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Proceedings of the
Proceedings of the Combustion Institute 32 (2009) 1165–1172
Combustion Institute www.elsevier.com/locate/proci
The impact of detailed multicomponent transport and thermal diffusion effects on soot formation in ethylene/air flames S.B. Dworkin a,*, M.D. Smookea, V. Giovangigli b a
Department of Mechanical Engineering, Yale University, Box 208284, New Haven, CT 06520-8284, USA b CMAP-CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France
Abstract As the drive toward greater accuracy in flame simulation continues, there is a need for more detail in the modeling of soot formation and its related phenomena. In this study we investigate computationally the effect of multicomponent transport and thermal diffusion on soot formation in ethylene/air flames. In the counterflow configuration, laminar diffusion flames and partially premixed flames are investigated using complex chemistry and detailed transport. The gas phase equations are coupled to a sectional soot model and the resulting set of partial differential equations admits a well-known similarity solution. Arc length continuation is used to compute flames for varying strain rates. In the coflow configuration, a modified vorticity–velocity formulation is used and the governing equations are solved on an adaptively refined grid using pseudo-transient continuation and Newton’s method nested with a Bi-CGSTAB iterative linear system solver. All transport coefficients, including thermal diffusion coefficients, are evaluated using costeffective, accurate algorithms derived from the kinetic theory of gases. The numerical results for the counterflow model provide a quantitative assessment of the effects of detailed multicomponent transport and thermal diffusion on soot concentrations as a function of strain rate for both a diffusion flame and partially premixed flame. The fidelity of the commonly used Fickian diffusion model is tested and it is shown that in certain cases, the impacts of detailed multicomponent transport and thermal diffusion modeling on soot concentrations are significant. The numerical results for the coflow model demonstrate that although minimal changes in flame shape and temperature profiles arise when transport models are varied, changes in the soot profiles can be seen. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Multicomponent transport; Thermal diffusion; Soot formation; Computation
1. Introduction The accurate quantitative characterization of soot formation in hydrocarbon flames is a crucial *
Corresponding author. Fax: +1 203 432 6775. E-mail address: [email protected] (S.B. Dworkin).
step in an ongoing drive toward cleaner and more efficient combustion. In addition to the significant health risks posed by the presence of combustiongenerated soot and polycyclic aromatic hydrocarbons in the atmosphere, there is growing concern with greenhouse gas emissions, much of which can be directly attributed to combustion processes. Acute control of soot formation in combustion
1540-7489/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2008.05.061
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processes has the potential to lead to an overall increase in combustion efficiency, allowing for the design of lower greenhouse gas emitting devices. With the goal of such accurate modeling of soot formation, attention is herein focused toward the effects of multicomponent transport and thermal diffusion processes and the combustion regimes for which these phenomena become non-negligible. A further understanding of pollutant formation in counterflow and coflow laminar flames is a natural part of the progression to complete theories of multidimensional turbulent flames. Progress has been made in recent years toward the modeling of various aspects of practical flames such as their time-dependent nature [1–9] and the accurate modeling of more practical higher hydrocarbons and surrogates [10–15]. As the shift toward modeling of higher hydrocarbons continues, there is more of a need, due to the associated molecular weight disparities of the various components of the gaseous mixtures, for more accurate transport modeling. Since, in some cases, accurate transport modeling begets accurate species concentration prediction, it is reasonable to assume that there may be associated effects on predicted soot concentrations, due to the roles played by some gaseous species in soot inception, growth and oxidation. The impact of multicomponent transport on counterflow methane/air and hydrogen/air flames has been well established [16–20]. Dixon-Lewis [16] and Greenberg [17] showed that for hydrogen/air flames, molar diffusion flux and thermal diffusion flux for hydrogen and nitrogen could be of the same order of magnitude. Bongers and de Goey [18] show that for larger equivalence ratios, transport modeling can have a significant effect on burning velocity in premixed methane and hydrogen flames. Ern and Giovangigli [20], provided a more comprehensive study detailing the impact of multicomponent transport on planar flame speed and extinction limits as a function of strain rate and equivalence ratio in both hydrogen/air and methane/air flames. Higher order transport effects have been studied in coflow diffusion flames experimentally in [21,22] by adding inert diluents of varying diffusivities and numerically in [23,24]. In [23] it is shown that with a simplified two-equation soot model, thermal diffusion affects soot formation in an ethylene/air flame when helium is added to the fuel stream. In [24] it is shown that thermal diffusion affects the structure of both lean and rich Bunsen hydrogen/air flames as well as rich methane/air diffusion flames and a methane/air jet diffusion flame. These studies [16–24] clearly show great interest in the combustion community in the ongoing task of fully quantifying higher order transport effects on flame properties and soot formation.
Fig. 1. Schematic illustration of a counterflow flame.
