Simulation of turbulent piloted methane non-premixed flame based on combination of finite-rate / eddy-dissipation model

Combustion is a complex phenomenon that is controlled by many physical processes including thermodynamics, buoyancy, chemical kinetics, radiation, mass and heat transfers and fluid mechanics. This makes conducting experiments for multi-species reacting flames extremely challenging and financially expensive. For these reasons, computer modeling of these processes is also playing a progressively important role in producing multiscale information that is not available by using other research techniques. In many cases, numerical predictions are typically less expensive and can take less time than similar experimental programs and therefore can effectively complement experimental programs. Computational models can help in predicting flame composition, regions of high and low temperature inside the burner, and detailed composition of byproducts being produced. Detailed computational results can also help us better predict the chemical structure of flames and understand flame stabilization processes. These capabilities make Computational Fluid Dynamics (CFD) an excellent tool to complement experimental methods for understanding combustion and thus help in designing and choosing better fuel composition according to the specific needs of a burner. With the advent of more and more powerful computing resources, better algorithms, and the numerous other computational tools in the last couple of decades, CFD has evolved as a powerful tool to study and analyze combustion. However, numerous challenges are involved in making CFD a reliable and robust tool for design and engineering purposes. The numerical simulation is a useful tool because it can easily employ various conditions by simply changing the parameters. Methane is the simplest hydrocarbon fuel available; several studies have focused on methane-air flames. The oxidation of methane is quite well understood and various detailed reaction mechanisms are reported in literature [1]. They can be divided into full mechanisms, skeletal mechanisms, and reduced mechanisms. The various mechanisms differ with respect to the considered species and reactions. However, considering the uncertainties and simplifications included in a turbulent flame calculation, the various mechanisms agree reasonably well [2]. In literature, several mechanisms of methane combustion exist. We can cite: for detailed mechanisms: Westbrook [3], Glarborg et al. [4], Miller and Bowman [5], and recently, Konnov et al. [6], Huges et al. [7], LCSR [8], Leeds v.1.5 [9], San Diego [10] and the standard GRI-Mech v.3.0 [11], GRI-Mech v.1.2 [12] for reduced mechanisms: Westbrook and Dryer [13], and Jones and Lindstedt [14] (more than 2 global reaction). For skeletal mechanisms: Kazakov and Frenklach [15], Yungster and Rabinowitz [16], Petersen and Hanson [17], Hyer et al. [18] and Li and Williams [19]. The objective of this study was to investigate how simplified kinetics mechanisms performed in comparison to the fast chemistry assumption when simulating a piloted CH4 nonpremixed flame with RANS and hybrid FiniteRate/Eddy Dissipation model. A one-step mechanism by Westbrook and Dryer [13] and the four-step mechanism suggested by Jones and Lindstedt [14] were chosen for comparison. Finally, Fluent is used for this study with some modifications of the usually adopted models for the representations of the turbulence-kinetics interaction are introduced (UDF for Hybrid Finite-Rate/Eddy Dissipation model). The results were compared with experimental data [20].


