Single-particle method for stochastic simulation of coagulation processes

A Monte Carlo stochastic simulation algorithm based on a single-particle method is suggested to describe steady-state particle coagulation processes. The method does not require any information on nearby particles; instead a fictitious coalescence partner with a given size is generated. The main drawback that limited applicability of this method in the past was that for each control volume the particle size distribution function had to be sampled and stored. In the present study we applied a discrete representation of the distribution function that requires only small memory resources and allows fast updating.     

Structural Properties and Filter Loading Characteristics of Soot Agglomerates

Semi-Lagrangian scheme (cubic polynomial semi-Lagrangian (CSL) or cubic interpolated propagation (CIP) scheme) is applied to condensation equation of cumulative number distribution, and simple correction using mathematical constrains of cumulative number distribution is applied after the semi-Lagrangian procedure at each time step. This solver exactly conserves total particle numberNand prevents overshoots of aerosol size distribution with steep gradients. The total surface areaSand the total volumeVare also easily conserved within small error. In the cases where the size distributions are not extremely sharp, CSL and CIP both can be utilized in the solver for the aerosol condensation. On the other hand, in the cases where the distributions are quite sharp, the solver using CIP gives the almost same result as the exact solution.     

Self-preserving theory of particulate systems

The similarity solution of the population balance equation for pure kinetic coagulation (coalescence) has proved to be a useful means of representing the evolution of size distributions of homogeneous particulate systems. The resulting solution is termed self-preserving since neither particle size nor time appears explicitly in the solution. In the present work similarity solutions are developed for a wide class of particulate processes that are governed by a general population balance equation for the inhomogeneous size distribution density function. The general theory has application to the analysis of size spectra of atmosphere aerosols undergoing simultaneous coagulation, turbulent diffusion, and growth by gas-to-particle condensation, to the analysis of size spectra during grinding processes, and to several other physical systems of interest.     

EVALUATION OF NUMERICAL METHODS FOR SIMULATING AN EVOLVING PARTICLE SIZE DISTRIBUTION IN GROWTH PROCESSES

The particle growth term renders hyperbolic the dynamic population balance equation. Problems associated with the numerical solution of hyperbolic partial differential equations with stationary grid methods are well known. Moreover in the common case of combined molecular particle growth and coagulation, the convolution integral of the coagulation terms makes the moving grid methods computationally intractable. To cope with practical problems, this work is focused on numerical solution methods, of the population balance equation, characterized by relatively small computational cost and fair accuracy - comparable to that of relevant experimental data. For this purpose, previous work on appropriate discretization of the coagulation terms is extended for the growth terms. Several numerical methods are systematically evaluated and further extended. Recommendations are made concerning the best method, by taking into account the nature of the problem, the prevailing physical conditions, and the main quantity of interest; i.e. certain specific moments, or the entire size distribution     

Analysis of particle contamination in plasma reactor by 2-sized particle growth model

Rapid particle growth in the silane plasma reactor by coagulation between 2-sized particles was analyzed for various process conditions. The particle coagulation rate was calculated considering the effects of particle charge distribution based on the Gaussian distribution function. The large size particles are charged more negatively than the small size particles. Some fractions of small size particles are in neutral state or charged positively, depending on the plasma conditions. The small size particle concentration increases at first and decreases later and reaches the steady state by the balance of generation rate and coagulation rate. The large size particles grow with discharge time by coagulation with small size particles and their size reaches the steady state, while the large size particle concentration increases with discharge time by faster generation rate and reaches the steady state by the balance of generation and disappearance rates. As the diameter of small size particles decreases, the diameter of large size particles increases more quickly by the faster coagulation with small size particles of higher concentration. As the residence time increases, the concentration and size of large size particles increase more quickly and the average charges per small size and large size particle decrease.     

