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Modeling of soot aggregate formation in a laminar ethylene/air coflow
diffusion flame with detailed PAH chemistry and an advanced sectional
aerosol dynamics model
Modeling of soot aggregate formation in a laminar ethylene/air coflow
diffusion flame with detailed PAH chemistry and an advanced sectional
aerosol dynamics model
Q. Zhang
a, H. Guo
b, F. Liu
b, G. J. Smallwood
b, M. J. Thomson
a*
a Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto,
Ontario, M5S 3G8, Canada; b Institute for Chemical Process and Environmental Technology, National Research Council of Canada, Building M-9, 1200 Montreal Road, Ottawa, Ontario, K1A 0R6, Canada
* Corresponding author: thomson@mie.utoronto.ca
1. Introduction
Soot formation is an important combustion research topic because of the environmental and health problems caused by the combustion-generated soot particles [1]. Compared to the global properties of soot particles such as their volume fraction, knowledge of the nanostructure and size distribution of soot particles is particularly helpful: (1) for better evaluating their health effects, optical properties, and contributions to flame radiative heat transfer; (2) for finding ways to eliminate unburned carbon; (3) for better understanding the individual sooting processes. As such, an interest to study soot nanostructure and size distribution persists in the literature [2-7]. It has long been observed that the flame-generated soot particles bear a fractal structure. Each fractal-like soot aggregate is formed by nearly spherical particles of almost equal size (primary particles) firmly connecting to each other [3].
Detailed soot models based on polycyclic aromatic hydrocarbons (PAHs) have been developed in the literature, e.g., [8,9]. The chemical kinetic mechanisms of this type of soot models describe the formation and growth of PAHs which are regarded as soot precursors. The Hydrogen-Abstraction/Carbon-Addition (HACA) mechanism along with the condensation of PAHs on the surface of soot particles are usually employed to describe the heterogeneous soot surface reactions. On the treatment of soot aerosol dynamics, there are several approaches: the moment method [8], the stochastic method [6], and the sectional model [10,11]. Sectional models can provide the mean properties and the size distribution of soot aerosols. Conventional sectional models solve for only one variable per section such as soot mass fraction [10], which is not adequate for modeling soot aggregate formation. To model the formation of the fractal-like soot aggregates, Park et al. [11] developed an advanced sectional model which solves two equations (number density of aggregates and number density of primary particles) per section. This sectional model was coupled to a PAH-based soot model [9] to study soot formation in plug flow reactors [11] and shock tubes [12]. Nevertheless, no such attempt has been made on multi-dimensional laminar diffusion flames.
Relatively comprehensive data on soot morphology has been experimentally obtained in the non-smoking laminar ethylene/air diffusion flame of Santoro el al. [13,14]. Megaridis and coworkers measured the size and number density of primary particles [2,3], and the fractal dimension of soot aggregates [5]. Puri et al. [4] obtained the aggregate number density. Recently, Iyer et al. [7] measured the number of primary particles per aggregate. In the current work, the advanced sectional aerosol dynamics model [11,12] and a recently published PAH-based soot model [9] are implemented in the diffusion flame of Santoro et al. The predicted results are validated against available experimental data in the literature. The objectives are: (1) to examine the performance of the sectional model in the multi-dimensional flame, and (2) to gain insights into the formation of soot aggregates. Considering that the computational load is very heavy, parallel computation is employed to speed up the calculation.
2. Methodology
2.1 Model description
The target flame of this work is the atmospheric pressure, non-smoking, laminar coflow ethylene/air diffusion flame of Santoro et al. [13,14] which has been extensively studied by many groups [2-5,7,15]. The burner is an 11.1 mm diameter fuel tube surrounded by a 102 mm diameter air annulus. The mean velocities of the fuel and air streams are 3.98 cm/s (flow rate 3.85 cm3/s) and 8.9 cm/s (flow rate 713.3 cm3/s), respectively. Further details of the burner setup and the measurement techniques can be found in the papers cited in Section 3. The fully coupled elliptic conservation equations for mass, momentum, gaseous species mass fractions, sectional soot aggregate and primary particle number densities, and energy in the two-dimensional axisymmetric cylindrical coordinate system are solved.
