vol1no4-9_debortolial

Engineering Applications of Computational Fluid Mechanics Vol. 1, No.4, pp. 337–349 (2007)

SIMULATION OF A CONFINED TURBULENT NONPREMIXED PILOTED METHANE JET FLAME

A. L. De Bortoli

UFRGS-IM/DMPA-Department of Pure and Applied Mathematics Bento Gonçalves 9500, P.O. Box 15080, Porto Alegre, Brazil

E-Mail: [email protected]

ABSTRACT: The present work develops a low cost numerical method for the solution of nonpremixed piloted methane jet flames. This method is based on the mixture fraction for fluid flow and on unsteady flamelet models, combined with the presumed probability density function, for the chemistry. Numerical tests, for the governing equations discretized by the finite difference and solved by the Gauss-Seidel scheme, were carried out for turbulent, nonpremixed, nonreacting propane-jet flow and for confined Sandia C and D flames for reasonable values of gaseous hydrocarbon chemistry. The methodology, developed for low Mach number flows and based on a density approximation, allows to decrease considerably the computational time while obtaining results which contributes to a better understanding of the complexity involved in the numerical solution of piloted methane jet diffusion flames.

Keywords: Sandia flames, low Mach number, finite difference, flamelet, LES

1. INTRODUCTION

Combustion theory is one of the most elegant areas of classical phenomenology, presenting a wide range of natural phenomena that can be deduced from a few fundamental principles (Buckmaster et al., 2005). In combustion there is a strong coupling among transport (heat transfer, molecular diffusion, convection, turbulent transport) and chemistry and hence is a multidisciplinary topic of research. Moreover, combustion models may turn very complex: the reaction mechanism of iso-octane oxidation includes 3600 elementary reactions among 860 chemical species, with 9 of 25 reaction classes sufficient to simulate many applications at high temperature (Curran et al., 2002). The mechanism of methane combustion has been identified as having more than 300 elementary reactions and over 30 species (Liu et al., 2003). For fuel jet A (80% n-decane and 20% 1,2,4-rrimethyl-benzene), the chemical mechanism contains approximately 1000 elementary reactions among 100 chemical species (Mahesh et al., 2006). However, it is hard to believe that all these species and elementary reactions are necessary to obtain a reasonable approximation of the flow inside a burner (Peters and Rogg, 1992). In this way, simplified mechanisms are usually adopted to describe the combustion process (Apte and Yang,

2002). Mahesh et al. (2006) and Pierce and Moin (2004) simulated flames of natural gas assuming it to be pure methane in the simulations, for example. Due to the necessity of simplifications, the development of appropriate models for burner's design becomes important. But, when the burners are improved, unexpected problems can appear (Poinsot and Selle, 2005). Most of the applications of technical interest are classified as nonpremixed and turbulent (Peters, 1997 & 2000; Warnatz, Maas and Dibble, 2001; Poinsot and Veynante, 2005); some of them include liquid fuel injection inside a chamber and to have high burning intensity, the fuel and the oxidizer must be well mixed. In practice, they are not perfectly premixed before burning and therefore, the combustion process turns less efficient. To model nonpremixed flames it is necessary to have a good understanding of the combustion process and of turbulent mixing because the reaction takes place when the fuel and the oxidizer mix in a molecular level (Pitsch and Fedotov, 2001; Veynante and Vervisch, 2003). Mixing is intensified by flame-vortex interactions (Renard et al., 2000 ) and the heat release distribution exerts a significant influence on the flame evolution and on turbulence, and in regions of high burning levels the eddy life-time is short (Peters, 2000; Warnatz, Maas and Dibble, 2001; Baurle, 2004).

Received: 7 May 2007; Revised: 11 Jul. 2007; Accepted: 13 Jul. 2007

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Engineering Applications of Computational Fluid Mechanics Vol. 1, No. 4 (2007)

Turbulent mixing plays an important role in nonpremixed combustion. It changes the density, the temperature, the heat capacity, the molar mass and also the mixture transport properties (Dimotakis, 2005). To understand the turbulent field it is necessary to have an accurate prediction of the turbulent velocity field. For nonpremixed combustion a jet flame is the most common configuration. In many of these applications the fuel is issued as a turbulent jet, with or without swirl (Peters, 1992). We will consider a jet issuing from a round nozzle with diameter d and exit velocity Uo into a surrounding air stream which may have a constant coflow velocity U∞< Uo, as shown in Fig. 1.

