Flamelet modeling of NO formation in laminar and turbulent diffusion flames

Chemosphere 42 (2001) 449±462

Flamelet modeling of NO formation in laminar and turbulent di€usion ¯ames A. Heyl, H. Bockhorn

*

Institut fur Chemische Technik, Universit at Karlsruhe (TH), Kaiserstrasse 12, 76128 Karlsruhe, Germany

Abstract The applicability of the laminar ¯amelet concept for the formation and destruction of nitric oxides in laminar and turbulent di€usion ¯ames has been studied. In a ®rst step, temperatures and species concentrations in an axisymmetric laminar di€usion ¯ame have been calculated (i) by solving the detailed conservation equations and (ii) by applying the laminar ¯amelet concept. The main purpose of this step was the identi®cation of di€erences between results from both approaches. It turned out that for highly temperature sensitive or relatively slow chemical processes, the inclusion of the full range of the prevailing scalar dissipation rates plays a major role for the calculated species concentrations. This behavior is obvious from the concept of the laminar ¯amelet model, where the scalar dissipation rate can be discussed in terms of the reciprocal of a residence time for attaining chemical equilibrium. In a second step, ¯amelet modeling of NOx formation was extended to a turbulent hydrogen di€usion ¯ame. In both the steps, the ¯ow ®elds of the ¯ames were calculated by solving the Navier±Stokes equations in axisymmetric formulation using the SIMPLER algorithm. For the turbulent ¯ow, Favre-averaged equations have been used and turbulence was modeled with the standard k± model including a correction term for axisymmetric systems. The averaging of the species concentrations was accomplished with presumed shape probability density functions (pdfs). The pdf of the mixture fraction was described with a b-function whereas that of the scalar dissipation rate was assumed to be log-normal. Buoyancy e€ects have been taken into account. The calculated temperatures and concentrations were compared with data from di€erent experiments. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: CFD; Combustion; Di€usion ¯ames; Flamelet model; Modeling; Nitric oxide

1. Introduction The knowledge of the ®elds of the major chemical species, pollutants and temperature is essential for the design of practical combustion devices such as industrial furnaces, gas turbines or internal combustion engines. The characteristic features of these systems are complex geometries and even more complex chemical reaction mechanisms. Their numerical simulation leads to a large

*

Corresponding author. Fax: +49-0721-608-48-20. E-mail address: [email protected] (H. Bockhorn).

set of coupled di€erential equations, that can be solved in acceptable computing time only for one-dimensional systems. One possible way to overcome this problem is the laminar ¯amelet concept (Peters, 1984, 1986). In laminar di€usion ¯amelets, scalar quantities (e.g., species mass fractions, temperature and density) are unique functions of the mixture fraction and the scalar dissipation rate that can be precalculated and stored for further use. One important assumption for this concept are reasonable fast chemical reactions compared with the transport of chemical species and enthalpy driven by convective or di€usive forces. In the recent past, the laminar ¯amelet concept has been applied to relatively slow chemical processes, e.g.,

0045-6535/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 6 5 3 5 ( 0 0 ) 0 0 2 1 7 - 4

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A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

the formation and destruction of soot in combustion systems (Balthasar et al., 1996; Pitsch et al., 1996; Mauû and Balthasar, 1997), as well as the oxidation of chlorinated hydrocarbons (Lentini and Puri, 1995). The laminar ¯amelet concept introduces deviations from chemical equilibrium by means of the scalar dissipation rate. The scalar dissipation rate can be interpreted as the reciprocal of a residence time for attaining chemical equilibrium. Therefore, relatively slow or highly temperature sensitive chemical processes are a€ected sensibly by small scalar dissipation rates. Small scalar dissipation rates reduce the di€usive transport of species and enthalpy from the reaction zone of the ¯amelet so that even slow chemical reactions are fast compared with di€usive transport. The application of the ¯amelet concept to the formation of nitric oxides is discussed controversially in literature. When modeling the formation of nitric oxides in turbulent di€usion ¯ames, Pfuderer et al. (1996) were lacking a consistent de®nition for the mixture fraction, that is suitable to model the di€erent di€usion velocities in the ¯ame region. Vranos et al. (1992) failed in modeling their turbulent methane-di€usion-¯ame satisfactorily with the stationary ¯amelet concept. Sanders et al. (1997) found a good agreement between ¯amelet-based simulations and experiments for the NOx concentration in the region of high Damk ohler numbers. The ¯amelet calculations of Chen and Chang (1996) show nitric oxide concentrations that surpass the experimentally determined values by about 200%. Chou et al. (1998) demonstrated for a Bunsen-type ¯ame good agreement between measurements and simulations using the ¯amelet concept. A further indication of the applicability of the ¯amelet concept to the formation and destruction of nitric oxide are the studies of Barlow and Carter (1996). They demonstrated that NOx concentrations in di€erent turbulent hydrogen di€usion ¯ames can be correlated uniquely to the mixture fraction.

