Solution of the population balance equation using constant-number Monte Carlo

We formulate a Monte Carlo simulation of the mean-field population balance equation by tracking a sample of the population whose size (number of particles in the sample) is kept constant throughout the simulation. This method amounts to expanding or contracting the physical volume represented by the simulation so as to continuously maintain a reaction volume that contains constant number of particles. We call this method constant-number Monte Carlo to distinguish it from the more common constant-volume method. In this work, we expand the formulation to include any mechanism of interest to population balances, whether the total mass of the system is conserved or not. The main problem is to establish connection between the sample of particles in the simulation box and the volume of the physical system it represents. Once this connection is established all concentrations of interest can be determined. We present two methods to accomplish this, one by requiring that the mass concentration remain unaffected by any volume changes, the second by applying the same requirement to the number concentration. We find that the method based on the mass concentration is superior. These ideas are demonstrated with simulations of coagulation in the presence of either breakup or nucleation.     

Modeling of aggregation kernel using Monte Carlo simulations of spray fluidized bed agglomeration

The present work attempts to consider the microscopic mechanisms of spray fluidized bed agglomeration while modeling the macroscopic kinetics of the process. A microscale approach, constant volume Monte‐Carlo simulation, is used to analyze the effects of micro‐processes on the aggregation behavior and identify the influencing parameters. The identified variables, namely the number of wet particles, the total number of particles, and the number of droplets are modeled and combined in the form of an aggregation kernel. The proposed kernel is then used in a one‐dimensional population balance equation for predicting the particle number density distribution. The only fitting parameters remaining in the population balance system are the collision frequency per particle and a success fraction accounting for the dissipation of kinetic energy. Predictions of the population balance model are compared with the results of Monte‐Carlo simulations for a variation of significant operating parameters and found to be in good agreement. © 2014 American Institute of Chemical Engineers AIChE J, 60: 855–868, 2014     

Monte Carlo simulation of terpolymerization: Optimizing the simulation and post-processing times

Monte Carlo simulations are a useful and easy way to understand a polymerization reaction process properly. However, achieving reliable results with Monte Carlo simulations can also lead to prohibitive computational times and a considerable amount of data to be processed afterward. The present study analyses the Monte Carlo simulation of a steady-state terpolymerization process to reduce the overall computational time of the simulation and the post-processing of its results. Different sorting algorithms (Bubble, Insertion, Selection, and Tim) and Python libraries (Joblib and Numba) were used. The chain composition distribution and the micro-structures resultant of different scenarios were assessed by processing the simulated mechanism results. The simulation time results indicate the Tim sorting algorithm as the best to use in the post-processing step and the Numba library as the best suited for both the simulation and the post-processing step.     

Part IV: Dynamic evolution of the particle size distribution in particulate processes. A comparative study between Monte Carlo and the generalized method of moments

The present work provides a comparative study on the numerical solution of the dynamic population balance equation (PBE) for batch particulate processes undergoing simultaneous particle aggregation, growth and nucleation. The general PBE was numerically solved using three different techniques namely, the Galerkin on finite elements method (GFEM), the generalized method of moments (GMOM) and the stochastic Monte Carlo (MC) method. Numerical simulations were carried out over a wide range of variation of particle aggregation and growth rate models. The performance of the selected techniques was assessed in terms of their numerical accuracy and computational requirements. The numerical results revealed that, in general, the GFEM provides more accurate predictions of the particle size distribution (PSD) than the other two methods, however, at the expense of more computational effort and time. On the other hand, the GMOM yields very accurate predictions of selected moments of the distribution and has minimal computational requirements. However, its main disadvantage is related to its inherent difficulty in reconstructing the original distribution using a finite set of calculated moments. Finally, stochastic MC simulations can provide very accurate predictions of both PSD and its corresponding moments while the MC computational requirements are, in general, lower than those required for the GFEM.     

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).     

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.     

