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.     

Dynamic prediction of the bivariate molecular weight-copolymer composition distribution using sectional-grid and stochastic numerical methods

In the present study, a two-dimensional fixed pivot technique (2-D FPT) and an efficient Monte Carlo (MC) algorithm are described for the calculation of the bivariate molecular weight-copolymer composition (MW-CC) distribution in batch free-radical copolymerization reactors. A comprehensive free-volume model is employed to describe the variation of termination and propagation rate constants as well as the variation of the initiator efficiency with respect to the monomer conversion. Simulations are carried out, under different reactor conditions, to calculate the individual monomer conversions, the leading moments of the ‘live’ and ‘dead’ polymer chain length distributions as well as the dynamic evolution of the distributed molecular properties (i.e., molecular weight distribution (MWD), copolymer composition distribution (CCD) and joint MW-CC distribution). The validity of the numerical calculations is examined via a direct comparison of the simulation results, obtained by the two numerical methods, with experimental data on the styrene-methyl methacrylate batch free-radical copolymerization. Additional comparisons between the 2-D FPT and the MC methods are carried out for different polymerization conditions. It is clearly shown that both numerical methods are capable of predicting the distributed molecular and copolymer properties, with high accuracy, up to very high monomer conversions. It is also shown that the proposed dynamic MC algorithm is less computationally demanding than the 2-D FPT.     

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.     

Modeling of High‐Pressure Ethene Homo‐ and Copolymerization

While reaction engineering of low‐molecular weight compounds mainly focuses on equilibria and selectivities, polymer properties are tremendously influenced by molecular weight distribution as well as branching structure. In order to determine the branching structure of low‐density polyethylene (LDPE) copolymers in dependence on chosen process conditions, a Monte‐Carlo approach was developed. By modeling the topology as well as the comonomer distribution in the polymer chains a deeper insight in the process‐microstructure‐properties relationship is gained.     

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.     

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.     

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.     

Modeling at Pore-Scale Isothermal Drying of Porous Materials: Liquid and Vapor Diffusivity

Numerical simulations of isothermal drying of non-hygroscopic liquid-wet rigid porous media are performed. Two- and three-dimensional pore networks represent pore spaces. Two types of mechanisms are considered: evaporation and hydraulic flow. The drying is considered to be a modified form of invasion percolation. Liquid in pore corners allows for a hydraulic connection throughout the network at all times. As drying progresses, liquid is replaced by vapor by two fundamental mechanisms: evaporation and pressure gradient-driven liquid flow. Using a Monte Carlo simulation, evaporation and drainage times are computed. The controlling mechanism is indicated by the shorter calculated time. Initially, the drying is governed by liquid flow, then by a combination of liquid flow and evaporation and finally by local evaporation. Reported here are the distributions of liquid and vapor with drying time, capillary pressure curves, liquid film saturation curves, and liquid diffusivity and vapor diffusivity as a function of liquid saturation.     

Modeling at Pore-Scale Isothermal Drying of Porous Materials: Liquid and Vapor Diffusivity

Numerical simulations of isothermal drying of non-hygroscopic liquid-wet rigid porous media are performed. Two- and three-dimensional pore networks represent pore spaces. Two types of mechanisms are considered: evaporation and hydraulic flow. The drying is considered to be a modified form of invasion percolation. Liquid in pore corners allows for a hydraulic connection throughout the network at all times. As drying progresses, liquid is replaced by vapor by two fundamental mechanisms: evaporation and pressure gradient–driven liquid flow. Using a Monte Carlo simulation, evaporation and drainage times are computed. The controlling mechanism is indicated by the shorter calculated time. Initially, the drying is governed by liquid flow, then by a combination of liquid flow and evaporation and finally by local evaporation. Reported here are the distributions of liquid and vapor with drying time, capillary pressure curves, liquid film saturation curves, and liquid diffusivity and vapor diffusivity as a function of liquid saturation.     

Modelling and simulation of suspension polymerization of vinyl chloride via population balance model

A detailed population balance model is presented for suspension polymerization of vinyl chloride in an isothermal batch reactor perfectly mixed on macrolevel. Coalescence and breakage of monomer droplets, as well as mass exchange of species between the droplets induced by collisions, termed micromixing, are also included into the model forming a complex three-scale system. The resulted population balance equation is solved by coupling the deterministic continuous time computation of polymerization reactions inside the droplets with the random coalescence and breakage events of droplets using Monte Carlo simulation. The results obtained by simulation revealed that aggregation, breakage and micro-mixing of species induced by droplet collisions affect the process significantly.     

