Aerosol Science and Technology, 38:963–971, 2004 c American Association for Aerosol Research Copyright ! ISSN: 0278-6826 print / 1521-7388 online DOI: 10.1080/027868290513847
Monte Carlo Simulation of Multicomponent Aerosols Undergoing Simultaneous Coagulation and Condensation Z. Sun,1 R. L. Axelbaum,1 and J. I. Huertas2 1 2
Department of Mechanical Engineering, Washington University in St. Louis, St. Louis, Missouri Department of Mechanical Engineering, University of Los Andes, Bogota, Colombia
A Monte Carlo method was developed to simulate multicomponent aerosol dynamics, specifically with simultaneous coagulation and fast condensation where the sectional method suffers from numerical diffusion. This method captures both composition and size distributions of the aerosols. In other words, the composition distribution can be obtained as a function of particle size. In this method, particles are grouped into bins according to their size, and coagulation is simulated by statistical sampling. Condensation is incorporated into the Monte Carlo method in a deterministic way. If bins with fixed boundaries are used to simulate the condensation process numerical dispersion occurs, and thus a moving bins approach was developed to eliminate numerical dispersion. The method was validated against analytical solutions, showing excellent agreement. An example of the usefulness of this model in understanding aerosol evolution is presented. The effects of the number of particles and number of bins on the accuracy of the numerical results are also discussed. It was found that with 20 bins per decade and 105 particles in the control volume results with less than 5% error can be obtained. The results are further improved to within 2% error by filtering the statistical noise with a cubic spline algorithm.
INTRODUCTION Traditional numerical methods to simulate aerosol dynamics, such as the moment method or sectional method, are powerful tools to understand or predict the evolution of particle size distribution. Nonetheless, these methods have limitations when more information about the aerosol is needed, for example, when the aerosol contains more than one component and information about the composition distribution is needed as a function of particle size. The multicomponent sectional method generally does not yield information on composition distribution within a secReceived 9 January 2004; accepted 22 July 2004. This work was supported by the NASA Microgravity Combustion Program under contracts NAG3-1910 and NAG3-1912. Address correspondence to Richard L. Axelbaum, Department of Mechanical Engineering, Washington University, St. Louis, MO 63130, USA. E-mail:
[email protected]
tion because it employs an assumption that all particles within a section have the same composition (Gelbard and Seinfeld 1980). This assumption is made for mathematical convenience. Without it the sectional method would be considerably more complicated and computationally limiting. In addition, sectional algorithms and code must be reconstructed when additional complexities are added to the problem. For example, to understand the formation of aggregates, Xiong and Pratsinis (1993) employed a “two-dimensional” sectional method where both surface area and volume distribution were determined and the surface fractal dimension was estimated from first principles. With the standard sectional method, fractal dimension is lost and modifying the scheme to retrieve it is not an easy task. A more robust method, and one that is more flexible with respect to adding complexity, would be valuable, and the Monte Carlo method is an excellent candidate. There are many approaches to applying Monte Carlo techniques to aerosols. Kourti and Schatz (1998) and Debry et al. (2003) employed a Monte Carlo approach to solve the General Dynamic Equation. Monte Carlo methods have also been used for direct simulation of aerosol dynamics. Bird (1976) developed a direct simulation Monte Carlo (DSMC) method where collision pairs are chosen through the known collision rates, not by the trajectory of the particles. By not considering the spatial position of the particles, the DSMC method consumes much less CPU time than the classical Monte Carlo method. Since this work, various versions of the Monte Carlo method have been applied to study particle size distribution undergoing coagulation and/or aggregation (Vemury et al. 1994; Smith and Matsoukas 1998; Tandon and Rosner 1999; Kruis et al. 2000; Rosner and Yu 2001; Benson et al. 2004), crystallization (van Peborgh Gooch and Hounslow 1996), fragmentation (Shah et al. 1977; Liffman 1992) and polymerization (Spouge 1985). Recently, Mitchell and Frenklach (1998, 2003) incorporated surface growth into the Monte Carlo method to study particle aggregation with simultaneous surface growth. Efendiev and Zachariah (2002, 2003) used a hierarchical hybrid Monte Carlo method to study aerosol 963
