Analysis of four Monte Carlo methods for the solution of population balances in dispersed systems

Monte Carlo (MC) constitutes an important class of methods for the numerical solution of the general dynamic equation (GDE) in particulate systems. We compare four such methods in a series of seven test cases that cover typical particulate mechanisms. The four MC methods studied are: time-driven direct simulation Monte Carlo (DSMC), stepwise constant-volume Monte Carlo, constant number Monte Carlo, and multi-Monte Carlo (MMC) method. These MC's are introduced briefly and applied numerically to simulate pure coagulation, breakage, condensation/evaporation (surface growth/dissolution), nucleation, and settling (deposition). We find that when run with comparable number of particles, all methods compute the size distribution within comparable levels of error. Because each method uses different approaches for advancing time, a wider margin of error is observed in the time evolution of the number and mass concentration, with event-driven methods generally providing better accuracy than time-driven methods. The computational cost depends on algorithmic details but generally, event-driven methods perform faster than time-driven methods. Overall, very good accuracy can be achieved using reasonably small numbers of simulation particles, O(10³), requiring computational times of the order 10²−10³ s on a typical desktop computer.     

Monte Carlo simulation of particle breakage process during grinding

The Monte Carlo method is quite useful in the modeling of particulate systems. It is used here to simulate the particle brekage process during grinding that can be represented by a population balance equation. The simulation technique is free from discretization of time or size. The results of simulation under restricted conditions of grinding compare very well with the available analytical solution of the population balance equation. The procedure is extended to simulate the grinding process in its entirety. This method provides an alternative to the modeling of the grinding process where the governing population balance equation cannot be readily solved.     

Accuracy and optimal sampling in Monte Carlo solution of population balance equations

Implementation of a Monte Carlo simulation for the solution of population balance equations (PBEs) requires choice of initial sample number (N₀), number of replicates (M), and number of bins for probability distribution reconstruction (n). It is found that Squared Hellinger Distance, H², is a useful measurement of the accuracy of Monte Carlo (MC) simulation, and can be related directly to N₀, M, and n. Asymptotic approximations of H² are deduced and tested for both one‐dimensional (1‐D) and 2‐D PBEs with coalescence. The central processing unit (CPU) cost, C, is found in a power‐law relationship, , with the CPU cost index, b, indicating the weighting of N₀ in the total CPU cost. n must be chosen to balance accuracy and resolution. For fixed n, M × N₀ determines the accuracy of MC prediction; if b > 1, then the optimal solution strategy uses multiple replications and small sample size. Conversely, if 0 < b < 1, one replicate and a large initial sample size is preferred. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2394–2402, 2015     

Two methods for sensitivity analysis of coagulation processes in population balances by a Monte Carlo method

We consider two stochastic simulation algorithms for the calculation of parametric derivatives of solutions of a population balance equation, namely, forward and adjoint sensitivity methods. The dispersed system is approximated by an N-particle stochastic weighted ensemble. The infinitesimal deviations of the solution are accounted for through infinitesimal deviation of the statistical weights that are recalculated at each coagulation. In the forward method these deviations of the statistical weights immediately give parametric derivatives of the solution. In the second method the deviations of the statistical weights are used to calculate a finite-mode approximation of the linearized version of the population balance equation. The linearized equation allows for the calculation of the eigenmodes and eigenvalues of the process, while the parametric derivatives of the solution are given by a Lagrange formalism.     

Monte Carlo simulation of microemulsion polymerization

Monte Carlo simulation of chemically reacting systems based on the master equation was used to describe the stochastic time evolution of the microemulsion polymerization system. A model was developed to demonstrate its applicability for hexyl methacrylate and styrene microemulsion polymerization. The properties of final latex, such as the particle size and molecular weight distributions were obtained simultaneously. The polymerization behavior and properties of final latexes were well reproduced. The model is valuable in confirming or elucidating the various mechanisms in the polymerization. The entry and desorption mechanism was well established to account for the polymerization kinetics. The general polymerization behavior of hydrophobic monomer in microemulsions was properly simulated by the model proposed.     

