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NUMERICAL MODELING OF A LAMINAR
AXISYMMETRIC COFLOW METHANE/AIR FLAME
AT PRESSURES BETWEEN 5 AND 20 ATM
F. LIU, K. A. THOMSON
†, H. GUO, and G. J. SMALLWOOD
Combustion Technology Group, Institute for Chemical process & Environmental Technology National Research Council, Building M9, 1200 Montreal Road, Ottawa, ON, Canada †Mechanical Engineering Department, University of Waterloo, Waterloo, ON, Canada
ABSTRACT
In this study, numerical calculations were conducted in a laminar axisymmetric coflow methane/air diffusion flame at pressures between 5 and 20 atm using detailed chemistry, thermal and transport properties, and radiation transfer model. Soot was modeled using a semi-empirical acetylene based model. Two surface growth sub-models were investigated and the predicted pressure dependence of soot yield was compared with available experimental data. It was found that the assumption of a square root dependence of the soot surface growth rate on 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 the linear dependence of the soot surface growth rate on soot surface area predicts a much faster increase in the soot yield with pressure than that observed experimentally.
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. To date, only several experimental studies have been carried out at elevated or high pressure with global soot properties (line-of-sight integrated soot volume fraction) measured [1-5]. These studies showed a significant increase in soot production with increasing pressure. More recent studies by Flower and Bowman [4] (1 to 10 atm) and Lee and Na [5] (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 the flame, ∞ f drv
−∞
∫
(fv is the soot volume fraction and r the radius), increases with pressure raised to a power of about 1.2 to 1.3. According to Glassman [6], such a pressure dependence of soot formation holds regardless of the type of fuel.Compared to the experimental studies of soot formation in laminar coflow diffusion flames at elevated or high pressure, even fewer numerical studies in this field have been reported in the literature. The paper of Zhang and Ezekoye [7] is perhaps the only such study. In their numerical study of soot formation in laminar methane/air jet diffusion flame at 1 and 4 atm using an 8-step reduced chemistry, Zhang and Ezekoye [7] suggested that the enhanced soot production at increased pressure can be simply attributed to the increased mixture density, which is proportional to pressure.
The present paper presents numerical results of the pressure dependence of some flame properties, such as visible flame shape (diameter and height), residence time, soot volume fraction, and flame diameter integrated and annular integrated soot volume fractions. A semi-empirical, global, acetylene based soot model was employed in the present study. The rationale for using such a simple soot model for the purpose of the present study is that the effect of pressure on the flame properties (including soot) is believed to be primarily a physical phenomenon rather than a chemical one. Besides, 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.
NUMERICAL MODEL
The steady-state governing equations of mass, momentum, energy, and species in axisymmetric cylindrical coordinates were solved. Low Mach number flow was assumed. The effect of buoyancy was accounted for by retaining the gravity term in the momentum equation in the flow direction (z, vertically upwards). The method of correction diffusion velocity was employed to ensure that the net diffusion flux of all species sums to zero in both 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 associated species. Only the thermal diffusion velocities of H2 and H were accounted for. Gas-phase chemistry was modeled using the GRI Mech3.0 [8] mechanism and associated database for species thermal and transport properties. The semi-empirical soot model used in this study is basically that described in [9]. In this soot model, the rates of nucleation and growth are given as R1=k T C H1( )[ 2 2] (kmol/m3/s) and 0.5 (kmol/m
2 2( ) s [ 2 2]
R =k T A C H 3/s), where
As=π(6/π)2/3ρC(S)−2/3 Ys2/3 ρN1/3 is the soot surface area per unit volume and [C2H2] is the mole concentration of acetylene. The density of soot ρC(S) is taken to be 1.9 g/cm3. Further details can be found in [9]. It is noted that the soot surface growth rate is assumed here to be proportional to the square root of the soot surface area. The frequently employed assumption in the literature is that the surface growth rate is proportional to As. It is worth pointing out that similar numerical results can be obtained under both assumptions by adjusting the constants in the surface growth rates for calculations at a given pressure. To investigate how the pressure dependence of soot yield is affected, calculations based on the linear relationship between R2 and As were also conducted assuming . The source term in the energy equation due to radiation heat transfer was also included and was calculated using the discrete-ordinates method coupled with a 9-band radiative property model [10]. The density of the mixture (including soot) was calculated using the ideal gas state equation.
