- Email: [email protected]

Numerical and experimental study of an axisymmetric coflow laminar methane–air diffusion flame at pressures between 5 and 40 atmospheres Fengshan Liu ∗ , Kevin A. Thomson, Hongsheng Guo, Gregory J. Smallwood Institute for Chemical Process & Environmental Technology, National Research Council, Building M-9, 1200 Montreal Road, Ottawa, ON, K1A 0R6 Canada Received 6 January 2005; received in revised form 5 February 2006; accepted 1 April 2006 Available online 3 July 2006

Abstract A numerical and experimental study of an axisymmetric coflow laminar methane–air diffusion flame at pressures between 5 and 40 atm was conducted to investigate the effect of pressure on the flame structure and soot formation characteristics. Experimental work was carried out in a new high-pressure combustion chamber described in a recent study [K.A. Thomson, Ö.L. Gülder, E.J. Weckman, R.A. Fraser, G.J. Smallwood, D.R. Snelling, Combust. Flame 140 (2005) 222–232]. Radially resolved soot volume fraction was experimentally measured using both spectral soot emission and line-of-sight attenuation techniques. Numerically, the elliptic governing equations were solved in axisymmetric cylindrical coordinates using the finite volume method. Detailed gas-phase chemistry and complex thermal and transport properties were employed in the numerical calculations. The soot model employed in this study accounts for soot nucleation and surface growth using a semiempirical acetylene-based global soot model with oxidation of soot by O2 , OH, and O taken into account. Radiative heat transfer was calculated using the discrete-ordinates method and a nine-band nongray radiative property model. Two soot surface growth submodels were investigated and the predicted pressure dependence of soot yield was compared with available experimental data. The experiment, the numerical model, and a simplified theoretical analysis found that the visible flame diameter decreases with pressure as Pa−0.5 . The flame-diameter-integrated soot volume fraction increases with pressure as Pa1.3 between 5 and 20 atm. The assumption of a square root dependence of the soot surface growth rate on the soot particle surface area predicts the pressure dependence of soot yield in good agreement with the experimental observation. On the other hand, the assumption of linear dependence of the soot surface growth rate on the soot surface area predicts a much faster increase in the soot yield with pressure than that observed experimentally. Although pressure affects the gas-phase chemistry, the increased soot production with increasing pressure seems primarily due to enhanced mixture density and species concentrations in the pressure range investigated. The increased pressure causes enhanced air entrainment into the fuel stream around the burner rim, leading to accelerated fuel pyrolysis. In the pressure range of 20 to 40 atm both the model and experiment show a diminishing sensitivity of sooting propensity to pressure with a greater decrease in the predicted sensitivity of soot propensity to pressure than the experimental results.

* Corresponding author. Fax: +1 613 957 7869.

E-mail address: [email protected] (F. Liu). 0010-2180/$ – see front matter Crown Copyright © 2006 Published by Elsevier Inc. on behalf of The Combustion Institute. All rights reserved. doi:10.1016/j.combustflame.2006.04.018

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Crown Copyright © 2006 Published by Elsevier Inc. on behalf of The Combustion Institute. All rights reserved. Keywords: Laminar diffusion flame; Soot formation; Pressure effect

1. Introduction Detailed knowledge of the pressure dependence of soot formation and other properties of laminar diffusion flames is of significant relevance to understanding the combustion process and soot emission in many systems operated at high pressures, such as diesel engines and gas turbine combustors. Although soot formation in laminar coflow diffusion flames at atmospheric pressure has been extensively investigated experimentally (e.g., [1–4]) and numerically (e.g., [5–8]), only a few experimental studies have been carried out at high pressure with global soot properties (line-of-sight integrated soot volume fraction) measured [9–15]. These studies showed a significant increase in soot production with increasing pressure. Miller and Maahs, who were primarily interested in NOx formation, provided some semiquantitative measurements of soot concentration in axisymmetric coflow laminar methane/air diffusion flames at pressures from 1 to 50 atm [10]. Their data are referred to in the literature as soot mass [16] or soot yield [12] with a wide range of pressure correlations (data ∝ Pan , where n is between 1.7 and 3). More recent studies by Flower and Bowman [12] (1 to 10 atm) and Lee and Na [13] (1 to 4 atm) conducted in axisymmetric coflow laminar ethylene/air diffusion flames utilized laser-based diagnostic techniques. They found that the peak integrated soot volume fraction across the diameter of ∞ fv dr (fv is the soot volume fraction the flame, −∞ and r the radius), increases with pressure raised to a power of about 1.2 to 1.3. Very recently, McCrain and Roberts [14] found that the flame-diameter-integrated soot volume fraction scales with pressure as Pa1.0 and Pa1.2 in laminar methane/air and ethylene/air diffusion flames, respectively, in their experimental study of laminar coflow diffusion flames at elevated pressures up to 25 atm. A very similar finding was made by Thomson et al. [15], who found that the maximum flame-diameter-integrated soot volume fraction increases with pressure as Pa1.3 for pressures between 5 and 20 atm and Pa0.9 for pressures between 20 and 40 atm for a methane/air flame. Therefore, there exists fairly good agreement on the pressure dependence of the flame-diameter-integrated soot volume fractions among these experimental studies. According to Glassman [17], such pressure dependence of soot formation holds regardless of the type of fuel. The mechanism of the pressure dependence of soot formation in laminar diffusion flames has been

discussed by several authors. Schalla and McDonald [9] suggested that the pressure dependence of soot formation is related to species diffusion processes, since the diffusion coefficient is inversely proportional to pressure. Miller and Maahs [10] suggested that pressure alters the reaction mechanism. Through measurements of soot particle size and particle number density at elevated pressures in coflow laminar ethylene/air diffusion flames stabilized in a Wolfhard–Parker burner, Flower and Bowman [11] concluded that the increased soot production with pressure is due to increases in both the surface growth and particle inception rates. While their findings are useful, they are short of revealing the more fundamental mechanism behind the pressure dependence of soot formation, since soot inception and surface growth rates are directly affected by concentration of inception and growth species, such as polycyclic aromatic hydrocarbons (PAH) and C2 H2 , and temperature. Besides the soot inception and surface growth rates, the soot yield in flames is also dependent on oxidation rate and residence time; i.e., the soot yield is the net effect of the two counteracting processes of production (inception and surface growth) and consumption (oxidation). Compared to the experimental studies of soot formation in laminar coflow diffusion flames at elevated or high pressures, even fewer numerical studies in this field have been reported in the literature. The paper of Zhang and Ezekoye [18] is perhaps the only such study. In their numerical study of soot formation in a laminar methane/air jet diffusion flame at 1 and 4 atm using an eight-step reduced gaseous chemistry and an acetylene-based global soot model, Zhang and Ezekoye [18] suggested that the enhanced soot production at increased pressure can be attributed simply to the increased mixture density, which is proportional to pressure. Besides the significantly increased soot production at elevated pressure, the visible flame diameter decreases considerably with pressure [10,12,14,15]. Miller and Maahs [10] suggested that the effect of pressure on the flame shape is due to a change in reaction mechanism with increasing pressure. Flower and Bowman [12] did not provide an explanation of the effect of pressure on the flame shape. Through scaling reasoning, Glassman [17] concluded that the flame cross-sectional area should decrease with pres−1/2 . However, recent experimental studsure as Pa ies by McCrain and Roberts [14] and Thomson et al. [15] found that the visible flame radius decreases