In this work, in a ®rst step, numerical results obtained from ¯amelet-based calculations for a laminar CH4 /air di€usion ¯ame are compared with results from the solution of the complete set of conservation equations for each species of the two-dimensional ¯ow ®eld. Particularly, the in¯uence of scalar dissipation rates on the predicted NO concentrations in ¯amelet-based simulations is emphasized. In a second step, the ¯amelet concept is used to predict the formation of NOx in a turbulent hydrogen di€usion ¯ame. The in¯uence of turbulent ¯uctuations on the chemical system is modeled with the k± model and presumed shape probability density functions (pdfs) for the mixture fraction and the scalar dissipation rate. The resulting velocities, temperatures and characteristic species concentrations are compared with experimental data from laser spectroscopic measurements. 2. Chemical mechanism The chemical model used in this study is the Gas Research Institute (GRI) mechanism (Bowman et al., 1995) in its version 2.11. It consists of 49 chemical species and 279 reactions describing the oxidation of methane and containing a comprehensive set of reactions for the formation and reduction of NO. Therefore, the ¯amelet calculations for the laminar CH4 /air-di€usion ¯ame presented below include the destruction of NO by convection into the fuel rich side of the ¯ame. The main processes for the formation of nitric oxides are summarized in Table 1. They can be distinguished in four di€erent mechanisms. The rate determining step in the thermal formation of NO (Zeldovich mechanism) is the breaking of the chemical bond in the N2 molecule. This process exhibits a high activation energy and is, therefore, highly sensitive to the local temperature.

Table 1 Sources of NOx in ¯ames Mechanism of formation Thermal NOx (Zeldovich, 1946)

oxygenÿrich

N ‡ NO N2 ‡ O N ‡ O2 NO ‡ O

Region/main factor of in¯uence

N ‡ OH NO ‡ H

Flame, exhaust-stream/residence time, O-concentration, temperature > 1600 K

Prompt NOx (Fenimore, 1970)

CH ‡ N2 HCN ‡ N HCN





NO

Flame/Cx -, O-, NH-radicals O2 -/fuel-concentration, temperature

NOx from N2 O mechanism (Roby and Bowman, 1987)

N2 ‡ O ‡ M N2 O ‡ M

Flame/temperature, pressure O-concentration

NOx from fuel molecules containing nitrogen (Warnatz et al., 1996)

Di€erent mechanisms

fuelÿrich

N2 O ‡ O 2NO NH ‡ NO N2 O ‡ H

Flame/O2 -concentration residence time, type of fuel

A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

Furthermore, this step is slow compared with the fast radical reactions in the oxidation zone of the ¯ame. The formation of prompt NO (Fenimore mechanism) is more complex than the thermal oxidation of nitrogen. It depends on the concentrations of various hydrocarbon radicals. The reactions mentioned in Table 1 are only examples for the variety of reactions that have to be considered. The activation energy for the reaction of a hydrocarbon radical with the nitrogen molecule is assumed to be small compared with that of the Zeldovich reaction path. The formation mechanism of N2 O is not written in detail. The concentration of N2 O increases with higher pressure in spite of the involved trimolecular reaction steps. For temperatures higher than 1500 K, N2 O is quickly destroyed and is, therefore, not detected in the exhaust of practical combustion devices. The quantity of NOx resulting from reactions of nitrogen contained in the fuel is characteristic for the fuel. The present work is based on gaseous fuels free from inherent nitrogen and, therefore, this source of NOx is not taken into account. Both the laminar reference ¯ame (CH4 /air) and the turbulent ¯ame (H2 /air) considered in this work have been selected such that the Zeldovich pathway to NO is the dominant one.

3. Numerical simulation 3.1. Flow ®eld The numerical simulation of the ¯ow ®eld includes the solution of the overall continuity equation and the Navier±Stokes equations in low Mach number formulation. For the turbulent ¯ame, the equations are written

Table 2 Conservation equations solved in the numerical simulations

a

451

in their Favre-averaged form and the turbulent ¯uctuations are taken into account with the standard k± model (Launder and Spalding, 1974) corrected for round jets (Pope, 1978). The ¯amelet concept necessitates the solution of a transport equation for the mixture fraction n, that is a normalized element mass fraction. For turbulent systems, an additional equation for the 002 has to be solved. variance of the mixture fraction nf The generalized conservation equation in cylindrical coordinates for a dependent variable / for axisymmetric problems can be written as o…q/† ot |‚‚{z‚‚}

Time dependence

‡

o…qu/† 1 o…rqv/† ‡ ox r or |‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚} Convection

    o o/ 1 o o/ C/ ‡ C/ r ˆ ‡ S/ |{z} ox ox r or or |‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚} Source

…1†

Diffusion

The source term S/ contains all parts of the conservation equation, which cannot be included in the convection or di€usion terms. For the ¯ames under consideration, the system of equations is setup by the conservation equations summarized in Table 2. For /  u and /  v, the di€usive transport coecient is the viscosity, C  l, and for scalars, the di€usive transport is assumed to be proportional to the exchange of momentum, C/  l=Sc or C/  l=Pr. For turbulent systems, the variable / has to be sub~ The additional stituted by its Favre-averaged value /. turbulent transport is included in the turbulent viscosity, lt ˆ Cl qk~2 =~  with Cl ˆ 0:09, and the turbulent transport coecient C/;eff . Again, for scalars such as species concentrations or enthalpy, the turbulent transport is assumed to be proportional to the exchange of

a

Equation for

Symbol

Source term

Continuity

1

0

Axial momentum

u

Radial momentum

v

ÿ
1 o ÿ ov
2 o h ou 1 o…rv†i ÿ op ‡ r or rl ox ÿ 3 ox l ox ‡ r or ‡ oxo l ou ox ox ÿ ou
1 o ÿ ov
2 o h ou 1 o…rv†i op o ÿ or ‡ ox l or ‡ r or rl or ÿ 3 or l ox ‡ r or