2 0.1 ( )2 s[ 2 ]
R = k T A C H2
Numerical calculations of the axisymmetric coflow laminar CH4/air diffusion flame stabilized on a burner of fuel pipe with 3.06 mm i.d. and 2.54 cm i.d. co-annular air pipe were conducted on a domain of 2.543 cm (z) × 2.458 cm (r). Both fuel and air inlet temperatures were 300 K. Non-uniform 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 (at r = 1.53 mm) in the radial direction with a grid size of 0.03825 mm. Outside the burner tip in r direction, the grid size became exponentially coarse. In the flow direction (z), very fine and non-uniform 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. 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 was applied to all variables. A fuel flow rate of 0.55 mg/s and an air flow rate of 0.4 g/s were maintained at all pressures.
THEORETICAL ANALYSIS OF THE PRESSURE DEPENDENCE OF
SOOT
In the following simplified analysis, it is assumed that the distributions of temperature and the mass fraction of C2H2 along a given streamline are independent of pressure. As such, the soot particle number density per unit mass N can be shown to be independent of pressure. Assuming surface growth is the dominant process of producing soot mass, it can be shown under the assumption of R2 ∝ As0.5 that the soot mass fraction along a streamline can be written as
2 2 1/ 3 1/ 2 ( , , ) s C H s DY F Y N T Y P Dt = (1) where is a function of acetylene mass fraction , soot particle number density N, and temperature only. Integration of Eq.(1) leads to . It can then be shown that the soot volume fraction increases with pressure as P
2 2 ( C H , , F Y N T ) 2 2 C H Y 3/ 4 s Y ∝P 1.75
based on its definition fv = ρYs/ρC(S). Under the assumption of R2 ∝ As, the soot mass fraction along a streamline can be written as
2 2 2 / 3 ( , , ) s C H s DY F Y N T Y P Dt = (2) Integration of Eq.(2) results in . The soot volume fraction is then expected to be proportional to P 3 s Y ∝P 4 .
RESULTS AND DISCUSSION
To assist the discussions of the numerical results, the assumptions of R2 ∝ As0.5 and R2 ∝ As are hereafter called Model I and Model II, respectively. Distributions of soot volume fraction calculated using Model I at pressures of 5, 10, and 20 atm are shown in Fig.1. Results based on Model II are qualitatively similar to those shown in Fig.1 with somewhat taller flames. It is evident that with increasing pressure the flame becomes narrower and the soot volume fraction increases significantly. Flame narrowing was previously shown experimentally by Miller and Maahs [2] and Flower and Bowman [4] and predicted theoretically by Glassman [6]. The calculated visible flame radiuses at several heights based on Model I are plotted in Fig.2. Model II produces almost identical flame radiuses as those shown in Fig.2. These results indicate that the flame radius decreases with pressure as r∝P−0.5, which implies that the cross-sectional area of the flame is inversely proportional to pressure. This result is expected based on the mass flux conservation within fixed streamlines, i.e. m =ρuA. In laminar flames where the flow is buoyancy controlled, the mean axial velocity u becomes independent of pressure at a very short distance from the burner exit (see below). As the mixture density ρ is proportional to P, the cross-sectional area A must be inversely proportional to P. It is noted that starting from the assumption that the axial velocity varies with pressure as P-1/2, rather than pressure independent,
Glassman [6] predicted that the cross-sectional area of the flame decreases as P-1/2, instead of P-1 derived in this study.