Reducing Statistical Noise and Extending the Size Spectrum by Applying Weighted Simulation Particles in Monte Carlo Simulation of Coagulation

The direct simulation Monte Carlo (DSMC) method is widely utilized to simulate microscopic dynamic processes in dispersed systems that give rise to the population balance equation. In conventional DSMC approaches, simulation particles are equally weighted, even for broad size distributions where number concentrations in different size intervals are significantly different. The resulting statistical noise and limited size spectrum severely restrict the application of these DSMC methods. This study proposes a new Monte Carlo (MC) method, the differentially weighted time-driven method, which captures the coagulation dynamics in dispersed systems with low noise and is simultaneously able to track the size distribution over the full size range. Key elements of this method include constructing a new jump Markov process based on a new coagulation rule for two differentially weighted simulation particles, and restricting the number of simulation particles in each size interval within prescribed bounds. The method is validated by using an ideal coagulation kernel with a known analytical solution and a real coagulation kernel for which an accurate solution can be found numerically (self-preserving particle size distribution in the continuum regime).     

Evolution of Particle Size Distributions due to Turbulent and Brownian Coagulation

The time evolution of particle size distribution due to Brownian and turbulent coagulation (using the kernel of Kruis and Kusters (1997)) was systematically investigated. Using a new definition of dimensionless size distribution parameters based on the geometric mean values, self-preserving particle size distributions for turbulent coagulation were found to exist. The width of such distributions depends on the initial size distribution as well as the turbulence intensity. When starting with submicron aerosols, however, only the turbulence intensity plays a role in determining the final self-preserving form, whereas the initial conditions have no influence. Typically, broad particle size distributions with σ g in the 1.5-1.9 range are obtained. Because of the importance of scavenging by the largest particles in the size distribution, the possibility of developing a "runaway mass" exists, for which some experimental indications in turbulent systems exist.     

Size Dependence of the Ratio of Aerosol Coagulation to Deposition Rates for Indoor Aerosols

A thorough understanding of the importance of aerosol coagulation and deposition relative to each other as modifiers of the particle size distribution plays an important role in the proper selection of conditions to estimate the deposition rate coefficient. In this work, a theoretical analysis was conducted for investigating the size-resolved ratio of coagulation to deposition for different types of size distributions using the Simpson integral method. The theoretical model was subsequently qualitatively validated by experiments in a completely mixed and ventilated aerosol chamber. Both experimental and theoretical studies show that the ratio of the rates of coagulation to deposition is strongly dependent on the total particle number concentration and the geometric mean diameter of the aerosol. A variation of the ratio of coagulation to deposition by several orders of magnitude for aerosols with differing size distributions was found. Thus the previously employed criterion for the negligence of coagulation based solely on the total particle number concentration was shown to be insufficient to accurately judge whether an aerosol is suited for the estimation of the deposition rate coefficient. Aerosols with wide size distributions are not recommended for use in the estimation of the deposition rate coefficient. The study provides a method to understand the role of coagulation and deposition for indoor aerosols.

Copyright 2013 American Association for Aerosol Research     

An Exact Solution of the Fragmentation Equation

The accuracy of the self-similarity assumption employed in the study of the grinding equation is examined in detail. This is made possible by obtaining an exact solution for any homogeneous breakup function, thereby enabling the asymptotic limit as time proceeds to be examined carefully. For the Randolph-Ranjan model of breakup, we have obtained some explicit results and these have been employed to highlight the limitations of current self-similar solutions. In particular, we note that use of a self-similar solution which depends only on the zeroth and first moment of the distribution cannot give any detailed information on the higher moments. Nevertheless, at times very soon after the start of grinding, self-similarity does lead to useful and practical asymptotic results for size distributions, since it appears that higher moments are then of less importance. Thus the reason for the success of similarity is explained and the rate of approach to this condition is given. We have also introduced a new model of breakup which assumes that the minimum particle size in a given breakup process is always a fixed fraction of the initial size. This has the advantage of eliminating the divergence of the total number of particles produced per grinding action while still allowing the equation to be dealt with analytically. Finally, we discuss some steady state grinding distributions that arise from the new model.     