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coagulation and phase segregation. In all these works, the probabilities of all possible collision pairs were calculated. For a simulation that contains 104 –106 particles, the calculation of all probabilities consumes a significant amount of computation time. Husar (1971) addressed this concern when simulating the coagulation process in aerosols. In this study the particles were grouped into bins according to their size. Results that compared favorably with experiments were obtained with a relatively small number of bins, but the error caused by the use of bins was not discussed in this work. Huertas (1997) followed the same approach and incorporated condensation into the Monte Carlo method and employed it to study the sodium/halide flame encapsulation (SFE) process (DuFaux and Axelbaum 1995; Axelbaum et al. 1997). Although valuable insight was obtained from this simulation, the results showed significant numerical dispersion due to the use of bins when condensation was considered. In this study, an improved Monte Carlo method, incorporating moving bins, is developed to simulate a process involving simultaneous condensation and coagulation. Moving bins, as opposed to fixed bins, ensure that numerical dispersion is avoided. This study also includes a validation of the Monte Carlo method for single and multicomponent aerosol dynamics by comparing the results with those of analytical solutions. In addition, the effects of the number of sample particles and bins on accuracy are discussed, and the associated errors are presented. The versatility of the code is demonstrated by an example that considers a two-component aerosol undergoing condensation and coagulation, and the simulation yields results in terms of size and composition distribution. This article is organized as follows: In the next section, we describe the Monte Carlo method for coagulation and the concept of bins, and we study how the accuracy of the results varies with the number of bins as well as the number of sample particles (i.e., sample volume). In the section that follows, we discuss the reason that numerical dispersion occurs when bin boundaries are fixed when simulating condensation, and then we introduce the moving bin method to eliminate numerical dispersion. The results are validated against analytical solutions. In the final section, the improved Monte Carlo method is used to study a two-component aerosol undergoing simultaneous coagulation and condensation. MONTE CARLO METHOD FOR COAGULATION The classical theory of coagulation is a population balance, which is essentially a scheme for keeping track of particle collisions as a function of particle size. Let Fij be the number of collisions occurring per unit time per unit volume between two particles of diameter di and d j . The collision frequency is given by Fij = β(di , d j )n i n j , where n i and n j are the number concentrations of particles with diameters di and d j , and β(di , d j ) is the collision frequency function, which depends on the size of the colliding particles and the mechanism of particle collision.
Thus, the net generation of particles of size k, given by ∞ ! 1 ! dnk Fij − Fik , = dt 2 i+ j=k i=1
can be written as ∞ ! dnk 1 ! β(vi , v j )n i n j − n k β(vi , vk )n i . = dt 2 i+ j=k i=1
The probabilistic nature of the coagulation process makes it straightforward to simulate the process by a Monte Carlo method. In the approach used here, position and velocity of the particle are not modeled, and only collisions are considered (Husar 1971; Huertas 1997). Let us see how collision pairs can be simulated by generating random numbers. If we only consider binary collisions, the collision rate between particles of size i and j is given by β(di , d j )n i n j , and the total collision rate is the sum of all collision rates for all collision pairs. So the probability of a collision of type i − j occuring is βij n i n j pij = " M " M , i=1 j=i βij n i n j
where M is the total number of size types. If all probabilities are represented along a probability vector ranging from 0 to 1, each type of collision will occupy a unique segment of a length that is proportional to its probability, as seen in Figure 1. A collision can be simulated by generating a random number between 0 and 1, and its location on the probability vector identifies the type of collision that “occurred.” The Monte Carlo simulation is carried out in the following way: First, a sample volume is chosen that is large enough to contain a statistically significant number of particles and small enough for affordable computation. The probabilities of all collision pairs are calculated and the results are stored in a probability vector. Then a time step, "t, is chosen and the number of collisions during that time step is determined by " M occurring "M h = "t · i=1 β (d j=i ij i , d j )Ni N j /V , where V is the sample volume and Ni and N j are the number of particles of diameter di and d j in the sample volume. Then, h collisions are randomly chosen by generating h random numbers, the time is advanced
Figure 1. Probability vector where the collision probabilities are aligned in a line from 0–1, such that a random number between 0 and 1 will uniquely identify the type of collision that occurs in the Monte Carlo simulation.