Turbulent energy

k~

Gk ÿ q~

Turbulent dissipation

~

…C1 Gk ÿ C2 q~† k~~ ‡ ~k~
CPope vPope

Mixture fraction

n

0

Chemical species

Yk

w_ k

Enthalpy

h

Variance mixture fraction

002 nf

0 for adiabatic systems (no radiation) h i ~ ~ 002 C1 lt oxonk oxonk ÿ 2C2 q k nf

2

Gk represents the generation of turbulence energy and w_ k stands for the net chemical reaction rate of the species k. For turbulent ¯ow, the variables have to be replaced by their Favre-averaged values.

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momentum in the ¯uid ¯ow. Therefore, C/;eff is derived from the turbulent viscosity lt as follows: C/;eff ˆ

lt rt

with rt ˆ Prt or rt ˆ Sct :

…2†

Sc stands for the Schmidt number and Pr for the Prandtl number. The resulting set of equations is solved with the twodimensional solver ComChe2D (Heyl and Bockhorn, 1998; Heyl, 1999) written for the simulation of reactive ¯ows containing Complex Chemistry in 2D geometries. The equations are discretized with the method of ®nite volumes. To suppress numerical di€usion, the higher order discretization scheme QUICK (Leonhard, 1979) is used. The system of non-linear equations with a block-pentadiagonal system matrix has been solved after resolving the block structure with an ADI method combined with the Thomas algorithm. The pressure correction has been obtained by the SIMPLER algorithm (Patankar, 1980).

Fig. 1. Iterative cycle for modeling of complex chemical kinetics with coupling functions.

density q and the temperature T are expressed as unique functions of a number of parameters Ki /k ˆ f …K1 ; K2 ; . . . ; Kn † with

3.2. Chemical model The reaction mechanism describing the formation of pollutants such as NOx adopted in this work contains 279 chemical reactions between 49 species. Table 3 shows the estimated number of conservation equations, that have to be solved for each grid node in the calculation domain. Additional equations for the thermodynamic properties and the transport coecients have to be solved. Furthermore, the in¯uence of turbulence has to be included, multiplying the numerical requirements. In three-dimensional geometries the number of grid points in the computational domain can be expanded easily to 100,000 and more. Considering a number of 50 and more conservation equations per grid point, the necessity for the reduction of the numerical e€ort by modeling assumptions is obvious. Clearly, the major part of the equations results from the chemical system, compare Table 3. A common way to reduce the numerical e€ort is the decoupling of the ¯ow ®eld from the chemical system. In this case, the species concentrations Yk , the laminar viscosity l, the Table 3 Estimated number of equations to solve for a single grid node in a discretized computational domain Property

Number of di€erential equations

Fluid dynamics (incl. turbulence) Enthalpy (incl. radiation) Chemistry

10±15 2±5 40±120

Sum

ca. 50±140

/k ˆ q; l; T ; Yk :

…3†

Examples for Ki are the mixture fraction or a set of reaction progress variables. Common models using this kind of relations are the ``fast chemistry model'' (Kuo, 1986), ``equilibrium models'' (Kuo, 1986), the ``ILDM model'' (Maas and Pope, 1992) and the ``¯amelet model'' (Peters, 1984, 1986) used in this work. If the coupling functions are known, the relations given by Eq. (3) can be determined in a preprocessing step. The results of this calculations are tabulated or ®tted to simple functions (e.g., splines) in dependence of Ki . The parameters Ki are calculated by means of the computational ¯uid dynamics (CFD)-code. In a following step, the tabulated information from the preprocessing step is used to determine the concentrations of the chemical species, the temperature and other properties. This properties are new input parameters for the next iteration step in the CFD-code. The iteration cycle is repeated until numerical convergence is achieved. Fig. 1 demonstrates the numerical procedure. The main advantage of this modeling strategy is the retaining of the full chemical information, standing in contrast to reduced mechanisms. The number of the coupling parameters Ki generally is small …i  1±10† compared with the number of species.Therefore, the computational time for numerical convergence can be reduced by 80% or more. Furthermore, the decoupling of the computation of the ¯ow ®eld and the chemical system give the intrinsic possibility to use parallel computing. 3.2.1. Flamelets in laminar di€usion ¯ames In laminar ¯amelets, all scalars are unique functions in the mixture fraction/scalar dissipation rate space.

