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Joint-scalar transported PDF modeling of soot formation and oxidation
Proceedings of the
Proceedings of the Combustion Institute 30 (2005) 775–783
Combustion Institute www.elsevier.com/locate/proci
Joint-scalar transported PDF modeling of soot formation and oxidation R.P. Lindstedt*, S.A. Louloudi Department of Mechanical Engineering, Imperial College London, South Kensington Campus, London SW7 2BX, UK
Abstract The ability of the transported probability density function (PDF) approach to reproduce the evolution of mean, rms fluctuations, and conditional PDFs of soot is explored in the context of two turbulent ethylene diffusion flames at Reynolds numbers of 11,800 and 15,600. The chemical similarity between surface reactions and PAH formation is explored on the basis of a second ring PAH analogy, and soot oxidation is accounted for through reactions with O, OH, and O2. The method of moments is used to account for coagulation and agglomeration in the coalescent and fractal aggregate limits. The soot model is coupled with a transported PDF approach closed at the joint-scalar level to directly account for interactions between turbulence, and the solid and gas phase chemistry. The latter is represented by a systematically reduced reaction mechanism for ethylene featuring 144 reactions, 15 solved and 14 steady-state species. Radiation from soot and gas phase species is accounted for through the RADCAL method and the inclusion of enthalpy into the joint-scalar PDF. Predicted temperature and soot statistics compare well with experimental data indicating the practical potential of the approach and the importance of turbulence-chemistry interactions in the context of soot formation and burnout. 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Particle dynamics; Soot; Transported PDF modelling
1. Introduction Emissions of particulates remain a concern in a wide range of applications, and increasingly detailed soot nucleation models have been proposed on the basis of indicative gas phase species. Significant uncertainties do prevail with respect to the chemical pathways to relevant poly-aromatic hydrocarbons (PAHs) involved in the soot nucleation process [1,2]. There is general agreement on the role of acetylene in the mass growth process, *
Corresponding author. Fax: +44 20 7589 3905. E-mail address: [email protected] (R.P. Lindstedt).
and the key experimental observation of Kent and Honnery [3] that the soot volume fraction cannot be modelled as a unique function of the mixture fraction has formed the basis for calculation methods based on perturbation approaches [4]. Kronenburg et al. [5] applied the CMC method and closed the chemistry using a detailed hydrocarbon mechanism and a simplified soot model [6]. Kollmann et al. [7] combined the soot model of Kennedy et al. [8] with a transported pdf closure at the joint mixture fraction, enthalpy, and soot volume fraction level. The chemistry was closed with a constrained equilibrium model. Zamuner and Dupoirieux [9] developed a soot model based on the HACA mechanism [10] along with a soot precursor model based on chemical lumping and
1540-7489/$ - see front matter 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2004.08.080
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applied the Probabilistic Eulerian Lagrangian (PEUL) approach in a post-processing mode to compute soot mass fractions in the turbulent jet diffusion flame of Coppalle and Joyeux [11]. Methods for addressing soot particle dynamics are typically based on increasingly accurate approximations of SmoluchowskiÕs master equation [12], nmax X dN 1 ¼ N 1 b1;j N j ; dt j¼1
proach. The work suggested that Brownian coagulation dominates at atmospheric pressure. The objective of the current work was to explore soot dynamics in turbulent flames via the transported PDF approach combined with (i) the method of moments, (ii) improved models for soot oxidation, and (iii) a PAH based analogy for surface reactions.
nmax i1 X dN i 1 X bj;ij N j N ij N i bi;j N j ¼ dt 2 j¼1 j¼1
The time evolution of the moments of the PSDF is readily obtained from SmoluchowskiÕs equation,
for i ¼ 2; 3; . . . ; nmax ;
ð1Þ
where Ni is the number density of particles in size class i, nmax is the total number of size classes, and bi, j is the collision frequency. Solutions may be obtained by approximating the particle size distribution through a discrete number of size classes/ bins. The majority of computational studies featuring this approach consider non-reacting cases (e.g., Jacobson and Turco [13]). However, Pope and Howard [14] applied sectional equations beyond a certain mass (400 amu) to describe the formation of soot in a stirred reactor. The time evolution of particles that coagulate due to turbulent shear motion has been reported by Koch and Pope [15], and the approach is a natural, but computationally demanding, way to include stochastic modelling of aerosol dynamics in turbulent reacting flows. An alternative approach, the method of moments (e.g., Frenklach [16]) is based on the solution of a small set of differential equations describing the evolution of the statistical moments (Mr) of the particle size distribution function (PSDF) derived from SmoluchowskiÕs equation. Frenklach and Harris [17] derived a particle size independent closure (bi, j = b) assuming coagulation in the free molecular regime and showed that b is only a function of the first two moments. The latter correspond to the mean number density and mass of the particle population. Knowledge of M0 and M1 thus provides a direct link to simplified soot models, e.g., Lindstedt [6], based on the solution of transport equations for the soot volume fraction and number density [18]. Pitsch et al. [19] simulated an ethylene diffusion flame using an unsteady flamelet approach for the gas phase chemistry closure coupled with the first two moments. Kazakov and Frenklach [20] extended the method by removing the assumption of particle size independent collision frequencies by including a description of particle coagulation and agglomeration via the formation and growth of fractal aggregates. Balthasar et al. [21] computed soot formation in a partially stirred plug flow reactor using four moments and a transported PDF ap-
2. Modelling of soot dynamics
dM 0 dt dM 1 dt dM 2 dt dM 3 dt .. .