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, rather than the visible flame with pressure as Pa cross-sectional area. The direct influence of pressure on a laminar diffusion flame and soot formation can be classified as (i) the physical effect and (ii) the chemical effect. The physical effect of pressure affects the flame and soot formation through the pressure dependence of the mixture density (ρ ∝ Pa ), the binary diffusion coefficient (Dij ∝ Pa−1 ), and the potential change in residence time due to change in the flame shape and possibly the processes of particle coagulation and agglomeration. The chemical effect of pressure includes its influence on the gas-phase combustion chemistry; i.e., it reduces the mass fractions of radicals H, O, and OH through enhancement of three-body recombination reactions, which affects soot kinetics according to the HACA mechanism [19], and potentially on the chemistry of soot particle inception, surface growth, and oxidation; i.e., the reaction rates of these processes might be pressure-dependent. Compared to other effects of pressure, the direct effect of pressure on the chemistry of soot has received little attention so far, which is not surprising given the fact that the chemistry of soot inception and growth is not fully understood even at atmospheric pressure. Nevertheless, this aspect of the pressure effect is beyond the scope of the present study. As a result of the rapid decrease in the flame diameter at elevated pressures and the requirement of optical access, it becomes very difficult to conduct experimental measurements of various quantities (such as velocity, temperature, species concentrations, and soot volume fraction) in laminar diffusion flames with adequate spatial resolution. Numerical simulation is a very useful tool to investigate the effects of pressure on the structure and soot formation characteristics in such flames, since it can provide detailed information on velocity, temperature, and species concentrations that otherwise cannot be fully available experimentally. The present paper presents experimental and numerical results of the pressure dependence of some flame properties, such as the visible flame shape (diameter and height), residence time, soot volume fraction, and flame-diameter-integrated and cross-sectional-area-integrated soot volume fractions. A semiempirical global soot model was employed in the present study. In this simplified soot model, acetylene is assumed to be the only species responsible for soot inception and surface growth. The reasons for using such a simple soot model for the purpose of the present study are twofold. First, the effect of pressure on the flame properties (including soot) is perhaps primarily a physical phenomenon rather than a chemical one, based on the findings of the previous numerical study of Zhang and Ezekoye [18]. This semiempirical soot model has been shown previously to predict

Fig. 1. Schematics of the high-pressure combustion chamber.

soot volume fractions in reasonably good agreement with experimental data in atmospheric pressure laminar ethylene and methane diffusion flames [20,21]. Second, use of a more detailed gas-phase reaction mechanism including PAH in the present 2D flame calculations drastically increases the computing time to an unacceptable extent. Numerical studies incorporating the contributions of PAH on soot inception and surface growth will be conducted in the future.

2. Experimental setup and numerical model 2.1. Experimental setup The pressure vessel used in this study was designed for working pressures up to 100 atm and is shown schematically in Fig. 1. The chamber is large, with an internal height of 600 mm and an internal diameter of 240 mm. Physical access to the chamber is possible through the upper and lower flanges. Optical access to the chamber is possible through three viewing ports oriented so that line-of-sight and 90◦ scatter measurements are possible. The chamber is mounted on an external 3-axis translation system. The nonpremixed annular flame burner is based on a design by Miller and Maahs [10], with which they achieved a stable flame over a pressure range of 1–50 atm. A schematic of the burner used in the present study is shown in Fig. 2. The fuel nozzle exit diameter is 3.06 mm and the air shroud diameter is 25.4 mm. Note that the inner diameter of the fuel pipe increases linearly close to the burner tip, rather than remaining constant. While this burner design helps flame stability, it imposes some uncertainty in the inlet velocity profiles of the fuel and air streams. In the original design, a cylindrical quartz tube was used around the

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to soot formation and oxidation, namely C2 H2 , CO, H2 , O2 , O, H, and OH. Only the thermal diffusion velocities of H2 and H were accounted for using the expression given in [25]. The source term in the energy equation due to radiation heat transfer was also included. A modified version of the semiempirical twoequation formulation of soot kinetics proposed by Leung et al. [26] was used to model soot nucleation, growth, and oxidation. The transport equations for the soot mass fraction and number density are given as ∂Ys ∂Ys + ρv ∂z ∂r ∂ 1 ∂ (rρVT,r Ys ) + Sm , (1) = − (ρVT,z Ys ) − ∂z r ∂r ∂N ∂N + ρv ρu ∂z ∂r ∂ 1 ∂ (rρVT,r N ) + SN , = − (ρVT,z N ) − (2) ∂z r ∂r where u and v are velocities in the streamwise (z) and radial (r) directions, ρ is the mixture density, Ys is the soot mass fraction, and N is the soot number density, defined as the particle number per unit mass of the mixture. The quantities VT,r and VT,z are the thermophoretic velocities of soot in the r and z directions, respectively, and are calculated as [20] ρu

Fig. 2. Details of the nonpremixed burner used in the experimental study.