Fig. 1 Burner sketch.

For longitudinal confined jet flames, the mixing process is influenced by the velocity ratio between the jet and the coflow, and the diameter of the fuel jet. For practical duct type combustors, usually one supplies more air than required to have a stoichiometric mixture so that the flame is short enough and the combustion is complete (Kanury, 1975). In the following section, some basic concepts related to the mixture fraction, the presumed probability density function, the scalar dissipation rate, the heat capacity as well as the low Mach number, incompressible and the compressible formulations are introduced.

2. MODEL FORMULATION

Usually for piloted methane jet flames, the Mach number is low, the pressure remains almost constant and the heat losses to the walls are small (Peters, 2000; Poinsot and Veynante, 2005). These allow some model simplifications. The flamelet concept, which covers a regime in turbulent combustion where chemistry is fast, can be employed in most practical situations, such as the nonpremixed combustion when the flame is almost in equilibrium. Flamelets are thin reactive-diffusive layers embedded within an otherwise nonreacting flow field (Peters, 2000). The stationary flamelet model 0=∂∂ t/Yi , with Yi being the mass fraction, has the advantage that flamelet profiles can be pre-computed and stored in a database called “flamelet library” containing all the required complex chemistry. The stationary laminar flamelet model has been applied to engineering calculations because of its simplicity. Such assumption indicates that the value of the scalar dissipation rate varies slowly; it is valid in a jet flame till x~30D, where x is the axial coordinate and D the jet diameter. The unsteady flamelet calculations can be performed using a separate code to solve the system of parabolic equations for mass fractions interacting with a CFD code. The main advantage of the flamelet concept is the fact that chemical time and length scales need not be resolved in a multidimensional CFD code (Peters, 1998). The mixture fraction is an important quantity in the theory of nonpremixed combustion since it is a conserved scalar (Peters, 2000). We write the global reaction equation for “complete” combustion of a hydrocarbon fuel as

[ ] [ ] [ ] [ ] heatOHvCOvOFv OHCOOF ++⇒+ 222 222v (1)

The reaction equation relates the fuel and the oxidizer mass fractions by

. For a homogeneous system this equation may be integrated resulting in

, where and subscript u corresponds to an unburnt quantity.

)()(222 FFFOOO Wv/dYWv/dY =

u,OuF,OF YvYYvY22

−=− )(22 FFOO Wv/Wvv =

Consider a system of two components: the subscript 1 denoting the fuel and 2 the oxidizer. The mixture fraction is defined as the local mass fraction of all elements within the mixture . The local mass fraction of the unburnt fuel is

)( 211 mm/mZ +=

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ZYY F,uF, 1= and that of the oxidizer is . The mixture fraction at any state of

combustion, after inserting the fuel and the oxidizer mass fractions into equation

and its corresponding value at stoichiometric condition, , is

given by

ZYY Ou,O 2,22=

u,OuF,OF YvYYvY22

−=−

2OF YvY =

2,1

2,

2

22

OF,

OOF

YvY

YYvYZ

+

+−= .

A probability density function pdf represents a probability distribution in terms of integrals. The presumed shape pdf approach seems to be the most appropriate method to be used together with the flamelet equations (Peters, 2000). There are many alternatives to a general class of shapes for the probability density functions of the conserved scalar, but it is desirable to have a small number of parameters (Williams, 1985). As the pdf provides the statistical information about the variables, it has the ability to treat finite-rate chemistry and the turbulence-chemistry interactions. Both the mixture fraction Z and the scalar dissipation rate χ, which measures the local diffusion zone thickness, fluctuate in turbulent flows and their statistical distribution needs to be considered. If the joint pdf (where χ)(~

stχZ,P st is χ at the stoichiometric condition) is known, the Favre mean of mass fraction iY~ can be obtained

from . ∫ ∫∞

=1

0 0

)(~)(~ dZdχtx,;χZ,PχZ,YY stststii

Inside a burner the Mach number is usually low; therefore, one needs numerical techniques that solve the original compressible flow equations, but which can also be efficiently used at low Mach numbers. The common low Mach number formulations of variable density are that when acoustic waves are filtered, the density change due to temperature variations remains independent of the pressure. Then, one can decouple the pressure field of the state equation from the pressure gradients of the momentum equations.