A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

Consequently, their balance equations are transformed into this space (Peters, 1984, 1986) q

n vst d2 T X hk ÿ w_ k ˆ 0; 2 2 dn c kˆ1 p

…4†

q

vst d2 Yk ‡ w_ k ˆ 0; 2 dn2

…5†

where the scalar dissipation rate vst is de®ned by "   2 # 2 on on ‡ : vst ˆ 2D ox or

…6†

Here Yi are the species mass fractions, T the temperature, and n is the mixture fraction. w_ k is the chemical production rate of species k, hk the species enthalpy, cp the constant pressure speci®c heat capacity and q the mass density. Eq. (4) includes no enthalpy loss by radiation. Although the in¯uence of radiation losses on the prediction of temperatures is crucial for the prediction of temperature-sensitive species such as NOx , radiation is not considered because of the optical thin systems investigated in the present work. The stationary solutions of Eqs. (4) and (5) were stored in tables containing the pro®les of temperature, mass density and mass fractions of all chemical species in dependence on mixture fraction and scalar dissipation rate. The coupling of chemistry and ¯ow ®eld was performed via the mixture fraction and the scalar dissipation rate, which were provided from the ¯ow ®eld

453

calculations. For every grid point in the ¯ow ®eld, the mass fractions of the gaseous species, temperature and the mass density are interpolated from tables corresponding to the ¯amelet perpendicular to the surface of stoichiometric mixture. For this speci®c ¯amelet, the value of the scalar dissipation rate vst is assumed to be constant along the mixture fraction coordinate. vst is determined at the point of stoichiometric mixture fraction. Fig. 2 shows the characteristic form of the mass fraction pro®le of an intermediate species Yintermed along the mixture fraction coordinate. 3.2.2. Flamelets in turbulent di€usion ¯ames The in¯uence of turbulent ¯uctuations on the chemical system can be described with help of pdfs. The mean value of a quantity / tabulated in the ¯amelet library is derived from the following integration: ~ r† ˆ U…x;

Z

1 0

Z

1 0

U…n; vst † pdf…n; vst ; x; r† dn dvst :

…7†

For the assumption of statistical independence pdf…n; vst ; x; r† can be written as pdf…n; v† ˆ pdf…n†
pdf…v†:

…8†

For the pdf of the mixture fraction, a b-function has been assumed pdf…n† ˆ R 1 0

ˆ

naÿ1 …1 ÿ n†bÿ1 naÿ1 …1 ÿ n†bÿ1 dn

C…a ‡ b† aÿ1 n …1 ÿ n†bÿ1 ; C…a†C…b†

where C…x† is Z 1 C…x† ˆ eÿt txÿ1 dt:

…9†

…10†

0

The parameters a and b are functions of the mean value and the variance of the mixture fraction ~ n~2 …1 ÿ n† ~ ÿ n; 002 nf ! 1 ÿ n~ ˆa : n~

aˆ

bˆ

~ ÿ n† ~2 n…1 ~ ÿ …1 ÿ n† 002 nf …11†

The pdf of the scalar dissipation rate pdf…vst † is assumed to have the shape of a logarithmic normal distribution ! …log vst ÿ llog †2 log e p   pdf…vst † ˆ : …12† exp ÿ 2r2log vst rlog 2p Fig. 2. Characteristic mass fraction pro®le for an intermediate chemical species for a ¯amelet perpendicular to the surface of stoichiometric mixture nst .

The parameters llog and rlog stand for the mean value and the variance of the transformed property

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fv ˆ log vst : They are derived from the mean value and the variance of the scalar dissipation rate as follows: ! r2log v~st ˆ exp llog ‡ ; 2     2 002 ˆ exp r2 ÿ 1 exp 2l vf log ‡ rlog : log st

…13†

The equation of the mean value of the scalar dissipation rate v~st is modeled as 002 v~st ˆ Cv nf

~ k~

with

Cv ˆ 2:

…14†

llog can be calculatedpwith  Eq. (14). rlog is assumed to have a value of rlog ˆ 2. The work of Liew et al. (1984) shows only little in¯uence of the exact value of rlog on the results obtained from ¯amelet calculations.

4. Laminar reference ¯ame

Fig. 3. Experimental setup for the laminar reference ¯ame taken from Feese and Turns (1998).

4.1. Experimental setup The boundary conditions (see Table 4) are similar to those adopted Feese and Turns (1998). The fuel consists of preheated CH4 (technical grade 98%) exiting into a coaxial ¯ow of preheated air. The burner geometry is shown in Fig. 3. The velocity pro®le of the fuel is assumed to be a fully developed pipe ¯ow. The outlet velocity of the oxidizer is assumed to be homogeneous. 4.2. Numerical setup for the ¯amelet calculations Three numerical studies (denoted runs 1±3) were performed. Run 1 adapts conservation equations for all chemical species and the enthalpy in the two-dimensional-CFD-code. Runs 2 and 3 are ¯amelet-based calculations, solving the species and energy conservation

Table 4 Experimental conditions for the CH4 ±air reference ¯ame Fuel (Index F) Inlet velocity uF Composition of fuel Inlet temperature TF Oxidizer (Index Ox) Inlet velocity uOx Composition of oxidizer Inlet temperature TOx

Fully developed pipe ¯ow with hui ˆ 7:8 cm/s 98.0 mole % CH4 ; 2.0 mole% N2 413 K Plug ¯ow with hui ˆ 3:5 cm/s 21.0 mole% O2 ; 79.0 mole% N2 398 K

Table 5 Common parameters for runs 1±3 Parameter Computational domain Number of grid points Discretization Pressure correction Chem. Mechanism Energy losses Lewis Numbers

105 mm  400 mm (using a symmetry boundary condition) 104  83 QUICK SIMPLER Bowman et al. (1995) O€, adiabatic All set to 1

equations in a separate one-dimensional code for prede®ned sets of the scalar dissipation rate and the mixture fraction. The coupling between the ¯ow ®eld and the ¯amelets is established via the tabulated density and viscosity. Runs 2 and 3 are di€erent in the used sets of scalar dissipation rates. Common to all three calculations are the grid resolution, the discretization, the pressure correction scheme and the chemical reaction mechanism (see Table 5). 4.2.1. Coupling of ¯ow ®eld and chemical reactions (run 1) All calculations in run 1 are done in the CFD-code. The species and the enthalpy are calculated following Eq. (1). With help of the caloric equation of state and