炭素材料粒度组成控制的理论与实践

简要介绍了Andreassen颗粒堆积理论，并结合生产实践，讨论了实际生产中所用的粒度组成与这一理论的联系与差异。与Andreassen方程所确定的粒度组成相比，实际配方中细颗粒和粗颗粒用量较大，中颗粒用量较小。     

Charge Distribution of Incipient Flame-Generated Particles

We report the size and electrical charge distributions of incipient nanoparticles generated in atmospheric pressure hydrocarbon/air premixed flames in conditions prior to the onset of soot particles. The particle size and charge distributions are measured by Differential Mobility Analysis (DMA) and compared to theoretical charge distributions predicted for flame conditions. The results show that the charge distribution attained in flames is well predicted by Boltzmann theory for all particles, including even the smallest incipient particles with diameters in the 1–3 nm size range. In flame conditions that produce only particles smaller than 3 nm, the charge fraction of particles agrees with that predicted by Boltzmann theory near the flame temperature (1700 K). In flame conditions with ‘bimodal’ particle size distributions, the charge fraction of the smallest particles agrees with the Boltzmann prediction at maximum flame temperature, while the charge fractions of larger particles agree with Boltzmann theory at temperatures that coincide with the local temperature near the probe surface (1000–1200 K). The results of this paper show that the temperature of the Boltzmann charge fraction that best agrees with the measured charge fraction for each particle size gives the local temperature of their last coagulation event. The smaller particles, which retain their charge fraction predicted by Boltzmann at the maximum flame temperature, do not thermalize by coagulation in the cool region near the probe evidencing low probability for charge transfer as well as for coagulation.     

Inverse Modeling of Aerosol Dynamics Using Adjoints: Theoretical and Numerical Considerations

In this paper we develop the algorithmic tools needed for inverse modeling of aerosol dynamics. Continuous and discrete adjoints of the aerosol dynamic equation are derived, as well as sensitivity coefficients with respect to the coagulation kernel, the growth rate, and the emission and deposition coefficients. Numerical tests performed in the twin experiment framework for a single component model problem show that the initial distributions and the dynamic parameters can be recovered from time series of observations of particle size distributions.     

A moment model for simulating raindrop scavenging of aerosols

The dynamics of a polydispersed aerosol size distribution, scavenged by precipitation, are numerically studied. The collision efficiency formula proposed by Slinn (Precipitation Scavenging in Atmospheric Sciences and Power Production—1979, Division of Biomedical Environmental Research, US Department of Energy, Washington, DC, USA, 1983, Chapter 11) and the moment method were introduced to represent the particle removal mechanism by raindrops and the aerosol size distribution, respectively. Consequently, the dynamics of the particle size distribution were reduced to a set of ordinary differential equations using the moment approach. A generalized raindrop distribution, including two widely used distributions; the Marshall–Palmer (MP) and Krigian–Mazin (KM) raindrop distributions, was adopted.Our model results have shown that raindrops with smaller diameters, and narrower distributions, collect aerosols more efficiently. Further, it was shown, in the small particle size region that the geometric mean diameter increases, while in the large particle region it decreases. For the two size ranges, the geometric standard deviations decrease with time, and a scavenging gap, the minimum particle removal efficiency region, exists between these particle size ranges.The dynamics of the particle size distributions, the MP and KM raindrop distributions, in the small particle range, show that the effects of the overestimation in the MP distribution were not as great as expected. Also, this study ascertained that the conventional parameterization of the constant collision efficiency introduces significant errors for estimating the particle size distribution dynamics by wet scavenging.     

二氧化硫凝聚MBS胶乳研究

提出了用二氧化硫凝聚MBS胶乳的新方法,研究了各种因素对凝聚时间、颗粒粒径分布和粉体堆积密度的影响,确定了最佳工艺条件:MBS胶乳的质量浓度为80～120g/L,凝聚温度50～60℃,搅拌速度200r/min,二氧化硫体积流量25 L/min.经8 m3凝聚釜生产试验,所得样品有较好的形貌,颗粒多为圆球;堆密度0.42;粒度分布集中,40和180目之间的颗粒含量达96%.PVC/MBS合金性能测试结果表明,由于塑化时间缩短,抗冲强度和透光率也得到了改善.     