The present study focuses on both diffusion and partially premixed sooting laminar ethylene/ air flames in the counterflow configuration (see Fig. 1) as well as a diffusion sooting laminar ethylene/air coflow flame (see Fig. 2). The counterflow flame model has been studied extensively, both computationally and experimentally for a variety of fuels for both premixed and non-premixed flames [10,25–27]. The coflow flame model has also been well characterized with recent publications employing the same modified vorticity– velocity formulation that is used in the present study [1,28]. The sectional soot model has also been studied extensively in both the counterflow and coflow configuration [29–33]. A crucial component to the multicomponent transport and thermal diffusion modeling in [20] as well as in the present work is the framework derived in [34] and [35]. It is this set of pioneering works that now permit, at moderate computational cost, in situ calculation of all the necessary transport coefficients of gas mixtures using rigorous and accurate expressions. To assess the effects of multicomponent transport and thermal diffusion for a
Fig. 2. Schematic illustration of a coflow flame.
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range of equivalence ratios and strain rates in counterflow flames, it is necessary to employ adaptive arc length continuation [36,37], which is described in the following sections.
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losses and scrubbing. The oxidation of soot occurs by O2 and OH as outlined in [40] and [41]. Surface growth is based on local C2H2 concentration [42,43]. Details of the soot model can be found in [29–32].
2. Problem formulation and numerical solutions 3. Solution procedure 2.1. Problem geometry
formulation for the
counterflow
In this section we present the problem formulation for a counterflow laminar ethylene/air flame. For both the diffusion flame and the partially premixed flame the goal is to predict the species mass fractions, temperature, and soot concentrations as a function of the independent spatial coordinate normal to the flame front (see Fig. 1). The model employs the low Mach number approximation and assumes a laminar, stagnation point flow. The system of two-dimensional, governing conservation equations for mass, momentum, energy and species mass admits a wellknown similarity solution, which is the solution of the one-dimensional boundary value problem and is valid along the centerline (see [25] and [38] for more details). 2.2. Problem formulation for the coflow geometry For the axisymmetric steady laminar diffusion flame. The goal here is to predict the species mass fractions, temperature, and soot concentrations as a function of the independent spatial coordinates r and z (see Fig. 2). The governing equations for mass, momentum, chemical species, soot size classes and energy are cast in vorticity–velocity form as in [39] with modifications according to [28]. The boundary conditions are defined to emulate a burner with an interior fuel tube and coaxial flowing air tube (see Fig. 2). The fuel consists of 32% C2H4 by mass in N2. The model employs the low Mach number approximation as well as negligible bulk viscosity and assumes laminar flow. 2.3. Soot modeling For the coflow flame and counterflow flames, the soot model employs a sectional representation for spheroid growth with 20 sections. The soot particle mass range is divided logarithmically into sections. For each section, coalescence, surface growth and oxidation are considered. Inception is considered for the smallest mass range and is based on the formation of two- and three- ringed aromatic species and depends on local acetylene, benzene, phenyl and molecular hydrogen concentrations [32]. Thermophoresis and effective section diffusion are considered. The gas phase equations are coupled through non-adiabatic radiative
3.1. Counterflow flame solution procedure The governing equations are discretized over a one-dimensional mesh using a second order centered difference scheme. The boundaries are taken to be adiabatic. Dirichlet boundary conditions are specified for the axial velocity, temperature, similarity variable, species and soot concentrations. For the diffusion flame, the fuel side boundary condition is set at 100% ethylene and the oxidizer side boundary condition contains only O2 and N2 in proportions appropriate for air. The strain rate is varied from 41 to 488 s1. For the partially premixed flame, the fuel side boundary condition is set at 87% C2H4 and 13% O2 and the oxidizer side boundary condition contains only O2 and N2 in proportions appropriate for air. The strain rate is varied from 41 to 300 s1. Ethylene chemistry is modeled using 66 species and 476 reversible reactions. Details of the chemical mechanism can be found in [44]. The solution method for the coflow flames combines a pseudo-arclength continuation procedure, damped modified Newton’s method and global adaptive gridding (see [45] and [46]). Details of the solution procedure can be found in [20,37] and [47]. An initial solution is generated using pseudo-transient continuation with adaptive step size selection to help bring an arbitrary starting estimate into the convergence domain of Newton’s method. This initial estimate is then used with the arclength continuation procedure where the solution branch is extended using a first-order Euler predictor followed by an implicit correction step involving Newton iterations. Within each Newton iteration, an optimized block-tridiagonal solver is used to solve the linear system. The Jacobian matrix is evaluated numerically using vector function evaluations and highly optimized libraries for thermochemistry [48] and multicomponent transport [49]. At each continuation step, a new adaptive mesh is generated by equidistributing a weight function involving the gradient and curvature of all the components of the numerical solution as in [37]. Such grid adaptation allows for the optimization of the number of grid points and the resolution of high activity regions. With varying strain rate, the position of local maxima of temperature and species concentrations can change rapidly. The adaptive meshing procedure permits prediction of such phenomena with relative ease.