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by "t and the size distribution is updated. To ensure convergence, the time step is reduced until the solution is not a function of the size of the time step. To minimize computation time, the probability vector need not be updated after every time step but instead only when the distribution has undergone significant change. Since the number of particles decreases √ during coagulation and the statistical error is proportional to 1/ N , the number of particles must be maintained above a minimum value to ensure accuracy. This is accomplished by continuously monitoring the number of particles, and when the number drops to a prescribed value the sample volume is increased and the total number of particles is increased proportionally to preserve number concentration. For this work, when the total number of particles in the sample volume N drops by 50% of its initial total number N0 , the sample volume is doubled, bringing N back to N0 . With the traditional Monte Carlo method, calculations are made for all collision pairs, and the computational time is proportional to M 2 . To save computational time and memory, we group the particles into bins according to their size, and all particles in bin i are assumed to have the same probability of colliding with particles of bin j. The finest set of bins can be obtained when bin i contains only particles made of i-mers. If the particle size covers a wide range, logarithmic binning, where the boundaries of the bins vary linearly on a log scale, is more useful. Thus, if we have a total of k bins, with diameter minima and maxima d0 and dk , respectively, the bin boundaries are set by di+1 /di = (dk /d0 )1/k . Results Without Condensation The Monte Carlo method with binning is first compared with the exact analytical solution for coagulation of a singlecomponent aerosol, following Smoluchowski (1917). The aerosol evolves from a monodisperse distribution with an initial total number concentration of N0 and a constant collision frequency function β0 . Figure 2 shows the relative error in total number concentration between the Monte Carlo simulation and the analytical solution, where τ = t/τcoll is the nondimensional time, τcoll = 2/N0 β is the characteristic coagulation time, and n is the number of particles used in the simulation. The binning of the size distribution in the Monte Carlo simulation is logarithmic, and 20 bins/decade are used. As seen in Figure 2, the error in all cases is below 0.1% for τ < 1000, showing excellent agreement between the Monte Carlo simulation and the exact solution. Furthermore, the error does not increase significantly with time. To study the predictive capability for particle size distribution, the Monte Carlo method is tested against the analytical solution of Gelbard and Seinfeld (1978) for a single-component coagulating aerosol. For all cases considered in this discussion the collision frequency function β0 is assumed to be constant. The solutions are expressed in terms of dimensionless groups, and analytical solutions are shown as solid lines, while Monte Carlo solutions are represented by discrete points.
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Figure 2. The relative error in total number concentration for the Monte Carlo simulation compared to the exact analytical solution for an initially monodisperse aerosol with constant collision frequency function. The numerical and analytical results are shown in Figure 3, where D¯ = D/D0 , D0 is the initial mean diameter, n the number density, N0 the total number of particles initially, and τ = N0 β0 t is the dimensionless time. In this run, 20 bins per decade and N0 = 104 particles are used. It can be seen that the Monte Carlo result, by its statistical nature, is scattered around the analytical solution. A cubic spline smoothing is performed to filter the statistical noise, and this result is shown by a dashed line. Defining the relative error as #" ( f i − f i,exact )2 " 2 , f i,exact
where f i is the Monte Carlo solution and f i,exact is the analytical solution, we find that there is a 7.24% error in Monte Carlo results, which improves to 3.52% after cubic spline smoothing. The error can be reduced by increasing the number of particles and number of bins, and the effects of these parameters are shown in Figure 4. A matrix of run conditions is performed with 5, 10, 20, 40, and 80 bins/decade and 104 , 105 , 106 , and 107 particles in the initial sample volume. Figure 4a shows the errors in the raw (unfiltered) Monte Carlo results. With the same number of bins per decade, a larger number of particles gives better results, as expected, due to improved statistics. For the same number of particles, with an increased number of bins the error first drops but then increases. The decrease in error when number of bins per decade increases from 5 to 20 is due to the increased resolution of the particle size distribution. The subsequent increase in the error when the number of bins increases further is because with more
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that the error can be controlled with a modest number of bins and particles. The results in Figure 4 suggest that an adaptive binning could be employed in the Monte Carlo simulation. For applications that initially have a narrow particle size distribution, a large number of bins should be used to resolve the particle size distribution, and thus a large number of particles are needed to control statistical error. The number of bins and number of particles in the simulation can be decreased when the particle size distribution becomes broader, thus reducing computational time without sacrificing accuracy.