flame to aid flame stabilization. To allow spatially resolved optical measurements in the present experiments, a chimney was designed that included three flat windows, aligned with the three viewports of the chamber. In this study, methane and air were respectively delivered through the central fuel pipe and the annular shroud region at room temperature. Radially resolved distributions of soot volume fraction were measured using the spectral soot emission diagnostic (SSE) [22] and the line-of-sight attenuation technique (LOSA). Further details on the experimental setup and the soot diagnostics can be found in Ref. [15]. Soot volume fraction measurements were obtained at pressures of 5, 10, 20, and 40 atm. The lower limit was set by the SSE technique, due to the low radiation emission intensities at pressures below 5 atm. The upper threshold was due to flame instabilities at pressures above 40 atm. For each pressure, measurements were obtained at height increments of 0.5 mm from the base to the tip of the flame at horizontal increments of 0.050 mm. A fuel flow rate of 0.55 mg/s and an air flow rate of 0.4 g/s were maintained at all pressures. 2.2. Numerical model The steady-state governing equations of mass, momentum, energy, and species in axisymmetric cylindrical coordinates and in the low-Mach-number limit given in Refs. [23,24] were solved. The effect of buoyancy was accounted for by retaining the gravity term in the momentum equation in the flow direction (z, vertically upward). The method of correction diffusion velocity described in Ref. [25] was employed to ensure that the net diffusion flux of all species sums to zero in both the r and z directions. Note that the correction velocity accounts for the thermophoretic velocity of soot. The interaction between the soot chemistry and the gas-phase chemistry was accounted for through the reaction rates of the species related

VT,x = −0.65

μ ∂T , ρT ∂x

x = r, z.

(3)

The source term Sm in Eq. (1) accounts for the effects of soot nucleation, surface growth, and oxidation. In these simplified soot nucleation and growth mechanisms, it is assumed that acetylene is the only soot nucleation and growth species and soot nucleation and surface growth proceed respectively via C2 H2 → 2C(S) + H2 and C2 H2 +nC(S) → (n+2) × C(S) + H2 . The rates of nucleation and surface growth are given as R1 = k1 (T )[C2 H2 ] (kmol/m3 /s) 3 and R2 = k2 (T )A0.5 s [C2 H2 ] (kmol/m /s), where −2/3 2/3

As = π(6/π )2/3 ρC(S) Ys ρN 1/3 is the soot surface area per unit volume and [C2 H2 ] is the mole concentration of acetylene (this surface growth model hereafter is called Model I). The nucleation and growth rate constants used in the present calculations were taken from our previous study [20]: k1 = 1000 exp(−16,103/T ) [1/s] and k2 = 1750 × exp(−10,064/T ) [m0.5 /s]. The density of soot, ρC(S) , is taken to be 1.9 g/cm3 . It is noted that the soot surface growth rate is assumed here to be proportional to the square root of the soot surface area, based on the recommendation of Leung et al. [26]. The more frequently employed assumption in the literature, e.g., [18], is that the surface growth rate is proportional to As . It is worth pointing out that similar numerical results could be obtained under both assumptions

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R5 = ϕO k5 (T )T −1/2 XO Pa ,

(7)

by adjusting the constants in the surface growth rates for calculations under a given ambient pressure. To investigate how the pressure dependence of soot yield is affected by the different assumptions of the R2 ∼ As relation, calculations based on the linear relationship between R2 and As were also conducted, assuming that R2 = 0.1k2 (T )As [C2 H2 ] (hereafter Model II). Although the square-root dependence of the soot surface growth rate on the soot surface area in Model I appears somewhat counterintuitive, the sublinear dependence of R2 on As has actually been extensively discussed in the literature and is linked to the socalled soot surface aging phenomenon [27,28] and active surface-site deactivation [29]. Soot surface deactivation in diffusion flames is likely to be related to the graphitization of carbon atoms in soot particles as they grow and are transported downstream in the flame. Graphitization refers to the process of rearrangement of carbon atoms from a disordered structure to an ordered (onionlike) structure. Since surface growth is believed to take place at active sites, such as defects or edges, graphitization reduces the number of active sites on the soot particle surface, leading to reduced dependence of surface growth rate on soot particle surface. Another related phenomenon in the soot surface growth process is soot aggregation, which forms fractal structured soot aggregates. Aggregated soot particles reduce the total accessible surface area. These two phenomena form the physical interpretation for the square root dependence of the soot surface growth rate on soot surface area suggested by Leung et al. [26]. The source term in the soot mass fraction equation can be written as

where XOH and XO denote the mole fractions of OH and O, Pa is the ambient pressure in atm, and φOH and φO are the collision efficiencies for OH and O attacking on soot particles. The rate of soot oxidation by O2 was based on the Nagle–Strickland-Constable model [30] with rate constants R3 and R4 taken from Ref. [31]. A constant collision efficiency of 0.2 was assumed for both OH and O. The rate constants R5 were taken from Ref. [32]. As such, the soot oxidation rate by O2 depends on pressure in a nonlinear way, while rates of soot oxidation by OH and O are proportional to pressure. It is noted from Eq. (4) that soot oxidation by all the three species considered, i.e., O2 , OH, and O, dissimilarly to the soot surface growth process in Model I, was assumed to be proportional to the soot surface area per unit volume. Although this assumption has been the common practice in the literature, there is no reason that the concept of active sites discussed above, which is the rationale for the soot surface growth Model I, should not be implemented in the soot oxidation process. The dependence of the rate of soot oxidation by O2 on the soot nanostructure has recently been experimentally demonstrated by Vander Wal and Thomasek [33]. Due to lack of better information, however, the rates of soot oxidation by O2 , OH, and O were assumed to be dependent on the soot surface area per unit volume in the present study. Clearly, further experimental and theoretical studies are required to develop improved soot oxidation rate expressions accounting for soot surface deactivation. The source term SN in Eq. (2) represents the production of the number density of soot particles due to nucleation,

Sm = 2MC(S) (R1 + R2 ) − (2R3 + R4 + R5 )As , (4)

SN =

where MC(S) is the molar weight of soot (assumed to be the same as carbon) and R3 , R4 , and R5 are the specific soot oxidation rates by O2 , OH, and O, respectively. Soot oxidation was assumed to proceed through the following reactions:

where NA is Avogadro’s constant (6.022 × 1026 kmol−1 ) and Cmin is the number of carbon atoms in the incipient soot particle (700, which gives a soot incipient particle diameter of about 2.4 nm). It is noted that the destruction of number density by agglomeration is neglected for the reasons discussed in [20]. The density of the mixture (including soot) was calculated using the ideal gas state equation

O2 + 0.5C(S) → CO, OH + C(S) → CO + H, O + C(S) → CO.