2.1 Governing equations

The set of governing equations in non-dimensionalized form can be derived by applying a spatial, density-weighted filter resulting in:

Momentum/Navier-Stokes

j

ijij

ejij

jii

xσ

τRμ

xxp

Mxuuρ

tuρ

∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

−=∂

∂+

∂∂ ~1)~~()~(

2 (2)

Mixture fraction

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

=∂

∂+

∂∂

jc

T

jj

j

xZ

Sμ

xxZuρ

tZρ ~)~~()~( (3)

Mixture fraction variance

χρ)Z(Dρx"Z

Sμ

xx"Zuρ

t"Zρ

Tjc

T

jj

j ~~2~)~~()~( 2

222−∇+⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

=∂

∂+

∂∂

(4) Enthalpy

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

=∂

∂+

∂∂

jr

T

jj

j

xh

Pμ

xxhuρ

thρ ~)~~()~( (5)

Species and mean species mass fraction

θ)α(1θ)(Z

OFai

e

ie

eYYDZY

Laχ

tY −−

−−

=∂∂

−∂∂ 1

1

2

2

2 (6)

dZt,xZ;PtZ,Yt,xY jiji )(~)()(~1

0∫= (7)

Mean temperature (which is obtained from the enthalpy)

∑=

=n

iii (T)hYh

1

~~~ (8)

where u is the velocity, ρ the density, t the time, p the pressure, τ the stress tensor, Z the mixture fracton, DT the turbulent diffusivity, h the enthalpy, Yi the mass fractions, μT the turbulent viscosity and T the temperature. Here the density is obtained

using the relation1~)1( +−

=Tα/α

pρ , 8.005.0 << α

(Rutland and Ferziger, 1991), which would decrease the computational time needed to obtain the results by an order of magnitude. The turbulent viscosity is given by the Smagorinsky model SΔ)(C2ρμ sT

~2= . The relation for 2"~Z , 222 ~"~ ZρΔCZρ Z ∇= , can also be used to determine

dynamically the Cs coefficient. In these equations (Eqs. 2–8), Re is the Reynolds, Sc the Schmidt, Pr the Prandtl, Da the Damköhler, Le

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the Lewis and Ze the Zel'dovich numbers;

8.0~T

TTα

b

ub −= and

ub

u

TTTTθ−−

= (Steiner and

Bushe, 1998). appears after the nondimensionalization of the species mass fraction equation.

( ) )1(Δ2 stst ZZZa −=

At low Mach numbers the spatial variations in pressure are small compared with the pressure itself and may be neglected in the equation of state, where p may be approximated by a constant (Liñán, 1991). The temperature is obtained from the enthalpy using a simple Newton iteration and the density comes from the state relation. Observe that the transformation introduced to obtain the flamelet equations (species

equations) Zt

Zτt ∂

∂∂∂

+∂∂

=∂∂ ,

ZxZ

x ∂∂

+∂∂

=∂∂

11 and

ZxZ

Zx kkk ∂∂

∂∂

+∂∂

=∂∂ eliminates the convective

(nonlinear) terms; besides it transforms the three-dimensional problem to a one-dimensional problem (Peters, 1984 & 1986).

2.2 Solution procedures

For the simulation of combustion flows of technical interest RANS, Reynolds Averaged Navier-Stokes, and LES, Large-Eddy Simulation, seem to be good alternatives. The DNS, Direct Numerical Simulation, on the other hand, is the most expensive technique because the number of grid points in each coordinate direction increases proportionally to the Reynolds value, limiting the DNS applications. The subgrid modeling of small-scale turbulence is important in LES especially at high Reynolds numbers and when relatively coarse grids are being used in strained flow regions. Eddy-viscosity, dynamic and hybrid models were developed for doing that (Riesmeier, 2003; Haworth, 1999). Among the subgrid models the Smagorinsky is still widely used, mainly because of its simplicity. It can be derived based on the assumption that in the SGS kinetic energy the production is equal to dissipation, and they have much larger magnitude than the transport term. The numerical application of Germano model can lead to numerical problems owing to the possible occurrence of negative turbulent viscosity, because the model parameter it produces is a rapidly varying function of the spatial coordinates and time. It has been observed that the model for Tμ becomes less important when better

mesh resolution is used; however, refining grid is restricted as a result of the rapid increase of the computational cost. As the turbulence can affect the flow in profound ways, it is common to find significant differences between the DNS and LES predictions. This leads to the possibility to use LES in coarse meshes as a tool for determining the gross features of the flow (Ferziger and Perić, 1999).

2.2.1 Pressure calculation using the Poisson's equation

It seems to be natural to derive a Poisson pressure equation from the momentum (Navier-Stokes) equations. The mass conservation equation can be used to simplify it, resulting in Cartesian coordinates (Ferziger and Perić, 1999; de Bortoli, 2003)