A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

455

NASA-polynomials for the speci®c enthalpy the local temperature is determined. 4.2.2. Coupling of ¯ow ®eld and chemical reactions (runs 2 and 3) For the given boundary conditions of the methane/ air ¯ame, two sets of ¯amelet libraries have been calculated and used in the CFD-code (see Table 6). Two di€erent sets of prescribed scalar dissipation rates in runs 2 and 3 were used, to elucidate the impact of low scalar dissipation rates on the concentration of species formed on slower time scales. With decreasing scalar dissipation rates, the di€usion terms in Eqs. (4) and (5) become smaller and ®nally approach zero. The equations then turn into the equations for the equilibrium state of the mixture. This transition is equivalent to ascending residence times and should have strong in¯uence on slower reaction channels such as the formation of NO via the Zeldovich mechanism. This is demonstrated in Fig. 4 for the mass fraction of NO, which increases considerably with lower scalar dissipation rates. The formation of thermal NO is very sensitive to residence time and temperature. This e€ect is ampli®ed by the increasing maximum temperatures with decreasing scalar dissipation rate, which is also a consequence of the higher residence time (see Fig. 5). Fig. 6 shows the mass fraction of the fuel CH4 versus mixture fraction. The in¯uence of the scalar dissipation rate is signi®cantly smaller than that on the NO-pro®les. This is a consequence of the relatively fast oxidation reactions, so that in Eq. (5), applied to CH4 , Table 6 Prescribed scalar dissipation rates v ‰sÿ1 Š for runs 2 and 3 Run 2 Run 3

1.0, 2.0, 5.0, 7.5, 10.0, 20.0, 30.0, extinction 0.003, 0.005, 0.01, 0.1, 0.5,1.0, 2.0, 5.0, 7.5, 10.0, 20.0, 30.0, extinction

Fig. 5. Pro®les of the temperature in the mixture fraction space in dependence on the scalar dissipation rate vst .

Fig. 6. Pro®les of the mass fraction of CH4 in the mixture fraction space in dependence on the scalar dissipation rate vst .

the second term on the left-hand side dominates at higher scalar dissipation rates than in the application of Eq. (5) to NO. 4.3. Results and discussion

Fig. 4. Pro®les of the mass fraction of NO in the mixture fraction space in dependence on the scalar dissipation rate vst .

The two-dimensional pro®les of the axial velocities and mixture fraction calculated in run 1 are shown in Figs. 7 and 8. The acceleration of the axial velocity from 7.8 cm/s to 2.4 m/s is caused by the steep temperature gradient and the corresponding thermal expansion and buoyancy e€ects. The stoichiometric mixture fraction is 0.057. This corresponds to the dashed line in Fig. 8. The scalar dissipation rates along the axis of this ¯ame vary from about 1:0 sÿ1 at 5 mm above the nozzle to 10ÿ6 sÿ1 at 250 mm above the nozzle. The stoichiometric mixture fraction on the axis is attained at 50 mm above the nozzle at scalar dissipation rates of about 10ÿ3 sÿ1 . Fig. 9 shows the radial temperature pro®les in an axial distance of 5 mm above the nozzle exit obtained in

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A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

Fig. 9. Comparison of the results from numerical calculations and measurements for the radial pro®les of the temperature 5 mm above the nozzle exit for the laminar reference ¯ame.

Fig. 7. Two-dimensional plot of the axial velocity u [m/s] for the laminar reference ¯ame.

Fig. 10. Comparison of the results from numerical calculations and measurements for axial pro®les of the temperature along the axis symmetry of the burner for the laminar reference ¯ame.

Fig. 8. Two-dimensional plot of the mixture fraction n for the laminar reference ¯ame. The dashed line of nst shows the contour of the ¯ame.

runs 1, 2 and 3 and corresponding thermocouple measurements. The measurements and the numerical results agree reasonably well. In the radial pro®les, di€erences between the ¯amelet-based calculations (runs 2 and 3) and the solution of the complete set of species conservation equations in the two-dimensional-CFD-code (run 1) can be seen in the fuel rich region of the ¯ame. This is a consequence of the constant scalar dissipation rate along the mixture fraction coordinate for every speci®c

¯amelet (Heyl, 1999). There are negligible di€erences between the two sets of ¯amelet ensembles, because in this height above the burner the scalar dissipation rate is not lower than v ˆ 1:0 sÿ1 which is the lower limit in run 2. The axial temperature pro®les (see Fig. 10) show little di€erences between runs 1 and 3. As expected, run 2 exhibits a considerable lower temperature maximum compared with run 3. This is a consequence of the different sets of scalar dissipation rates used for the ¯amelet calculations. The missing scalar dissipation rates from 0:003 to 0:5 sÿ1 in run 2 reduce the range of residence times.At larger heights, above the nozzle, the axial temperatures are somewhat overpredicted. This e€ect may be traced back to the exclusion of radiative heat losses from the ¯ame in the model. As can be seen from Fig. 11, the cut o€ of the lower scalar dissipation rates (i.e., long residence times) leads

A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

Fig. 11. Comparison of the results from di€erent numerical calculations for the axial pro®les of the NO mass fraction for the laminar reference ¯ame.