ρ=

The reaction rates per unit surface area of these three reactions (kg m−2 s−1 ) are given as kA XO2 Pa χ R3 = 120 + kB XO2 Pa (1 − χ ) , 1 + kZ XO2 Pa −1 kT χ = 1+ (5) , kB XO2 Pa R4 = ϕOH k4 (T )T −1/2 XOH Pa ,

(6)

2 NA R1 , Cmin

Ru T

Pa KK

k=1 Yk /Wk

(8)

,

(9)

where Ru is the universal gas constant and KK the number of gas-phase species. The source term due to thermal radiation in the energy equation was obtained using the discreteordinates method in axisymmetric cylindrical geometry, described in [21]. The statistical narrow-band correlated-K (SNBCK) based nine-band model recently developed by Liu and Smallwood [34] was

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employed to obtain the absorption coefficients of the combustion products containing CO, CO2 , H2 O, and soot at each band and each quadrature point. Gas-phase combustion chemistry was modeled using the GRI-Mech 3.0 mechanism [35], which was optimized for methane combustion, with the removal of species and reactions related to NOx formation (except N2 ). This simplified GRI-Mech 3.0 mechanism contains 36 species and 219 reactions. Thermal and transport properties of species and the mixture were obtained from CHEMKIN subroutines and the GRIMech 3.0 database. The transport equations for mass, momentum, energy, gas-phase species, soot mass fraction, soot number density, and radiation intensity are closed with the equation of state and appropriate boundary conditions on each side of the computational domain. These equations were discretized using the control volume method [36]. Diffusion and convective terms in the conservation equations are discretized respectively using the central and upwind difference scheme. The SIMPLE numerical algorithm [36] was used to treat the pressure and velocity coupling. Governing equations of momentum, energy, soot mass fraction, and number density were solved using the tridiagonalmatrix algorithm (TDMA). Governing equations of the gas-phase species were solved in a fully coupled fashion at every computational control volume using a direct solver to speed up the convergence process [37]. Numerical calculations of the axisymmetric coflow laminar CH4 /air diffusion flame stabilized on the burner of central fuel pipe of i.d. 3.06 mm and the i.d. 2.54 cm coannular air pipe shown in Fig. 2 were conducted on a domain of 2.543 cm (z) × 2.458 cm (r). The mass flow rates specified for the fuel and air streams were consistent with those of the experiments. Both fuel and air inlet temperatures were 300 K. Nonuniform grids were used in both the r and z directions to provide greater resolution in the large-gradient regions without an excessive increase in the computing time. Very fine and uniform grids were placed within the burner tip (1.53 mm) in the radial direction with a grid size of 0.03825 mm. Outside the burner tip in the r direction, the grid size became gradually coarser. In the flow direction (z), very fine and nonuniform grids were used in the burner exit region up to 10 mm (grid size less than 0.075 mm). Further downstream, uniform but coarser grids were used. The computational domain was divided into 301 (z) × 95 (r) grid lines. A uniform velocity profile was assigned to both the fuel and the air inlets. Along the centerline (r = 0 cm) and the outer boundary (r = 2.458 cm), v = 0 cm/s and zero gradient for all other variables are assumed. At the top boundary (z = 2.543 cm), a zero-gradient condition

461

was applied to all variables. The convergence criterion used in the calculations was that relative changes of the peak soot volume fraction and the peak temperature were to be less than 1 × 10−5 . Numerical calculations were carried out on a 3.2-GHz PC. Each run took about 1 week to achieve convergence.

3. Theoretical analysis of the pressure dependence of soot In the following simplified analysis, it is assumed that the distributions of temperature and mass fraction of C2 H2 along a given streamline are independent of pressure. As such, the soot particle number density per unit mass N can be easily shown to be independent of pressure as well along a given streamline. Since surface growth is the dominant process for producing soot mass, it can be shown from Eqs. (1) and (4) and under the assumption of Model I, i.e., R2 ∝ A0.5 s , that the soot mass fraction along a streamline can be written as DYs 1/3 1/2 = F (YC2 H2 , N, T )Ys Pa , Dt

(10)

where F (YC2 H2 , N, T ) is a pressure-independent function of acetylene mass fraction YC2 H2 , soot particle number density N , and temperature only. Integra3/4 tion of Eq. (10) along a streamline leads to Ys ∝ Pa . It can then be shown that the soot volume fraction increases with pressure as Pa1.75 , based on its definition, fv = ρYs /ρC(S) . Under the assumption of Model II, i.e., R2 ∝ As , the soot mass fraction along a streamline can be written as DYs 2/3 = F (YC2 H2 , N, T )Ys Pa . Dt

(11)

Integration of Eq. (11) results in Ys ∝ Pa3 . Thus the soot volume fraction is expected to be proportional to Pa4 . This simplified theoretical analysis indicates that the assumption of linear dependence of the soot surface growth rate on the soot surface area (Model II) leads to much greater pressure dependence of soot yield than the assumption of square root dependence (Model I). Clearly, these theoretical results are approximately valid only in regions not too far away from the burner exit surface. At higher locations, however, the assumptions made in this analysis, such as pressure-independent C2 H2 mass fraction and temperature distributions along a streamline and negligible soot oxidation, become invalid due to different degrees of C2 H2 depletion at different pressures, different degrees of temperature drop through thermal radiation heat loss, and increasing soot oxidation.

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4. Results and discussion 4.1. Visible flame shape and soot volume fraction Photographs of the visible flame shape of the flame taken at a series of pressures from 1 to 80 atm were presented in [15]. Note that those photographs were taken under the condition of a higher fuel flow rate of 0.66 mg/s (20% higher than that specified in the present study). The visible flame height increases somewhat significantly when the pressure is increased from 1 to 5 atm. Between 5 and 20 atm, the visible flame height holds almost constant at about 10 mm. At higher pressures, the experimentally observed visible flame height starts to drop roughly linearly with pressure. In addition, the flame becomes progressively narrower with increasing pressure. The calculated (using soot surface growth Model I) and experimentally observed visible flame heights and visible flame radius at several heights between z = 3 and 7 mm are compared in Fig. 3. In the analysis of the numerical results, the visible flame radius and height are defined as the radial and axial locations where the soot volume fractions drop to 0.05 ppm. Choice of smaller values of soot volume fraction in defining the visible flame shape has a negligible impact on the results, since the decrease of the soot volume fraction near the edge of the visible flame shape is very rapid. The calculated visible flame height is roughly about 1 mm taller than the experimental one between 5 and 20 atm. However, the model predicts a slight increase in the flame height between 5 and

Fig. 3. Comparison of the experimentally observed and numerically calculated (surface growth Model I) visible flame radii at several heights and the visible flame heights. Filled symbols with dotted line: experiment. Open symbols with solid line: model.