Figure 3. Comparison of particle size distribution obtained from the Monte Carlo simulation with that of an analytical solution for a single-component aerosol undergoing coagulation. The data are also shown for the initial distribution and the Monte Carlo solution after cubic spline smoothing. Twenty bins/decade and 104 particles are used in this simulation. bins the average number of particles in each bin decreases, which increases the statistical fluctuation. A more accurate particle size distribution can be recovered by smoothing. This is clearly shown in Figure 4b, where the error after smoothing is plotted. The error monotonically decreases with increasing number of bins. With 105 particles and 20 bins/decade, a Monte Carlo result with 1.5% error can be obtained and this can be further improved to within 0.5% by increasing N0 and the number of bins. Defining the statistical error as #"
( f i − f i,smooth )2 " 2 , f i,smooth
where f i,smooth is the solution after cubic spline smoothing, the variation of the statistical error with average number of particles per bin can be evaluated, as shown in Figure 4c. The number of particles per bin is calculated from the total number of particles divided by the number of bins per decade. In general, the statistical error decreases as the average number of particles per bin increases. In this study, the midsize of each bin is used to calculate β i and the probability vector. Calculations using the mass-mean size of the particles in each bin were also performed, and the results showed no appreciable difference. The change in error with time is also considered. From Figure 4b the relative error with 105 particles and 20 bins/decade is 1.5% at τ = 20. For a two-order of magnitude increase in τ to 2000, the error increases to 1.9%. Recalling Figure 2, where the evolution of error was considered in greater detail, it is clear
INCORPORATING CONDENSATION INTO THE MONTE CARLO METHOD A deterministic approach is now employed to incorporate condensation into the Monte Carlo method. In principle, the condensation equation can be solved for each particle of arbitrary size. The new size of the particle after δt can be calculated and the size distribution updated. To save computational time, we assume that all particles inside each bin have the same size, and that size is the mass-mean size of the particles in that bin. Thus, the condensation equation needs to be solved only once for each bin. As we have seen, employing bins with fixed boundaries and variable mass mean particle size yields accurate results for coagulation. For condensation, this approach can result in numerical dispersion. Numerical dispersion results in artificial peaks and valleys in the distribution. The reason for numerical dispersion is illustrated in Figure 5. Figure 5a shows the true evolution of the particle size distribution during time interval δt, when the particles are subject to condensation. In the Monte Carlo simulation, seen in Figure 5b, particles in bin i grow large enough to fall into the next larger bin, i + 1. Since all of the particles in bin i have the same size, they will be removed out of bin i and fall into bin i + 1. At the same time, the particles in the smaller bin, i − 1, may not grow large enough to fall into bin i. This is true for two reasons. First, the condensation rate for bin i can be different from that of bin i + 1. Second, to conserve both mass and number, the mean size of the particles in the bin varies, i.e., it is not assumed to be the mean bin size. Thus, even for identical condensation rates, one bin of particles can grow out of its bin while another can stay within its bin. If this occurs the number of particles in bin i will become zero and create an artificial valley in the distribution. Similarly, the particles that were in bin i + 1 may not grow out of bin i + 1, while those of bin i grow into i + 1, creating an artificial peak in the distribution. One remedy for this dispersion is to employ moving bins. When condensation occurs, the boundaries of the bins grow in accordance with the growth of a particle of the same size. In this way no mass passes through the boundaries during condensation, and dispersion is avoided without entailing significant complexity. In Figure 5c, where moving bins are illustrated, we see that all of the particles stay in their original bin during the time interval δt but the mass mean particle size of each bin grows.