Table 7 Calculated and measured emission index NOx (EINOx ) Experiment

3.5

Run 1 Run 2 Run 3

3.2 1.7 3.3

to a considerable underprediction of the NO mass fractions. This ®nding is con®rmed by the integral Emission Index NOx (EINOx -value), which stands for the quantity of NO in grams emitted for 1 kg burned fuel. Table 7 compares the values for the EINOx from the three di€erent runs with measurements. Both runs 1 and 3 agree very well with the measured emission index of NOx . Run 2 underpredicts the EINOx value because of its limited set of scalar dissipation rates. As mentioned before, the small di€erences between the reference run 1 and the ¯amelet-based results of run 3 can be traced back to omitting a distribution of scalar dissipation rate in the mixture fraction space (Heyl, 1999). The cut-o€ of the lower scalar dissipation rates obviously has much more e€ect than the slight overprediction of the temperature.

5. Turbulent ¯ame 5.1. Experimental setup The turbulent ¯ame investigated in this work consists of diluted hydrogen injected into a coaxial air ¯ow. There are experimental values for the ¯uid velocities, the temperature and various species concentrations. The complete data set is published in Tacke et al. (1998). Details for the experimental setup, the diagnostics and numerical simulations of this ¯ame setup

457

Fig. 12. Experimental setup taken from Tacke et al. (1998)

are published in Pfuderer et al. (1996), Tacke et al. (1996, 1998) and Neuber et al. (1998). Fig. 12 shows the burner setup.The outlets for the pilot ¯ames (see Fig. 12) are not activated in the considered experiments. Table 8 summarizes the boundary conditions in consideration. The non-piloted jet ¯ame exhibits scalar dissipation rates in the region of stoichiometric mixture fraction lower than 10 sÿ1 , which is far below the extinction scalar dissipation rate. The ¯ame is operated in non-lift-o€ mode. 5.2. Numerical setup for the ¯amelet calculations As discussed in Section 4, the scalar dissipation rate has a tremendous impact on the prediction of products of slow chemical processes. As a consequence, di€erent distribution functions for v should also in¯uence the predicted mass fraction of NOx in turbulent di€usion ¯ames. Therefore, two numerical experiments (denoted runs 4 and 5) were performed to study the in¯uence of the distribution function of the scalar dissipation rate on the formation of NOx . The pdf of the mixture fraction is in all cases a b-function. In run 4, the pdf(v) is modeled as logarithmic normal distribution as described in Section 3.2.2. In run 5, the presumed pdf of v is assumed to be a Dirac-d-function, standing for one mean value of the scalar dissipation rate. This is a common modeling assumption used for example in the representative interactive ¯amelet concept (Pitsch et al., 1996). All calculations are based on a set of 17 ¯amelets (see Table 9). The numerical parameters and methods for the CFD-model are kept constant in all runs. Table 10 shows the characteristic parameter settings. Common to all calculations are the grid resolution, the discretization, the pressure correction scheme, the turbulence model and the chemical reaction mechanism (see Table 10).

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Table 8 Experimental conditions for the turbulent H2 ±air ¯ame Fuel (Index F) Inlet velocity u~F Composition of fuel Inlet temperature TF

Fully developed turbulence pipe ¯ow with h~ ui ˆ 34:5 m/s 50.0 mole% H2 ; 50.0 mole% N2 300 K

Oxidizer (Index Ox) Inlet velocity u~Ox Composition of oxidizer Inlet temperature TOx

Plug ¯ow with h~ ui ˆ 0:2 m/s 21.0 mole% O2 ; 79.0 mole% N2 300 K

5.3. Results and discussion Table 9 Prescribed scalar dissipationrates v ‰sÿ1 Š for runs 4 and 5 v [1/s]

0.01, 0.05, 0.1, 0.5,1.0, 2.0, 5.0, 7.5, 10.0, 25.0, 50.0, 100.0, 250.0,500.0, 1000.0, 2500.0, extinction

Table 10 Common parameters for runs 4 and 5 Parameter Computational domain Number of grid points Turbulence model Discretization Pressure correction Chem. mechanism Energy losses Lewis numbers

120 cm  12 cm (using a symmetry boundary condition) 54  46 k±-model with Pope-correction QUICK SIMPLER (Bowman et al., 1995) O€, adiabatic All set to 1

Fig. 13 shows the two-dimensional ®elds of the mean axial velocity and the mean mixture fraction from run 5. The results are typical for turbulent di€usion jet ¯ames. The axial velocity and the mixture fraction decrease rapidly and the contours expand in radial direction. The expansion of the ¯ame induces a low pressure region near the outlet leading to a recirculating ¯ow with small negative axial velocities. Fig. 14 gives a comparison between the results from runs 4 and 5 and measurements for the mean axial velocity, the mean mixture fraction and the mean temperature. The results of the numerical calculations agree very well with the experimental results. The differences between runs 4 and 5 are negligible. The form of the distribution function of v has only little in¯uence on the results for the ¯amelet calculations for the ®eld of the mean temperature. The ®elds of the mean axial velocity and the mean mixture fraction show the analogous behavior. Both ®elds are mainly in¯uenced by the mean temperature and its back coupling via the mean density. Flamelets with a scalar dissipation rate

Fig. 13. Two-dimensional ®elds of the mean axial velocity and the mean mixture fraction in the turbulent hydrogen ¯ame.