¼ RN;0 RA;0 ; ¼ RN;1 þ RG;1 RO;1 ; ¼ RN;2 þ RA;2 þ RG;2 þ RO;2 ;
ð2Þ
¼ RN;3 þ RA;3 þ RG;3 þ RO;3 ;
where the index N indicates nucleation (inception), A agglomeration, G mass growth, and O oxidation for the different moments. Particle growth is here considered in the free, continuum, and transition regimes. Agglomeration may be regarded as the limit where coalescent growth is replaced by the formation of chain-like structures composed of similarly sized primary particles. Ko¨ylu¨ et al. [22] have shown that aggregates obey a fractal relationship with a dimension Df 1.8. The regimes are determined by the value of the Knudsen number, which in the latter case becomes a function of the mean aggregate diameter rather than the mean diameter of the primary particles. Agglomeration in the free molecular regime is given by ð0;0Þ
RA;0 ¼ 12K f M 20 f1=2 ; RA;r
r1 X 1 ¼ K f M 20 2 k¼1
r ðk;rkÞ f k 1=2
ð3Þ for r ¼ 2; 3; . . . ð4Þ
where Kf is the free molecular regime coagulation ðx;yÞ follows constant [20]. The grid function fl Kazakov and Frenklach [20] who also derived expressions for agglomeration in the continuum regime, 1=Df 1=3 f RcA;0 ¼ K c ½1 þ hm1=3 ihm1=3 N 1=31=D i A Np A p 1=Df 1=3 f N 1=31=D i þ hm1=3 i þ K 0c ðhm1=3 A A Np p
hm2=3 N p2=Df 2=3 iÞM 20 : A
ð5Þ
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The free molecular and continuum terms are harmonically averaged to give the agglomeration term in the transition regime [18]. There is a lack of physical models describing the transition from coalescent growth to aggregation. Kazakov and Frenklach [20] switched to the aggregation limit when the mean particle diameter reached values of 25 or 30 nm, and 0here 27.5 nm has been used. The terms hmrA N rp i in Eq. (5) require knowledge of the joint pdf of the mass (mA) and the number density (Np) of the aggregate, and the following simplification has been suggested [20], 0 hmrA N rp i ¼ lr pr0 ð6Þ with pr = Pr/P0, where Pr represents the size distribution of the primary particles in the aggregate and is defined in a fashion similar to Mr (e.g., P0 = M0) [18,20]. Following Frenklach [16], the normalized fractal-order moments (lr and pr) appearing in the source terms are computed by Lagrange interpolation among logarithms of the whole-order moments. The remaining terms to be modelled for the closure of the above equations include particle nucleation, surface growth, and oxidation. The present work features a simple nucleation step that has been shown to work well in laminar ethylene diffusion flames [6] and further developments that account for the dynamics of the PAH formation process are desirable, particularly in the context of premixed flames and for fuels with aromatic content. NA RN;r ¼ 2 k N ðT Þ½C2 H2 ; r ¼ 0; 2; 3; . . . ð7Þ C min where NA is AvogadroÕs Number (6.022 · 1026 kmol1) and Cmin corresponds to naphthalene. Assuming spherical particles and a linear dependence of the surface growth and oxidation rates on the particle surface area, the following expressions result (e.g. [16]): RG;r ¼ AG N A pd 2p;min M 0 ; r1 X r ðrkÞ 2 lkþ2=3 ; k k¼0
ð8Þ
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Table 1 Reaction rate constants for soot formation and oxidation presented in the form Ai ai T bi expðEi =RT Þ [6,18] Ai
ki
4
0.63 · 10 0.75 · 103 8.82 9.09 6.43
kN kG kOH kO k O2
ai
bi
Ei/R
1 1 0.05 0.20 0.723
0 0 1/2 1/2 1/2
21,000 12,100 0 0 11,250
Units are in K, kmol, m3, and s.
The parameter vs is discussed below, and dp, min is the diameter of the first incipient soot particle assigned a value of 0.3 nm corresponding to a naphthalene molecule. The collision efficiencies (e.g., aOH = 0.05 and aO = 0.20) for the oxidation terms have been obtained from the work of Roth et al. [23]. The surface and oxidation terms are affected by particle aggregation as the aggregate surface area is larger than that of a spherical particle with the same mass, and Eqs. (8) and (9) are accordingly modified with the binary moments determined via Eq. (6). RG;r ¼ AG N A pd 2p;min M 0 ;
ð12Þ
r1 X r ðrkÞ kþ2=3 1=3 2 hmA N p i; k k¼0
RO;r ¼ AO N A pd 2p;min M 0 ; r1 X r k¼0
k
ð13Þ
ð2ÞðrkÞ hmAkþ2=3 N 1=3 p i:
The reaction rate constants kN, kG, kOH, kO, and k O2 are given in Table 1. For reference purposes, the simplified soot mechanism of Lindstedt [6] is used to provide comparisons. The model is readily recovered from the above equations by consideration of the two first moments and the introduction of a spherical particle shape.
r ¼ 0; 2; 3; . . . 3. Soot surface chemistry
RO;r ¼ AO N A pd 2p;min M 0 ; r1 X k¼0
r ðrkÞ lkþ2=3 ; ð2Þ k
ð9Þ r ¼ 0; 2; 3; . . .
The terms AG and AO are related to the gas phase chemistry and correspond to soot mass growth via acetylene addition and soot oxidation. AG ¼ k G ðT Þ½C2 H2 vs ;
ð10Þ
AO ¼ ðk OH ðT Þ½OH þ k O ðT Þ½O þ k O2 ðT Þ½O2 Þ: ð11Þ
The hypothesis of chemical similarity between soot surface chemistry and that of PAHs is promising [24], and the current work features an analogy based on naphthalene. The full reaction mechanism has been evaluated with reference to plug flow reactor data and shown to produce good agreement [25]. C10 H7 þ H ¼ C10 H8
ðIÞ
C10 H7 þ H2 ¼ C10 H8 þ H
ðIIÞ
C10 H8 þ O ¼ C10 H7 O þ H
ðIIIÞ
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C10 H8 þ OH ¼ C10 H7 þ H2 O
ðIVÞ
C10 H7 þ O2 ¼ C10 H7 O þ O
ðVÞ
C10 H7 þ O2 ¼ C10 H7 OO
ðVIÞ
C10 H7 OO ! C9 H7 þ CO2
ðVIIÞ
C10 H7 OO ! C10 H6 O2 þ H
ðVIIIÞ
C10 H7 O ! C9 H7 þ CO
ðIXÞ
C10 H7 þ C2 H2 ! C12 H8 þ H
ðXÞ
The rate constants are given in Table 2. Reaction (X), by analogy, corresponds to the soot mass growth step. RG ¼ k fX ðT Þ½C10 H7 ½C2 H2 :
ð14Þ
The naphthyl radical concentration ([C10H7]) is obtained via a truncated steady-state approximation based on reactions (I), (II), (IV), (V), (VI), and (X). Lð½C10 H7 Þ ¼
xrI
þ
xfI
xrII
þ
xfII
xfIV
þ
xrIV
xrV
þ
xrVI
xfV xfVI
xfX :
ð15Þ
The naphthalene concentration is expressed as a function of the number of the active sites [6,10]. v ½C10 H8 ¼ as sh As : ð16Þ NA The remaining molar concentrations are also obtained by considering the corresponding truncated steady-state approximations. RG ¼ k fX ðT Þvs As ½C2 H2 ; vs ¼
ð17Þ
K 1K7 þ K 2K 6 v as K 9 sh ; K 4K 7K 9 K 2K 5K 9 K 3K 7K 8 NA ð18Þ
K 1 ¼ k rI þ k rII ½H þ k fIV ½OH;
ð19Þ
K 2 ¼ k rV ½O;
K 3 ¼ k rVI ;
K 4 ¼ k fI ½H þ k fII ½H2 þ k rIV ½H2 O þ k fV ½O2 þ k fVI ½O2 þ k fX ½C2 H2 ; K 5 ¼ k fV ½O2 ;
K 6 ¼ k fIII ½O;
K 7 ¼ k rIII ½H þ k rV ½O þ k fIX ; K 8 ¼ k fVI ½O2 ;
K 9 ¼ k rVI þ k fVII þ k fVIII :
Frenklach and Wang [10] proposed that the number of active sites (vs h) is approximately equal to 2.32 · 1019 sites/m2. A strong sensitivity of soot predictions to as has been reported, and a range of values have been explored (e.g., 0.10–1.00) [6,26,27]. 4. Closure considerations The transported PDF approach of Lindstedt and Louloudi [28] is here extended to include the soot model outlined above, and the joint-scalar PDF may be written as the following random vector f~ / ð/a ; f ; H ; M 0 ; M 1 ; M 2 ; M 3 ; P 1 ; P 2 ; xÞ; ð20Þ where /i with i = 1, . . . , a are the species mass fractions of the gas phase and H is the enthalpy of the mixture. The systematically reduced mechanism for ethylene [29] features 144 reactions, 15 solved (H, O, OH, HO2, H2O, H2, O2, CO, CO2, CH3, CH4, C2H2, C2H4, C2H6, and N2) and 14 steady-state (C, CH, 1CH2, 3CH2, CHO, CH2OH, CH3O, C2, C2H, C2H3, C2H5, C2HO, C2H2O, and CH2O) species. Mixing is treated via the modified CurlÕs model [30] with a constant time-scale ratio C/ = 2.3. The radiative loss term is expressed on the basis of the optically thin assumption accounting for radiation from H2O, CO, CO2, CH4, and soot [18,31]. QRAD ¼ 4rSB
K X
pi ap;i ðT 4 T 4b Þ
i¼1
Table 2 Reaction rate constants for the second ring based PAH analogy [25] in the form Ai ai T bi expðEi =RT Þ Step
Ai
ai
bi
Ei/R
I II III IV V VI VII VIII IX X
0.783 · 1011 0.444 · 102 0.25 · 1011 0.17 · 106 0.215 · 1011 0.25 · 1010 0.2272 · 109 0.2272 · 109 0.18 · 1012 0.357 · 1022
1 1 1 1 1 1 1 1 1 1
0 2.43 0 1.42 0 0.15 0 0 0 3.176
0 3158 2347 729 3076 78 0 0 22,062 7471
Units are in K, kmol, m3, and s.