20 atm, while the experiment shows a slight decrease. At higher pressures, the predicted flame height starts to drop only slightly, consistent with the experimental observation, but the model-predicted magnitude of the drop is much smaller than the experimental drop. The discrepancy in the flame height between the model prediction and experiment can be primarily attributed to the following two factors. First, it is difficult to assign the velocity profiles of the fuel and air streams at the burner exit surface in the present numerical simulation. Second, heat conduction from the flame base to the burner tip was not taken into account in the present numerical modeling. The second factor becomes increasingly pronounced as the pressure increases. As the flame base moves closer to or even slightly inside the burner fuel pipe, the fuel and the air in the vicinity of the fuel pipe are preheated. It is believed that the neglect of this preheat effect is primarily responsible for the large discrepancy between the calculated visible flame height and the observed one at high pressures. The predicted flame radius is also somewhat larger than the measured one, especially at higher locations, which is associated with the greater flame height predicted by the model. Besides the neglect of the preheat effect, the discrepancy between the calculated and experimentally observed visible flame shape (radius and height) can also be attributed to the uncertainty in the soot oxidation model used in the present calculations, especially at elevated pressures. In general, the model qualitatively captures the variation of the flame radius with pressure. Both the experimental data and the numerical results indicate that the visible flame radius decreases with pressure roughly as r ∝ Pa−0.5 , consistent with the recent experimental finding of McCrain and Roberts [14]. In addition, the flame radius defined in terms of the CH concentration at different pressures was also analyzed from the numerical results. It was found that the flame radius based on the CH radical is only about 10% larger than the visible flame radius shown in Fig. 3 and also varies with pressure as r ∝ Pa−0.5 . These results demonstrate that the pressure dependence of the visible flame radius, obtained either experimentally or numerically, can be used to represent the pressure dependence of the flame shape. The predicted distributions of soot volume fraction at various pressures using Model I and II are respectively shown in Figs. 4 and 5. Experimentally measured soot volume fraction distributions using LOSA are displayed in Fig. 6. Although the soot volume fractions predicted by Model I are significantly lower than the experimental data, the calculated soot volume fraction distributions qualitatively reproduce the overall experimental observations (compare Figs. 4 and 6). On the other hand, although the peak soot volume fractions predicted by Model II are comparable

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Fig. 4. Distributions of soot volume fractions calculated numerically using Model I at 5, 20, and 40 atm.

Fig. 5. Distributions of soot volume fractions calculated numerically using Model II at 5, 20, and 40 atm.

to those of the experiment, Model II fails to predict the soot volume fraction distributions in the centerline region qualitatively to such a degree that the flame tip is open rather than closed at pressures of 20 atm and higher, Fig. 5. The narrowing of the flames with increasing pressure and the dramatic increase of the soot volume fraction are evident from both numerical results and the experiment. Flame narrowing with increasing pressure was also shown experimentally by Miller and Maahs [10], Flower and Bowman [12], and McCrain and Roberts [14] and predicted theoretically by Glassman [17]. Experimentally observed

and numerically calculated visible flame radii at several heights, Fig. 3, indicate that the flame radius decreases with pressure as r ∝ Pa−0.5 , which implies that the cross-sectional area of the flame is inversely proportional to pressure. While this conclusion is consistent with the experimental observations of McCrain and Roberts [14], it is inconsistent with the theoretical analysis of Glassman [17]. In fact, the variation of the visible flame radius with pressure as r ∝ Pa−0.5 is expected based on the mass flux conservation within fixed streamlines; i.e., m ˙ = ρ uA. ¯ In laminar flames where the flow is buoyancy-controlled, the mean ax-

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Fig. 6. Distributions of soot volume fractions measured experimentally using LOSA at 5, 20, and 40 atm.

Fig. 7. Variation of the peak soot volume fraction with pressure.

ial velocity u¯ quickly becomes almost independent of pressure at a very short distance above the burner exit (see Fig. 8 below). As the mixture density ρ is proportional to Pa , the cross-sectional area A must be inversely proportional to Pa or r ∝ Pa−0.5 . It is noted that starting from the assumption that the ax−1/2 , rather ial velocity varies with pressure as Pa than being pressure-independent, Glassman [17] predicted that the cross-sectional area of the flame de−1/2 , instead of Pa−1 as found in this creases as Pa study. Variation of the calculated and measured peak soot volume fractions with pressure is shown in Fig. 7. The measured peak soot volume fractions exhibit a fv,max ∝ Pa2 relation between 5 and 20 atm. Model I predicts that fv,max ∝ Pa1.86 between 5 and 20 atm, which is in reasonably good agreement with experi-

ment. However, Model II predicts a much faster increase of fv with pressure with fv,max ∝ Pa3.9 between 5 and 10 atm and a slower rise above 10 atm. Clearly, the sensitivity of peak soot volume fraction to pressure diminishes above 20 atm in both the experiment and the numerical results (in both soot models), with the trend from the results of Model I in better agreement with the experimental data than that of Model II. Note that the pressure dependences of the peak soot volume fraction on pressure based on the numerical results are in reasonable agreement with the simplified theoretical analysis discussed above, i.e., Pa1.75 for Model I and Pa4 for Model II. It is important to recall that the predicted or measured soot volume fractions are the net effect of soot production (inception and surface growth) and soot oxidation. Consequently, it is of interest to quantity the effect of soot oxidation on the calculated soot volume fractions. An attempt was made for this purpose for pressures of Pa = 5, 10, and 20 atm and using Model I. In such calculations, the soot oxidation terms were removed by setting terms R3 , R4 , and R5 to zero in Eq. (4) and the soot transport equations, Eqs. (1) and (2), and all other transport equations were solved in a fully coupled way. This is necessary to adequately evaluate the effect of soot oxidation, since the increase in the calculated soot volume fraction, as a result of removing soot oxidation, in turn affects the flame temperature through enhanced thermal radiation loss and the concentration of acetylene through enhanced consumption. In the absence of soot oxidation the flame at all the pressures considered becomes smoking, as expected. Compared to the results with soot oxidation shown in Fig. 4, the peak soot volume

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Fig. 8. Distributions of the predicted axial velocity and temperature along the flame centerline at pressures between 5 and 40 atm.