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(a)
(b)
(c) Figure 4. The effect of the number of bins and the number of particles on the error of the Monte Carlo method: (a) the error of the raw Monte Carlo data, (b) the error of the data after cubic spline smoothing, and (c) the statistical noise versus the average number of particles per bin. In this way the true distribution of the particles after time interval δt is reproduced (Figure 5a). Results with Condensation The numerical dispersion that results from fixed bins can be clearly seen in Figure 6. Both coagulation and condensation are considered in this simulation. The initial distribution is the
same as in Figure 3. The growth rate is dv/dt = σ v, where σ/N0 β0 = 105 and v is the volume of the particle. Figure 6 shows the results at τ = 10 with 40 bins/decade and 105 particles. The relative error for the raw Monte Carlo results with fixed bins is 87%. The improved results obtained by employing moving bins are also shown in Figure 6. The relative error is within 4%. Excellent agreement is achieved without numerical diffusion or
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Figure 6. Elimination of numerical dispersion by employing moving bins when aerosols are undergoing simultaneous coagulation and condensation. The results at τ = 10 with fixed bins and moving bins are shown. The initial distribution and coagulation rate are the same as in Figure 2, and the condensation rate is σ = 105 N0 β0 . Forty bins/decade and 105 particles are used. noticeable increase in the error is observed for the other run conditions of the run matrix of Figure 4. Since the growth rate equation is only applied on the mean size of the particles in each bin and on the bin boundaries, if the number of bins per decade is not excessively large, the computational overhead to employ condensation and moving bins is small.
Figure 5. Illustration of numerical dispersion and the effect of moving bins when an aerosol undergoes condensation: (a) time evolution of particle size distribution after δt, (b) artificial valley and peak due to fixed bins, (c) reproduction of true distribution with moving bins. The broken tic marks on the abscissa of (c) are the boundaries of fixed bins, and the solid tic marks are the boundaries of moving bins. dispersion, while the aerosol has substantially evolved and the mean size has increased by a factor of 40. Moving bins can reproduce the exact solution when the aerosol undergoes only condensation, and they do not introduce excessive errors when coagulation is included. With initially 20 bins per decade and 104 particles, for the run conditions of Figure 6 and τ = 20 the relative error of the raw Monte Carlo data is 7.7%. This is only slightly higher than the 7.2% for the conditions of Figure 4, which has the same coagulation rate. No
MONTE CARLO SIMULATION OF MULTICOMPONENT AEROSOLS The Monte Carlo method is flexible and including more than one component does not add significant complexity to the problem. Compared to the single-component case, an extra array dimension is needed to store the number density and mass for each additional component. The composition domain must be discretized as well. Thus, the computational domain has multiple dimensions: one in size and the other in composition, and the particles are classified in bins with respect to their size and composition. Since the collision frequency function β can often be assumed to be only a function of size, the procedure to sample random collision is the same as for the single-component aerosol. Once the collision pair is identified, a similar Monte Carlo procedure is used to determine the composition of each particle of the collision pair. In other words, the composition distribution for each size bin leads to a probability vector based on composition. By generating a random number, the composition of the particle is identified and the composition and distribution are updated accordingly. With this approach, we retain n(di , Y1 j , . . . , Ykj ), and m(di , Y1 j , . . . , Ykj ), where k is the number of components and Yk is the mass fraction of component k.