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3.2. Coflow flame solution procedure The governing equations are discretized over a two-dimensional mesh using a second order centered difference scheme. The inlet boundary conditions include a parabolic velocity profile in the fuel tube with an average velocity of 35 cm/s and a plug flow velocity profile in the coflow with a velocity of 35 cm/s. The temperature is set to 298 K along the entire inflow and the species mass fractions are set to zero or appropriate values for C2H4 and N2 in the fuel tube, and O2 and N2 in the coflow. It is assumed that all variables are fully developed at the outflow and in the far field radial direction, i.e., zero axial derivatives are imposed at the outflow and zero radial derivatives for large radial values. Since the problem is axisymmetric, the centerline is taken as an axis of symmetry with zero radial derivatives of temperature, species and axial velocity and zero radial velocity and vorticity. Ethylene chemistry is modeled using the same 66 species mechanism as in the case of the counterflow flame. The computational model solves the full set of elliptic, partial differential conservation equations for mass, momentum, species and energy [50]. A modified vorticity–velocity formulation [28] is used to compute the velocity field as it is more effective at conserving mass than the formulation in [39]. The system is closed with the ideal gas law and appropriate boundary conditions on each side of the computational domain, according to the framework for mass conservation in [28]. The gas is assumed Newtonian and the flow’s small Mach number implies that the pressure field can be obtained via the ideal gas law. The divergence of the net radiative flux is calculated using an optically-thin radiation submodel for the soot and the gas phase radiating species (H2O, CO, and CO2). The details can be found in [51,52]. The set of coupled, non-linear, partial differential equations is solved using a damped, modified Newton’s method [46]. At each adaptively chosen timestep [53] the linear Newton equations are solved using Bi-CGSTAB [54,55] with a block Gauss-Seidel preconditioner. Calculations were performed on a 2.0 GHz AMD Opteron processor with 8 GB RAM. 3.3. Transport modeling The transport coefficients arising in the governing equations for both the counterflow and coflow flame models include the diffusion matrix D = (Dij)i, j 2 [1,N], the rescaled thermal diffusion ratio of the ith species vi (the thermal diffusion ratio divided by the mole fraction of that species), the thermal conductivity k and the shear viscosity g. These transport coefficients depend on the local state of the mixture (i.e., the temperature and species mass fractions) and need to be evaluated at
each point in the computational domain. Kinetic theory does not provide explicit expressions for the transport but rather linear systems, which need to be solved. These linear systems are formed and solved using highly efficient and optimized algorithms. Details on the derivation and implementation of these algorithms can be found in [20], [34] and [35]. To investigate the impact of thermal diffusion and multicomponent transport on soot formation, three different models for species diffusion will be tested and compared. The first model is the most simplified and most commonly used. It considers only the diagonal projection of the mass diffusion matrix and no thermal diffusion. Specifically, for the first model (from this point referred to as M1), the diffusion velocities are evaluated as V iz ¼
K X
½0
Dij
j¼1
dX j ; dz
i ¼ 1; 2; . . . ; K;
ð1Þ
½0
where Dij is the diagonal projection of the diffusion matrix Dij and Xj is the mole fraction of the jth species. When using the diagonal projection of the diffusion matrix, it is possible to evaluate the diffusion velocities in terms of the species mass fractions and a correction velocity as in [18]. Results did not change when using the diagonal projection whether the diffusion velocities were evaluated in terms of mass fraction or mole fraction. The second model considers full multicomponent mass diffusion but, as in the first model, it does not consider thermal diffusion. The principal purpose of inclusion of this model is to quantify the errors associated with the simplified diagonal projection, which is used in the first model. Specifically, for the second model (from this point referred to as M2), the diffusion velocities are evaluated as V iz ¼
K X j¼1
Dij
dX j ; dz
i ¼ 1; 2; . . . ; K:
ð2Þ
The third model (from this point referred to as M3), which is the most physically accurate, considers the full diffusion matrix as well as thermal diffusion effects. In this case, diffusion velocities are K X dX j d V iz ¼ þ X j vj ðlog T Þ ; Dij dz dz j¼1 i ¼ 1; 2; . . . ; K:
ð3Þ
The comparison of predicted soot formation in the ethylene/air system for a range of equivalence ratios will allow for a better understanding of the errors associated with neglecting thermal diffusion and diagonalization of the species diffusion in these systems. It is important to stress here that the modeling of transport of the soot sections
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remains unchanged throughout this study. It is only the effect on soot formation of changing the transport models for the chemical species that is studied here. 4. Results and discussion 4.1. Counterflow flame An ethylene/air diffusion flame is computed with pure ethylene flowing on the fuel side, air on the oxidizer side and a strain rate of 41 s1. For the three transport models, M1, M2 and M3, the temperature profiles and major species profiles lay coincident indicating no major accuracy loss associated with the less accurate transport models M1 and M2 (see Fig. 3a). For some of the lighter species such as H, there is a slight increase in peak mass fraction in the flame due to the thermal diffusive effects, pushing the light species into the hotter regions (see Fig. 3b). Although some of these lighter species can play
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a major role in predicting soot volume fraction, the effects of multicomponent species diffusion and thermal diffusion are negligible on soot volume fraction for the case of the ethylene/air diffusion flame at a strain rate of 41 s1 (see Fig. 3c). Further calculations were done to determine if predictions varied for the diffusion flame for increasing strain rate. Results for strain rates of 200 and 488 s1 are shown in Fig. 4. It can be seen that the M3 model, which includes thermal diffusion, predicts slightly lower soot volume fractions in the flame region than the two other transport models. This effect was consistently independent of strain rate. For the case of a partially premixed flame with 87% C2H4 and 13% O2 by mass on the fuel side and air on the oxidizer side and a strain rate of 41 s1, the soot volume fraction is lowest for the M1 transport model compared to the other two models. Both multicomponent transport model computations predict higher soot concentrations in the flame region. Thermal diffusion, which is only included in the M3 model, has a negligible
Fig. 3. (a) Temperature profile for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3). (b) H mass fraction profiles for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3). (c) Soot volume fraction profiles for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3).