fraction without soot oxidation increases significantly by a factor of 4.2 at 5 atm, a factor of 3.6 at 10 atm, and a factor of 2.8 at 20 atm. The decreased effect of soot oxidation on the peak soot volume fraction with increasing pressure is anticipated, due to significant decrease in the concentration of C2 H2 as more soot is produced. These results reveal that soot oxidation plays a very important role in the predicted soot volume fraction, both qualitatively and quantitatively. The peak soot volume fractions calculated without soot oxidation are also plotted in Fig. 7 for comparison. In the absence of soot oxidation, Model I predicts fv,max ∝ Pa1.84 between 5 and 10 atm and fv,max ∝ Pa1.57 between 10 and 20 atm. The pressure dependence of the peak soot volume fraction with or without soot oxidation is very similar, i.e., fv,max ∝ Pa1.86 vs fv,max ∝ Pa1.84 , implying that soot oxidation does not alter the pressure dependence of the soot volume fraction at relatively low pressures between 5 and 10 atm. 4.2. Temperature and residence time The calculated distributions of temperature and axial velocity along the flame centerline between 5 and 40 atm are shown in Fig. 8. In the near burner exit region, up to about 3 mm, the temperature increases with increasing pressure. This is caused by the enhanced air entrainment into the fuel stream as pressure increases, leading to partial premixing and accelerated fuel pyrolysis. Further downstream, in the centerline region, the flame temperature decreases with increasing pressure, which is attributed to increased thermal radiation heat loss through increased soot volume fraction. Although the temperature distribution along the flame centerline does exhibit some dependence on pressure, the variation seen is not dramatic. Therefore, the assumption made in the simplified theoretical analysis that the temperature along a given streamline is independent of pressure (Section 3) is

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Fig. 9. Variation of the flame-diameter-integrated soot volume fraction with height at 5, 10, and 20 atm.

Fig. 10. Variation of the cross-sectional-area-integrated soot volume fraction with height at 5, 10, and 20 atm.

considered reasonable. As a consequence of maintaining the fuel mass flow rate, the inlet fuel stream velocity decreases with increasing pressure. However, the centerline axial velocity distribution shows negligibly small dependence on pressure. This is expected since the flow is buoyancy-controlled and the effect of burner exit velocity quickly vanishes. An examination of the numerical results confirmed that the axial velocity exhibits small variation (i.e., <5%) in the radial direction within the annular sooting region. Therefore, the nearly pressure-independent axial and radial velocity distribution within the sooting region implies that soot particles experience almost the same residence time at different pressures. 4.3. Integrated soot volume fraction The calculated flame-diameter and cross-sectional-area-integrated soot volume fractions are compared with the experimental data at 5, 10, and 20 atm in Figs. 9 and 10. It is noted first that the experimental data obtained from two different techniques (LOSA and SSE) agree fairly well. Although there are large quantitative discrepancies between the model predic-

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Fig. 11. Variations of the calculated and measured peakpath-integrated and area-integrated soot volume fraction with pressure between 5 and 20 atm.

tion and the experimental data, it is evident ∞ that Model fv dr and II predicts a much faster increase of −∞ ∞ 0 fv 4π r dr with pressure than Model I and the experimental data. Results of Model I are consistently lower than the experimental data. However, results of Model II are first lower than the experimental data at 5 atm and then increase rapidly to exceed the experimental data at 10 atm and beyond. Both the numerical results and the experimental data indicate that the peak flame-diameter-integrated soot volume fraction occurs at a lower height with increasing pressure. These observations made in the flame-diameterintegrated soot volume fraction, Fig. 9, also essentially hold for the flame area integrated soot volume fraction, Fig. 10. ∞ fv dr and Variations of the peak values of −∞ ∞ 0 fv 4π r dr with pressure are compared in Fig. 11. Both the experimental and the numerical results of Model I indicate that the peak flame-diameterintegrated soot volume fraction increases with pressure roughly as Pa1.3 , Fig. 11a, for pressures between 5 and 20 atm. Such pressure dependence is consistent with that experimentally observed by Flower and Bowman [12] (1 to 10 atm), Lee and Na [13] (1 to 4 atm). However, the more recent experimental study of McCrain and Roberts [14] reported a different pressure dependence for the laminar methane/air flame (Pa1.0 between 1 and 25 atm) and for the ethylene/air flame (Pa1.2 between 1 and 16 atm). On the other hand, Model II predicts a much faster rise of the flame-diameter-integrated soot volume fraction with pressure than Model I and the experiment

with Pa3.2 between 5 and 10 atm and Pa1.14 between 10 and 20 atm. Therefore, it can be seen that Model I correctly predicts the pressure dependence of the path-integrated soot volume fraction, though the calculated values are significantly lower than the experimental data. Also plotted in Fig. 11 are the results calculated using Model I without soot oxidation. Although the peak integrated soot volume fractions are much higher in the absence of soot oxidation, their slopes in Fig. 11 differ only slightly from those with soot oxidation at relatively low pressures between 5 and 10 atm. These results suggest once again that soot oxidation does not significantly alter the pressure dependence of soot volume fraction. The numerical ∞(from Model1.3I) and experimental refv dr ∝ Pa relation for pressures sults of the −∞ between 5 and 20 atm is consistent with the simplified theoretical analysis, which predicts that the pathintegrated soot volume fraction scales with pressure ∞ fv dr ∝ as Pa1.75 (fv ) × Pa−0.5 (r) = Pa1.25 . The −∞ Pa3.2 relation from the numerical results of Model II between 5 and 10 atm is also consistent with the theoretical analysis, which shows that the path-integrated soot volume fraction should scale with pressure as Pa4.0 (fv ) × Pa−0.5 (r) = Pa3.5 . Both the results of Model I and the experiment show that the area-integrated soot volume fraction increases linearly with pressure between 5 and 20 atm, Fig. 11b. Without soot oxidation, Model I also predicts a nearly linear pressure dependence of the peak area integrated soot volume fraction between 5 and 10 atm. Such pressure dependence is somewhat different from what is expected theoretically. Based on the theoretical results of fv ∝ Pa1.75 and r ∝ Pa−0.5 , it is anticipated that the area-integrated soot volume fraction scales with pressure as Pa0.75 . It is seen from Fig. 11b that Model II significantly overpredicts the pressure dependence of the area-integrated soot volume fraction with Pa3.16 between 5 and 10 atm. This numerical pressure dependence is quite close to the theoretical prediction of Pa3.0 . Overall, comparison between the numerical results and the experimental data indicates that the assumption of the square root dependence of the soot surface growth rate on the soot surface area is much better than the assumption of the linear dependence. The possible explanations for the sublinear dependence of the soot surface growth rate on the soot surface area are soot surface deactivation and soot particle agglomeration, discussed in Section 2.2. In the early stage of soot particle formation low in the flame, soot particles appear as unaggregated, individual spherical primary particles and the surface of soot particles is highly reactive. Under such conditions, it appears more reasonable to assume the linear rela-