The sectional soot equations are introduced below while the other governing equations can be found in [16]. It is noted that the interactions between soot formation and gas-phase chemistry through soot nucleation, surface growth and oxidation processes and the interaction between soot formation and flame temperature through radiative heat transfer are accounted for in the governing equations.
In the current sectional model, each aggregate is assumed to be comprised of equally-sized spherical primary particles and a constant fractal dimension of 1.8 is assumed for the soot aggregates, consistent with the findings in [4,5]. The mass range of soot aggregates is divided into a number of discrete sections, each with a prescribed representative mass. According to their mass, soot aggregates are assigned to individual sections. The nucleation step connects the gaseous incipient species with the solid soot phase. By coagulation or surface growth, lower section aggregates migrate to higher sections. Conversely, higher section aggregates move to lower sections or become gaseous products by oxidation. Further details of the current sectional model can be found in [11,12]. In a convective-diffusive-reactive system with the gradient of temperature being high, the transport equations for the soot aggregates and primary particles in each section are assumed to have components attributed to flow convection, normal diffusion, thermophoresis, nucleation (nu), coagulation (co), surface growth (sg), oxidation (ox), and surface condensation (sc). Mathematically, the transport equations are:
( ),( ) ( ),( ) ( ),( ) ( ),( ) ( ),( ) ( ),( ) , , ( ),( ) ( ),( ) ( ),( ) ( ),( ) ( ),( ) 1 1 ( ) ( ) ( ) ( ) ( ) ( A P A P A P A P A A A P A P i i i i i i i Tr s i Tz s A P A P A P A P A P i i i i i nu co sg ox sc N N N N v u r D D r N V N V r z r r r z z r r z N N N N N t t t t t ρ ρ ρ ρ ρ ρ ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = + − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + + + ∂ ∂ ∂ ∂ ∂ i=1, 2,...,SN) (1)
In equation (1), r, z, v, u, are the radial and axial coordinates, the radial and axial velocities and the mixture density, respectively. ( ),( )A P
i
N is the number of either ith sectional aggregates (superscript A) or primary particles (superscript P)
per unit mass of the gaseous mixture; VTr,s and VTz,s are the radial and axial thermophoretic velocities of soot
aggregates and are calculated according to Talbot et al. [17]; SN is the total number of soot sections; A i
D is the ith
sectional aggregate diffusivity, which is calculated by the method in [18]. Note that the same A i
D appears in the
transport equation of the ith sectional primary particles.
Nucleation rate is calculated by the collision rate of two pyrene molecules in the free-molecular region with a van der Waals enhancement factor of 2.2 [8]. Coagulation terms are calculated using the same method in [12]. The collision kernel of two aggregates in the entire Knudsen number regime is calculated based on [18]. It is noted that this method assumes unitary coagulation efficiency, i.e., once two aggregates collide they stick together. The rates of the surface growth and oxidation of soot particles are calculated according to the HACA mechanism in [9]. All parameters are kept original, except the parameter — the fraction of the reactive soot surface. Here, the of Xu et al. [19] is used, i.e., = 0.004exp(10800/T). A similar correlation was used by Guo et al. [16] in the modeling of an ethylene/air diffusion flame. Using the original in [9], we found that the soot level could not be well reproduced in this flame. Unfortunately, there is no universal currently in the literature that works well under all circumstances. Further systematic investigations on should be pursued. PAH-soot surface condensation accounts for the growth of soot particles due to the collision of pyrene molecules with soot aggregates and the resulting condensation of pyrene molecules on the surface of soot aggregates [9]. The condensation rate in the current model is calculated by the collision theory between pyrene molecules and aggregates [11,12]. Not all collisions lead to successful condensation [20]. To account for the probability of sticking in each collision, the PAH-soot condensation efficiency is
introduced and assumed to be 0.5. This value of offers the results that agree best with the experimental data. A detailed sensitivity study of the parameter on soot formation showed that the predicted soot level and number density of primary particles both decrease as increases, because the PAH-soot condensation process competes with the nucleation process for pyrene molecules. The larger consumption of pyrene in the surface condensation process leads to a smaller nucleation rate and consequently a smaller HACA surface growth rate by which most of the soot mass is gained. Neglecting the condensation of pyrene on soot surface by setting = 0 significantly overpredicts the soot level and primary particle number density. Nevertheless, the calculated results are not sensitive to if it is larger than 0.5. For instance, assuming perfect sticking by setting = 1 only moderately to slightly underpredicts the soot level and primary particle number density.The source term in the energy equation due to the nongray radiative heat transfer by soot and gaseous species H2O, CO2 and CO is calculated using the Discrete-Ordinates Method (DOM) in
obtain the absorption coefficients of H2O, CO2 and CO at each band. More details of the radiation model can be
found in [15].