Fig. 4. (a) Soot volume fraction profiles for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 200 s1. (b) Soot volume fraction profiles for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 488 s1.
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Fig. 5. (a) Soot volume fraction profiles for the counterflow ethylene/air partially premixed flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 41 s1. (b) Soot volume fraction profiles for the counterflow ethylene/air partially premixed flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 100 s1. (c) Soot volume fraction profiles for the counterflow ethylene/air partially premixed flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 200 s1. (d) Soot volume fraction profiles for the counterflow ethylene/air partially premixed flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 300 s1.
effect on predicting soot volume fraction in this flame as the M2 and M3 concentrations lie coincident (see Fig. 5a). When the strain rate is increased to 100 s1, the trend remains but is less pronounced (see Fig. 5b). Here, the Fickian diffusion model used in M1, still underpredicts the soot volume fraction but the discrepancy is less dramatic. The trends for the partially premixed flame are different as strain rate is continually increased to 200 and 300 s1 (see Fig. 5c and d, respectively). Here again, the inclusion of multicomponent transport in M2 increases the predicted soot volume fraction as compared with the Fickian diffusion model M1, but the subsequent inclusion of thermal diffusion in M3 has the inverse effect. For the strain rate of 200 s1, the predicted soot volume fractions from M3 lay between the predic-
tions of M1 and M2, and for the strain rate of 300 s1, the predicted soot volume fractions from M3 lay nearly coincident with the predictions from M1, indicating that the effects of inclusion of multicomponent transport and thermal diffusion in this case act in opposition to each other. 4.2. Coflow flame A coflow ethylene/air diffusion flame is computed with 32% C2H4 (mole fraction) in N2 flowing in an inner fuel tube surrounded by a coflowing air tube. The flame was computed using all three transport models and the results for the soot volume fraction are shown in Fig. 6. Figure 6a shows a comparison between soot volume fraction contours for the Fickian diffusion model (left half of Fig. 6a), denoted M1, and the model with
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[32] shows that in a lower sooting flame, such as in the present study, inception dominates soot formation. For this reason, movement of the larger species (those involved in inception) due to thermal gradients has a greater effect here than in [23]. 5. Concluding remarks
Fig. 6. Soot volume fraction contours for the coflow ethylene/air diffusion flame. (a) Comparison of the Fickian diffusion model (M1) with the model containing full species diffusion (M2). (b) Comparison of the Fickian diffusion model (M1) with the model containing full species diffusion and thermal diffusion (M3).
full multicomponent transport, denoted M2 (right half of Fig. 6a). It can be seen that the overall soot distribution is preserved as much of the soot is formed on the wings of the flame for both the M1 and the M2 model and the peak soot volume fraction changes by less than 2%. Figure 6b shows a comparison between soot volume fraction contours for the Fickian diffusion model (left half of Fig. 6b), denoted M1, and the model with full multicomponent transport and thermal diffusion denoted M3 (right half of Fig. 6b). It can be seen that the overall soot distribution has changed significantly. The location where soot has formed along the centerline has moved down and the region of highest concentration has moved from the wings to the centerline of the flame. The peak soot volume fraction has decreased from 1.01 to 0.946 ppm, a decrease of more than 8%. Moreover, if one interrogates the effects of the individual thermal diffusion velocities on the soot distribution, the dominant contributions are due to the thermal diffusion velocities of heavier species as opposed to conventional thinking that only the light species (H and H2) play a role. It is these heavier species which play a key role in soot inception and thus affect the soot distribution. This result does not contradict the findings of [23] which does not note significant effects of transport on methane/air flames (without diluents). For the operating conditions in [23], it has been shown that most soot is formed on the wings through a surface growth mechanism [32] and thus it is less sensitive to movement of heavy species.
This work investigates the effect of transport modeling on soot formation in counterflow laminar diffusion and partially premixed ethylene/air flames, as well as a coflow diffusion flame. At varying strain rates, it is found that the inclusion of thermal diffusion causes lower predictions of soot volume fractions in the counterflow diffusion flame. For the case of the partially premixed flame, the effects of transport modeling change over the range of strain rates studied. For lower strain rates, the lack of inclusion of thermal diffusion caused a moderate overprediction of soot volume fraction but as the strain rate was increased, this effect was dampened as the effects of the inclusion of multicomponent transport and thermal diffusion tended toward cancellation. For the coflow diffusion flame, the transport model can impact peak soot concentration and flame position but peak temperature was largely unchanged. Whereas the inclusion of multicomponent transport had a minimal effect on the coflow diffusion flame, the inclusion of thermal diffusion moved the peak soot concentration from the wings of the flame toward the centerline and caused a significant downward shift in the location of peak soot concentrations.
Acknowledgments This work is supported by the US Department of Energy Office of Basic Energy Sciences (grant no. DE-FG02-88ER13966), the National Science Foundation (grant no. CTS-0328296) and the National Science Foundation NIRT (grant no. EEC0506968).