( )2

2

tρ

xρb

)τuu(ρxxx

px i

iijji

jiii ∂∂

+∂

∂+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ (9)

Such equation is frequently solved by explicit methods; the implicit methods are usually adopted in the case of steady flows. Based on this equation, an algorithm to solve the Navier-Stokes equations could be written as:

obtain the velocities using the momentum equations (without the pressure term)

solve the Poisson equation for the pressure

correct the velocity field

Other possibility is to employ a fractional step method. In this case, the approximated velocities (ui)* are advected using the pressure field from the previous time-step. Then, half of the old pressure gradient is removed and one calculates a new approximation for velocities (ui)** as follows (where δ means a finite difference approximation):

i

nii

δxδp

t*uρ**uρ

21

Δ)()(

=−

(10)

and in order to satisfy the mass conservation a Poisson's equation is needed.

i

**i

i

n

i δxρuδ

tδxδp

δxδ 21 )(

Δ2

=⎟⎟⎠

⎞⎜⎜⎝

⎛ +

(11)

Then the velocities can be recalculated using the new pressure field. This procedure is preferred to

340


solve unsteady flows, while SIMPLE type methods are more frequently used to solve steady flow problems (Ferziger and Perić, 1999). Other possibility (Pitsch and Steiner, 2000) is to obtain the velocities (without the pressure term) first. Then one solves the Poisson's equation for the pressure (Boersma, 1998) as

⎥⎦

⎤⎢⎣

⎡∂∂

+∂

∂=∇

t*ρ

x*ρutp

i

in )(Δ2 (12)

and correct the velocities with the pressure gradient. Observe that here the pressure changes are due to mass conservation variations. It seems that the employment of a Poisson's equation for the pressure is permissible. The dynamic pressure is very small (less than 0.01% of the total pressure) and the corresponding pressure gradients are even smaller, but they are responsible for local effects. Reynolds (Reynolds and Kassinos, 1995) regarded the non-local effect of pressure a greater challenge to turbulence modeling than the non-linearity of the Navier-Stokes equations.

2.2.2 Finite difference approximation in Cartesian meshes

The ability to model flows through/over complex geometries is one of the main challenges of CFD, Computational Fluid Dynamics, since the majority of flows involve some kind of complex geometry. As an alternative to the boundary fitted method, the virtual boundary technique maintains the efficiency of the Cartesian solution procedure (Verzicco et al., 1998; von Terzi et al., 2001) and allows transferring the influence between each boundary point to its neighboring points. The force field can be smoothed in the neighborhood of the boundary grid nodes by interpolation (Saiki and Biringen, 1996). The finite difference approximations of second order, for the first and second space derivatives, considering a one-dimensional problem, are indicated in the following:

For the first derivative: hyy~y iii 2)/(´ 11 −+ −

For the second derivative: 2

11 )/2(´´ hyyy~y iiii −+ +−

where h corresponds to the grid size. The equations approximated using the finite difference can be integrated using the Gauss-Seidel

scheme after an adequate implementation of the boundary conditions.

2.2.3 Boundary conditions

Consider the longitudinal section of the burner as shown in Fig. 2. The boundary conditions can be summarized as follows:

For the solid walls the condition results in (n corresponds to the normal direction):

0"~~~~~

2=

∂∂

=∂∂

=∂∂

=∂∂

=∂∂

=∂∂

=n

ZnZ

nh

nY

nρ

npu i

i

For the inflow it results in:

1,1~,1~0~~~~~222

====∂∂

===== ρTY;xpYYYwv OOHCOF

except at fuel injection point where 1~ =u

(parabolic), and the other conditions are given by Schefer for the nonreacting flow or by Barlow and Frank for Sandia flames C and D (Barlow and Frank, 2003).

TChYY POF~~,0~,1~

2===

For the outflow the convective condition results in:

0=∂∂

+∂∂

xu

t cφφ

with 1=p , where Ti TYhZZwvuρ ]~~~"~~~~~[ 2=φ and

is the convective velocity. cu

Fig. 2 Boundary conditions.

solid wall

solid wall

outflow inflow

coflow

fuel jet

hot pilot

2.2.4 Stiffness

The loss of computational efficiency can be seen as a stiffness problem since it is a direct consequence of flow and acoustic speeds being widely different (Wang and Trouvé, 2004). The numerical methods which are better for solving stiff problems do more

341


