed chemistry� and it is well known thatthe strain�rate modi�es the chemical pathways through reduced �ame temperaturereduction and residence times �Hahn and Wendt �� Haworth et al� �� Drakeand Blint �� Chelliah et al� �� Mauss et al� �� Vranos et al� �� Takenoet al� �� Egolfopoulos ��a� ��b� Nishioka et al� �� Samaniego et al�� Egolfopoulos � Campbell �� In such a geometry� the �ow �eld is that ofa strained �ame where a jet of oxidizer impinges upon a jet of fuel� The oxidizeris a mixture of oxygen and nitrogen� Various mixtures are used �� O� and ��N�� O� and � N�� O� and �� N�� The fuel is a mixture of methaneand nitrogen� Two di�erent mixtures are used �� CH� and �� N�� CH�

and � N��The numerical simulation is conducted by solving the steady and unsteady equa�

tions of mass� momentum� energy� and species concentrations along the streamline�Details on the set of equations and numerical method can be found in Egolfopoulos��a and ��b�� and in Egolfopoulos � Campbell �� The chemical schemethat is used for combustion of methane is the latest GRI �� mechanism� whichaccounts for �� species and � reactions� Calculations are performed without ra�diative losses as it has been demonstrated that radiative losses from CO�� H�O�CO� and CH� play a negligible role in counter�ow oxygen �ames �Samaniego et al��

�� Results

The objective of this study is to check whether the Zeldovich mechanism aloneis capable of predicting with su�cient precision the rate of formation of NO in anoxygen �ame� In order to do so� two sets of calculations have been carried out� onewith the full mechanism which includes all NO formation routes� and one with thethree reactions of the Zeldovich mechanism� namely


1.00.80.60.40.20.00.0-40

-20

0

20

40

60

80

Distance from oxygen-feed nozzle, cm

Vel

oci

ty (

m/s

)

Figure �� Computed velocity pro�le along the stagnation streamline of a coun�ter�ow O��CH� �ame for a strain�rate of � s��

Distance from oxygen-feed nozzle, cm1.00.80.60.40.20.00.0

0

500

1000

1500

2000

2500

3000

0.0

0.2

0.4

0.6

0.8

1.0

Tem

per

atu

re (

K)

O2 an

d C

H4 m

ass fraction

s

Figure �� Computed pro�les of temperature �� oxygen mass fraction �� andmethane mass fraction �M� in a counter�ow O��CH� �ame for a strain�rate of �s��


O �N� � NO �N �R��

N �O� � NO �O �R��

N �OH � NO �H �R��

For both mechanisms� a series of about �� steady strained �ames is calculatedfor strain�rates varying from �� to �� s�� The strain�rate is de�ned as themaximum velocity gradient in the oxidant stream� Figures � and � depict the �amestructure for a strain�rate of � s�� which is typical of all the strain�rates computedin this study� This result is obtained with the full GRI �� mechanism� and it isalmost identical for the Zeldovich calculations since the NO chemistry does notsigni�cantly impact the fuel chemistry� The horizontal axis is the co�ordinate alongthe stagnation streamline� with oxidant being fed at x � � cm and fuel being fed atx � �� cm� The feed velocities in this case are V � �� cm�s for both streams� Thevelocity pro�le exhibits a stagnation point at x � �� cm �Fig� �� The velocitydecreases from the oxygen nozzle feed� then increases due to thermal expansionand decreases again until crossing the stagnation point� The velocity pro�le has asimilar behavior on the fuel side� and asymmetries are due to the di�erent molecularweights of O� and CH� and to chemical e�ects� Figure � depicts the pro�les of massfractions of O�� CH�� and temperature� The maximum temperature is reached atx � �� cm which corresponds to the �ame zone� The consumption of oxygen andmethane are almost complete at x � �� cm� a location slightly on the fuel siderelative to the maximum temperature� This can be attributed to the fact that somerecombination reactions� which are exothermic� occur on the oxygen side� Figure �depicts the pro�le of NO mass fraction� YNO� in the case of the full and Zeldovichmechanisms� and in this case both mechanisms give similar maximum values butslightly di�erent pro�les in the fuel side�

To allow comparison with well known results� a series of strained unpreheated air�ames also has been calculated� Figure � depicts the maximum NO mass fractions�YNO�max� as a function of strain�rate for the oxygen and air �ames� and for theZeldovich and full mechanisms� In all cases� YNO�max decreases with increasingstrain�rate� and this can be attributed to reduced residence times� In the case ofthe oxygen �ame� YNO�max is well predicted by the Zeldovich mechanism alone� Incontrast� in the case of the air �ame� YNO�max is underpredicted by the Zeldovichmechanism by one to three orders of magnitude� The di�erence in behavior betweenthe air and oxygen �ames can be explained on the basis of the �ame temperatureas indicated in Samaniego et al� �� Oxygen �ames are much hotter than air�ames and� as a consequence� the Zeldovich mechanism� which is highly sensitive totemperature� is predominant over the prompt mechanism� in contrast� air �ames arenot as hot and the prompt mechanism is predominant over the Zeldovich mechanismat all strain�rates �Drake � Blint �� Nishioka et al� ��

A further comparison between full and Zeldovich mechanisms for the oxygen �ameis performed by plotting the integral of the N� consumption rate�

R��d�N��dt�dx�

as a function of strain�rate for both calculations� The reason for looking at this


Distance from oxygen-feed nozzle, cm1.00.80.60.40.20.00.0

0.0e+0

1.0e-3

2.0e-3

3.0e-3

4.0e-3

5.0e-3

6.0e-3

NO

mas

s fr

acti

on

Figure �� Computed pro�les ofNO mass fraction in a counter�owO��CH� �amewith � N� addition in both streams for a strain�rate of � s�� Filled symbols full GRI �� mechanism� Open symbols Zeldovich mechanism alone�

Aerodynamic Strain Rate, 1/s120010008006004002000

10 -8

10 -7

10 -6

10 -5

10 -4

10 -3

10 -2

Max

imu

m N

O m

ass

frac

tio

n

Figure �� E�ect of strain�rate on maximum NO mass fraction� � Full GRI�� mechanism for a O��CH� counter�ow di�usion �ame with � N� addition inboth streams� � Zeldovich mechanism alone for a O��CH� counter�ow di�usion�ame with � N� addition in both streams� M Full GRI �� mechanism for anunpreheated Air�CH� counter�ow di�usion �ame� N Zeldovich mechanism alonefor an unpreheated Air�CH� counter�ow di�usion �ame�


Aerodynamic strain rate, 1/s1200100080060040020000

2.0e-7

4.0e-7

6.0e-7

8.0e-7

1.0e-6

Inte

gra

ted

co

nsu

mp

tio

nra

te o

f N

2 (

mo

l/cm

2/s

)

Figure �� E�ect of strain�rate on integrated consumption rate of N�� Full GRI�� mechanism for a O��CH� counter�ow di�usion �ame with � N� addition inboth streams� � Zeldovich mechanism alone for a O��CH� counter�ow di�usion�ame with � N� addition in both streams�

Distance from oxygen-feed nozzle, cm0.80.70.60.50.40.3

-1.0e-5

-5.0e-6

0.0e+0

5.0e-6

1.0e-5

Co

nsu

mp

tio

n r

ate

of

N2 (

mo

l/cm

3/s

)

Figure �� Pro�le of N� consumption rate in a strained O��CH� counter�owdi�usion �ame with � N� addition in both streams for a strain�rate of � s�� Full GRI �� mechanism� � Zeldovich mechanism alone�


Distance from oxygen-feed nozzle, cm0.680.660.640.620.600.580.560.54

-1.0e-5

-8.0e-6

-6.0e-6

-4.0e-6

-2.0e-6

0.0e+0

2.0e-6

Co

nsu

mp

tio

n r

ate

of

N2 (

mo

l/cm

3/s

)