þ 4rSB fv ðT 5 T 5b Þ:
ð21Þ
In the above equation, rSB is the Steffan–Boltzmann constant (rSB = 5.669 · 108 W/m2 K4), pi is the partial pressure of species i in atmospheres, ap, i is the Planck mean absorption coefficient of species ‘‘i’’ in m1 atm1, T is the local flame temperature, Tb is the background temperature equal to 298.15 K, and C = 1.307 · 103 m1 K1 [18]. 5. Results and discussion The two non-premixed turbulent sooting ethylene–air flames investigated experimentally by
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Coppalle and Joyeux [11] and Kent and Honnery [3] are computed using an implicit parabolic x x formulation. The flames are fairly close in terms of physical conditions, and a greater variation, particularly in terms of the Re number, would have been preferred. The cross-stream directions are discretized by means of 105 and 63 computational cells, respectively, with each containing on average 100 particles. More than 20,000 axial steps were used to cover the flame lengths of 170 and 220 jet diameters. The flames considered are not piloted, and a conserved scalar approach [4] is applied up to axial locations of x/ D = 10 and 39 followed by a switch to the transported PDF approach. The results for the flame investigated by Coppalle and Joyeux [11] are unaffected by a further reduction in the switch-over point, while a switch at x/D = 20 has a modest (in the overall context) influence ( 6 40%) at the first two measuring stations for the flame of Kent and Honnery [3]. Results are unaffected further downstream. The calculation featuring the first two moments of the PSDF (M0 and M1) is functionally equivalent to the method of Lindstedt [6], and results obtained with this model are shown for comparison purposes. Figure 1 depicts the axial and radial temperature profiles for the flame of Kent and Honnery [3] obtained with four moments (M0–M3) and with coagulation in the coalescent limit. Calculations obtained with fewer moments suggest that differences are small up to x/D 80, whereas further downstream an increase in the number of moments solved results in an increase in the peak temperature. The observation is consistent with the radial profiles. Figure 2 shows the evolution of the radial profiles of the soot volume fraction in physical space at x/D = 46, 80.5, 115, and 161.
Fig. 1. Axial and radial profiles of temperature for the flame of Kent and Honnery [3]. Symbols represent experimental data [3] and the line predictions obtained with the method of moments.
779
Fig. 2. Radial profiles of soot volume fraction for the flame of Kent and Honnery [3]. Symbols as in Fig. 1.
An increase in the number of moments solved leads to a consistent decrease of the peak soot levels at the centre-line, and an excellent agreement is obtained at x/D = 115 and 161. For reasons of clarity, results are shown for M0–M1 and M0– M3 only. Predictions using M0–M2 are close to those obtained with four moments (M0–M3). Calculations have also been performed for the flame of Coppalle and Joyeux [11], and the corresponding axial profiles are shown in Fig. 3. The radial profiles of the mean soot volume fraction and its rms (Fig. 4) show similar trends, and an increase in the number of moments solved again leads to lower soot levels at the centre-line. The normalized [11] conditional PDFs of temperature and soot volume fraction at four different axial locations are presented in Fig. 5. The shape of the temperature PDF departs significantly from the Gaussian shape suggested by the measure-
Fig. 3. Axial profiles of soot volume fraction and its rms, and temperature and its rms for the flame of Coppalle and Joyeux [11]. Symbols represent experimental data and the line predictions obtained with the method of moments.
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Fig. 4. Radial profiles of soot volume fraction and its rms for the flame of Coppalle and Joyeux [11]. Notation as in Fig. 3.
Fig. 5. Measurements and computations of the normalized conditional PDFs of temperature and soot volume fraction for the flame of Coppalle and Joyeux [11]. Notation as in Fig. 3.
ments. The reasons for the differences are not clear, and the computed PDF appears rather more sensitive to soot concentrations than that measured. Related work featuring the non-sooting SANDIA flames shows an excellent agreement with temperature PDFs using the current methodology [18]. Furthermore, the experimental temperatures were obtained using two-colour pyrometry, and the investigators suggested that the technique is not accurate at temperatures 61000 K [11]. It is hence possible that the shape of the measured temperature PDF is subjected to significant bias, though a further possibility is that the currently applied optically thin approximation is questionable. The normalized conditional PDF of the soot volume fraction shows much better agreement, and the downstream evolution is comparatively well captured, which lends further support to the
current approach. It is also interesting to note that an increase in the number of solved moments has a positive impact at the first measuring station. Comparisons between the simplified ‘‘standard’’ method [6] and the method of moments (M0– M1) yield almost identical results for the soot volume fraction and its rms as shown in Fig. 6. To provide a more complete description of particle dynamics, the fractal aggregate limit of coagulation has also been incorporated within the framework of the method of moments. Comparison between the computed soot volume fractions for the flame of Kent and Honnery [3] (Fig. 7) shows that the inclusion of agglomeration leads to higher soot concentrations at the centreline though the differences are not large in the present case. Predictions obtained with the simplified method [6] for the Kent and Honnery [3] flame are
Fig. 6. Radial profiles of soot volume fraction and its rms for the flame of Coppalle and Joyeux [11]. Symbols represent experimental data and line predictions obtained solving for M0–M1.
Fig. 7. Radial profiles of soot volume fraction for the flame of Kent and Honnery [3]. Symbols represent experimental data and the line predictions obtained with M0–M3 accounting for coagulation and coagulation with agglomeration.