2.2 Numerical method
The computational domain covers 15.24 cm (z) × 4.71 cm (r) and is divided into 210 (z) × 88 (r) control volumes. Non-uniform mesh is used to save computational time while resolving the large gradients. Very fine grids are placed in the r-direction (resolution 0.2 mm up to r = 0.8 cm) and near the burner exit in the z-direction (resolution 0.5 mm up to z = 8 cm). It has been checked that further refinement of the mesh has negligible effect on the results. As in [15], a parabolic profile is assumed for the inlet fuel velocity and a boundary layer profile is assumed for the air stream velocity. The inlet temperatures for fuel and air are both assumed to be 300K. Symmetry, free-slip and zero-gradient conditions are enforced at the centerline, outer radial boundary and the exit boundary respectively. In total, 35 sections are used for the sectional model with a section spacing factor of 2.35 (i.e. the representative mass of one section is 2.35 times that of the preceding section). As in previous work [15,16,21], the finite volume method is used to discretize the governing equations. SIMPLE algorithm with the staggered mesh is used to handle the pressure and velocity coupling [22]. The diffusive terms are discretized by the second order central difference scheme while the convective terms are discretized by the power law scheme [22]. The gaseous species equations are solved
simultaneously to effectively deal with the stiffness of the system and speedup the convergence process [23]. The sectional soot equations are solved in the same manner as the species equations due to the stiffness of the system. The remaining governing equations are solved by the Tri-Diagonal Matrix Algorithm. The thermal and transport properties of gaseous species and chemical reaction rates are obtained by CHEMKIN subroutines [24] and the database associated with the selected reaction mechanism [9]. To speed up the calculations, the whole code is developed in the parallel mode with the domain decomposition method [21]. The whole computational domain is divided uniformly in the z-direction into 16 sub-domains and each sub-domain is assigned to one CPU for the calculation. It has been checked that the load balance of the code by such parallelization method is good. And a speedup of about 13 is achieved.
3. Results and discussion
3.1 Temperature and soot volume fraction
From left to right:
Fig. 1 Calculated temperature field in the unit of K with soot Fig. 2 Calculated temperature field in the unit of K without soot Fig. 3 Calculated soot volume fraction field in the unit of ppm
Fig. 4 Radial profiles of (a) temperature and (b) soot volume fraction at various axial heights above the burner. Line: calculated; symbol: measured [14].
The calculated two-dimensional temperature field is shown in Fig. 1. An annulus of high temperature can be found in the lower portion of the flame (z < 5 cm). The peak flame temperature (2045K) appears in the annular region instead of the centerline, which is consistent with previous modeling results [15]. To study the effect of soot
formation on the flame temperature, the whole model was rerun with the soot formation subroutine turned off. Fig. 2 shows the calculated temperature field without soot. A dramatically different shape of the temperature field is seen in Fig. 2. From Fig. 2, an annular region of high temperature is found in the lower portion of the flame, which is similar to Fig. 1. The high temperature regime, however, gradually shifts to the centerline as z increases and
eventually closes at the centerline. The peak flame temperature (2078K) is found in the centerline rather than in the annular regime. From the comparative study of the temperature fields, it can be concluded that soot formation and flame temperature are closely coupled in this ethylene/air coflow diffusion flame. It is worth noting that this close coupling was not found in our previous study [21] of the lightly sooting methane/air coflow diffusion flame of Smooke et al. [10]. The reason appears to be the different levels of soot volume fraction formed in these two flames. The peak soot volume fraction of the current ethylene/air flame is about 25 times of that of the methane/air flame. The primary coupling between soot formation and flame temperature is via radiative heat transfer. The large amount of soot formed in the current ethylene/air flame caused significant radiative heat loss, whereas the very small amount of soot formed in the methane/air flame contributed negligibly to the radiative heat loss. The close coupling of the temperature field and the soot volume fraction field of the current flame highlights the merit of using the detailed DOM radiation model in this study. It also indicates that soot formation in the current ethylene/air flame cannot be treated as a post process. Instead, the soot equations must be solved in a coupled fashion with the remaining governing equations.