Monte Carlo simulation of polyampholyte-nanoparticle complexation

The complexation between a polyampholyte and a charged particle was investigated via Monte Carlo simulations on an off-lattice. The simulations revealed that there are three regions for the conformation of the complex formed between a positively charged particle and a polyampholyte chain. When the charge density and the size of the particle were small, the chain adsorbed on the particle surface maintained to a large extent its configuration from the bulk (spherical, dumb-bell, necklace or rod-like). By increasing the charge density and the particle size, the polyampholyte chain has collapsed on the surface. Further increases of the charge density and size of the particle caused a segregation of the beads of the sub-chains with mostly positive charges from those with mostly negative charges, those richer in positive charges being repelled by the particle. The simulation results are compared with some analytical results.     

Fast Monte Carlo methodology for multivariate particulate systems—I: Point ensemble Monte Carlo

A fast Monte Carlo methodology for particulate processes is introduced. The proposed methodology combines concepts from discrete population balance equations and dynamic Monte Carlo simulations of chemical kinetics to construct a new jump Markov model that approximates the population balance dynamics. The Markov model consists of a definition of a new type of reaction channel, in which the reaction product is a stochastic process by itself. One feature of this model is that, although a coarse view of the process is taken, it still conserves the history of individual particles. This is a very important aspect for effective modeling of multivariate models, especially when part of the goal is to study the evolution of the internal states of the particles (i.e., composition, phase behavior, etc.).Numerical experiments show that this algorithm can improve the computational load of the exact method by orders of magnitude without sacrificing computational accuracy. The methodology is useful especially in stochastic optimization applications where many function calls (simulations) are required. Possible applications are optimization and dynamic optimization using an artificial chemical process algorithm, genetic algorithm, or simulated annealing among others.     

Monte Carlo simulation of nano-particle sintering

This paper uses a multi-state Potts model to simulate the sintering of nano-particles by boundary migration and evaporation-condensation. The variables in this simulation are the reduced temperature, a next-nearest neighbour weighting and the ratio of interfacial to surface energy. The effect of these parameters on simulation of sintering of two and three-particle clusters is systematically explored as a basis for the study of more complex aggregates.     

利用Monte Carlo方法求解粒数衡算方程分析影响气泡分布的各种因素

精馏塔板上气泡尺寸分布是计算精馏塔板效率以及设计和操作精馏塔的关键参数 ,为了对它进行更好的预测 ,文中以Kolmogoroff各向同性湍流理论为基础并结合概率理论 ,利用计算机图形处理技术对气泡的聚并和破裂现象进行观察研究 ,分析气泡的聚并和破裂机理 ,采用MonteCarlo模拟技术求解粒数衡算方程 ,对影响精馏塔板上气泡尺寸分布的各种因素进行研究分析。求解结果与实验数据相吻合并显示气泡尺寸分布为对数正态分布 ,这与其他研究者的结论相一致     

Monte Carlo simulation of the PEMFC catalyst layer

The performance of the polymer electrolyte membrane fuel cell (PEMFC) is greatly controlled by the structure of the catalyst layer. Low catalyst utilization is still a significant obstacle to the commercialization of the PEMFC. In order to get a fundamental understanding of the electrode structure and to find the limiting factor in the low catalyst utilization, it is necessary to develop the mechanical model on the effect of catalyst layer structure on the catalyst utilization and the performance of the PEMFC. In this work, the structure of the catalyst layer is studied based on the lattice model with the Monte Carlo simulation. The model can predict the effects of some catalyst layer components, such as Pt/C catalyst, electrolyte and gas pores, on the utilization of the catalyst and the cell performance. The simulation result shows that the aggregation of conduction grains can greatly affect the degree of catalyst utilization. The better the dispersion of the conduction grains, the larger the total effective area of the catalyst is. To achieve higher utilization, catalyst layer components must be distributed by means of engineered design, which can prevent aggregation.     