A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

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Fig. 14. Axial pro®les of the mean axial velocity, the mean temperature and the mean mixture fraction along the axis of symmetry. The lower right plot shows the pro®les of temperature in ¯amelet space in dependence on the scalar dissipation rate.

below 100 sÿ1 show similar temperature pro®les in the mixture fraction space (see Fig. 14). As the scalar dissipation rates in that ¯ame are not higher than 100 sÿ1 , the computed results for the temperature are independent of the distribution function of the scalar dissipation rate. Fig. 15 shows analogous plots for the mass fractions of H2 , O2 and H2 O reacting all in fast reaction channels. It can be seen that hydrogen is consumed quantitatively at a height of x=D ˆ 40. At this point, the corresponding mean mass fraction of water shows its expected maximum. At x=D > 30, oxygen coexists with water and hydrogen. At smaller heights, O2 is fully consumed. The agreement between the results from measurements and numerical simulation is very good. The ¯amelet calculations for runs 4 and 5 give identical results. This ®nding is a result of the independence of the mass fraction pro®les of the fast reacting species of the scalar dissipation rate (see theH2 ¯amelet at the lower right of Fig. 15) in the prevailing order of magnitude of scalar dissipation rates. The reactions of this species can be summarized under ``fast chemistry''-assumptions and show high Damk ohler numbers. Fig. 16 compares the mass fractions of NO and OH reacting on slower chemical time scales. There are considerable deviations between the calculations and the

measurements. Furthermore, there exists a di€erence between runs 4 and 5. This is a result of the strong dependencies of the NO and OH mass fractions on the scalar dissipation rate for dissipation rates in the prevailing order of magnitude as shown for nitric oxide in the lower part of Fig. 16. The form of the distribution function of v determines which ¯amelets are weighted more or less in the averaging procedure according to Eq. (7). Di€erences in the ¯amelets lead to di€erent mean values for this species. Interestingly, run 5, assuming a monodisperse distribution for the scalar dissipation rate, shows a slightly better agreement with the experimental results. To improve the quality of the predictions for the slow chemical processes in ¯amelet calculations the following points have to be emphasized: · Inclusion of radiative heat losses in the model (either in the ¯amelet calculation or CFD-code). · Implementation of di€erent di€usion velocities in one ¯amelet leading to a dependence of vst ˆ f …n†. · An experimental proof of the postulated statistical independence of the probability functions of vst and n. · Calculations based on ®rst principle assumptions or experiments to determine the correct distribution function for the scalar dissipation rate.

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A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

Fig. 15. Axial pro®les of the mean mass fraction of H2 , the mean mass fraction of O2 and the mean mass fraction of H2 O along the axis of symmetry. The lower right plot shows the pro®les of the mass fraction of H2 in ¯amelet space in dependence on the scalar dissipation rate.

Fig. 16. Axial pro®les of the mean mass fraction of OH and the mean mass fraction of NO along the axis of symmetry. The lower plot shows the pro®les of the mass fraction of NO in ¯amelet space independence on the scalar dissipation rate.

A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

· Implementation of a conservation equation for vst (Fox, 1992). 6. Conclusions Numerical modeling of a laminar axisymmetric CH4 / air di€usion ¯ame has been performed using the ¯amelet model and the full set of conservation equations for all species and energy in a two-dimensional-CFD-code. The comparison between these models shows reasonable agreement. In ¯amelet modeling, the consideration of low scalar dissipation rates is essential. Insucient resolution of the ¯amelet library in the low scalar dissipation rate region has a tremendous impact on the prediction of relatively slow reacting, temperature sensitive species. When applying the ¯amelet model to a turbulent hydrogen di€usion ¯ame, good agreement between the measured mean velocities, the mean temperature and the mean mass fractions of the species reacting on fast time scales and the predictions can be achieved. The calculated mean mass fractions for NO and OH, however, show some deviations from the experimental ones. These di€erences are results of the in¯uence of the form of the distribution function of the scalar dissipation rate on the predicted concentrations of slowly reacting species such as NO and OH. Both the Diracd-function and the logarithmic distribution as well, lead to correct mass fractions for the chemical species reacting on fast paths. Possible ways to re®ne the ¯amelet modeling of relatively slow chemical processes are listed and are object of future investigations. Acknowledgements The authors acknowledge ®nancial support of this research by the VW-foundation in their program `Modeling of complex systems in process engineering'. References Balthasar, M., Heyl, A., Mauû, F., Schmitt, F., Bockhorn, H., 1996. Flamelet modeling of soot formation in laminar ethyne/air di€usion ¯ames. In: 26th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, p. 2369. Barlow, R.S., Carter, C.D., 1996. Relationship among nitric oxide temperature and mixture fraction in hydrogen jet ¯ames. Combust. Flame 104, 288. Bowman, C.T., Hanson, R.K., Davidson, D.F.,Gardiner, W.C. Jr., Lissianski, V., Smith, G.P., Golden, D.M., Frenklach, M.,Wang, H., Goldenberg, M., 1995. GRI-Mech Version 2.11,http://www.gri.org/.