Constant number Monte Carlo simulation of population balances with multiple growth mechanisms

We present a complete simulation scheme for particulate processes based on the constant number Monte Carlo methodology. Specifically, the proposed scheme can be applied towards the solution of population balances that include nucleation, coagulation and surface deposition, coupled to chemical reactions. The synthesis of titania (TiO₂) by flame oxidation of TiCl₄ is employed as a comparison basis of the relative advantages and weaknesses of Monte Carlo against more classical numerical approaches. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Molecular-Level Characterization of Heterogeneous Catalytic Systems by Algorithmic Time Dependent Monte Carlo

Monte Carlo algorithms and codes, used to study heterogeneous catalytic systems in the frame of the computational section of the NANOCAT project, are presented along with some exemplifying applications and results. In particular, time dependent Monte Carlo methods supported by high level quantum chemical information employed in the field of heterogeneous catalysis are focused. Technical details of the present algorithmic Monte Carlo development as well as possible evolution aimed at a deeper interrelationship of quantum and stochastic methods are discussed, pointing to two different aspects: the thermal-effect involvement and the three-dimensional catalytic matrix simulation. As topical applications, (i) the isothermal and isobaric adsorption of CO on Group 10 metal surfaces, (ii) the hydrogenation on metal supported catalysts of organic substrates in two-phase and three-phase reactors, and (iii) the isomerization of but-2-ene species in three-dimensional supported and unsupported zeolite models are presented.     

Dynamic Monte Carlo Simulations of Oscillatory Reactions

The dynamic Monte Carlo method has been used to simulate the 2 A + B₂ → 2 AB reaction catalyzed by a reconstructing substrate. Oscillatory behavior and spatio-temporal is studied as a function of grid size. Spatio-temporal pattern formation has been simulated in various forms: cellular patterns, target patterns, rotating spirals, and turbulent patterns. Cellular patterns are a manifestation of a local synchronization mechanism in which all reaction fronts periodically extinguish each other. This illustrates that dynamic Monte Carlo simulations form a promising technique and can be used to predict macroscopic kinetic phenomena on a molecular basis.     

Population Balance-Monte Carlo Simulation for Gas-to-Particle Synthesis of Nanoparticles

The process simulation of nanoparticle synthesis via the gas-phase method is essential to understanding the detailed dynamic evolution of nanoparticles within a very short time period under high temperature. The task is, however, very challengeable up to now as the conversion of the gaseous precursor to the end-use nanoparticle is a complex physicochemical process involving nucleation of the particulate phase, agglomeration between particles and sintering under industrial production conditions. In this article, we extended the differentially weighted Monte Carlo method for population balance to simulate the dynamic evolution of titania (TiO₂) nanoparticles synthesized by gas-to-particle conversion in a single aerosol reactor, considering simultaneous nucleation, agglomeration, and sintering. The simulated size distribution of TiO₂ agglomerate and primary particles produced by the thermal decomposition of titanium tetraisoproxide agreed well with the experimental data. In the simulation, the fast population balance-Monte Carlo method was utilized to accelerate the process simulation on a desktop PC. Results were obtained up to 178 times faster than that of a normal Monte Carlo method. The inhomogeneous internal structure of primary particles was considered through solving population balance of polydisperse primary particles within agglomerate. It was found the polydisperse model could predict the primary particle size distribution better. Simulation results revealed a complex competition relation among nucleation, agglomeration and sintering.

Copyright 2013 American Association for Aerosol Research     

Direct simulation Monte Carlo for simultaneous nucleation, coagulation, and surface growth in dispersed systems

A Monte Carlo simulation technique is developed to describe dispersed phase systems. The method is formulated for simultaneous coagulation, nucleation and surface growth, but can be extended to include other processes. These processes are considered in an initially constant simulation volume. The changes in the particle ensemble are determined by a random choice procedure, while the particle number in the simulation volume changes according to the chosen events. Every time, when the particle number in the simulation volume increases or decreases by a factor of two of its initial value, the simulation volume is halved or doubled, respectively. Therefore, this method is called a stepwise constant-volume Monte Carlo simulation. It allows to use only several thousands simulation particles, even if the particle number concentration experiences changes of several orders of magnitude. The simulations are validated through a comparison with the exact mathematical solutions for several simple cases. An example of simultaneous nucleation and coagulation in the free-molecular region demonstrates, that the stepwise constant-volume Monte Carlo simulations lead to more accurate results than the constant-number Monte Carlo simulations.     