Monte Carlo simulation studies of the catalytic combustion of methane

A Monte Carlo (MC) simulation is carried out of the catalytic oxidation of methane based on the pseudo-Langmuir-Hinshelwood (LH) mechanism proposed by Iglesia et al., that attempts to interpret the behavior of this reaction, which from experiments is assumed to be of the Mars-van Krevelen type. The results interpret reasonably some experimental findings (reaction order with respect to H₂O, CH₄, activity and structural sensitivity) if certain criteria based on the experiment and some laboratory data for the kinetic and thermodynamic parameters from the literature are considered.     

Monte Carlo simulation of chain extension using bisoxazolines as coupling agent

The chain extension using bisoxazolines (OO) as coupling agent was studied by means of the Monte Carlo (MC) method, focusing on the reaction kinetics. A comparison between simulated results and those calculated by an improved kinetic model was made. The coupling efficiency of the chain extender, average molecular weight and molecular weight distributions (MWDs) were investigated. The results show that the biggest coupling efficiency, the highest average molecular weight and the narrowest MWDs can be obtained when the initial concentrations of carboxyl and oxazoline groups are equal. The results indicate that higher activity difference between the oxazoline group in OO and that in blocked CA can lead to higher average molecular weight and narrower MWDs. Those results are in good agreement with the experiments. Besides above factors, diffusion effect and degradation effect are important factors for the average molecule weight and MWDs. And lower reaction kinetics constants of diffusion effect and degradation effect both result in a higher number-average molecular weight . The results also indicate that two peaks, at the different molecular weight, appear in the curve of molecular weight distributions during chain extension reaction. The variation of these two peaks corresponds to different polydispersities.     

Monte Carlo simulation of xylene isomerization over zeolite catalysts

The isomerization reaction of xylene was simulated by means of the Monte Carlo method based on the experimentally observed parameters, including the diffusivity, equilibrium adsorption constant and intrinsic rate constant. The dependence of the product selectivity upon the Thiele modulus was examined and the results were satisfactorily consistent with those of the continuous model as well as the experiments. This suggests that the Monte Carlo method is helpful for investigating the nature of shape selectivity in zeolite-catalyzed reactions.     

Monte Carlo simulation of radiation-induced solid state polymerization

A Monte Carlo method was used for a computer simulation of radiation-induced solid state polymerization. The propagation of polymer chains was simulated by means of self-avoiding random walks on a tetrahedral lattice. The initiation and termination of the chains were modelled by pseudorandom processes. The influence of conditions of the in-source process on the post-polymerization kinetics and on the degree of polymerization of the polymers was studied.     

均匀性试验抽样误差的蒙特卡罗模拟

对现行均匀性试验抽样误差和扩大样本容量后的抽样误差,应用蒙特卡罗方法进行数值模拟。结果表明,当总体标准偏差为1.5MPa,样本容量为10,对应水泥28d抗压强度分别为40MPa、50MPa和60MPa时,以变异系数表示的抽样误差分别为1.7%、1.4%和1.1%。这个误差很大,可能导致对水泥均匀性的错误评价。根据扩大样本容量后的抽样误差模拟结果,提出了均匀性试验抽样方法的修改建议。     

Monte Carlo simulation of electron-solid interactions in cement-based materials

Knowledge of the size of the electron-solid interaction volume and the sampling volume of various signals within it is important for interpretation of images and analytical results obtained from electron microscopy. In this study we used a Monte Carlo technique to simulate electron trajectories in order to investigate the shape and size of the interaction volume, the spatial and energy distribution of backscattered electrons and characteristic X-rays in cement-based materials. We found that the maximum penetration depth of the electron trajectories ranges from 0.75 to 1.5 μm at 10 keV and from 2.5 to 5.0 μm at 20 keV. For backscattered electrons, the maximum sampling depth is about 30% of the interaction volume depth and its lateral dimension is close to the interaction volume depth. The sampling volume size of characteristic X-rays is a substantial fraction of the interaction volume. For ettringite, the amount of material analysed in X-ray microanalysis is in the order of 1 to 100 μm³ at conventional SEM accelerating voltages of 10 to 20 keV.     