Introduction
Combustion is a complex phenomenon that is controlled by many physical processes including thermodynamics, buoyancy, chemical kinetics, radiation, mass and heat transfers and fluid mechanics.This makes conducting experiments for multi-species reacting flames extremely challenging and financially expensive.For these reasons, computer modeling of these processes is also playing a progressively important role in producing multiscale information that is not available by using other research techniques.In many cases, numerical predictions are typically less expensive and can take less time than similar experimental programs and therefore can effectively complement experimental programs.Computational models can help in predicting flame composition, regions of high and low temperature inside the burner, and detailed composition of byproducts being produced.Detailed computational results can also help us better predict the chemical structure of flames and understand flame stabilization processes.These capabilities make Computational Fluid Dynamics (CFD) an excellent tool to complement experimental methods for understanding combustion and thus help in designing and choosing better fuel composition according to the specific needs of a burner.With the advent of more and more powerful computing resources, better algorithms, and the numerous other computational tools in the last couple of decades, CFD has evolved as a powerful tool to study and analyze combustion.However, numerous challenges are involved in making CFD a reliable and robust tool for design and engineering purposes.The numerical simulation is a useful tool because it can easily employ various conditions by simply changing the parameters.Methane is the simplest hydrocarbon fuel available; several studies have focused on methane-air flames.The oxidation of methane is quite well understood and various detailed reaction mechanisms are reported in literature [1].They can be divided into full mechanisms, skeletal mechanisms, and reduced mechanisms.The various mechanisms differ with respect to the considered species and reactions.However, considering the uncertainties and simplifications included in a turbulent flame calculation, the various mechanisms agree reasonably well [2].In literature, several mechanisms of methane combustion exist.We can cite: for detailed mechanisms: Westbrook [3], Glarborg et al. [4], Miller and Bowman [5], and recently, Konnov et al. [6], Huges et al. [7], LCSR [8], Leeds v.1.5[9], San Diego [10] and the standard GRI-Mech v.3.0 [11], GRI-Mech v.1.2[12] for reduced mechanisms: Westbrook and Dryer [13], and Jones and Lindstedt [14] (more than 2 global reaction).For skeletal mechanisms: Kazakov and Frenklach [15], Yungster and Rabinowitz [16], Petersen and Hanson [17], Hyer et al. [18] and Li and Williams [19].
The objective of this study was to investigate how simplified kinetics mechanisms performed in comparison to the fast chemistry assumption when simulating a piloted CH 4 nonpremixed flame with RANS and hybrid Finite-Rate/Eddy Dissipation model.A one-step mechanism by Westbrook and Dryer [13] and the four-step mechanism suggested by Jones and Lindstedt [14] were chosen for comparison.Finally, Fluent is used for this study with some modifications of the usually adopted models for the representations of the turbulence-kinetics interaction are introduced (UDF for Hybrid Finite-Rate/Eddy Dissipation model).The results were compared with experimental data [20].

Mathematical formulation
The numerical model solves the independent time equations for conservation of mass, momentum, energy and individual species: where:

Heat radiation
The radiative heat flux q r is expressed as follows: where σ is the Stefan-Boltzmann constant and p i the partial pressure of species i. a p,i is the Plank mean absorption coefficients for radiating species i [22] and P the static pressure, g term of gravity, Y i is the mass fraction, D i,m the thermal diffusivity, T the temperature, ρ the density, h the specific enthalpy, m i the mass flux of fuel.Radiation/absorption in gas is neglected.

Turbulence modeling
The standard k- model (include a correction for round jets performed by using the Pope formulation) turbulence closure model is adopted.In the k- model the Reynolds stress is closed using mean velocity gradients employing Boussinesq hypothesis [22].In the case of a jet flame, a correction is necessary to accurately predict the spreading rate of the jet.This is performed by using the Pope correction, P pc , as an additional term in the equation of turbulence dissipation rate () [22]: The term S  can be written as (Pope, 1978): where 1 2 1 2 where C 3 = 0.79, the standard model constants have been chosen.As an option in the formulation of the k- model, enhanced wall functions were selected in accordance with the grid design.This option ensured that appropriate modeling occurred to resolve the viscous sub-layer.

Problem description
The flame simulations were performed and compared with experimental data from literature [20].The flame is a turbulent non-premixed piloted methane flame.A co-flowing air was placed around the flame to avoid the influence of air flow in radial direction.The boundary conditions of the measurements are shown in Table 1.The geometrical configuration of the burner for predicted CH 4 /air turbulent combustion is that reported in (Fig. 1).The geometry for this test is a cylindrical combustor with coaxial injectors, where the natural gas is injected by the primary tube and the air piloted flow through the secondary annulus.In the fuel stream, the uniforme inlet gas velocity is 49.6 m/s with a temperature of 294 K.The Co-flowing air parallel to the flame was about 0.9 m/s with a preheated temperature of 291 K.The Reynolds number of the jet exit is 22400 with a low probability of localized flame extinction.The pilot flame burns a mixture of gases having the same composition and enthalpy as a CH 4 /air mixture at 0.27 mixture fraction.The experiment was performed at pressure atmospheric.