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Thus, we obtain a result where particles of the same size can have different compositions. The moving bins method can be applied to condensation for multicomponent aerosols. If the condensation rate is a function of particle size only, the approach discussed in the previous section can be employed without modification. If the condensation rate is a function of composition, the growth rate of the larger boundary of the bin can be calculated based on the mean composition of the particles in that bin. With this approach, there is a possibility that some of the particles will grow out of the larger boundary of the bin, or the smaller boundary of the bin will grow too fast and surpass some of the particles in that bin. Adjustment on the growth rate of the boundary can be made to prevent some of these situations. For the rare case where particle growth out of bins cannot be avoided, the resulted numerical dispersion is much less severe than that of the fixed bin approach. For the two cases we will discuss below, no adjustment on the growth rate of the boundaries is made and no numerical dispersion is observed. To validate the Monte Carlo method for multicomponent aerosols, results are tested against the analytical-numerical solution of Katoshevski and Seinfeld (1997). In this case, the Monte Carlo method is examined by comparing the solution for a twocomponent aerosol undergoing coagulation and condensation. The components have different condensation rates in the form dmi /dt = αi m iδ for the ith component, where α and δ are constants. Brownian coagulation is represented by the Fuchs form of the coagulation coefficient. The results are presented in terms of mass distribution, qi (m, t) = m i n(m, t). Figure 7 shows the initial mass distribution and the mass distribution after 6 h of evolution for each component. For this
Figure 7. Comparison of results from the Monte Carlo simulation with an analytical-numerical solution of a two-component aerosol undergoing coagulation and condensation. The Fuchs form of the Brownian coagulation coefficient is used, and the condensation growth rate is described in the text.
Figure 8. Particle size and composition distributions for a twocomponent aerosol undergoing simultaneous coagulation and condensation: (a) initial distribution, (b) τ = 10, and (c) τ = 70, i where τ = N0 β0 t, β = β0 , dm = G i m i , and G 1 = G 2 = 0.01 dt N0 β0 .
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calculation there are 40 bins/decade in size and 10 bins in composition, N0 = 105 particles, and moving size bins are employed. Note that the initial mass distribution is different for the two components and the two components also have different condensation rates. Parameter values in the growth law are: α1 = 14 × 10−10 g1−δ s−1 , α2 = 8 × 10−10 g1−δ s−1 , and δ = 2/3. For the purpose of plotting the distribution, particle mass is converted to particle diameter D p by assuming spherical particles with ρp = 1 g cm−3 . Noticeable growth of both components is observed in the figure, and component 1 has a greater growth rate. Excellent agreement is obtained between the analytical-numerical solution and the Monte Carlo solution. An important benefit of this Monte Carlo method is that information on the composition distribution of particles with the same size can be easily obtained. To illustrate this, a simple run is performed with an initially lognormal particle size distribution where half of the particles are composed of only component 1 and the other half are composed of only component 2, as shown in Figure 8a. In practice, this could be two streams of particles coming together, and each stream of particles has a lognormal distribution and is composed of different materials. The aerosols are experiencing simultaneous coagulation and condensation. For this illustration the growth rate for the ith component of the i particles is assumed to be of the form dm = G i m i . Figure 8b dt shows the results at τ = 10 and G 1 = G 2 = 0.01 N0 β 0 . Average particle size has grown from 5 nm to about 10 nm. Collisions between particles on opposite ends of the composition space result in new particles with a composition of about 50% of both components, and this is why we see the particle composition distribution peaking at 0.5. Since the time is short, the composition distributions are still broad. Figure 8c shows the particle size and composition distribution at τ = 70. As expected, the composition distributions are narrower and the particles are largely composed of 50% of each component, because random collisions between particles eventually redistribute the mass of the two components evenly among particles.