work per step, but can have larger steps, such as the implicit methods; an alternative is the Gauss-Seidel with some relaxation. The exponential source term, of the Arrhenius type approximation e-Z/T, depends strongly on the temperature, which in combustion varies considerably. Such variation must be transferred to the density, which affects the stability of the numerical code. Therefore, some relaxation have to be employed in the density evaluation. Besides, the dynamic pressure variations are very small and yet, important and need to be correctly evaluated. Decreasing reaction exponents, as often done in many reduced schemes, leads to increased stiffness. In the reaction rate expression the exponential term

can be replaced by , where

/TTae− )]1(1/[)1(/ θαθβαβ −−−−− ee75.0~/)( 212 TTT −=α and 8~/ 2TTaαβ =

(Poinsot and Veynante, 2005).

3. NUMERICAL RESULTS

The jet flame was chosen because it seems to be a representative of the class of nonpremixed flames. When showing the results (where Z, χ fluctuate) it is common to show the mean values, and hence

they are employed to present some of the following results.

3.1 Results for turbulent nonpremixed, nonreacting, propane-jet flow

The schematic diagram of the burner longitudinal section is shown in Fig. 3. The duct has a square cross section with H=1 (which corresponds to 30 cm) and the jet of propane is injected from a tube with d=0.025H; the length of the combustor is L=11H.

Fig. 3 Geometry for nonpremixed, nonreacting

propane jet-flow.

H

coflow

fuel jet d

L/100

L

Fig. 4 Comparison of the mean mixture fraction and its variance (left) and the velocity (right) profiles along the duct centerline for a turbulent, nonpremixed, nonreacting propane-jet flow with experimental data from Schefer and with the theoretical solution for a turbulent jet diffusion flame (Peters, 2000).

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Fig. 4 shows the mean velocity (right) and the mixture fraction (left) comparisons with experimental data of Schefer and theoretical/analytical values (Peters and Donnerhack, 1981; Peters, 1992 & 2000) for turbulent jet diffusion flame. It is a round turbulent jet of propane into coflowing air. Although the theoretical/analytical solution was not obtained for exactly the same situation of the experiment, it helps to understand the general solutions behavior. The numerical/theoretical results show small

discrepancies when compared to the experimental data. Fig. 5 presents the comparison among the experimental/numerical/theoretical mean velocity profiles at positions x/D = 15 and 30. The discrepancies, which were already indicated on the axial profiles, become more evident in the radial profiles.

Fig. 5 Comparison of the mean radial velocity profiles for x/D = 15 and 30 for a turbulent, nonpremixed, nonreacting propane-jet flow with experimental data from Schefer and with the theoretical solution for a turbulent jet diffusion flame (Peters, 1992 & 2000).

3.2 Results for the burner based on Sandia flames C and D

Sandia flames C and D consist of a main jet with a mixture of 25% of methane and 75% of air. This jet is placed in a coflow of air and the flame is stabilized by a pilot. The fuel is premixed with air in order to minimize the formation of polyciclic aromatic hydrocarbons and soot (Pitsch, Riesmeier and Peters, 2000). The jet velocites are 29.7 and 49.6 m/s for flames C and D, respectively (Barlow and Frank, 2003; Schneider et al., 2003). The pilot bulk velocities are 6.8 m/s for flame C and 11.4 m/s for flame D.

Consider the longitudinal section of the burner as shown in Fig. 6. The duct has a square cross section with H = 1 and a cylindrical tube which injects fuel with d = 0.025; the tube of the coflow has a diameter of D = 0.0267 and the burner length is L = 11. The number of grid points was taken as 41x41x149 in the (y, z , x) directions, respectively.

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Fig. 6 Longitudinal section of the burner.

Fig. 7 displays the comparison between the experimental and numerical instantaneous mixture fraction profiles for flames C and D along the burner centerline. Fig. 8 shows the comparison between the experimental and numerical instantaneous temperature profiles for flames C and D along the burner centerline. The discrepancies are due to the model simplification.

coflow

fuel jet d

L/100

L

hot pilot H D

Fig. 7 Comparison of the mixture fraction profiles for Sandia flames C (left) and D (right) along the burner centerline

with experimental data (Barlow and Frank, 2003; Schneider et al., 2003).