R.P. Lindstedt, S.A. Louloudi / Proceedings of the Combustion Institute 30 (2005) 775–783
compared with those obtained with the method of moments featuring the solution of M0–M3. In both cases, the more complete description of particle dynamics, considering coagulation and agglomeration, is implemented. Predictions were obtained with the two methods featuring identical representations of the physics governing particle collisions. Elevated soot levels are obtained with the method of moments at all axial locations though the differences between the two methods are not large ( 650%). The final aspect investigated in the present work is the treatment of surface growth on the basis of the above PAH analogy. The computed soot volume fractions for the Kent and Honnery [3] flame are shown in Fig. 8. All predictions presented feature the method of moments (M0–M3) and the more complete description of particle coagulation in the coalescence and aggregate limits. Figure 8 reveals a very strong sensitivity of soot levels to as, which is consistent with computational studies of laminar flames. The observed overprediction at the first two measurement locations vanishes further downstream resulting in an arguably excellent agreement at x/D = 115 and 161. Figure 9 compares the radial profiles of the mean soot volume fraction and rms for the flame of Coppalle and Joyeux [11], and the observations are consistent with those made above. Given the strong sensitivities and uncertainties associated with the functional dependencies of as, it is very encouraging that the application of ‘‘standard’’ values combined with a PAH analogy results in an acceptable behaviour without the need for a re-scaling of reaction rate constants. However, it may be pointed out that as = 0.1, arguably more typical of the post-flame region, is not compatible with the currently assumed site density [10].
781
Fig. 9. Radial profiles of soot volume fraction and its rms for the flame of Coppalle and Joyeux [11]. Symbols represent experimental data. Notation as in Fig. 8. See Fig. 6 for results obtained with the standard model.
6. Conclusions The method of statistical moments [20] has been implemented within a transported PDF methodology and combined with improved soot oxidation and PAH analogy based models for the soot surface chemistry. The results obtained show encouraging agreement with respect to the evolution of the conditional PDF of soot, and the mean and rms of the soot volume fraction. It is shown that an increase in the number of moments results in an improved agreement but that differences are comparatively modest downstream. The use of a comparatively simple soot model [6] also results in a surprisingly good agreement when combined with a transported PDF approach. The results highlight the importance of accurate modelling of turbulence-chemistry interactions in the context of soot dynamics in turbulent flames. A remaining principal uncertainty is, however, associated with the temporal evolution and functional dependencies of soot surface site densities and reactivities.
Acknowledgments The authors thank Fabian Mauss and Michael Frenklach for valuable discussions and the European Union for financial support through Grant G4RD-CT-1999-00075. The support of Dr. G. Roy and the Office of Naval Research under Grant N00014-2-1-0664 is also gratefully acknowledged. Fig. 8. Radial profiles of soot volume fraction for the flame of Kent and Honnery [3]. Symbols represent experimental data and line predictions obtained with M0–M3 and the second ring based PAH analogy with as = 1.00 and as = 0.75.
References [1] R.P. Lindstedt, L.Q. Maurice, M.P. Meyer, Faraday Discuss. 119 (2001) 409–432.
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[2] H. Richter, J.B. Howard, Prog. Energy Combust. Sci. 26 (2000) 565–608. [3] J.H. Kent, D. Honnery, Combust. Sci. Technol. 54 (1987) 383–397. [4] M. Fairweather, W.P. Jones, S.H. Ledin, R.P. Lindstedt, Proc. Combust. Inst. 24 (1992) 1067–1074. [5] A. Kronenburg, R.W. Bilger, J.H. Kent, Combust. Flame 121 (2000) 24–40. [6] R.P. Lindstedt, in: H. Bockhorn (Ed.), Soot Formation in Combustion: Mechanisms and Models. Springer-Verlag, Berlin, 1994, pp. 417–441. [7] W. Kollmann, I.M. Kennedy, M. Metternich, J.-Y. Chen, in: H. Bockhorn (Ed.), Soot Formation in Combustion: Mechanisms and Models. SpringerVerlag, Berlin, 1994, pp. 503–526. [8] I.M. Kennedy, W. Kollmann, J.-Y. Chen, Combust. Flame 81 (1990) 73–85. [9] B. Zamuner, F. Dupoirieux, Combust. Sci. Technol. 158 (2000) 407–438. [10] M. Frenklach, H. Wang, Proc. Combust. Inst. 23 (1990) 1559–1566. [11] A. Coppalle, D. Joyeux, Combust. Flame 96 (1994) 275–285. [12] M. Smoluchowski, Phys. Chem. 92 (1917) 144. [13] M.Z. Jacobson, R.P. Turco, Aerosol Sci. Technol. 22 (1995) 73–92. [14] C.J. Pope, J.B. Howard, Aerosol Sci. Technol. 27 (1997) 73–94. [15] D.L. Koch, S.B. Pope, Phys. Fluids 14 (7) (2002) 2447–2455. [16] M. Frenklach, Chem. Eng. Sci. 57 (2002) 2229–2239. [17] M. Frenklach, S.J. Harris, J. Colloid Interf. Sci. 118 (1987) 252–261.
[18] S.A. Louloudi, Transported Probability Density Function Modelling of Turbulent Jet Flames, Ph.D. thesis, Imperial College London, 2003. [19] H. Pitsch, E. Riesmeier, N. Peters, Combust. Sci. Technol. 158 (2000) 389–406. [20] A. Kazakov, M. Frenklach, Combust. Flame 114 (1998) 484–501. [21] M. Balthasar, F. Mauss, A. Knobel, M. Kraft, Combust. Flame 128 (2002) 395–409. [22] U.O. Ko¨ylu¨, G.M. Faeth, T.L. Farias, M.G. Carvalho, Combust. Flame 100 (1995) 621–633. [23] P. Roth, O. Brandt, S. von Gersum, Proc. Combust. Inst. 23 (1990) 1485–1491. [24] M. Frenklach, Phys. Chem. Chem. Phys. 4 (2002) 2028–2037. [25] M.L. Potter, Detailed Chemical Kinetic Modelling of Propulsion Fuels, Ph.D. thesis, Imperial College London, 2003. [26] J. Appel, H. Bockhorn, M. Frenklach, Combust. Flame 121 (2000) 122–136. [27] M. Frenklach, H. Wang, in: H. Bockhorn (Ed.), Soot Formation in Combustion: Mechanisms and Models. Springer-Verlag, Berlin, 1994, pp. 165–192. [28] R.P. Lindstedt, S.A. Louloudi, E.M. Va´os, Proc. Combust. Inst. 28 (2000) 149–156. [29] M.P. Meyer, The Application of Detailed and Systematically Reduced Chemistry to Transient Laminar Flames, Ph.D. thesis, Imperial College London, 2001. [30] J. Janicka, W. Kolbe, W. Kollmann, J. Non-Equilib. Thermodyn. 4 (1) (1979) 47–66. [31] W.L. Grosshandler, RADCAL: A Narrow-band Model for Radiation Calculations in a Combustion Environment, NIST Technical Note 1402, 1993.
Comments Heinz Pitsch, Stanford University, USA. The comparison with the experimental data looks much better than what we often see in predictions of laminar flames. Did you compare predictions of your chemistry model with laminar flames? What is the level of agreement for these? A second question I have is related to the transport of soot. In principle, there is a preferential transport between soot and the gas-phase, since particles do not diffuse. We have shown in a numerical study [1] that this effect could substantially affect the soot volume fraction. Differential diffusion effects are not included in your model. Do you conclude from your results that these effects are unimportant?
ers have explored the use of moment methods extensively (Ref. [24] in paper) in laminar premixed flames and the naphthalene oxidation chemistry has been independently validated using stirred reactor data (Ref. [25] in paper). The potential consequences of a lack of diffusive transport of soot particles have also been raised by Kronenburg et al. (Ref. [5] in paper) and it would be premature to conclude that such effects are generally unimportant. However, it is reasonable to suggest that the influence will diminish with increasing Reynolds number and that it is unlikely that the effect is predominant in the current flames.