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Proceedings of the
Proceedings of the Combustion Institute 32 (2009) 1165–1172
Combustion Institute www.elsevier.com/locate/proci
The impact of detailed multicomponent transport and thermal diffusion effects on soot formation in ethylene/air flames S.B. Dworkin a,*, M.D. Smookea, V. Giovangigli b a
Department of Mechanical Engineering, Yale University, Box 208284, New Haven, CT 06520-8284, USA b CMAP-CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France
Abstract As the drive toward greater accuracy in flame simulation continues, there is a need for more detail in the modeling of soot formation and its related phenomena. In this study we investigate computationally the effect of multicomponent transport and thermal diffusion on soot formation in ethylene/air flames. In the counterflow configuration, laminar diffusion flames and partially premixed flames are investigated using complex chemistry and detailed transport. The gas phase equations are coupled to a sectional soot model and the resulting set of partial differential equations admits a well-known similarity solution. Arc length continuation is used to compute flames for varying strain rates. In the coflow configuration, a modified vorticity–velocity formulation is used and the governing equations are solved on an adaptively refined grid using pseudo-transient continuation and Newton’s method nested with a Bi-CGSTAB iterative linear system solver. All transport coefficients, including thermal diffusion coefficients, are evaluated using costeffective, accurate algorithms derived from the kinetic theory of gases. The numerical results for the counterflow model provide a quantitative assessment of the effects of detailed multicomponent transport and thermal diffusion on soot concentrations as a function of strain rate for both a diffusion flame and partially premixed flame. The fidelity of the commonly used Fickian diffusion model is tested and it is shown that in certain cases, the impacts of detailed multicomponent transport and thermal diffusion modeling on soot concentrations are significant. The numerical results for the coflow model demonstrate that although minimal changes in flame shape and temperature profiles arise when transport models are varied, changes in the soot profiles can be seen. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Multicomponent transport; Thermal diffusion; Soot formation; Computation
1. Introduction The accurate quantitative characterization of soot formation in hydrocarbon flames is a crucial *
Corresponding author. Fax: +1 203 432 6775. E-mail address: [email protected] (S.B. Dworkin).
step in an ongoing drive toward cleaner and more efficient combustion. In addition to the significant health risks posed by the presence of combustiongenerated soot and polycyclic aromatic hydrocarbons in the atmosphere, there is growing concern with greenhouse gas emissions, much of which can be directly attributed to combustion processes. Acute control of soot formation in combustion
1540-7489/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2008.05.061
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processes has the potential to lead to an overall increase in combustion efficiency, allowing for the design of lower greenhouse gas emitting devices. With the goal of such accurate modeling of soot formation, attention is herein focused toward the effects of multicomponent transport and thermal diffusion processes and the combustion regimes for which these phenomena become non-negligible. A further understanding of pollutant formation in counterflow and coflow laminar flames is a natural part of the progression to complete theories of multidimensional turbulent flames. Progress has been made in recent years toward the modeling of various aspects of practical flames such as their time-dependent nature [1–9] and the accurate modeling of more practical higher hydrocarbons and surrogates [10–15]. As the shift toward modeling of higher hydrocarbons continues, there is more of a need, due to the associated molecular weight disparities of the various components of the gaseous mixtures, for more accurate transport modeling. Since, in some cases, accurate transport modeling begets accurate species concentration prediction, it is reasonable to assume that there may be associated effects on predicted soot concentrations, due to the roles played by some gaseous species in soot inception, growth and oxidation. The impact of multicomponent transport on counterflow methane/air and hydrogen/air flames has been well established [16–20]. Dixon-Lewis [16] and Greenberg [17] showed that for hydrogen/air flames, molar diffusion flux and thermal diffusion flux for hydrogen and nitrogen could be of the same order of magnitude. Bongers and de Goey [18] show that for larger equivalence ratios, transport modeling can have a significant effect on burning velocity in premixed methane and hydrogen flames. Ern and Giovangigli [20], provided a more comprehensive study detailing the impact of multicomponent transport on planar flame speed and extinction limits as a function of strain rate and equivalence ratio in both hydrogen/air and methane/air flames. Higher order transport effects have been studied in coflow diffusion flames experimentally in [21,22] by adding inert diluents of varying diffusivities and numerically in [23,24]. In [23] it is shown that with a simplified two-equation soot model, thermal diffusion affects soot formation in an ethylene/air flame when helium is added to the fuel stream. In [24] it is shown that thermal diffusion affects the structure of both lean and rich Bunsen hydrogen/air flames as well as rich methane/air diffusion flames and a methane/air jet diffusion flame. These studies [16–24] clearly show great interest in the combustion community in the ongoing task of fully quantifying higher order transport effects on flame properties and soot formation.
Fig. 1. Schematic illustration of a counterflow flame.
The present study focuses on both diffusion and partially premixed sooting laminar ethylene/ air flames in the counterflow configuration (see Fig. 1) as well as a diffusion sooting laminar ethylene/air coflow flame (see Fig. 2). The counterflow flame model has been studied extensively, both computationally and experimentally for a variety of fuels for both premixed and non-premixed flames [10,25–27]. The coflow flame model has also been well characterized with recent publications employing the same modified vorticity– velocity formulation that is used in the present study [1,28]. The sectional soot model has also been studied extensively in both the counterflow and coflow configuration [29–33]. A crucial component to the multicomponent transport and thermal diffusion modeling in [20] as well as in the present work is the framework derived in [34] and [35]. It is this set of pioneering works that now permit, at moderate computational cost, in situ calculation of all the necessary transport coefficients of gas mixtures using rigorous and accurate expressions. To assess the effects of multicomponent transport and thermal diffusion for a
Fig. 2. Schematic illustration of a coflow flame.