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tionship between the soot surface growth rate and the total soot surface area. Further downstream, primary soot particles form mass fractal aggregates of different sizes and the surface of soot particles becomes less reactive due to increased degree of graphitization. As such, the surface growth rate of soot particles depends sublinearly on their surface area. Both the model and experiment indicate a slower increase in the integrated soot volume fractions beyond 20 atm (not shown here). For example, in the ∞ fv dr ∝ Pan , pressure range of 20 to 40 atm, −∞ where n = 0.22 for the model results (Model I) and n = 0.75 for the experimental results. Although the model predicts a relatively small increase when the pressure is increased from 20 to 40 atm, the experiment shows an almost 90% increase. It should be noted that the experimental results include not only contribution from soot, but also contribution from PAH, which near the nozzle tip becomes more significant as pressure increases [12]. However, it is unlikely that the different pressure-dependent behavior of the model and the experiment between 20 and 40 atm can be attributed solely to the contribution of PAH in the experimental data, since the PAH concentration decays quite rapidly [12]. One possible explanation for the model predicting a much weaker pressure dependence than the experiment at very high pressures (above 20 atm) is the neglect of the burner tip preheat effect. With increasing pressure, significant chemical reactions occur in the immediate region above the burner exit (and perhaps even inside the fuel tube). The lowering of the reaction zone leads to enhanced heat transfer to the fuel tube, which in return preheats the fuel and prompts its pyrolysis, leading to a larger soot production. This phenomenon was not taken into account in the present model. The departure of the numerical pressure dependence of soot yield with the theoretical prediction at pressures beyond 20 atm in Model I and 10 atm in Model II is actually anticipated. First, the assumption of pressure-independent distribution of the mass fraction of C2 H2 along a streamline gradually breaks down with increasing pressure. This is because, as more soot is formed, more acetylene is consumed. Second, the assumption of pressure-independent temperature distribution along a streamline also gradually breaks down as the pressure increases due to the enhanced partial premixing in the immediate burner exit region and increased radiation heat loss further downstream; see Fig. 8. 4.4. Effect of pressure on gas-phase chemistry It is well known that increased pressure has a significant influence on the gas-phase chemistry through enhancing the reaction rate of reactions that are

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weakly dependent on or even independent of temperature and through pressure-dependent third-body reactions. To gain a quantitative understanding of the effect of pressure on the gas-phase chemistry in the methane/air coflow investigated in this study, further numerical calculations were conducted without soot formation (by simply setting the nucleation rate to zero). Note that radiation heat loss from gases species (CO, CO2 , and H2 O) was included in these calculations. Some representative results of interest to the present context are shown in Figs. 12–14. The striking effect of pressure on the shape of the flame is evident. The peak flame temperature first increases slightly when the ambient pressure is increased from 5 to 10 atm then holds almost constant between 10 and 20 atm; see Fig. 12. The peak flame temperature drops significantly when the pressure is further increased beyond 20 atm. Since the adiabatic flame temperature of methane/air mixtures has a very weak dependence on pressure, the effect of pressure on the flame temperature shown in Fig. 12 is likely to be the result of the complex interactions among thermal radiation, gas-phase chemistry, and species diffusion. The effect of pressure on the peak mass fraction of C2 H2 shown in Fig. 13 exhibits a trend similar to that for the peak temperature in Fig. 12. The results shown in Figs. 12 and 13 support the assumptions made in the simplified theoretical analysis in Section 3 at pressures below about 20 atm and also provide additional explanations for the reduced dependence of soot concentration on pressure at pressures above 20 atm, besides the C2 H2 consumption mentioned earlier. Fig. 14 shows that the mass fraction of OH radical, which is the dominant species for soot oxidation, decreases rather significantly with increasing pressure. This, however, does not necessarily mean that the relative importance of soot oxidation by OH is reduced at higher pressures. Nevertheless, results shown in Figs. 12–14 indicate that pressure has a significant influence on the gasphase chemistry. 4.5. Air entrainment and accelerated fuel pyrolysis An interesting phenomenon observed from the numerical results is that the increased pressure enhances air entrainment into the fuel stream around the burner tip. This phenomenon is clearly shown in the calculated distributions of O2 mass fraction and the velocity vector in the burner near field at three pressures, 5, 20, and 40 atm, displayed in Fig. 15. Although there is significant leakage of O2 into the flame at the flame base at the lowest pressure considered (5 atm), examination of the velocity vector near the flame base reveals a negligible radial component. With increasing pressure, however, air is strongly entrained into the flame near the burner tip as clearly indicated

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Fig. 12. Effect of pressure on the calculated temperature distribution without soot.

Fig. 13. Effect of pressure on the calculated distribution of mass fraction of C2 H2 without soot.

by the significant inward radial velocity component, Figs. 15b and 15c. With increasing pressure the enhanced air entrainment into the fuel stream and the increased mixture density give rise to accelerated fuel pyrolysis. To demonstrate this aspect of the pressure effect, the calculated distributions of CH4 mass fraction at 5, 20, and 40 atm are compared in Fig. 16. It is clearly shown that the mass fraction of CH4 in the centerline region decays more rapidly as the pressure increases. Besides its impact on the rate of fuel pyrolysis, the enhanced air entrainment also plays an important role in oxidizing the soot particle in the lower portion of the flame. The enhanced air entrainment

into the fuel stream low in the flame with increasing pressure suggests that it is questionable to specify pure fuel at room temperature at the fuel pipe exit surface.

5. Conclusions A combined numerical and experimental study of axisymmetric coflow laminar methane–air diffusion flames at pressures between 5 and 40 atm was conducted. The visible flame diameter decreases with pressure as Pa−0.5 . The visible flame height

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Fig. 14. Effect of pressure on the calculated distribution of mass fraction of OH without soot.