Variation of the calculated peak soot volume fraction with pressure is shown in Fig.3. Model I predicts that
max
1.9 v
f ∝P between 5 and 20 atm, while Model II predicts a much faster increase of fv with pressure with
max
3.9 v
f ∝P between 5 and 10 atm and a slower rise between 10 and 20 atm. Note that these numerical results are in reasonable agreement with the above simple theoretical analysis, i.e. P1.75 for Model I and P4 for Model II. It is observed from the experimental data [11] that the peak soot volume fraction increases as
max
2 v
f ∝P over the pressure range of 5 to 20 atm. The calculated flame diameter and cross sectional area integrated soot volume fractions are compared with the experimental data [11] in Figs. 4 and 5. Although there are large discrepancies between the model prediction and the experimental data, it is evident that Model II predicts a much stronger increase of ∞ f drv
−∞
∫
and0 fv4πrdr ∞
∫
with pressure than Model I and the experimental data. Variations of the peak values of ∞ f drv−∞
∫
and0 fv4πrdr ∞
∫
with pressure are compared in Fig.6. Both the experimental and the numerical results of Model I indicate that the peak flame diameter integrated soot volume fraction increases with pressure roughly as Pa1.25 for pressures between 5 and 20 atm. Such pressure dependence is consistent with those experimentally observed by Flower and Bowman [4] (1 to 10 atm) and Lee and Na [5] (1 to 4 atm) in laminar annular coflow ethylene/air diffusion flames. On the other hand, Model II predicts a much faster rise of soot yield with pressure than that experimentally observed.It is notable that the oxidation of soot by O2 entrained into the fuel stream low in the flame has an influence on the calculated soot volume fraction and causes the numerical pressure dependence of the peak soot volume fraction to be somewhat different from the theoretical one. As the flame diameter varies with pressure as shown earlier, the flame diameter integrated soot volume fraction should depend on pressure as P
0.5 a r∝P−
a1.25 for Model I and Pa3.5 for Model II, which are reasonably confirmed by the numerical results. It is also expected that these theoretical predictions hold only within a certain range of pressure for the following two reasons. First, the assumption of pressure independent distribution of the mass fraction of C2H2 along a streamline gradually breaks down with increasing pressure. This is because as more soot is formed, more acetylene is consumed. Secondly, 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.
The better agreement between the pressure dependence of the calculated soot volume fraction and the integrated soot volume fractions based on Model I and the experimental data supports the assumption that the soot surface growth rate is proportional to the square root of the soot particle surface area per unit volume. The assumption of R2 ∝ As leads to a much faster soot yield increase with pressure than that experimentally observed. However, it can be seen from Figs.4 and 5 that Model II predicts a better trend of the variation of the integrated soot volume fractions with pressure than Model I low in the flame. This observation might be related to the soot
particle agglomeration process. At the early stage of soot particle formation which occurs low in the flame, soot particles appear as unaggregated, individual spherical particles. Under this condition, it is more reasonable to assume that the surface growth rate is proportional to soot particle surface area. Further downstream, soot particles form aggregates of fractal structure as a result of agglomeration. As such, the effective soot surface area is reduced due to the so-called ‘shielding effect’. It is then anticipated that the soot surface growth rate dependents sub-linearly on the soot particle surface area.
CONCLUSIONS
Numerical calculations were conducted in a axisymmetric coflow laminar CH4/air diffusion flame at pressures between 5 and 20 atm using detailed gas-phase chemistry, detailed radiation model, a simplified semi-empirical soot model with two different surface growth sub-models. The present numerical results reveal great details on the effect of pressure on the flame structure, distributions of species concentration, temperature, and velocity. The predicted soot yield is sensitive to the assumed relationship between the soot surface growth rate and the soot surface area. Comparison between the numerical results and experimental data suggests that the assumption of the soot surface growth rate is proportional to the square root of soot surface area is more reasonable than the linear relationship. It is recommended that a detailed aerosol dynamics model to be incorporated into the soot model in order to the taken into account of the effect of soot morphology on the soot surface growth rate.