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range of equivalence ratios and strain rates in counterflow flames, it is necessary to employ adaptive arc length continuation [36,37], which is described in the following sections.
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losses and scrubbing. The oxidation of soot occurs by O2 and OH as outlined in [40] and [41]. Surface growth is based on local C2H2 concentration [42,43]. Details of the soot model can be found in [29–32].
2. Problem formulation and numerical solutions 3. Solution procedure 2.1. Problem geometry
formulation for the
counterflow
In this section we present the problem formulation for a counterflow laminar ethylene/air flame. For both the diffusion flame and the partially premixed flame the goal is to predict the species mass fractions, temperature, and soot concentrations as a function of the independent spatial coordinate normal to the flame front (see Fig. 1). The model employs the low Mach number approximation and assumes a laminar, stagnation point flow. The system of two-dimensional, governing conservation equations for mass, momentum, energy and species mass admits a wellknown similarity solution, which is the solution of the one-dimensional boundary value problem and is valid along the centerline (see [25] and [38] for more details). 2.2. Problem formulation for the coflow geometry For the axisymmetric steady laminar diffusion flame. The goal here is to predict the species mass fractions, temperature, and soot concentrations as a function of the independent spatial coordinates r and z (see Fig. 2). The governing equations for mass, momentum, chemical species, soot size classes and energy are cast in vorticity–velocity form as in [39] with modifications according to [28]. The boundary conditions are defined to emulate a burner with an interior fuel tube and coaxial flowing air tube (see Fig. 2). The fuel consists of 32% C2H4 by mass in N2. The model employs the low Mach number approximation as well as negligible bulk viscosity and assumes laminar flow. 2.3. Soot modeling For the coflow flame and counterflow flames, the soot model employs a sectional representation for spheroid growth with 20 sections. The soot particle mass range is divided logarithmically into sections. For each section, coalescence, surface growth and oxidation are considered. Inception is considered for the smallest mass range and is based on the formation of two- and three- ringed aromatic species and depends on local acetylene, benzene, phenyl and molecular hydrogen concentrations [32]. Thermophoresis and effective section diffusion are considered. The gas phase equations are coupled through non-adiabatic radiative
3.1. Counterflow flame solution procedure The governing equations are discretized over a one-dimensional mesh using a second order centered difference scheme. The boundaries are taken to be adiabatic. Dirichlet boundary conditions are specified for the axial velocity, temperature, similarity variable, species and soot concentrations. For the diffusion flame, the fuel side boundary condition is set at 100% ethylene and the oxidizer side boundary condition contains only O2 and N2 in proportions appropriate for air. The strain rate is varied from 41 to 488 s1. For the partially premixed flame, the fuel side boundary condition is set at 87% C2H4 and 13% O2 and the oxidizer side boundary condition contains only O2 and N2 in proportions appropriate for air. The strain rate is varied from 41 to 300 s1. Ethylene chemistry is modeled using 66 species and 476 reversible reactions. Details of the chemical mechanism can be found in [44]. The solution method for the coflow flames combines a pseudo-arclength continuation procedure, damped modified Newton’s method and global adaptive gridding (see [45] and [46]). Details of the solution procedure can be found in [20,37] and [47]. An initial solution is generated using pseudo-transient continuation with adaptive step size selection to help bring an arbitrary starting estimate into the convergence domain of Newton’s method. This initial estimate is then used with the arclength continuation procedure where the solution branch is extended using a first-order Euler predictor followed by an implicit correction step involving Newton iterations. Within each Newton iteration, an optimized block-tridiagonal solver is used to solve the linear system. The Jacobian matrix is evaluated numerically using vector function evaluations and highly optimized libraries for thermochemistry [48] and multicomponent transport [49]. At each continuation step, a new adaptive mesh is generated by equidistributing a weight function involving the gradient and curvature of all the components of the numerical solution as in [37]. Such grid adaptation allows for the optimization of the number of grid points and the resolution of high activity regions. With varying strain rate, the position of local maxima of temperature and species concentrations can change rapidly. The adaptive meshing procedure permits prediction of such phenomena with relative ease.