Kinetic Monte Carlo Simulation of Molecular Processes on Supported Metal Particles

A general model is proposed to describe the kinetics of molecular reactions taking place on supported metal particles, which are deformed by the effect of temperature, through kinetic Monte Carlo simulations. The model is applied to the study of the CO oxidation reaction. The effects of adsorbate?Cadsorbate and adsorbate?Cmetal interactions and of CO and metal atoms diffusion on the reaction window and the overall reaction rate are determined.     

Original Monte Carlo Methodology Devoted to the Study of Sintering Processes

A Monte Carlo methodology different from others based on the Potts model has been developed to solve capillary-driven mass transport involved in the sintering process. The addition of a cohesive energy term to the energetic model enables quantification of the stress gradients within grains, as well as the induced mass fluxes. The study of two-particle sintering involving volume diffusion has been chosen as a first example. The morphological evolution of the two-particle system to a single particle is followed. Real-time-dependent neck growth and shrinkage rates are defined. The rheological behavior of the particles induced by the Monte Carlo procedure is discussed.     

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 simulations of colloidal particle coagulation and breakup under turbulent shear

Both a Monte Carlo model and an algorithm were presented to simulate the particle coagulation and breakup phenomena taking place in a colloidal solution under turbulent fluid shear. The model is represented by the probability density functions that describe the stochastic coagulation and breakup phenomena taking place among numerous particles. From a dimensional analysis of the model two dimensionless groups,K _c andK _b , were derived that represent the relative intensity of the coagulation and breakup phenomena. In order to overcome the memory problem in saving the sizes of a large number of particles, the model was converted to a form suitable for carrying out a sectional mass balance. Detailed simulation steps were presented and applied to acrylonitrile-butadiene-styrene (ABS) latex coagulation. Numerical simulations revealed that the steady state particle size distribution does not depend on the initial distributions but on theK _c /K _b ratio. Setting the operation variables to increase the ratio was found to shift the particle size distribution toward larger particles.     

Comparison of Monte Carlo and quasi-Monte Carlo technique in structure and relaxing dynamics of polymer in dilute solution

Structure and dynamics of polymer in solvent solution is an important area of research since the functional properties of polymer are largely dependent on the morphology of the polymers in solution. This structure related properties are especially important in case of surface science where the phase-separated morphology in the micro/nano scale dictates the properties of the product. Modeling polymers in solution is an efficient way to determine the morphology and thus the properties of the products. It saves time as well as helps to design novel materials with desired properties. Polymers in solution systems are generally modeled with bead spring model and Monte Carlo or importance sampling Monte Carlo simulations is used to find the optimal configuration where the energy of the system is minimized. Often in these simulations, random numbers are used in the Monte Carlo steps. Normally random numbers try to form clusters and do not cover the entire dimension of the system. Thus the minimum energy structures obtained from simulations with random numbers are not optimal configuration of the system. In the present work a lattice-based model is used for polymer solution system and importance sampling Monte Carlo is used for simulation. Quasi-random numbers generated from Hammersley sequence sampling (HSS) are used in the simulation steps for stochastic selection polymers and its movements. Quasi-random numbers obtained from HSS are random in nature and they have n-dimensional uniformity. They do not form clusters and the structural configuration obtained using quasi-random numbers are optimal in nature. The optimal configurations of the polymers as obtained from random number and quasi-random number are compared. The result shows that simulation with HSS attains a lower energy state after initial quench. At the late stage of spinodal decomposition, the structure factor decrease-showing Ostwald ripening which is not observed from simulation with random numbers.     