461

Chen, J.-Y., Chang, W.-C., 1996. Flamelet and PDF-modeling of CO and NOx emission from a turbulent, methane hydrogen jet nonpremixed ¯ame. In: 26th Symposium (International) on Combustion, The Combustion Institute,Pittsburgh, p. 2207. Chou, C.-P., Chen, J.-Y., Yam, C.G., Marx, K., 1998. Numerical modeling of no formation in laminar bunsen ¯ames ± a ¯amelet approach. Combust. Flame 114, 420. Feese, J.J., Turns, S., 1998. Nitric oxide emissions from laminar di€usion ¯ame: e€ects of air-side versus fuel-side dilutent addition. Combust. Flame 113, 66. Fenimore, C.P., 1970. Formation of nitric oxide in premixed hydrocarbon ¯ames. In: 13th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, p. 373. Fox, R.O., 1992. The Fokker±Planck closure for turbulent molecular mixing: passive scalars. The Phys. Fluids 4, 1230. Heyl, A., 1999. Modeling complex kinetics in turbulent ¯ows, Ph.D. Theses, University of Karlsruhe. Heyl, A., Bockhorn, H., 1998. Modeling of pollutant formationin complex geometries under consideration of detailed chemical mechanisms. In: Papailiou, K.D., Tsahalis, D., Periaux, J., Kn orzer, D. (Eds.), Computational Fluid Dynamics'98, vol. 2. Wiley, Chichester, p. 808. Kuo, K.K, 1986. Principles of Combustion. Wiley, New York. Launder, B.E., Spalding, D.B., 1974. The numerical computation of turbulent ¯ows. Comput. Meth. Appl. Mech. Eng. 3, 269. Lentini, D., Puri, I., 1995. Stretched laminar ¯amelet modeling of turbulent chloromethane-air nonpremixed jet ¯ames. Combust. Flame 103, 328. Leonhard, B.P., 1979. A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Comput. Meth. Appl. Mech. Eng. 19, 59. Liew, S.K., Bray, K.N.C., Moss, J.B., 1984. A stretched laminar ¯amelet model of turbulent nonpremixed combustion. Combust. Flame 56, 199. Maas, U., Pope, S.B., 1992. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combust. Flame 88, 239. Mauû, F., Balthasar, M., 1997. Simpli®cation of a detailed kinetic soot model for the application in 3D-programs. In: Fourth International Conference on Technologies and Combustion for a Clean Environment, Lissabon, p. 17. Neuber, A., Krieger, G., Tacke, M., Hassel, E.P., Janicka, J., 1998. Finite rate chemistry and no mole fraction in nonpremixed turbulent ¯ames. Combust. Flame 113, 198. Patankar, S.V., 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington, DC. Peters, N., 1984. Laminar di€usion ¯amelet models in nonpremixed turbulent combustion. Prog. Energy Combust. Sci. 10, 319. Peters, N., 1986. Laminar ¯amelet concepts in turbulent combustion. In: 21st Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, p. 1231. Pfuderer, D.G., Neuber, A.A., Fr uchtel, G., Hassel, E.P., Janicka, J., 1996. Turbulence modulation in jet di€usion ¯ames: modeling and experiments. Combust. Flame 106, 301. Pitsch, H., Barths, H., Peters, N., 1996. Three-dimensional modeling of NOx and soot formation in DI-diesel engines

462

A. Heyl, H. Bockhorn / Chemosphere 42 (2001) 449±462

using detailed chemistry based on the interactive ¯amelet approach, SAE-paper 962057. Pope, S.B., 1978. An explanation of the turbulent round-jet/ plane-jet anomaly. AAIA J. 16, 179. Roby, R.J., Bowman, C.T., 1987. Formation of N2 O in laminar premixed fuel-rich ¯ames. Combust. Flame 70, 119. Sanders, J.P.H., Chen, J.-Y., G okalp, I., 1997. Flamelet-based modeling of no formation in turbulent hydrogen jet di€usion ¯ames. Combust. Flame 111, 1. Tacke, M.M., Cheng, T.C., Hassel, E.P., Janicka, J., 1996. Experimental study of swirling recirculating hydrogen ¯ame using UV Raman spectroscopy. In: 26th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, p. 169. Tacke, M.M., Linow, S., Geiss, S., Hassel, E.P., Janicka, J., 1998. H3 Flame Data ± Release 1998, TechnischeHochsch-

ule Darmstadt, Fachgebiet Energie- and Kraftwerkstechnik,http://www.tu-darmstadt.de/fb/mb/ekt/¯amebase.html. Tacke, M.M., Linow, S., Geiss, S., Hassel, E.P., Janicka, J., Chen, J.Y., 1998. Experimental and numerical study of a highly diluted turbulent di€usion ¯ame close to blow-out. In: 27th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, p. 307. Vranos, A., Knight, B.A., Proscia, W.M., Chiappetta, L., Smooke, M.D., 1992. Nitric oxide formation and di€erential di€usion in a turbulent methane±hydrogen di€usion ¯ame. In: 24th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, p. 377. Warnatz, J., Maas, U., Dibble, R.W., 1996. Combustion. Springer, Berlin. Zeldovich, Y.B., 1946. The oxidation of nitrogen in combustion and explosions. Acta Physicochim. U.R.S.S., vol. 21. p. 579.