Reference [1] H. Pitsch, et al., Comb. Sci. Technol. 158 (2000) 389–406.
J.P. Gore, Purdue University, USA. There is considerable self-absorption of radiation in the flames being considered. Is the radiation model capable of adequately treating the self-absorption process?
Reply. The basic soot model has been extensively validated (Ref. [6] in paper) in laminar counterflow diffusion flames with the rate constants applied unchanged in the current work. The level of agreement was found to be satisfactory. Frenklach and co-work-
Reply. The radiation treatment is based on a narrow band model (Refs. [18,31] in paper) that has been found to work satisfactorily for non-sooting flames (Ref. [18] in paper). The additional term (proportional to the soot volume fraction) exerts a significant influence in the cur-
d
R.P. Lindstedt, S.A. Louloudi / Proceedings of the Combustion Institute 30 (2005) 775–783 rent flames and the computed temperature profiles show fair agreement with measurements. However, it is possible that a refinement in the treatment of self-absorption
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will prove beneficial and the strong coupling of different model elements suggests a need for more detailed experimental data as part of such an evaluation.
Proceedings of the Combustion Institute 30 (2005) 775–783
Combustion Institute www.elsevier.com/locate/proci
Joint-scalar transported PDF modeling of soot formation and oxidation R.P. Lindstedt*, S.A. Louloudi Department of Mechanical Engineering, Imperial College London, South Kensington Campus, London SW7 2BX, UK
Abstract The ability of the transported probability density function (PDF) approach to reproduce the evolution of mean, rms fluctuations, and conditional PDFs of soot is explored in the context of two turbulent ethylene diffusion flames at Reynolds numbers of 11,800 and 15,600. The chemical similarity between surface reactions and PAH formation is explored on the basis of a second ring PAH analogy, and soot oxidation is accounted for through reactions with O, OH, and O2. The method of moments is used to account for coagulation and agglomeration in the coalescent and fractal aggregate limits. The soot model is coupled with a transported PDF approach closed at the joint-scalar level to directly account for interactions between turbulence, and the solid and gas phase chemistry. The latter is represented by a systematically reduced reaction mechanism for ethylene featuring 144 reactions, 15 solved and 14 steady-state species. Radiation from soot and gas phase species is accounted for through the RADCAL method and the inclusion of enthalpy into the joint-scalar PDF. Predicted temperature and soot statistics compare well with experimental data indicating the practical potential of the approach and the importance of turbulence-chemistry interactions in the context of soot formation and burnout. 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Particle dynamics; Soot; Transported PDF modelling
1. Introduction Emissions of particulates remain a concern in a wide range of applications, and increasingly detailed soot nucleation models have been proposed on the basis of indicative gas phase species. Significant uncertainties do prevail with respect to the chemical pathways to relevant poly-aromatic hydrocarbons (PAHs) involved in the soot nucleation process [1,2]. There is general agreement on the role of acetylene in the mass growth process, *
Corresponding author. Fax: +44 20 7589 3905. E-mail address: [email protected] (R.P. Lindstedt).
and the key experimental observation of Kent and Honnery [3] that the soot volume fraction cannot be modelled as a unique function of the mixture fraction has formed the basis for calculation methods based on perturbation approaches [4]. Kronenburg et al. [5] applied the CMC method and closed the chemistry using a detailed hydrocarbon mechanism and a simplified soot model [6]. Kollmann et al. [7] combined the soot model of Kennedy et al. [8] with a transported pdf closure at the joint mixture fraction, enthalpy, and soot volume fraction level. The chemistry was closed with a constrained equilibrium model. Zamuner and Dupoirieux [9] developed a soot model based on the HACA mechanism [10] along with a soot precursor model based on chemical lumping and
1540-7489/$ - see front matter 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2004.08.080
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applied the Probabilistic Eulerian Lagrangian (PEUL) approach in a post-processing mode to compute soot mass fractions in the turbulent jet diffusion flame of Coppalle and Joyeux [11]. Methods for addressing soot particle dynamics are typically based on increasingly accurate approximations of SmoluchowskiÕs master equation [12], nmax X dN 1 ¼ N 1 b1;j N j ; dt j¼1
proach. The work suggested that Brownian coagulation dominates at atmospheric pressure. The objective of the current work was to explore soot dynamics in turbulent flames via the transported PDF approach combined with (i) the method of moments, (ii) improved models for soot oxidation, and (iii) a PAH based analogy for surface reactions.
nmax i1 X dN i 1 X bj;ij N j N ij N i bi;j N j ¼ dt 2 j¼1 j¼1
The time evolution of the moments of the PSDF is readily obtained from SmoluchowskiÕs equation,
for i ¼ 2; 3; . . . ; nmax ;
ð1Þ
where Ni is the number density of particles in size class i, nmax is the total number of size classes, and bi, j is the collision frequency. Solutions may be obtained by approximating the particle size distribution through a discrete number of size classes/ bins. The majority of computational studies featuring this approach consider non-reacting cases (e.g., Jacobson and Turco [13]). However, Pope and Howard [14] applied sectional equations beyond a certain mass (400 amu) to describe the formation of soot in a stirred reactor. The time evolution of particles that coagulate due to turbulent shear motion has been reported by Koch and Pope [15], and the approach is a natural, but computationally demanding, way to include stochastic modelling of aerosol dynamics in turbulent reacting flows. An alternative approach, the method of moments (e.g., Frenklach [16]) is based on the solution of a small set of differential equations describing the evolution of the statistical moments (Mr) of the particle size distribution function (PSDF) derived from SmoluchowskiÕs equation. Frenklach and Harris [17] derived a particle size independent closure (bi, j = b) assuming coagulation in the free molecular regime and showed that b is only a function of the first two moments. The latter correspond to the mean number density and mass of the particle population. Knowledge of M0 and M1 thus provides a direct link to simplified soot models, e.g., Lindstedt [6], based on the solution of transport equations for the soot volume fraction and number density [18]. Pitsch et al. [19] simulated an ethylene diffusion flame using an unsteady flamelet approach for the gas phase chemistry closure coupled with the first two moments. Kazakov and Frenklach [20] extended the method by removing the assumption of particle size independent collision frequencies by including a description of particle coagulation and agglomeration via the formation and growth of fractal aggregates. Balthasar et al. [21] computed soot formation in a partially stirred plug flow reactor using four moments and a transported PDF ap-
2. Modelling of soot dynamics
dM 0 dt dM 1 dt dM 2 dt dM 3 dt .. .