Simulating the early stage of high-shear granulation using a two-dimensional Monte-Carlo approach

A two-dimensional (2-D) model of a granulation process is presented in this paper. It aims to simulate an entire granulation batch without the use of an initial experimental or fictitious 2-D density function, by taking the experimental operating conditions into account. The mass of liquid and solid in the granules are the two predicted internal variables. The 2-D population balance equation is solved by a Constant Number Monte-Carlo method. This is a stochastic technique tracking the evolution of a population, whilst performing the calculations with a fixed number of particles. This is achieved by reducing or increasing the sample volume when an event results in a net production or a net decrease in the number of particles, respectively. An original multi-population approach is developed to describe the early stage of the process, where small numbers of granules are formed amongst a large number of primary particles. It consists of separating the primary particles from the granule population. A specific intensive variable is introduced to keep track of the repartition of masses. The overall density function is reconstructed a posteriori from the combination of the two populations. This approach allows the simulation to commence from the initial addition of liquid at the start of the process, rather than to start from an early granule size distribution. The early stage of the granulation process, frequently referred as nucleation, can therefore be studied numerically. Four different mechanisms are implemented. Nucleation and re-wetting describe the addition of liquid to the system. The interactions between liquid and solid phases are modelled by a layering process. An aggregation model is also included to simulate the growth of particles undergoing frequent collisions. Finally, the relevance of this new model is demonstrated by confronting the simulations to real experimental data.     

蒙特卡洛优化法在炼焦配煤中的应用

应用蒙特卡洛优化法配煤,只需做一定量的试验就可确定目标函数或约束条件函数中的某些系数,并根据需要,建立数学模型,编制程序,求出最优解。     

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     

Method of moments with interpolative closure

The article summarizes the principal details of a method of moments with interpolative closure. This is a mathematically rigorous yet numerically economical approach to particle dynamics, describing time evolution of a particle ensemble undergoing simultaneous nucleation, coagulation, and surface growth. The method was introduced some time ago and since then has undergone further development as well as extensive testing in reactive flow simulations of practical systems. These results, scattered over quite diverse literature, are presented here in a unified form, focussing on logical development rather than on chronological order. In addition, the validity of the numerical approach is addressed on rigorous mathematical grounds. Also discussed are method shortcomings along with possible directions to their resolution.     

Stochastic Modeling of Fluidized Bed Agglomeration: Determination of Particle Moisture Content

The present study describes the theoretical modeling of particle formation by agglomeration in fluidized beds. The model is mathematically solved by applying a stochastic Monte Carlo method. It takes into account the complete droplet history (predrying during flight, spreading, solidification and penetration after deposition on a particle) so that thermal effects such as the increase in fluidization gas mass flow or gas temperature have a physically based impact on process kinetics. Contradictions between measured and simulated particle moisture contents were found. Thus, we present a modified drying model and compare results with experimental data.     

Modeling and Optimization of Two Stage AC Electrostatic Desalter

Electrostatic desalters are commonly used to separate water from crude oil emulsions. In this study, a novel mathematical model based on a bivariate population balance equation has been proposed to model the steady state distribution of water droplet sizes and salt concentration along an Alternative Current (AC) electrostatic desalter. Pilot plant data for two heavy crude oils are used to validate the model. Effect of different operational parameters on the outlet dehydration efficiency and salt content of crude oil have been studied. Furthermore, the effect of two-stage desalting on the outlet Basic Sediment and Water (BS&W) and salt content of oil have been studied and the optimized parameters of such configuration have been found by differential evolution method to get the best performance of the electrostatic desalters.     

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     

Solution of bivariate population balance equations with high-order moment-conserving method of classes

In this work the high-order moment-conserving method of classes (HMMC) (Alopaeus et al., 2006) is extended to solve the bivariate Population Balance Equation (PBE). The method is capable of guaranteeing the internal consistency of the discretized equations for a generic moment set, including mixed-order moments of the distribution. The construction of the product tables in the case of aggregation, breakage and convection in internal coordinate space are discussed. Eventually, several test cases are considered to assess the accuracy of the method. The application to a realistic mass transfer problems in a liquid–liquid system is preliminarily discussed. The comparison with analytical solutions of pure aggregation problems shows that the proposed method is accurate with only a limited number of categories.