高分子构象模拟研究中的Monte Carlo方法

介绍了Monte Carlo方法及其特点,进而分析了Monte Carlo用于高分子模拟的优势,并描述了两类模拟模型。论文重点综述了近年来Monte Carlo方法在高分子构象模拟中的一些研究与应用,并展望了Monte Carlo方法在高分子构象模拟中的发展趋势和前景。     

A generalized framework for solving dynamic optimization problems using the artificial chemical process paradigm: Applications to particulate processes and discrete dynamic systems

The solution of optimal control problems (OCPs) becomes a challenging task when the analyzed system includes non-convex, non-differentiable, or equation-free models in the set of constraints. To solve OCPs under such conditions, a new procedure, LARES-PR, is proposed. The procedure is based on integrating the LARES algorithm with a generalized representation of the control function. LARES is a global stochastic optimization algorithm based on the artificial chemical process paradigm. The generalized representation of the control function consists of variable-length segments, which permits the use of a combination of different types of finite elements (linear, quadratic, etc.) and/or specialized functions. The functional form and corresponding parameters are determined element-wise by solving a combinatorial optimization problem. The element size is also determined as part of the solution of the optimization problem, using a novel two-step encoding strategy. These building blocks result in an algorithm that is flexible and robust in solving optimal control problems. Furthermore, implementation is very simple.The algorithm's performance is studied with a challenging set of benchmark problems. Then LARES-PR is utilized to solve optimal control problems of systems described by population balance equations, including crystallization, nano-particle formation by nucleation/coalescence mechanism, and competitive reactions in a disperse system modeled by the Monte Carlo method. The algorithm is also applied to solving the DICE model of global warming, a complex discrete-time model.     

A stochastic simulation for the collection process of fly ashes in single-stage electrostatic precipitators

The dynamic evolution of particle size distribution (PSD) along the longitudinal length of electrostatic precipitator (ESP) quantitatively describes the collection process of ESP, and then is capable of estimating the performance of ESP. The details of the evolution of PSD are obtained by the solutions of population balance equation (PBE) for electrostatic collection. The paper promoted a stochastic method to solve the PBE. The method is based on event-driven technique and introduces the concept of weighted fictitious particles. The halving procedure of number weight of fictitious particle is adopted to restore the statistical samples of the stochastic approach and maintain the computational domain. The method, which is named event-driven constant volume (EDCV) method, is used to simulate the collection process of particles in a small-scale single-stage wire-plate ESP, considering simultaneously the electrostatic force, the convection force and the transverse particle diffusion in the model of collection kernel. The agreement among the results of the Monte Carlo method, the experimental data and the results of method of moments is good.     

Stochastic simulation of population balance models with disparate time scales: Hybrid strategies

The Monte Carlo methods have been an effective tool for the numerical solution of population balance models (PBMs). They are particularly useful for complex multidimensional problems. Less attention has been paid to solving population balance models where some species are away from the thermodynamic limit (very dilute or finite) and other species can be considered deterministic (high concentration). These types of problem often result in a stochastic system with rates spanning orders of magnitude for different mechanisms. Using the exact Monte Carlo solution to solve these types of problem is very inefficient because of the simulation time spent sampling fast events. These fast events are associated with species with large populations for which a single event does not change the population appreciably. This frequent sampling of fast events becomes a bottleneck during a simulation in which many single MC steps are required to make an appreciable change in the population.In this work, a hybrid solution strategy is developed to effectively solve this type of problem. The method implements the self-consistent fast/slow partitions used to solve stochastic equations in chemical kinetics. One strategy is found on the capacity of a coarse-grained Markov model called particle ensemble random product (PERP) to accelerate the simulation of fast events of PBMs (Chem. Eng. Sci. 63, 7649–7664; Chem. Eng. Sci. 63, 7665–7675). A second strategy approximates the fast events using mass conservation equations. These models are coupled with the exact MC simulation of slow events. Two extreme cases of heterocoagulation are studied to demonstrate these hybrid strategies.