Fig. 15. Distributions of the calculated O2 mass fraction and velocity vector in the burner near field at 5, 20, and 40 atm.

maintains a near-constant value up to about 20 atm. The numerical model successfully reproduces experimental observations/data in terms of the effect of pressure on the visible flame shape (diameter and height) and the pressure dependence of the soot volume fraction and the integrated soot volume fraction across the flame diameter at pressures up to about 20 atm. With increasing pressure, the flame temperature in the region immediately above the burner exit along the flame centerline increases due to enhanced air entrainment into the fuel stream, leading to ac-

celerated fuel pyrolysis. Further downstream, flame temperature decreases with increasing pressure due to increased radiation heat transfer associated with larger amount of soot produced. The effects of pressure on the flame structure and soot formation can be largely explained in terms of its direct impact on the mixture density. The enhanced air entrainment into the fuel stream with increasing pressure also plays an important role in affecting the flame structure and soot formation and oxidation. In the pressure range of 20 to 40 atm both the model and

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Fig. 16. Distributions of the calculated CH4 mass fraction at 5, 20, and 40 atm.

experiment show a diminishing sensitivity of sooting propensity to pressure, with the experiments showing a higher sensitivity than the numerical results. The agreement between the numerical results of Model I and experimental data supports the model assumption that the soot surface growth rate depends on the square root of the soot particle surface area. This study demonstrated that variation of the pressure is an effective way to probe the relationship between the soot surface growth rate and soot surface area. The enhanced air entrainment into the fuel stream around the burner rim at high pressures implies that the numerical inlet boundary should be placed some distance below the burner exit surface to accurately specify the inlet conditions and account for the fuel preheat effect in the simulation. Further study on the effects of pressure on soot formation and gas-phase chemistry in laminar diffusion flames should include PAH chemistry and more detailed mechanisms for soot nucleation and surface growth.

References [1] I. Glassman, P. Yaccarino, Proc. Combust. Inst. 18 (1981) 1175–1183. [2] J.H. Kent, H.G.G. Wagner, Proc. Combust. Inst. 20 (1984) 1007–1015. [3] R.J. Santoro, T.T. Yeh, J.J. Horvath, H.G. Semerjian, Combust. Sci. Technol. 53 (1987) 89–115. [4] C.M. Megaridis, R.A. Dobbins, Combust. Sci. Technol. 66 (1989) 1–16. [5] I.M. Kennedy, C. Yam, I.M. Rapp, R.J. Santoro, Combust. Flame 107 (1996) 368–382.

[6] M.D. Smooke, C.S. McEnally, L.D. Pfefferle, R.J. Hall, M.B. Colket, Combust. Flame 117 (1999) 117–139. [7] C.S. McEnally, A.M. Schaffer, M.B. Long, L.D. Pfefferle, M.D. Smooke, M.B. Colket, R.J. Hall, Proc. Combust. Inst. 27 (1998) 1497–1505. [8] A. D’Anna, A. D’Alessio, J. Kent, Combust. Flame 125 (2001) 1196–1206. [9] R.L. Schalla, G.E. McDonald, Proc. Combust. Inst. 5 (1955) 316–324. [10] I.M. Miller, H.G. Maahs, High-Pressure Flame System for Pollution Studies with Results for Methane–Air Diffusion Flames. NASA TN D-8407, 1977. [11] W.L. Flower, C.T. Bowman, Proc. Combust. Inst. 20 (1984) 1035–1044. [12] W.L. Flower, C.T. Bowman, Proc. Combust. Inst. 21 (1986) 1115–1124. [13] W. Lee, Y.D. Na, JSME Int. J. Ser. B 43 (4) (2000) 550– 555. [14] L.L. McCrain, W.L. Roberts, Combust. Flame 140 (2005) 60–69. [15] K.A. Thomson, Ö.L. Gülder, E.J. Weckman, R.A. Fraser, G.J. Smallwood, D.R. Snelling, Combust. Flame 140 (2005) 222–232. [16] H. Matzing, H.G. Wagner, Proc. Combust. Inst. 21 (1986) 1047–1055. [17] I. Glassman, Proc. Combust. Inst. 27 (1998) 1589– 1596. [18] Z. Zhang, O.A. Ezekoye, Combust. Sci. Technol. 137 (1998) 323–346. [19] M. Frenklach, H. Wang, Proc. Combust. Inst. 23 (1990) 1559–1566. [20] F. Liu, H. Guo, G.J. Smallwood, Ö.L. Gülder, J. Quant. Spectrosc. Radiat. Transfer 73 (2002) 409–421. [21] F. Liu, H. Guo, G.J. Smallwood, Combust. Flame 138 (2004) 136–154.

F. Liu et al. / Combustion and Flame 146 (2006) 456–471 [22] D.R. Snelling, K.A. Thomson, G.J. Smallwood, Ö.L. Gülder, E.J. Weckman, R.A. Fraser, AIAA J. 40 (9) (2002) 1789–1795. [23] K.K. Kuo, Principles of Combustion, Wiley, New York, 1986, pp. 172–205. [24] H. Guo, F. Liu, G.J. Smallwood, Ö.L. Gülder, Int. J. Comput. Fluid Dynam. 18 (2) (2004) 139–151. [25] R.J. Kee, J.F. Grcar, M.D. Smooke, J.A. Miller, A FORTRAN Program for Modeling Steady Laminar One-Dimensional Premixed Flames. SANDIA Report, SAND85-8240, Reprint, 1994. [26] K.M. Leung, R.P. Lindstedt, W.P. Jones, Combust. Flame 87 (1991) 289–305. [27] S.J. Harris, Combust. Sci. Technol. 72 (1990) 67–77. [28] M. Frenklach, Phys. Chem. Chem. Phys. 4 (2002) 2028–2037. [29] M.D. Smooke, M.B. Long, B.C. Connelly, M.B. Colket, R.J. Hall, Combust. Flame 143 (2005) 613– 628.

471

[30] J. Nagle, R.F. Strickland-Constable, Oxidation of carbon between 1000 and 2000 ◦ C, in: Proceedings of the Fifth Conference on Carbon, 1962, pp. 154–164. [31] J.B. Moss, C.D. Stewart, K.J. Young, Combust. Flame 101 (1995) 491–500. [32] D. Bradley, G. Dixon-Lewis, S.E. Habik, E.M.J. Mushi, Proc. Combust. Inst. 20 (1984) 931–940. [33] R.L. Vander Wal, A.J. Thomasek, Combust. Flame 134 (2003) 1–9. [34] F. Liu, G.J. Smallwood, J. Quant. Spectrosc. Radiat. Transfer 84 (2004) 465–475. [35] G.P. Smith, D.M. Golden, M. Frenklach, N.W. Moriarty, B. Eiteneer, M. Goldenberg, C.T. Bowman, R.K. Hanson, S. Song, W.C. Gardiner Jr., V.V. Lissianski, Z. Qin, available at http://www.me.berkeley.edu/ gri_mech/. [36] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980. [37] Z. Liu, C. Liao, C. Liu, S. McCormick, Multigrid Method for Multistep Finite Rate Combustion, AIAA Paper 95-0205, 1995.