¼ RN;0 RA;0 ; ¼ RN;1 þ RG;1 RO;1 ; ¼ RN;2 þ RA;2 þ RG;2 þ RO;2 ;
ð2Þ
¼ RN;3 þ RA;3 þ RG;3 þ RO;3 ;
where the index N indicates nucleation (inception), A agglomeration, G mass growth, and O oxidation for the different moments. Particle growth is here considered in the free, continuum, and transition regimes. Agglomeration may be regarded as the limit where coalescent growth is replaced by the formation of chain-like structures composed of similarly sized primary particles. Ko¨ylu¨ et al. [22] have shown that aggregates obey a fractal relationship with a dimension Df 1.8. The regimes are determined by the value of the Knudsen number, which in the latter case becomes a function of the mean aggregate diameter rather than the mean diameter of the primary particles. Agglomeration in the free molecular regime is given by ð0;0Þ
RA;0 ¼ 12K f M 20 f1=2 ; RA;r
r1 X 1 ¼ K f M 20 2 k¼1
r ðk;rkÞ f k 1=2
ð3Þ for r ¼ 2; 3; . . . ð4Þ
where Kf is the free molecular regime coagulation ðx;yÞ follows constant [20]. The grid function fl Kazakov and Frenklach [20] who also derived expressions for agglomeration in the continuum regime, 1=Df 1=3 f RcA;0 ¼ K c ½1 þ hm1=3 ihm1=3 N 1=31=D i A Np A p 1=Df 1=3 f N 1=31=D i þ hm1=3 i þ K 0c ðhm1=3 A A Np p
hm2=3 N p2=Df 2=3 iÞM 20 : A
ð5Þ
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The free molecular and continuum terms are harmonically averaged to give the agglomeration term in the transition regime [18]. There is a lack of physical models describing the transition from coalescent growth to aggregation. Kazakov and Frenklach [20] switched to the aggregation limit when the mean particle diameter reached values of 25 or 30 nm, and 0here 27.5 nm has been used. The terms hmrA N rp i in Eq. (5) require knowledge of the joint pdf of the mass (mA) and the number density (Np) of the aggregate, and the following simplification has been suggested [20], 0 hmrA N rp i ¼ lr pr0 ð6Þ with pr = Pr/P0, where Pr represents the size distribution of the primary particles in the aggregate and is defined in a fashion similar to Mr (e.g., P0 = M0) [18,20]. Following Frenklach [16], the normalized fractal-order moments (lr and pr) appearing in the source terms are computed by Lagrange interpolation among logarithms of the whole-order moments. The remaining terms to be modelled for the closure of the above equations include particle nucleation, surface growth, and oxidation. The present work features a simple nucleation step that has been shown to work well in laminar ethylene diffusion flames [6] and further developments that account for the dynamics of the PAH formation process are desirable, particularly in the context of premixed flames and for fuels with aromatic content. NA RN;r ¼ 2 k N ðT Þ½C2 H2 ; r ¼ 0; 2; 3; . . . ð7Þ C min where NA is AvogadroÕs Number (6.022 · 1026 kmol1) and Cmin corresponds to naphthalene. Assuming spherical particles and a linear dependence of the surface growth and oxidation rates on the particle surface area, the following expressions result (e.g. [16]): RG;r ¼ AG N A pd 2p;min M 0 ; r1 X r ðrkÞ 2 lkþ2=3 ; k k¼0
ð8Þ
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Table 1 Reaction rate constants for soot formation and oxidation presented in the form Ai ai T bi expðEi =RT Þ [6,18] Ai
ki
4
0.63 · 10 0.75 · 103 8.82 9.09 6.43
kN kG kOH kO k O2
ai
bi
Ei/R
1 1 0.05 0.20 0.723
0 0 1/2 1/2 1/2
21,000 12,100 0 0 11,250
Units are in K, kmol, m3, and s.
The parameter vs is discussed below, and dp, min is the diameter of the first incipient soot particle assigned a value of 0.3 nm corresponding to a naphthalene molecule. The collision efficiencies (e.g., aOH = 0.05 and aO = 0.20) for the oxidation terms have been obtained from the work of Roth et al. [23]. The surface and oxidation terms are affected by particle aggregation as the aggregate surface area is larger than that of a spherical particle with the same mass, and Eqs. (8) and (9) are accordingly modified with the binary moments determined via Eq. (6). RG;r ¼ AG N A pd 2p;min M 0 ;
ð12Þ
r1 X r ðrkÞ kþ2=3 1=3 2 hmA N p i; k k¼0
RO;r ¼ AO N A pd 2p;min M 0 ; r1 X r k¼0
k
ð13Þ
ð2ÞðrkÞ hmAkþ2=3 N 1=3 p i:
The reaction rate constants kN, kG, kOH, kO, and k O2 are given in Table 1. For reference purposes, the simplified soot mechanism of Lindstedt [6] is used to provide comparisons. The model is readily recovered from the above equations by consideration of the two first moments and the introduction of a spherical particle shape.
r ¼ 0; 2; 3; . . . 3. Soot surface chemistry
RO;r ¼ AO N A pd 2p;min M 0 ; r1 X k¼0
r ðrkÞ lkþ2=3 ; ð2Þ k
ð9Þ r ¼ 0; 2; 3; . . .
The terms AG and AO are related to the gas phase chemistry and correspond to soot mass growth via acetylene addition and soot oxidation. AG ¼ k G ðT Þ½C2 H2 vs ;
ð10Þ
AO ¼ ðk OH ðT Þ½OH þ k O ðT Þ½O þ k O2 ðT Þ½O2 Þ: ð11Þ
The hypothesis of chemical similarity between soot surface chemistry and that of PAHs is promising [24], and the current work features an analogy based on naphthalene. The full reaction mechanism has been evaluated with reference to plug flow reactor data and shown to produce good agreement [25]. C10 H7 þ H ¼ C10 H8
ðIÞ
C10 H7 þ H2 ¼ C10 H8 þ H
ðIIÞ
C10 H8 þ O ¼ C10 H7 O þ H
ðIIIÞ
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C10 H8 þ OH ¼ C10 H7 þ H2 O
ðIVÞ
C10 H7 þ O2 ¼ C10 H7 O þ O
ðVÞ
C10 H7 þ O2 ¼ C10 H7 OO
ðVIÞ
C10 H7 OO ! C9 H7 þ CO2
ðVIIÞ
C10 H7 OO ! C10 H6 O2 þ H
ðVIIIÞ
C10 H7 O ! C9 H7 þ CO
ðIXÞ
C10 H7 þ C2 H2 ! C12 H8 þ H
ðXÞ
The rate constants are given in Table 2. Reaction (X), by analogy, corresponds to the soot mass growth step. RG ¼ k fX ðT Þ½C10 H7 ½C2 H2 :
ð14Þ
The naphthyl radical concentration ([C10H7]) is obtained via a truncated steady-state approximation based on reactions (I), (II), (IV), (V), (VI), and (X). Lð½C10 H7 Þ ¼
xrI
þ
xfI
xrII
þ
xfII
xfIV
þ
xrIV
xrV
þ
xrVI
xfV xfVI
xfX :
ð15Þ
The naphthalene concentration is expressed as a function of the number of the active sites [6,10]. v ½C10 H8 ¼ as sh As : ð16Þ NA The remaining molar concentrations are also obtained by considering the corresponding truncated steady-state approximations. RG ¼ k fX ðT Þvs As ½C2 H2 ; vs ¼
ð17Þ
K 1K7 þ K 2K 6 v as K 9 sh ; K 4K 7K 9 K 2K 5K 9 K 3K 7K 8 NA ð18Þ
K 1 ¼ k rI þ k rII ½H þ k fIV ½OH;
ð19Þ
K 2 ¼ k rV ½O;
K 3 ¼ k rVI ;
K 4 ¼ k fI ½H þ k fII ½H2 þ k rIV ½H2 O þ k fV ½O2 þ k fVI ½O2 þ k fX ½C2 H2 ; K 5 ¼ k fV ½O2 ;
K 6 ¼ k fIII ½O;
K 7 ¼ k rIII ½H þ k rV ½O þ k fIX ; K 8 ¼ k fVI ½O2 ;
K 9 ¼ k rVI þ k fVII þ k fVIII :