Fig. 1 Geometry of the coaxial combustor 20
In the present computation, the reaction rate are computed by combination of Finite-Rate and Eddy-dissipation model for turbulent flow, both the Arrhenius rate and mixing rate are computed and smaller of the two is used.The specific heat values for the species are defined as piecewise-polynominal function of temperature.

The laminar finite rate model (source of species)
The laminar finite-rate model computes the chemical source terms using Arrhenius expressions, and ignores the effects of turbulent fluctuations.
The model is exact for laminar flames, but is gen-erally inaccurate for turbulent flames due to highly nonlinear Arrhenius chemical kinetics.The net source of chemical species i due to reaction am computed the sum of the Arrhenius reaction sources over the N R reactions that the species may participate in: where M w,i is the molecular mass of species i and , ik R is the molar rate of creation/destruction of species i in reaction k.Reaction may occur in the continuous phase between continuous phase species only, or at resulting in the surface deposition or evolution of a continuous-phase species.The reaction rate, , ik R is controlled either by an Arrhenius kinetic rate expression or by mixing of the turbulent eddies containing fluctuating species concentrations.

The arrhenius rate (chemical kinetics)
Chemical kinetic governs the behavior of reacting chemical species.As explained earlier, a combustion reaction proceeds over many reaction steps, characterized by the production and consumption of intermediate reactants.Several conditions determining the rate of reaction are the concentration of reactants and the temperature.The concentration of the reactants affects the probability of reactant collision, while the temperature determines the probability of the reaction occurring given a collision.In general, a chemical reaction can be written in the form as follows: where N is number of chemical species in the system; Eq. ( 18) is valid for both reversible and nonreversible reactions.For non-reversible reactions, the backward rate constant k b,k is simply omitted.The summations in Eq. ( 18) are for all chemical species in the system, but only species involved as reactants or products will have non-zero stoichiometric coefficients.Hence, species that are not involved will drop of the equation except for thirdbody reaction species.

Reaction rate
The molar rate of creation/destruction of species i' where j C is molar concentration of each reactant or prod- uct species j, Kmol m -3 ; , jk '  is rate exponent for reactant j' in reaction k; , jk "  is rate exponent for product j' in reaction k;  is represents the net effect of third bodies on the reaction rate.This term is given by: where , jk γ is the third-body efficiency of the th j' species in the k-th reaction.The forward rate constant for reaction k, k f,k , is computed using the Arrhenius expression where A k is pre-exponential factor (consistent units);  k is temperature exponent (dimensionless); E k is activation energy for the reaction, J Kgmol -1 ; R is universal gas constant (8313), J Kmol  can be provided the problem definition.If the reaction is reversible, the backward rate constant for reaction k, k b,k , is computed from the forward rate constant using the following relation: where k K is the equilibrium constant for the k-th reaction.Computed from: where P atm denotes atmospheric pressure (101325 Pa).The term within the exponential represents the change in Gibbs free energy, and its components are computed as follows: where 0 i' S and 0 i' h are, respectively, the standard-state entropy and standard-state enthalpy including heat of formation.These values are specified in Fluent.

The eddy dissipation model (EDM)
Fluent provides a turbulence-chemistry interaction model, based on the work of Magnussen and Hjertager [21], called the eddy-dissipation model.The rate of production of species i due to reaction, R i , is given by the smaller of the two expressions below: where A is Magnussen constant for reactants (default 4.0); B is Magnussen constant for products (default 0.5); M is molecular mass of species; (R), (P) is reactants, products; ' i , '' j are stoichiometric coefficient for reactant i and product j in reaction.