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3.2. Coflow flame solution procedure The governing equations are discretized over a two-dimensional mesh using a second order centered difference scheme. The inlet boundary conditions include a parabolic velocity profile in the fuel tube with an average velocity of 35 cm/s and a plug flow velocity profile in the coflow with a velocity of 35 cm/s. The temperature is set to 298 K along the entire inflow and the species mass fractions are set to zero or appropriate values for C2H4 and N2 in the fuel tube, and O2 and N2 in the coflow. It is assumed that all variables are fully developed at the outflow and in the far field radial direction, i.e., zero axial derivatives are imposed at the outflow and zero radial derivatives for large radial values. Since the problem is axisymmetric, the centerline is taken as an axis of symmetry with zero radial derivatives of temperature, species and axial velocity and zero radial velocity and vorticity. Ethylene chemistry is modeled using the same 66 species mechanism as in the case of the counterflow flame. The computational model solves the full set of elliptic, partial differential conservation equations for mass, momentum, species and energy [50]. A modified vorticity–velocity formulation [28] is used to compute the velocity field as it is more effective at conserving mass than the formulation in [39]. The system is closed with the ideal gas law and appropriate boundary conditions on each side of the computational domain, according to the framework for mass conservation in [28]. The gas is assumed Newtonian and the flow’s small Mach number implies that the pressure field can be obtained via the ideal gas law. The divergence of the net radiative flux is calculated using an optically-thin radiation submodel for the soot and the gas phase radiating species (H2O, CO, and CO2). The details can be found in [51,52]. The set of coupled, non-linear, partial differential equations is solved using a damped, modified Newton’s method [46]. At each adaptively chosen timestep [53] the linear Newton equations are solved using Bi-CGSTAB [54,55] with a block Gauss-Seidel preconditioner. Calculations were performed on a 2.0 GHz AMD Opteron processor with 8 GB RAM. 3.3. Transport modeling The transport coefficients arising in the governing equations for both the counterflow and coflow flame models include the diffusion matrix D = (Dij)i, j 2 [1,N], the rescaled thermal diffusion ratio of the ith species vi (the thermal diffusion ratio divided by the mole fraction of that species), the thermal conductivity k and the shear viscosity g. These transport coefficients depend on the local state of the mixture (i.e., the temperature and species mass fractions) and need to be evaluated at
each point in the computational domain. Kinetic theory does not provide explicit expressions for the transport but rather linear systems, which need to be solved. These linear systems are formed and solved using highly efficient and optimized algorithms. Details on the derivation and implementation of these algorithms can be found in [20], [34] and [35]. To investigate the impact of thermal diffusion and multicomponent transport on soot formation, three different models for species diffusion will be tested and compared. The first model is the most simplified and most commonly used. It considers only the diagonal projection of the mass diffusion matrix and no thermal diffusion. Specifically, for the first model (from this point referred to as M1), the diffusion velocities are evaluated as V iz ¼
K X
½0
Dij
j¼1
dX j ; dz
i ¼ 1; 2; . . . ; K;
ð1Þ
½0
where Dij is the diagonal projection of the diffusion matrix Dij and Xj is the mole fraction of the jth species. When using the diagonal projection of the diffusion matrix, it is possible to evaluate the diffusion velocities in terms of the species mass fractions and a correction velocity as in [18]. Results did not change when using the diagonal projection whether the diffusion velocities were evaluated in terms of mass fraction or mole fraction. The second model considers full multicomponent mass diffusion but, as in the first model, it does not consider thermal diffusion. The principal purpose of inclusion of this model is to quantify the errors associated with the simplified diagonal projection, which is used in the first model. Specifically, for the second model (from this point referred to as M2), the diffusion velocities are evaluated as V iz ¼
K X j¼1
Dij
dX j ; dz
i ¼ 1; 2; . . . ; K:
ð2Þ
The third model (from this point referred to as M3), which is the most physically accurate, considers the full diffusion matrix as well as thermal diffusion effects. In this case, diffusion velocities are K X dX j d V iz ¼ þ X j vj ðlog T Þ ; Dij dz dz j¼1 i ¼ 1; 2; . . . ; K:
ð3Þ
The comparison of predicted soot formation in the ethylene/air system for a range of equivalence ratios will allow for a better understanding of the errors associated with neglecting thermal diffusion and diagonalization of the species diffusion in these systems. It is important to stress here that the modeling of transport of the soot sections
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remains unchanged throughout this study. It is only the effect on soot formation of changing the transport models for the chemical species that is studied here. 4. Results and discussion 4.1. Counterflow flame An ethylene/air diffusion flame is computed with pure ethylene flowing on the fuel side, air on the oxidizer side and a strain rate of 41 s1. For the three transport models, M1, M2 and M3, the temperature profiles and major species profiles lay coincident indicating no major accuracy loss associated with the less accurate transport models M1 and M2 (see Fig. 3a). For some of the lighter species such as H, there is a slight increase in peak mass fraction in the flame due to the thermal diffusive effects, pushing the light species into the hotter regions (see Fig. 3b). Although some of these lighter species can play
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a major role in predicting soot volume fraction, the effects of multicomponent species diffusion and thermal diffusion are negligible on soot volume fraction for the case of the ethylene/air diffusion flame at a strain rate of 41 s1 (see Fig. 3c). Further calculations were done to determine if predictions varied for the diffusion flame for increasing strain rate. Results for strain rates of 200 and 488 s1 are shown in Fig. 4. It can be seen that the M3 model, which includes thermal diffusion, predicts slightly lower soot volume fractions in the flame region than the two other transport models. This effect was consistently independent of strain rate. For the case of a partially premixed flame with 87% C2H4 and 13% O2 by mass on the fuel side and air on the oxidizer side and a strain rate of 41 s1, the soot volume fraction is lowest for the M1 transport model compared to the other two models. Both multicomponent transport model computations predict higher soot concentrations in the flame region. Thermal diffusion, which is only included in the M3 model, has a negligible
Fig. 3. (a) Temperature profile for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3). (b) H mass fraction profiles for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3). (c) Soot volume fraction profiles for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3).