The solution indicates the axial decreasing behavior of the mixture fraction; the temperature increases in the reaction zone, as expected. The temperature is overpredicted mainly in the range x/D = 40–60; the global agreement seems to be reasonable. Fig. 9 compares the fuel CH4 mass fraction profiles with the experimental data for Sandia flames C and D. The fuel consumption is very well captured in both flames; the oscillations refer to the instantaneous values. Fig. 10 shows the oxidizer O2 and the product CO2 mass fractions along the burner centerline. Although these mass fractions are well predicted in the rich part of the flame, the oxidizer mass fraction is underpredicted in the lean part of the flame, which leads to an overprediction of the products mass fractions. It seems that such behavior is

amplified by some type of dissipation which can also be influenced by the time-step employed. Observe that products such as NO, H2, OH, CO, etc, are not considered in the present model. Finally, Fig. 11 compares the radial velocity profiles for flame D at x/D = 15 and 45. The numerical solution is dissipative at the base of the profile for x/D = 15; for x/D = 60, the dissipation occurs in the proximity of its tip. The longitudinal velocity profile agrees reasonably with the experimental data, similar to the mixture fraction agreement shown in Fig. 7. The results seem to be reasonable compared to other works found in the literature (Demiraydin, 2002; Riesmeier, 2003; Sheikhi et al., 2005). It has been observed that for low Mach number flows the error can contaminate the pressure

344


gradient. Accurate gradients of reacting species may require a grid spacing several orders of magnitude finer than that necessary to resolve other characteristic flow structures. Moreover, the

existence of high-frequency acoustic waves acts as a severe restriction on the time-stepping increments used to advance the fully compressible equations in time (McMurtry et al., 1986).

Fig. 8 Comparison of the temperature profiles for Sandia flames C (left) and D (right) along the burner centerline with

experimental data (Barlow and Frank, 2003; Schneider et al., 2003).

Fig. 9 Comparison of the mass fraction CH4 for Sandia flames C (left) and D (right) along the burner centerline with experimental data (Barlow and Frank, 2003; Schneider et al., 2003).

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Fig. 10 Comparison of the O2 and CO2 mass fraction along the burner centerline for flame D with experimental data

(Barlow and Frank, 2003).

Fig. 11 Comparison of the velocity profiles at x/D=15 and 45 for flame D with experimental data (Barlow and Frank,

2003).

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4. CONCLUSIONS

The present work developed a low cost efficient numerical method for the solution of nonpremixed piloted methane jet flames. The formulation was based on the flamelet equations for the chemistry and on the mixture fraction for the flow. The numerical results for the nonpremixed, nonreacting, propane jet flow as well as for the confined Sandia flames C and D compare reasonably with the available data found in the literature. At low Mach numbers the dynamic pressure is very small (less than 0.01% of the total pressure) and the corresponding pressure gradients are even smaller compared with the pressure itself and may be neglected in the equation of state, where p may be approximated as a constant (Liñán, 1991). The pressure gradient comes from the Poisson equation (Eq. (12)), which corresponds to a pressure correction by mass conservation. The temperature is obtained from the enthalpy using a simple Newton iteration and the density comes from the relation ]1~)1/(/[ +−= Tααpρ , which helps to increase the time-step by an order of magnitude when compared to the time-step necessary using the relation Tpρ ~/= , due to high temperature gradients. The above are the main contributions of this work. The developed method, based on the low Mach number formulation, helps to obtain reasonable results at low cost for confined jet diffusion flames. Such results contribute to a better understanding of the complexity involved in the numerical solution of piloted methane jet diffusion flames.

ACKNOWLEDGEMENTS

This research started at ITV/RWTH-Aachen under the sponsorship of CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - MCT/Brasil under process 0272/06-0 and continues under the sponsorship of CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico under process 304600/2006-7. The author gratefully acknowledges the financial support from CAPES and CNPq and the opportunity to stay at the Institute für Technische Verbrennung-RWTH/Aachen, Germany, from August/2006 to February/2007.

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