REFERENCES
1. Schalla, R. L., and McDonald, G. E., (1955) Proc. Combust. Inst. 5:316-324.
2. Miller, I. M., and Maahs, H. G., (1977) High-pressure flame system for pollution studies with results for methane-air diffusion flames. NASA TN D-8407.
3. Flower, W. L., and Bowman, C. T., (1984) Proc. Combust. Inst. 20:1035-1044. 4. Flower, W. L., and Bowman, C. T., (1986) Proc. Combust. Inst. 21:1115-1124. 5. Lee, W., and Na, Y. D., (2000) JSME Int. J. Series B 43(4):550-555.
6. Glassman, I., (1998) Proc. Combust. Inst. 27:1589-1596.
7. Zhang, Z., and Ezekoye, O. A., (1998) Combust. Sci. Tech. 137:323-346.
8. Smith, G. P., Golden, D. M., Frenklach, M., Moriarty, N. W., Eiteneer, B., Goldenberg, M., Bowman, C. T., Hanson, R. K., Song, S., Gardiner Jr., W. C., Lissianski, V. V., and Qin, Z.,
http://www.me.berkeley.edu/gri_mech/.
9. Liu, F., Guo, H., Smallwood, G. J., and Gülder, Ö. L., (2002) J. Quant. Spectros. Radiative Transfer 73:409-421.
10. Liu, F., and Smallwood, G. J., (2004) J. Quant. Spectros. Radiative Transfer 84:465-475. 11. Thomson, K. A., Gülder, Ö. L., Weckman, E., Fraser, R., Smallwood, G. J., and Snelling, D. R., (2004) paper presented at this meeting.
Pressure, atm 1 10 100 Vi si bl e f lam e r a di us , m m 0.01 0.1 1 z = 4 mm z = 5 mm z = 6 mm z = 7 mm 0.5 r∝P− -0.1 0 0.1 0.2 r, cm 0
Fig.1 Distributions of soot volume fraction Fig.2 Variation of the flame diameter with
calculated using Model I. pressure at several flame heights.
Pressure, atm 1 10 100 Peak so ot vo lume frac tion , pp m 0.1 1 10 100 Model I, slope 1.9 Model II, slope 3.9
z, mm 0 1 2 3 4 5 6 7 8 9 10 Flam e d ia. int e g rat ed fv , mm p p m 0 5 10 15 20 25 30 Exp.: SSE Exp.: LOSA Model II Model I 20 atm 10 atm 5 atm
Fig.3 Variation of the peak soot volume fraction Fig.4 Variation of the flame diameter with pressure. integrated soot volume fraction with pressure.
z, mm 0 1 2 3 4 5 6 7 8 9 10 Cross sectio n a rea int . fv , mm 2 ppm 0 5 10 15 20 25 30 35 40 45 50 Exp.: SSE Exp.: LOSA Model II Model I 20 atm 10 atm 5 atm Pressure, atm 1 10 100 P e a k in tegrat e d f v , mm pp m o r mm 2 pp m 1 10 100
Area int. SSE Area Int. LOSA Area int. Model II Area int. Model I Line int. SSE Line int. LOSA Line int. Model II Line int. Model I
slope 3.2
slope 1.3 slope 1.0
slope 1.17
Fig.5 Variation of the cross sectional area Fig.6 Variation of the peak integrated soot integrated soot volume fraction with pressure. volume fractions with pressure.
0.2 4 6 8 0. 0. 0. 1 z, c m (a) 5 atm eak v .11 pm P f : 1 p -0.1 0 0.1 0.2 r, cm 0 0.2 0.4 0.6 0.8 1 z, cm (b) 10 atm eak v: .62 P f 4 ppm -0.1 0 0.1 0.2 r, cm 0 0.2 0.4 0.6 0.8 1 z, c m (c) 20 atm ak v: 15.27 ppm Pe f