Figure �� Pro�le of N� consumption rate in a strained O��CH� counter�owdi�usion �ame with � N� addition in both streams for a strain�rate of �� s�� Full GRI �� mechanism� � Zeldovich mechanism alone�

quantity rather than the �ux of NO is that reburn� i�e� production of N� fromNO and other nitrogen�containing species� can be clearly identi�ed� Figure de�picts that� in both cases� the consumption rate of N� is positive and decreases withstrain�rate� Furthermore� the Zeldovich approach shows two weaknesses �� at lowstrain�rates� it overpredicts the consumption of N�� and �� at high strain�rates� itunderpredicts it� Overprediction of N� consumption at low strain�rates is evidencethat reburn through the hydrocarbon chemistry is active in these cases� and under�prediction at high strain�rates indicates that N� consumption through the promptmechanism is signi�cant�

The discrepancies between the full and Zeldovich mechanisms may be explainedby analyzing the pro�le of d�N��dt at low and high strain�rates� Figure � depictsthe pro�les of d�N��dt for both mechanisms and the pro�les of temperature� oxygenmass fraction� and methane mass fraction for a strain�rate of � s�� For both mech�anisms� d�N��dt is negative on the oxygen side and positive on the fuel side� On theoxygen side� N� is consumed to produce NO and other nitrogen�containing speciesand the two mechanisms give similar results� This indicates that N� consumptionis mainly due to the Zeldovich mechanism via the reaction O�N� � NO�N andthat the prompt and reburn mechanisms are not active� On the fuel side� N� is pro�duced� which is evidence of reburn� and there is a signi�cant discrepancy betweenthe full and Zeldovich mechanisms� In this case� the Zeldovich pathway predictssome reburn through the reverse reaction �N � NO � O � N�� but misses thereburn through hydrocarbon chemistry which is accounted for by the full GRI ��reaction mechanism� The picture is di�erent at higher strain�rates� Figure depictsthe pro�les of d�N��dt for both mechanisms and the pro�les of temperature� oxygen


mass fraction and methane mass fraction for a strain�rate of �� s�� In this case�d�N��dt is always negative everywhere in the �ame for both the Zeldovich and fullmechanisms� indicating that there is no reburn� However� the full GRI �� schemepredicts higher consumption rates of N� everywhere in the �ame zone both on theoxygen side and in the fuel side� On the oxygen side� most of the N� is consumedvia the Zeldovich mechanism� but additional reactions are also taking place� Onthe fuel side� the full mechanism exhibits a �bump� that is missed by the Zeldovichmechanism and which is due to the �prompt� reaction� CH �N� � N �HCN �Therefore� the following picture can be drawn for oxygen �ames �� the amount

of N� consumption and� consequently� NO production decreases by about �� forstrain�rates varying between �� and �� s�� in this range of strain�rates� theZeldovich mechanism predicts N� consumption within � �� compared to the fullGRI mechanism� �� at low strain�rates� the Zeldovich mechanism overpredicts N�

consumption by about �� because it does not account for reburn through thehydrocarbon chemistry� �� at high strain�rates� the Zeldovich mechanism under�predicts the N� consumption by about �� because it does not account for N�

consumption through the prompt mechanism�

�� An overall one�step model for N� consumption

The next step consists in deriving an overall one�step reaction mechanism for N�

consumption based on the Zeldovich mechanism alone� The proposed model �model�� includes reburn and considers the following reactions

NO �N � N� �O �R��

N �O� � O �NO �R��

N �OH � H �NO �R��

H �O� � OH �O �R��

H �OH �M � H�O �M �R�

O �O �M � O� �M �R��

The model is based on the following �� reaction � plays a negligible role �its netreaction rate is � to � orders of magnitude less than those of reactions � and �� atomic nitrogen is in steady�state �d�N ��dt � �� reactions � to � are assumed inpartial equilibrium� After some algebra and considering N�atom conservation� one�nds

d�NO�

dt� ��d�N��

dt� �

k��k��H��NO�� k�� k

�

� �N��OH��O�

k�� NO� � k�� OH��

with

�H� �

��H�O��O�

K�K��O��

��


Aerodynamic strain rate, 1/s100010010

2.0e-7

4.0e-7

6.0e-7

8.0e-7

1.0e-6

1.2e-6

1.4e-6

1.6e-6

Inte

gra

ted

co

nsu

mp

tio

nra

te o

f N

2 (

mo

l/cm

2/s

)

Figure �� E�ect of strain�rate on integrated consumption rate of N� in a O��CH�

counter�ow di�usion �ame with � N� addition in both streams� Solid line FullGRI �� mechanism� � Zeldovich mechanism alone� M model with no reburn�N model with reburn�

�OH� �

�K��H�O��O��

K��O�

��

�O� �

��O��

K�

��

where bracketed quantities refer to concentrations� and where k is the speci�c re�action rate� K is the equilibrium constant� subscripts refer to reaction numbers�superscript �� refers to forward rate of reaction� and superscript �� refers to back�ward rate of reaction�

An alternative model �model �� with no reburn e�ects and with the partial equi�librium assumption for atomic oxygen can be derived by assuming k�� Oneobtains

d�NO�

dt

��no reburn

� ��d�N��

dt

��no reburn

� �k��

K��

�O��N��

These global one�step models are compared with the full GRI �� and the Zel�dovich mechanisms by focusing on the evolution of the spatially�integrated N� con�sumption rate with strain�rate �Fig� �� This �gure also depicts the values ofR��d�N��dt�dx of the full GRI �� and Zeldovich mechanisms� As in Fig� � all

models lead to a decrease of the integrated consumption rate of N� with strain�rate� Model � has a behavior similar to the Zeldovich mechanism except that atstrain�rates greater than �� s�� it underpredicts N� consumption� This is due


to the fact that for the higher strain�rates the partial equilibrium assumption foratomic oxygen fails� Compared to the full GRI �� scheme� model � overpredictsN� consumption at low strain�rates by about �� due to the fact that it does notaccount for reburn through the hydrocarbon chemistry� At high strain�rates� it un�derpredicts N� consumption by about �� due to the fact that it does not accountfor the prompt mechanism and for non�equilibrium e�ects of atomic oxygen� Model� overpredicts signi�cantly N� consumption at low strain�rates and behaves likemodel � at high strain�rates� The overprediction of N� consumption is due to thefact that no reburn mechanism is included� neither that of the Zeldovich mechanismalone� nor that of the full GRI �� scheme� This shows that inclusion of the reversestep� N�NO � N��O� is essential in getting correct estimates of N� consumptionat low strain�rates�

� Eect of turbulence

�� Numerical approach

Concerning turbulence� the closure method is investigated by post�processing the�D data base of a turbulent di�usion �ame developed by Vervisch �� This database was the result of a direct numerical simulation of a turbulent non�premixed�ame in decaying turbulence� The �ow �eld was resolved accurately by solvingthe full Navier�Stokes equations� The �ame was modeled by a one�step irreversiblereaction� O � F � P � where O is the oxidizer� F is the fuel� and P is the product�and the reaction rate follows an Arrhenius formulation

�w � k�YO�YF exp��E�RT � ��

where �w is the reaction rate� k is the pre�exponential factor� � is the density� YO isthe oxidizer concentration� YF is the fuel concentration� E is the activation energy�R is the gas constant� and T is the temperature� The formalism that was usedwas that of Williams �� the quantities were non�dimensionalized in such away that the heat of reaction was expressed in terms of a temperature jump� � ��Tb�Tu��Tb and the activation energy was expressed in terms of a Zeldovich number�� E�Tb�Tu��RTb

�� The values of these parameters are � � �� and � � �� Thepre�exponential factor was selected such that the initial global Damk ohler numberwas equal to one� This number is de�ned in Chen et al� �� as

Da �ltu�

��

�fl

Z�fl

�wdx

��

where lt is the integral scale of turbulence� u� is the rms velocity� and �fl is the�ame thickness� The Damk ohler number expresses the ratio between the initialeddy turnover time and a characteristic chemical time� Furthermore� the molecularweights of the fuel and the oxidizer were taken to be equal�

At time t � �� the �ow �eld was initialized with a given spectrum for turbulenceand with a planar laminar di�usion �ame located at the center of the computational


domain� The calculations were carried out on a �� grid� The resultingdata base is composed of a series of eight di�erent time intervals they correspondto times �� where time is non�dimensionalized bythe acoustic time� ta � L�c �c is the speed of sound in the fresh reactants and L isa reference length��

The approach for testing various modeling formulations consists of computingat each time�step the volume�average of d�N��dt and comparing this value withthat given by the various closure models� To compute d�N��dt� a chemical modelis needed� and we use model � in this section� The reason for not using model �is that it is necessary to have the NO concentration for this model� However� thedata base does not provide this information� but provides the oxidizer mass fractionand the temperature �eld� and� therefore� only model � can be used� Consequently�reburn is not accounted for� Since the objective is to assess closure formulations� itis expected that model � is su�cient� In this case� the consumption rate of NO isexpressed from Eq� � as follows

d�NO�

dt�rYOT

�

Texp��Ta�T � ��

where YO�T is assumed to be proportional to �O�� T is assumed to be pro�portional to �N�� and Ta is the activation temperature� In this data base� T isnon�dimensional and varies between �� and �� therefore� Ta is non�dimensionaland is set equal to

Ta �EEquilibrium�cal�mol�

R�cal�mol�K��

Tb�K��

Here� we take Tb � �� K in order to reproduce the temperature sensitivity ofN� consumption occurring in an oxygen�methane �ame� and this leads to Ta �� Consequently� the exact turbulent production of NO in the direct numericalsimulation is taken to be

d�NO�

dtjexact � �

NxNyNz!Nx�Ny�Nz

pYO

T ��exp��Ta�T � ��

whereNx� Ny and Nz are the number of points in directions x� y and z� respectively�

�� Results

The PDF closure models of Eq� �a� referred to as JPDF for joint PDF� and Eq��b� referred to as SPDF for single PDF� are assessed by comparing their predictionswith the actual NO production rates as estimated from Eq� �� The predictionsby JPDF and SPDF are computed as follows

d�NO�

dt

��JPDF

�


dNO�dt

time

Figure � Time�evolution of turbulent NO production term� The terms arenon�dimensionalized by the exact value at t � �� Exact term� N Joint PDF�M Single scalar PDF�

�

Nx!Nx

��

NyNz!Ny�Nz

�

T

��

NyNz!Ny�Nz

rYOT

��

NyNz!Ny�Nz

exp�Ta�T �

�

��a�

d�NO�

dt

��SPDF

�

�

Nx!Nx

��

NyNz!Ny�Nz

�

T

��s�

NyNz!Ny�Nz

YOT

��

NyNz!Ny�Nz

exp�Ta�T �

�

��b�Figure � shows a comparison between the predictions of JPDF� SPDF� and the

exact d�NO��dt� In all cases� the production rate of NO starts increasing and thendecreases at later times� This can be explained as follows at initial times� the �amesurface is wrinkled by turbulence� leading to an increase of �ame surface area and�therefore� to an increase in NO production� at later times� the �ame is strainedby the turbulent �ow �eld and is locally quenched� as can be seen in Chen et al�� thereby reducing the rate of production of NO� The JPDF and SPDFmodels overestimate signi�cantly the amount of NO production rates� with JPDFbeing slightly better than SPDF� The reason for this overestimation is that �O��and T are assumed to be uncorrelated� This can be seen as follows �for simplicity�we consider �N�� as constant�

d�NO�

dt� �O�� exp�Ta�T � � ��O��

��exp��Ta�T � ��

where ��O�� is a weighted average of �O�� with

��O��

ZV

�O��

�exp��Ta�T �R

Vexp��Ta�T �dV

�dV ��


where V is the volume on which is conducted the averaging procedure� here� thecomputational domain� Equation �� is a weighted average which is strongly biasedtowards the high temperatures and� therefore� strongly biased to low values of �O��due to chemical reaction and dilatation� Consequently� we obtain

��O�� O��

Furthermore� we have

�O�� O��

��

By combining Eq� �� and �� and assuming that �N�� inhomogeneities do notmodify the relationships� one �nds

ZV

�N��O�� exp��Ta�T �dV � �N��

�ZV

�O��dV

��ZV

exp��Ta�T �dV�

� �N�� O��

ZV

exp��Ta�T �dV ��

Therefore� not accounting for the correlation between �O�� and T leads necessarilyto an overestimation of the turbulent production term of NO� In addition� the erroris dependent on the averaging volume as is apparent in Eq� �� In the case of Fig�� the averaging volumes over which are computed mean quantities are yz planesas indicated in Eq� ��a and ��b� and this corresponds to a case that minimizesthe error� Additional computations have been carried where a single averagingvolume corresponding to the whole computational domain has been used� and thecorresponding estimates of d�NO��dt by the SPDF and JPDF models are �� timeshigher�

�� A model based on laminar �ame structure

To improve the predictive capabilities of PDF formulations for NO formation� acorrelation between �O�� and T is sought assuming a laminar �ame structure� Thegoal is to express �O�� as a function of temperature� Figure �� shows the correlationbetween YO and the reduced temperature � at time t � � in the data base� where� is de�ned by

� �T � TuTb � Tu

��

The relationship has two branches� one with high values of YO corresponding tothe oxidizer side� and one with low values of YO corresponding to the fuel side�Matching the oxidizer branch leads to the following �t

YO � ��

and the fuel branch is �t by YO � ��


Y�

�T � Tu��Tb � Tu�

Figure �� Relationship between � and YO across the planar laminar di�usion�ame at time t � � in the data base�

dNO�dt

time

Figure �� Time�evolution of turbulent NO production term� The terms arenon�dimensionalized by the exact value at t � �� Exact term� � proposedsingle scalar PDF model�

Combining Eq� �� and �� leads to the following alternate single PDF model

d�NO�

dt

��model

��

NxNyNz!Nx�Ny�Nz

�

�

�p��

T ��

p��

T ��

�exp��Ta�T � ��

Figure �� shows the evolution of the turbulent production term of NO for the newsingle PDF model� The agreement is very good and the predictions of this newmodel are insensitive to the choice of the averaging volume�

This test shows that this new model is promising and can be applied in practicalcodes that use single PDF formulations for NO production� In the case of anoxygen�natural gas �ame� reburn should be accounted for and concentrations ofO�� H�O� and NO should be included� Therefore� correlations between �O�� and


T � �H�O� and T �and �NO� and T should be derived from� for example� a library oflaminar strained �ames�

In practice� this PDF model would be able to predict the amount of NO producedin the �ame zone� however� it would not be appropriate in the post��ame zone since�O�� and T are no longer correlated� Therefore� two models should be used� one forthe reaction zone� similar to that developed in this paper� and one for the post��amezone of the kind of the SPDF or JPDF models�

�� Conclusion

The e�ects of chemistry and turbulence on the rate of production of nitric oxidein oxygen�natural gas di�usion �ames has been investigated�

It has been shown that� due to the high temperatures encountered in these �ames�the Zeldovich mechanism is dominant and accounts for more than � of the rate ofproduction of NO over a large range of strain�rates� The limitations of predictionsbased on the Zeldovich mechanism alone have been identi�ed at low strain�rates� itoverpredictsN� consumption� which is the source of NO� due to the fact that it doesnot account for reburn� and at high strain�rates it underpredicts N� consumptiondue to the fact that it does not account for the prompt mechanism� Despite theseshortcomings� the Zeldovich mechanism is satisfactory for predicting NO formationrates within �� and an overall one�step mechanism that accounts for reburn andwhich assumes partial equilibrium of atomic oxygen with O� has been developed�This simple model gives the same results as the Zeldovich mechanism at low strain�rates but leads to further underprediction of N� consumption due to the fact thatit cannot reproduce non�equilibrium e�ects�

The e�ect of turbulence is to generate �uctuations of temperature and speciesconcentrations as well as straining e�ects� It has been shown by postprocessing the�D DNS data base of Vervisch �� that usual closure models based on singlescalar PDF�s or joint PDF�s lead to signi�cant overpredictions of NO formationrates� The main reason is that these models do not account for the intrinsic cor�relation between oxygen concentration and temperature in the reaction zone of adi�usion �ame� A new single scalar PDF model incorporating such a correlationhas been developed� The model assumes that the relationship between oxygen andtemperature is that of a laminar di�usion �ame� The predictions of this model arein excellent agreement with the results from the DNS� and it is expected that it canbe implemented in practical codes that use single scalar PDF�s for NO formation�

Acknowledgments

The �rst author would like to thank Prof� P� Moin for giving the opportunity toparticipate to this summer program� He would also like to thank Greg Ruetsch andNigel Smith for their help� The second author would like to express his gratitudeto Air Liquide and CTR for their partial �nancial support�

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