Combustion model
The turbulent non-premixed combustion process is simulated using the hybrid Finite-Rate/Eddy-Dissipation model [22].The fuel is assumed to burn by a 1-step and 4-step chemical reaction.For the 1-step reaction, the global chemical kinetics scheme of Westbrook and Dryer (Table 2) was used to describe the combustion process in terms of 5 species (CH 4 , CO 2 , H 2 O, O 2 and N 2 as inert).For the 4-step reaction, the reduced chemical kinetics scheme of Jones and Lindstedt (Table 3) was used to describe the combustion process in terms of 7 species (CH 4 , CO 2 , H 2 O, O 2 , CO, H 2 and N 2 as inert).

Finite-rate/eddy-dissipation model (FR-EDM)
The Eddy-Dissipation Model assumes that reactions are fast and that the system is purely mixing limited and when that is not the case, it can be combined with finite-rate chemistry.In that case, the kinetic rate is calculated in addition to the reaction rate predicted by the eddydissipation model.The slowest reaction rate is then used: if turbulence is low, mixing is slow and this will limit the reaction rate.If turbulence is high, but the kinetic rate is low, this will limit the reaction rate.This model can be used for a variety of systems, but with the following caveats: the model constants A and B need to be empirically adjusted for each reaction in each system.The default values of 4.0 and 0.5 respectively were determined for one and two-step combustion processes.The model (FR-EDM) always requires some product to be present for reactions to proceed.If this is not desirable, initial mass fractions of product of 1E-10 can be patched and a model constant B = 1E10 used.

Simulation detail
The governing equations are solved using the ANSYS-FLUENT CFD package modified with User-Defined Functions (UDF) in order to integrate the reaction rate formula proposed by Jones and Lindstedt [14].In computational fluid dynamics, the differential equations govern the problem are discretized into finite volume and then solved using algebraic approximations of differential equations.These numerical approximations of the solution are then iterated until adequate flow convergence is reached.The chemical kinetics information is then coupled into fluid dynamics equations to allow both phenomena to be incorporate into a single problem.SIPMLE algorithm [22] was chosen for the coupling between the velocity and the pressure.For all simulations presented in this paper, a First Order Upwind scheme was used for all the conservation equations (U, V. E, Y i , k,  and P1).The Standard scheme [22] was used for interpolation methods for pressure.This means that the solution approximation in each finite volume was assumed to be linear.This saved on computational expense.In order to properly justify using a first order scheme, it was necessary to show that the grid used in this work had adequate resolution to accurately capture the physics occurring within the domain.In other words, the results needed to be independent of the grid resolution.This was verified by running simulations with higher resolution grids.In a reacting flow such as that studied in this work, there are significant time scale differences between the general flow characteristics and the chemical reactions.In order to handle the numerical difficulties that arise from this, the STIFF Chemistry Solver was enabled in Fluent.For more information about this technique refer to [22].The criterion of convergence is the summation of residual mass sources less than 10 -3 for the other terms of the transport equations and is 10 -6 for energy equation.The criterion of convergence is the summation of residual mass sources less than 10 -3 for the other terms of the transport equations and is 10 -6 for energy equation.The options used in this work are presented in Table 4.The computational space seen in Fig. 1 given a finite volume mesh is divided by a staggered non-uniform quadrilateral cell (Fig. 2).A total number of 640 (32  20) quadrilateral cells were generated using non-uniform grid spacing to provide an adequate resolution near the jet axis and close to the burner where gradients were large.The grid spacing increased in the radial and axial directions since gradients were small in the far-field.The combustion will be modeled using 1-step global mechanism and reduced 4-step reaction mechanism and the radiative heat transfer of the piloted flames is calculated with the P1 model [22].The interaction between turbulence and chemistry is oftenhandled through the Finite-Rate/Eddy-Dissipation Model.
Table 2 Westbrook and dryer global chemical kinetics mechanism for CH 4 /air combustion and reaction rate coefficients [13] No. Reaction