Fig. 4. (a) Soot volume fraction profiles for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 200 s1. (b) Soot volume fraction profiles for the counterflow ethylene/air diffusion flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 488 s1.
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Fig. 5. (a) Soot volume fraction profiles for the counterflow ethylene/air partially premixed flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 41 s1. (b) Soot volume fraction profiles for the counterflow ethylene/air partially premixed flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 100 s1. (c) Soot volume fraction profiles for the counterflow ethylene/air partially premixed flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 200 s1. (d) Soot volume fraction profiles for the counterflow ethylene/air partially premixed flame, comparing the three transport models (M1, M2, and M3) for a strain rate of 300 s1.
effect on predicting soot volume fraction in this flame as the M2 and M3 concentrations lie coincident (see Fig. 5a). When the strain rate is increased to 100 s1, the trend remains but is less pronounced (see Fig. 5b). Here, the Fickian diffusion model used in M1, still underpredicts the soot volume fraction but the discrepancy is less dramatic. The trends for the partially premixed flame are different as strain rate is continually increased to 200 and 300 s1 (see Fig. 5c and d, respectively). Here again, the inclusion of multicomponent transport in M2 increases the predicted soot volume fraction as compared with the Fickian diffusion model M1, but the subsequent inclusion of thermal diffusion in M3 has the inverse effect. For the strain rate of 200 s1, the predicted soot volume fractions from M3 lay between the predic-
tions of M1 and M2, and for the strain rate of 300 s1, the predicted soot volume fractions from M3 lay nearly coincident with the predictions from M1, indicating that the effects of inclusion of multicomponent transport and thermal diffusion in this case act in opposition to each other. 4.2. Coflow flame A coflow ethylene/air diffusion flame is computed with 32% C2H4 (mole fraction) in N2 flowing in an inner fuel tube surrounded by a coflowing air tube. The flame was computed using all three transport models and the results for the soot volume fraction are shown in Fig. 6. Figure 6a shows a comparison between soot volume fraction contours for the Fickian diffusion model (left half of Fig. 6a), denoted M1, and the model with
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[32] shows that in a lower sooting flame, such as in the present study, inception dominates soot formation. For this reason, movement of the larger species (those involved in inception) due to thermal gradients has a greater effect here than in [23]. 5. Concluding remarks
Fig. 6. Soot volume fraction contours for the coflow ethylene/air diffusion flame. (a) Comparison of the Fickian diffusion model (M1) with the model containing full species diffusion (M2). (b) Comparison of the Fickian diffusion model (M1) with the model containing full species diffusion and thermal diffusion (M3).
full multicomponent transport, denoted M2 (right half of Fig. 6a). It can be seen that the overall soot distribution is preserved as much of the soot is formed on the wings of the flame for both the M1 and the M2 model and the peak soot volume fraction changes by less than 2%. Figure 6b shows a comparison between soot volume fraction contours for the Fickian diffusion model (left half of Fig. 6b), denoted M1, and the model with full multicomponent transport and thermal diffusion denoted M3 (right half of Fig. 6b). It can be seen that the overall soot distribution has changed significantly. The location where soot has formed along the centerline has moved down and the region of highest concentration has moved from the wings to the centerline of the flame. The peak soot volume fraction has decreased from 1.01 to 0.946 ppm, a decrease of more than 8%. Moreover, if one interrogates the effects of the individual thermal diffusion velocities on the soot distribution, the dominant contributions are due to the thermal diffusion velocities of heavier species as opposed to conventional thinking that only the light species (H and H2) play a role. It is these heavier species which play a key role in soot inception and thus affect the soot distribution. This result does not contradict the findings of [23] which does not note significant effects of transport on methane/air flames (without diluents). For the operating conditions in [23], it has been shown that most soot is formed on the wings through a surface growth mechanism [32] and thus it is less sensitive to movement of heavy species.
This work investigates the effect of transport modeling on soot formation in counterflow laminar diffusion and partially premixed ethylene/air flames, as well as a coflow diffusion flame. At varying strain rates, it is found that the inclusion of thermal diffusion causes lower predictions of soot volume fractions in the counterflow diffusion flame. For the case of the partially premixed flame, the effects of transport modeling change over the range of strain rates studied. For lower strain rates, the lack of inclusion of thermal diffusion caused a moderate overprediction of soot volume fraction but as the strain rate was increased, this effect was dampened as the effects of the inclusion of multicomponent transport and thermal diffusion tended toward cancellation. For the coflow diffusion flame, the transport model can impact peak soot concentration and flame position but peak temperature was largely unchanged. Whereas the inclusion of multicomponent transport had a minimal effect on the coflow diffusion flame, the inclusion of thermal diffusion moved the peak soot concentration from the wings of the flame toward the centerline and caused a significant downward shift in the location of peak soot concentrations.
Acknowledgments This work is supported by the US Department of Energy Office of Basic Energy Sciences (grant no. DE-FG02-88ER13966), the National Science Foundation (grant no. CTS-0328296) and the National Science Foundation NIRT (grant no. EEC0506968).
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