The calculated two-dimensional soot volume fraction field is shown in Fig. 3. It is clearly seen that soot volume fraction peaks in a annular regime. The predicted peak soot volume fraction is 9.9 ppm, which is close to the measured peak value of 9 ppm [14,15]. A sharp decay of soot volume fraction is found on the tip of the annulus at z = 8.2 cm indicating a luminous flame height of 8.2 cm, which agian is close to the experimentally observed flame height of 8.8 cm [4,14]. A detailed image analysis of Figs. 1 and 3 reveals that the soot volume fraction annulus is enclosed inside the high temperature annulus. This indicates that soot tends to form in the fuel-rich high temperature regime, which is consistent with previous findings [2].
Fig. 4a compares the radial profiles of the calculated and measured temperatures at three axial heights above the burner. Overall, the temperature field is well reproduced. The only evident discrepancy is the centerline temperature at z = 2.0 cm, which is underpredicted by about 150 K. This might be due to the neglect of the fuel preheat effect by heat conduction from the base of the flame to the fuel pipe as shown in [15]. Fig. 4b shows the predicted radial profiles of soot volume fraction fv and the measurements by Santoro et al. [14] at two axial heights. It is observed
that the trend of the distribution and the peak level are both well captured by the current model. 3.2 Aggregate structure
In the current sectional model, all primary particles in the same section are assumed identical. However, inter-sectional primary particles are allowed to have different diameters. Thus, a good measure of primary particle size is the nominal diameter dP defined as (6fv/ NP)1/3 where NP is the number density of primary particles summed over all
sections in the unit of particles/cm3. Fig. 5a shows the calculated and measured distributions of dP along the gas
streamline exhibiting maximum soot volume fraction. It is noted that although the current model tends to overpredict dP in upper flame region, the trend of the dP distribution and the absolute level of dP are both well captured. dP first
increases owing to surface growth and then decreases due to soot oxidation.
Fig. 5b compares the calculated and measured distributions of NP (number density of primary particles summed over
all sections) along the maximum soot volume fraction streamline. It can be found that the absolute level of NP is
reasonably predicted by the current model. NP first rapidly increases due to nucleation. Then it slightly decreases
with increasing z. This is the region where coagulation and surface growth dominate. A similar phenomenon was observed by Puri et al. [4].