CONCLUSIONS The Monte Carlo method with moving size bins was demonstrated to be suitable for simulating multicomponent aerosol dynamics with simultaneous coagulation and condensation. Binning reduces computational time while maintaining accuracy. For the case of only coagulation, results with less than 5% error can be obtained with 20 bins per decade and 105 particles in the sample volume. The results can be improved further to within 2% error by filtering the statistical noise with a cubic spline smoothing. The accuracy of the Monte Carlo results improves with increased number of particles in the sample volume and increased number of bins. Condensation can be incorporated into the Monte Carlo method in a deterministic way. If bins with fixed boundaries are used to simulate condensation processes, numerical dispersion is present, but this can be eliminated by employing moving bins. The computational overhead needed to
include condensation is small and the accuracy of the Monte Carlo method is maintained. The primary benefit of the Monte Carlo approach presented is that the composition distribution of particles of a given size can be obtained for a multicomponent aerosol process. REFERENCES Axelbaum, R. L., Dufaux, D. P., Frey, C. A., and Sastry, S. M. L. (1997). A Flame Process for the Synthesis of Unagglomerated, Low Oxygen Nanoparticles: Application to Ti and TiB2, Metall. Mater. Trans. B, 28B:1199–1211. Benson, C. M., Levin, D. A., Zhong, J., Gimelshein, S. F., and Montaser, A. (2004). Kinetic Model for Simulation of Aerosol Droplets in HighTemperature Environments, J. Thermophys. Heat Transfer 18(1):122– 134. Bird, G. A. (1976). Molecular Gas Dynamics, Clarendon Press, Oxford. Debry, E., Sportisse, B., and Jourdain, B. (2003). A Stochastic Approach for the Numerical Simulation of the General Dynamics Equation for Aerosols, J. Computat. Phys. 184(2):649–669. DuFaux, D. P., and Axelbaum, R. L. (1995). Nanoscale Unagglomerated Nonoxide Particles from a Sodium Coflow Flame, Combustion and Flame, 100: 350. Efendiev, Y., and Zachariah, M. R. (2002). Hybrid Monte Carlo Method for Simulation of Two-Component Aerosol Coagulation and Phase Segregation, J. Colloid Interface Sci. 249:30–43. Efendiev, Y., and Zachariah, M. R. (2003). Hierarchical Hybrid Monte-Carlo Method for Simulation of Two-Component Aerosol Nucleation, Coagulation and Phase Segregation, J. Aerosol Sci. 34:169–188. Gelbard, F., and Seinfeld, J. H. (1978). Numerical Solution of the Dynamic Equation for Particulate Systems, J. Comput. Phys. 28:357–375. Gelbard, F., and Seifeld, J. H. (1980). Simulation of Multicomponent Aerosol Dynamics, J. Colloid Interface Sci. 78(2):485–501. Husar, R. B. (1971). Coagulation of Knudsen Aerosols, Ph.D. Dissertation, University of Minnesota, Twin Cities, MN. Huertas, J. I. (1997). Monte Carlo Simulation of Two-Component Aerosol Processes, D.Sc. Dissertation, Washington University in St. Louis, St. Louis, MO. Katoshevski, D., and Seinfeld, J. H. (1997). Analytical-Numerical Solution of the Multicomponent Aerosol General Dynamic Equation—with Coagulation, Aerosol Sci. Technol. 27:550–556. Kourti, N., and Schatz, A. (1998). Solution of the General Dynamic Equation (GDE) for Multicomponent Aerosols, J. Aerosol Sci. 29(1–2):41– 55. Kruis, F. E., Maisels, A., and Fissan, H. (2000). Direct Simulation Monte Carlo Method for Particle Coagulation and Aggregation, AIChE J. 46(9):1735– 1742. Liffman, K. (1992). A Direct Simulation Monte-Carlo Method for Cluster Coagulation, J. Comput. Phys. 100:116. Mitchell, P., and Frenklach, M. (1998). Monte Carlo Simulation of Soot Aggregation with Simultaneous Surface Growth—Why Primary Particles Appear Spherical, Proc. Combust. Inst. 27:1507. Mitchell, P., and Frenklach, M. (2003). Particle Aggregation with Simultaneous Surface Growth, Phys. Rev. E 67(6):061407. Rosner, D. E., and Yu, S. Y. (2001). Monte Carlo Simulation of Aerosol Aggregation and Simultaneous Spheroidization, AICHE J. 47(3):545–561. Shah, B. H., Ramkrishna, D., and Borwanker, J. D. (1977). Simulation of Particle Systems Using the Concept of the Interval of Quiescence, AIChE J. 23:897– 904. Smith, M., and Matsoukas, T. (1998). Constant-Number Monte Carlo Simulation of Population Balances, Chem. Eng. Sci. 53:1777–1786. Smoluchowski, M. V. (1917). Versuch einer Mathematischen Theorie der Koagulations—Kinetik kolloider Losungen, Z. Physik. Chemie. 92:129. Spouge, J. L. (1985). Monte Carlo Results for Random Coagulation, J. Colloid Interface Sci. 107:38.
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