Frenklach and Wang [10] proposed that the number of active sites (vs h) is approximately equal to 2.32 · 1019 sites/m2. A strong sensitivity of soot predictions to as has been reported, and a range of values have been explored (e.g., 0.10–1.00) [6,26,27]. 4. Closure considerations The transported PDF approach of Lindstedt and Louloudi [28] is here extended to include the soot model outlined above, and the joint-scalar PDF may be written as the following random vector f~ / ð/a ; f ; H ; M 0 ; M 1 ; M 2 ; M 3 ; P 1 ; P 2 ; xÞ; ð20Þ where /i with i = 1, . . . , a are the species mass fractions of the gas phase and H is the enthalpy of the mixture. The systematically reduced mechanism for ethylene [29] features 144 reactions, 15 solved (H, O, OH, HO2, H2O, H2, O2, CO, CO2, CH3, CH4, C2H2, C2H4, C2H6, and N2) and 14 steady-state (C, CH, 1CH2, 3CH2, CHO, CH2OH, CH3O, C2, C2H, C2H3, C2H5, C2HO, C2H2O, and CH2O) species. Mixing is treated via the modified CurlÕs model [30] with a constant time-scale ratio C/ = 2.3. The radiative loss term is expressed on the basis of the optically thin assumption accounting for radiation from H2O, CO, CO2, CH4, and soot [18,31]. QRAD ¼ 4rSB
K X
pi ap;i ðT 4 T 4b Þ
i¼1
Table 2 Reaction rate constants for the second ring based PAH analogy [25] in the form Ai ai T bi expðEi =RT Þ Step
Ai
ai
bi
Ei/R
I II III IV V VI VII VIII IX X
0.783 · 1011 0.444 · 102 0.25 · 1011 0.17 · 106 0.215 · 1011 0.25 · 1010 0.2272 · 109 0.2272 · 109 0.18 · 1012 0.357 · 1022
1 1 1 1 1 1 1 1 1 1
0 2.43 0 1.42 0 0.15 0 0 0 3.176
0 3158 2347 729 3076 78 0 0 22,062 7471
Units are in K, kmol, m3, and s.
þ 4rSB fv ðT 5 T 5b Þ:
ð21Þ
In the above equation, rSB is the Steffan–Boltzmann constant (rSB = 5.669 · 108 W/m2 K4), pi is the partial pressure of species i in atmospheres, ap, i is the Planck mean absorption coefficient of species ‘‘i’’ in m1 atm1, T is the local flame temperature, Tb is the background temperature equal to 298.15 K, and C = 1.307 · 103 m1 K1 [18]. 5. Results and discussion The two non-premixed turbulent sooting ethylene–air flames investigated experimentally by
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Coppalle and Joyeux [11] and Kent and Honnery [3] are computed using an implicit parabolic x x formulation. The flames are fairly close in terms of physical conditions, and a greater variation, particularly in terms of the Re number, would have been preferred. The cross-stream directions are discretized by means of 105 and 63 computational cells, respectively, with each containing on average 100 particles. More than 20,000 axial steps were used to cover the flame lengths of 170 and 220 jet diameters. The flames considered are not piloted, and a conserved scalar approach [4] is applied up to axial locations of x/ D = 10 and 39 followed by a switch to the transported PDF approach. The results for the flame investigated by Coppalle and Joyeux [11] are unaffected by a further reduction in the switch-over point, while a switch at x/D = 20 has a modest (in the overall context) influence ( 6 40%) at the first two measuring stations for the flame of Kent and Honnery [3]. Results are unaffected further downstream. The calculation featuring the first two moments of the PSDF (M0 and M1) is functionally equivalent to the method of Lindstedt [6], and results obtained with this model are shown for comparison purposes. Figure 1 depicts the axial and radial temperature profiles for the flame of Kent and Honnery [3] obtained with four moments (M0–M3) and with coagulation in the coalescent limit. Calculations obtained with fewer moments suggest that differences are small up to x/D 80, whereas further downstream an increase in the number of moments solved results in an increase in the peak temperature. The observation is consistent with the radial profiles. Figure 2 shows the evolution of the radial profiles of the soot volume fraction in physical space at x/D = 46, 80.5, 115, and 161.
Fig. 1. Axial and radial profiles of temperature for the flame of Kent and Honnery [3]. Symbols represent experimental data [3] and the line predictions obtained with the method of moments.
779
Fig. 2. Radial profiles of soot volume fraction for the flame of Kent and Honnery [3]. Symbols as in Fig. 1.
An increase in the number of moments solved leads to a consistent decrease of the peak soot levels at the centre-line, and an excellent agreement is obtained at x/D = 115 and 161. For reasons of clarity, results are shown for M0–M1 and M0– M3 only. Predictions using M0–M2 are close to those obtained with four moments (M0–M3). Calculations have also been performed for the flame of Coppalle and Joyeux [11], and the corresponding axial profiles are shown in Fig. 3. The radial profiles of the mean soot volume fraction and its rms (Fig. 4) show similar trends, and an increase in the number of moments solved again leads to lower soot levels at the centre-line. The normalized [11] conditional PDFs of temperature and soot volume fraction at four different axial locations are presented in Fig. 5. The shape of the temperature PDF departs significantly from the Gaussian shape suggested by the measure-
Fig. 3. Axial profiles of soot volume fraction and its rms, and temperature and its rms for the flame of Coppalle and Joyeux [11]. Symbols represent experimental data and the line predictions obtained with the method of moments.
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Fig. 4. Radial profiles of soot volume fraction and its rms for the flame of Coppalle and Joyeux [11]. Notation as in Fig. 3.
Fig. 5. Measurements and computations of the normalized conditional PDFs of temperature and soot volume fraction for the flame of Coppalle and Joyeux [11]. Notation as in Fig. 3.
ments. The reasons for the differences are not clear, and the computed PDF appears rather more sensitive to soot concentrations than that measured. Related work featuring the non-sooting SANDIA flames shows an excellent agreement with temperature PDFs using the current methodology [18]. Furthermore, the experimental temperatures were obtained using two-colour pyrometry, and the investigators suggested that the technique is not accurate at temperatures 61000 K [11]. It is hence possible that the shape of the measured temperature PDF is subjected to significant bias, though a further possibility is that the currently applied optically thin approximation is questionable. The normalized conditional PDF of the soot volume fraction shows much better agreement, and the downstream evolution is comparatively well captured, which lends further support to the
current approach. It is also interesting to note that an increase in the number of solved moments has a positive impact at the first measuring station. Comparisons between the simplified ‘‘standard’’ method [6] and the method of moments (M0– M1) yield almost identical results for the soot volume fraction and its rms as shown in Fig. 6. To provide a more complete description of particle dynamics, the fractal aggregate limit of coagulation has also been incorporated within the framework of the method of moments. Comparison between the computed soot volume fractions for the flame of Kent and Honnery [3] (Fig. 7) shows that the inclusion of agglomeration leads to higher soot concentrations at the centreline though the differences are not large in the present case. Predictions obtained with the simplified method [6] for the Kent and Honnery [3] flame are
Fig. 6. Radial profiles of soot volume fraction and its rms for the flame of Coppalle and Joyeux [11]. Symbols represent experimental data and line predictions obtained solving for M0–M1.