Self-condensing vinyl polymerization in the presence of multifunctional initiator with unequal rate constants: Monte Carlo simulation

Monte Carlo method was applied to simulate self-condensing vinyl polymerization (SCVP) in the presence of multifunctional initiator with unequal reactive rate constants. Simulations showed that, with the increase of reactivity of multifunctional initiator, the molecular weight distribution decreases, the fraction of multi-armed hyperbranched polymers increases, and the best reactivity ratio of multifunctional initiator to inimor is 10/1; the reactivity difference of two active sites in inimer has strong effects on the molecular weight distribution and degree of branching. The branch degree and fraction of branch-points depend on both the conversion of double bonds and reactivity difference of two active centers in inimer, and have no relation with the adding of multifunctional initiator. The density distribution of degree of branching further shows common self-similarity in SCVP.     

Investigating the effect of shape on particle segregation using a Monte Carlo simulation

A Monte Carlo simulation has been used to investigate the segregation potential of a range of particulate systems under conditions in which the particles undergo high amplitude low frequency shaking. These systems involve a wide range of binary powder mixtures in which complex particle shapes have been investigated, including plates and rods which represent the real world materials encountered in pharmaceutical systems such as those which include crystalline components. Previous simulations on the segregation propensity of systems with different shapes were limited to spheres and spherocylinders, with relatively low vibrational amplitude drops. A commercial computer application for particle packing—called MacroPac—has been successfully employed here, as it has been able to model systems that are more complex where the shape variation is much wider. These simulations apply to the case of macroscopic particles, in the absence of air resistance and inter-particle forces. For non-spherical shapes, an ‘effective size’ which relates to the radius of gyration of the particles is determined. Our studies indicate that with high amplitude low frequency shaking, in a mixture of particles with different shapes but with equal volumes, the particles with the larger ‘effective size’, which tend to have a lower packing fraction, segregate to the top.     

Monte Carlo simulation of pneumatic tribocharging in two-phase flow for high-inertia particles

Pneumatic transport, triboelectrostatic experimentation studying the removal of carbon from fly ash has shown the importance of particle charging on process performance. To further elucidate the influence of particle charging, a tribo-electrification model for charging spherical particles in vertical pipes is proposed. Unlike previous numerical approaches, the charging mechanism is studied in the framework of a charge relaxation process in which the equilibrium charge for uniform and localized charge distributions on the particle surfaces are determined. A recently proposed statistical model was also adopted to evaluate particle-wall collisions for wall-bounded flows, taking into account the influence of particle-particle collisions. Effects of particle concentrations, along with particle size and turbulence intensity, are also investigated. Model predictions are compared with experimental results, with good agreement found if the charge is assumed to be distributed on small surface sector of the particles. After defining suitable probability density functions for the variables considered, a Monte Carlo analysis is employed for simulating particle charge distributions and compared with published measurements to identify the important parameters during particle tribocharging.     

Micro-kinetic analysis and Monte Carlo simulation in methane partial oxidation into synthesis gas

The mechanism outlined in literature for methane steam reforming over nickel catalysts is reviewed. Activation energy of this set of elementary steps for methane conversion into synthesis gas is obtained on the basis of bond order conservation Morse potential (BOC-MP), and the pre-exponential factors of the rate expressions of these elementary steps are then estimated according to the reported ranges in literature. A real-time Monte Carlo approach is developed to simulate the micro-kinetic trends for this catalytic process.