Results
We begin by comparing the computational cost of the two kinetic models in terms of the average CPU (execution) time per time step (s).The relative elapsed CPU times are compared in Table 5.In the 4-step mechanism, more reaction equations are computed, them more CPU time is spent and more difficult it is to convergence.That in general the computational cost increases with the number of reaction-step and species and more difficult it is to convergence.3 show the contour plot of the temperature for Westbrook and Dryer 'WD' (Fig. 3, a) and Jones and Lindstedt 'JL' mechanism (Fig. 3, b), respectively.It is observed that the temperature is over predicted with the 'WD' mechanism and the temperature profile is quite flat for a wide region.In case of the 'JL' mechanism the temperature are also over predicted.Is noticed that the smallest flame is predicted by the 1-step model 'WD', whereas the largest flame is predicted by the 4-step model 'JL'.The axial distribution of the mean temperature is represented in (Fig. 4).One-step 'WD' and four-step 'JL' model all predict the peak of temperature distribution and the predicted peaks are all near the position of the experimental one: The radial distribution of the mean temperature is represented in (Figs.5-7) using two mechanisms have the same right trend; however, quantitatively the 4-step scheme (J-L mechanism) gives the best agreement with experimental data and along the radial coordinate, the temperature profiles move from the highest values, characteristic of the hot core of the flame to the decrease and then finally to the asymptotic value which refers to the recirculating exhausted gas.The concentration of CH 4 is good estimated with 1-step and 4-step reaction scheme in the inner recirculation zone x / D jet = 7.5 (Fig. 9), where fuel is mixed into the flow.Once again, the 4-step global mechanism gives the best results to predicted CH 4 concentration (Figs. 10 and  11).Using the 1-step is much larger that measured, since the fuel is much less consumed under the much lower reaction rate provided by the 1-step mechanism model.Similar results are obtained for oxygen concentration profiles (Figs.11 and 12).

Conclusion
In the present work the applicability of two different kinetic mechanisms: global mechanism of Westbrook and Dryer 'WD' and reduced mechanism of Jones and Lindstedt, 'JL' on the predictions on a piloted nonpremixed turbulent flame to account for an accurate prediction of the flow and temperature field of a jet flame was investigated.The 7 species reduced mechanism was successfully implemented and tested with the global mechanism into the CFD solver Fluent.The precompiled mechanism was linked to the solver by the means of User Defined Function (UDF).This implemented was tested with the Barlow Piloted CH 4 /air flame.In general, in the 4-step mechanism, the presence of CO and H 2 lowers the total heat release and the adiabatic flame temperature is below the values predicted by the 1-step global mechanism and the smallest flame is predicted by the global reaction, whereas the largest flame is predicted by the 4-step mechanism.

SIMULATION OF TURBULENT PILOTED METHANE NONPREMIXED FLAME BASED ON COMBINATION OF FINITE-RATE/EDDY DISSIPATION MODEL S u m m a r y
Two chemical kinetic mechanisms of methane combustion were tested and compared using a piloted axisymmetric turbulent non-premixed flame: 1-step and 4-step mechanism, to predict the temperature and species distributions.The numerical results are presented and compared with the experimental data.A 4-step methane mechanism was successfully implanted into CFD solver Fluent.The numerical solution is in very good agreement with previous numeral of 4-step mechanism and the experimental data.

Fig. 2
Fig. 2 Computational mesh and the boundarys conditions

Fig. 3 Fig. 4
Fig. 3 Instantaneous temperature in Kelvin contour plot with the 'WD' mechanism a) and 'JL' mechanism b)

Fig. 8
Fig. 8 Center line profile the concentration of H 2 O 5.3.CH 4 and O 2 concentrations profiles Figs.9-13 shows CH 4 and O 2 concentration profiles.The axial distribution of the concentration of CH 4 is represented in (Fig. 8).

Table 1
Experimental and simulation condistions [20] Jet diameter, D jet , mm Pilot inner diameter, D P,I , mm Pilot diameter, D P,O , mm Burner outer wall diameter, D B,O , mm Jet Reynolds number, Re U coflow , m/s, T coflow , K Jet composition CH 4 /air (volume fraction) U jet , m/s, T jet , K U p , m/s, T P , K

Table 3
[14]s and Lindstedt reduced multi-step chemical kinetics mechanism for CH 4 /air combustion and reaction rate coefficients[14]

Table 5
Average execution time per time step