Fig. 5c demonstrates the calculated distribution of the average number of primary particles per aggregate nP defined
as NP / NA where NA is the number density of aggregates summed over all sections in the unit of particles/cm3. Also
shown is the measured nP distribution of Iyer et al. [7]. Although the first measured point agrees perfectly with the
model, this data point is most likely a measurement error according to Iyer et al. [7]. Thus, this point is neglected in the following discussion. From Fig. 5c, the calculated nP monotonically increases with z due to coagulation. Such an
increasing trend is especially pronounced at the burner exit region, i.e., z 1.5 cm where nucleation mainly takes place and produces a large amount of small primary particles. Those small particles have large mobility and thus coagulate strongly. As z further increases, soot aggregates grow larger and their mobility decreases. Meanwhile, as a result of the initial rapid coagulation, the number density of aggregates decreases. Thus, the coagulation rate decreases and the nP curve plateaus. The increasing trend of nP with z is qualitatively consistent with the
experimental observation [7]. The calculated nP level, however, is significantly higher than the measurement by
(indicated by nP) of primary soot particles which are the building blocks of soot aggregates, although the average
size and the total number of the primary particles are reasonably predicted (see Figs. 5a and 5b). One possible reason for this discrepancy is the assumption of the unitary coagulation efficiency of soot aggregates (i.e., once two
aggregates collide they always stick together forming a larger aggregate) in the current sectional model, which neglected particle-particle interactions and particle-fluid interactions. For example, D’Alessio and coworkers [25,26] observed that the coagulation efficiency of soot particles could be well below unity, especially for the small particles less than 3 nm. Due to the ‘thermal rebound effect’ of the small particles, their coagulation efficiency could be 2~3 orders of magnitude lower than that of the large soot particles whose coagulation efficiency is on the order of unity [25,26]. The neglect of the ‘thermal rebound effect’ [25], which essentially is a particle-particle interaction, could lead to the over-clustering of the primary particles. Kellerer et al. also observed that the soot coagulation efficiency was less than unity in their shock tube studies [27]. With an assumption of unitary coagulation efficiency, no satisfactory agreement could be found between their simulation and experiment [27]. Another possible reason for the overprediction of np is the neglect of the fragmentation of soot aggregates in the current sectional model. Soot aggregate fragmentation can occur for several reasons associated with particle-particle interactions and particle-fluid interactions. Aggregate fragmentation can occur if the impact energy of the colliding particle is greater than the bond strength of some weakly attached primary particles. Aggregate fragmentation can also occur due to the hydrodynamic effect [28]. As aggregates grow by coagulation, their loose fractal structure expands, which causes them to be more susceptible to the local strain rate of fluid motion. The fluid strain tends to tear apart the large aggregate. When the strength of this hydrodynamic effect overcomes the bond strength of some weakly attached primary particles, fragmentation occurs. Lastly, aggregate fragmentation can occur due to particle oxidation [29]. Soot aggregate fragmentation, which can reduce the clustering of primary soot particles, is neglected in the current aerosol dynamics model. If the fragmentation is taken into account, it is expected that the average number of primary particles per aggregate nP would be lower. Further study on the coagulation efficiency and fragmentation of
soot aggregates should be conducted in the future and more measurements on the nanostructure of soot aggregates are needed to further validate the sectional model and improve our understanding of the soot aggregation process.
4. Conclusions
A PAH-based soot model, an advanced sectional aerosol dynamics model and a detailed radiation model are successfully implemented in a two-dimensional laminar ethylene/air diffusion flame to model the formation of the fractal-like soot aggregates. The model is validated against available experimental data in the literature. The flame temperature, soot volume fraction, the average size and number density of primary particles are reasonably well reproduced. The average number of primary particles per aggregate is significantly overpredicted by almost one order of magnitude. This discrepancy is presumably due to the assumption of the unitary coagulation efficiency of soot aggregates in the current sectional model and the neglect of the fragmentation of soot aggregates. It is very challenging to predict the nanostructure of soot aggregates. To effectively do so, the average size of the primary soot particles, which are the building blocks of soot aggregates, must be reasonably well predicted firstly. Then, the clustering of primary soot particles must be modeled appropriately by taking into account various particle-particle interactions and particle-fluid interactions that can lead to a non-unitary coagulation efficiency of soot aggregates or the fragmentation of soot aggregates. More experimental data on aggregate structure is needed for further validating and improving the current sectional soot model and better understanding the soot aggregation process.
Fig. 5 Distributions of (a) nominal primary particle diameter, (b) primary particle number density, and (c) primary particle number per aggregate along the maximum soot volume fraction streamline. Line: calculated; symbol: measured. Measurements are from [3], [2] and [7] respectively.
Acknowledgements
The authors gratefully acknowledge AUTO21TM for the financial support of this project. Dr. S.H. Park, Dr. J.Z. Wen and Dr. S.N. Rogak are acknowledged for the helpful discussions of the sectional aerosol dynamics model. Last but not the least, Mr. R. Jerome and Dr. C.P.T. Groth are acknowledged for the help with the NRC/ICPET and
UTIAS/HPACF high performance parallel computing clusters respectively.
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