Optimizing the balance between viscosity/modulus and impact in particulate composites

The objective of this study was to develop some new concepts of importance when trying to optimize the viscosity/modulus and impact relative to the particle‐size distribution in suspensions and particulate composites. The results of this study appear to indicate that, conceptually, it is possible to significantly improve the viscosity versus the impact balance for material formulations by optimizing the particle‐size distribution. For binary particle‐size distributions, the influence of the preferred particle‐size distribution, as determined using a square‐root distribution, did not yield the most desirable particle‐size distribution if the particle‐to‐particle component of the interaction coefficient was high. However, if three or more particles were utilized in the distribution, then the optimum particle‐size distribution utilized can apparently be characterized using the square‐root distribution even when the particle–particle component, σ_pc, of the interaction coefficient, σ, was found to be quite high. In addition, this same square‐root particle‐size distribution can also satisfactorily predict a probability of impact that can remain consistently high as long as the particles utilized are well chosen and not too close in size. Thus, this preferred particle‐size distribution can be utilized to predict at least one of the preferred distributions to optimize the balance of properties between impact and the viscosity/modulus. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 83: 291–304, 2002     

Solution of population balance equation with pure aggregation in a fully developed turbulent pipe flow

This paper presents our preliminary effort in predicting particle size distributions in particulate processes in turbulent flow systems. The focus has been on processes of pure aggregation, occurring in a turbulent environment. A remarkably simple strategy has been used to solve the population balance equation (PBE) for spatially dependent pure aggregation with insignificant diffusive transport of particles in turbulent flow systems. The method makes use of the solution of a batch PBE through a mathematical transformation linking time to spatial variables. Furthermore, we investigate the self-similar solution of batch aggregation to show scaling behavior of particle size distributions in such flow systems using spatially dependent average particle sizes. Average particle sizes across the pipe cross section have been computed using both averaged frequencies as well as spatially varying frequencies. Comparison of the two solutions shows significant differences between them, establishing the sheer inappropriateness of the use of average aggregation frequencies in the prediction of absolute particle size distribution as done in the past.     

凝并和冷凝/蒸发对颗粒尺度分布的影响

利用多重Monte Carlo算法对13种典型工况进行数值模拟,考察不同类型的凝并核和冷凝/蒸发核对多分散性颗粒尺度分布时间演变的影响。发现常凝并核要比线性和二次方凝并核对小颗粒的影响大一些、对大颗粒的影响小一些,线性和二次方凝并核对颗粒尺度分布的时间演变影响则取决于具体情况;常冷凝核要比线性冷凝核对小颗粒的影响大一些、对大颗粒的影响小一些;连续区布朗凝并核类似于常凝并核;扩散冷凝核对颗粒尺度分布的影响界于常冷凝核和线性冷凝核之间。     

A simple bimodal model for the evolution of non-spherical particles undergoing nucleation,coagulation and coalescence

A simple and efficient particle dynamics model is developed accounting for simultaneous nucleation, coagulation, and coalescence or sintering of non-spherical particles. In this model two discrete monodisperse modes are used to represent the non-spherical particle size distributions approximately: a size-fixed nucleation mode and a moving accumulation mode. The size-fixed nucleation mode accounts for the introduction of newly generated particles and the moving accumulation mode characterizes the particle growth by coagulation and coalescence. The simulation results for titania particle formation and growth using the proposed bimodal model are compared with those using the previous monodisperse non-spherical particle dynamics model and non-spherical polydisperse sectional model. The present bimodal model results in a very good agreement with the polydiserse sectional model even when particle nucleation coexists with coagulation process while the monodisperse model shows significant differences. It successfully predicts the morphological change of the non-spherical particles by coalescence. The present model is also shown to be capable of predicting the polydispersity of non-spherical particle distribution. The present non-spherical bimodal model requires the same level of the computation time that the simple monodisperse model does.