Fig. 7. Radial profiles of soot volume fraction for the flame of Kent and Honnery [3]. Symbols represent experimental data and the line predictions obtained with M0–M3 accounting for coagulation and coagulation with agglomeration.
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compared with those obtained with the method of moments featuring the solution of M0–M3. In both cases, the more complete description of particle dynamics, considering coagulation and agglomeration, is implemented. Predictions were obtained with the two methods featuring identical representations of the physics governing particle collisions. Elevated soot levels are obtained with the method of moments at all axial locations though the differences between the two methods are not large ( 650%). The final aspect investigated in the present work is the treatment of surface growth on the basis of the above PAH analogy. The computed soot volume fractions for the Kent and Honnery [3] flame are shown in Fig. 8. All predictions presented feature the method of moments (M0–M3) and the more complete description of particle coagulation in the coalescence and aggregate limits. Figure 8 reveals a very strong sensitivity of soot levels to as, which is consistent with computational studies of laminar flames. The observed overprediction at the first two measurement locations vanishes further downstream resulting in an arguably excellent agreement at x/D = 115 and 161. Figure 9 compares the radial profiles of the mean soot volume fraction and rms for the flame of Coppalle and Joyeux [11], and the observations are consistent with those made above. Given the strong sensitivities and uncertainties associated with the functional dependencies of as, it is very encouraging that the application of ‘‘standard’’ values combined with a PAH analogy results in an acceptable behaviour without the need for a re-scaling of reaction rate constants. However, it may be pointed out that as = 0.1, arguably more typical of the post-flame region, is not compatible with the currently assumed site density [10].
781
Fig. 9. Radial profiles of soot volume fraction and its rms for the flame of Coppalle and Joyeux [11]. Symbols represent experimental data. Notation as in Fig. 8. See Fig. 6 for results obtained with the standard model.
6. Conclusions The method of statistical moments [20] has been implemented within a transported PDF methodology and combined with improved soot oxidation and PAH analogy based models for the soot surface chemistry. The results obtained show encouraging agreement with respect to the evolution of the conditional PDF of soot, and the mean and rms of the soot volume fraction. It is shown that an increase in the number of moments results in an improved agreement but that differences are comparatively modest downstream. The use of a comparatively simple soot model [6] also results in a surprisingly good agreement when combined with a transported PDF approach. The results highlight the importance of accurate modelling of turbulence-chemistry interactions in the context of soot dynamics in turbulent flames. A remaining principal uncertainty is, however, associated with the temporal evolution and functional dependencies of soot surface site densities and reactivities.
Acknowledgments The authors thank Fabian Mauss and Michael Frenklach for valuable discussions and the European Union for financial support through Grant G4RD-CT-1999-00075. The support of Dr. G. Roy and the Office of Naval Research under Grant N00014-2-1-0664 is also gratefully acknowledged. Fig. 8. Radial profiles of soot volume fraction for the flame of Kent and Honnery [3]. Symbols represent experimental data and line predictions obtained with M0–M3 and the second ring based PAH analogy with as = 1.00 and as = 0.75.
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[18] S.A. Louloudi, Transported Probability Density Function Modelling of Turbulent Jet Flames, Ph.D. thesis, Imperial College London, 2003. [19] H. Pitsch, E. Riesmeier, N. Peters, Combust. Sci. Technol. 158 (2000) 389–406. [20] A. Kazakov, M. Frenklach, Combust. Flame 114 (1998) 484–501. [21] M. Balthasar, F. Mauss, A. Knobel, M. Kraft, Combust. Flame 128 (2002) 395–409. [22] U.O. Ko¨ylu¨, G.M. Faeth, T.L. Farias, M.G. Carvalho, Combust. Flame 100 (1995) 621–633. [23] P. Roth, O. Brandt, S. von Gersum, Proc. Combust. Inst. 23 (1990) 1485–1491. [24] M. Frenklach, Phys. Chem. Chem. Phys. 4 (2002) 2028–2037. [25] M.L. Potter, Detailed Chemical Kinetic Modelling of Propulsion Fuels, Ph.D. thesis, Imperial College London, 2003. [26] J. Appel, H. Bockhorn, M. Frenklach, Combust. Flame 121 (2000) 122–136. [27] M. Frenklach, H. Wang, in: H. Bockhorn (Ed.), Soot Formation in Combustion: Mechanisms and Models. Springer-Verlag, Berlin, 1994, pp. 165–192. [28] R.P. Lindstedt, S.A. Louloudi, E.M. Va´os, Proc. Combust. Inst. 28 (2000) 149–156. [29] M.P. Meyer, The Application of Detailed and Systematically Reduced Chemistry to Transient Laminar Flames, Ph.D. thesis, Imperial College London, 2001. [30] J. Janicka, W. Kolbe, W. Kollmann, J. Non-Equilib. Thermodyn. 4 (1) (1979) 47–66. [31] W.L. Grosshandler, RADCAL: A Narrow-band Model for Radiation Calculations in a Combustion Environment, NIST Technical Note 1402, 1993.
Comments Heinz Pitsch, Stanford University, USA. The comparison with the experimental data looks much better than what we often see in predictions of laminar flames. Did you compare predictions of your chemistry model with laminar flames? What is the level of agreement for these? A second question I have is related to the transport of soot. In principle, there is a preferential transport between soot and the gas-phase, since particles do not diffuse. We have shown in a numerical study [1] that this effect could substantially affect the soot volume fraction. Differential diffusion effects are not included in your model. Do you conclude from your results that these effects are unimportant?
ers have explored the use of moment methods extensively (Ref. [24] in paper) in laminar premixed flames and the naphthalene oxidation chemistry has been independently validated using stirred reactor data (Ref. [25] in paper). The potential consequences of a lack of diffusive transport of soot particles have also been raised by Kronenburg et al. (Ref. [5] in paper) and it would be premature to conclude that such effects are generally unimportant. However, it is reasonable to suggest that the influence will diminish with increasing Reynolds number and that it is unlikely that the effect is predominant in the current flames.
Reference [1] H. Pitsch, et al., Comb. Sci. Technol. 158 (2000) 389–406.
J.P. Gore, Purdue University, USA. There is considerable self-absorption of radiation in the flames being considered. Is the radiation model capable of adequately treating the self-absorption process?
Reply. The basic soot model has been extensively validated (Ref. [6] in paper) in laminar counterflow diffusion flames with the rate constants applied unchanged in the current work. The level of agreement was found to be satisfactory. Frenklach and co-work-
Reply. The radiation treatment is based on a narrow band model (Refs. [18,31] in paper) that has been found to work satisfactorily for non-sooting flames (Ref. [18] in paper). The additional term (proportional to the soot volume fraction) exerts a significant influence in the cur-
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R.P. Lindstedt, S.A. Louloudi / Proceedings of the Combustion Institute 30 (2005) 775–783 rent flames and the computed temperature profiles show fair agreement with measurements. However, it is possible that a refinement in the treatment of self-absorption
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will prove beneficial and the strong coupling of different model elements suggests a need for more detailed experimental data as part of such an evaluation.