Formulation of a non-linear framework for population balance modeling of batch grinding: Beyond first-order kinetics

Population balance models (PBMs) for batch grinding are based on the concepts of specific breakage rate and breakage distribution. In the traditional PBMs, the breakage rate is assumed first-order, thus neglecting the effects of the temporally evolving material properties and multi-particle interactions. As an attempt to explain some of the above effects, a time-dependent specific breakage rate was introduced in the literature. The time-variant PBMs are inadequate to explain the multi-particle interactions explicitly and thoroughly. In this paper, we formulate a non-linear population balance framework to explain the non-first-order breakage rates that originate from multi-particle interactions. Based on this framework, four size-discrete non-linear models with varying complexity have been derived. A simple non-linear model with non-uniform kinetics assumption, Model B, was used to simulate the slowing-down phenomenon commonly observed in dry grinding processes. Not only does the model explain the effects of the fines accumulation on the specific breakage rate of the coarse, but also it is capable of predicting the significant influence of the initial population density. Identification of the proposed models, i.e., the solution of the inverse problem is also discussed.     

A mixed Weibull model for size reduction of particulate and fibrous materials

This paper describes a simple alternative to the classical population balance breakage model, which characterizes and controls the size distribution of particles submitted to a reduction process. The new approach is based on cumulative distribution functions of mixed random variables. Results indicate that a Weibull mixture distribution function adequately models the size of particles submitted to various breakage processes. The model was further applied to experimental reduction processes with apparently random breakage probability and yielded good estimates of the unbroken particle and fragment distributions. Use of these results for direct and indirect prediction of the size alteration under dimensional reduction processes is discussed.     

基于群体平衡理论的水合物聚集动力学模型

水合物颗粒间的聚集是导致油气输送管道堵塞的重要原因,建立聚集模型能更好地进行水合物聚集的防治。首先,假定管道中水合物颗粒粒径连续分布并忽略对流扩散的影响,建立了以群体平衡模型为基本框架的水合物聚集动力学模型。重点考虑水合物的聚集和破碎,该模型的核心主要包括聚集核和破碎核两部分。其中,聚集核包括水合物颗粒间的碰撞频率和碰撞后的聚集效率,破碎核包括水合物颗粒的破碎频率和破碎后子颗粒的粒径分布函数。接着,以前人研究为基础并结合水合物颗粒的自身特性,对模型聚集核和破碎核的计算方法进行了选取和改进。最后,采用计算流体力学方法对模型进行了求解并将求解结果与相关实验数据进行了对比分析。该模型可为管道流动安全保障提供技术支持。     

AGGREGATION AND BREAKAGE OF SOLID PARTICLES IN SUSPENSIONS AGITATED IN A VIBRATING MIXER: A NON-FRACTAL APPROACH

Aggregation and breakage of solid particles or aggregates suspended in liquid can be found in numerous industrial processes. It is crucially important to predict the evolution of aggregates at any moment during the process. This aim can be achieved by modeling based on solid phase population balances. A lumped discrete population balance model has been selected for the verification of the experimental data. The experiments have been carried out in a laboratory-scale vibrating mixer equipped with a disc-type reciprocating impeller. A reasonably good agreement between the computational and experimental results of PSD data has been obtained. Dependencies between aggregation and breakage rate coefficients and the average energy dissipation rate in the mixer have been shown.     

AGGREGATION AND BREAKAGE OF SOLID PARTICLES IN SUSPENSIONS AGITATED IN A VIBRATING MIXER: A NON-FRACTAL APPROACH

Aggregation and breakage of solid particles or aggregates suspended in liquid can be found in numerous industrial processes. It is crucially important to predict the evolution of aggregates at any moment during the process. This aim can be achieved by modeling based on solid phase population balances. A lumped discrete population balance model has been selected for the verification of the experimental data. The experiments have been carried out in a laboratory-scale vibrating mixer equipped with a disc-type reciprocating impeller. A reasonably good agreement between the computational and experimental results of PSD data has been obtained. Dependencies between aggregation and breakage rate coefficients and the average energy dissipation rate in the mixer have been shown.     

Modelling of flocculation using a population balance equation

In this paper, a model based on a population balance equation (PBE) is developed. It aims at reproducing experimental floc size distributions obtained at steady state in a jar-test. The objective is to develop a simple model, based on physical phenomena, and that does not contain any adjustable parameters. Floc size distributions obtained using a part of a particle image velocimetry (PIV) device and image analysis are used to develop mathematical expressions for the aggregation and breakage kernels. A critical volume beyond which breakage is of significant importance is identified and related to the hydrodynamics. Hydrodynamic sequencing allows the distribution of the daughter particles resulting from a breakage event to be established. The model is finally successfully validated against experimental results.     

Simulation of Turbulent Emulsification in a Sonolator Mixer: Interpretation of Drop Breakage Time

The simulation of dilute emulsions in a model ACIP2 Sonolator is investigated using computational fluid dynamics and population balance methods. Two breakage frequency models are used that differ in the expression of the drop breakage time. Drop breakage modeling based on homogenous isotropic turbulence (HIT) shows poor agreement of the Sauter mean diameter when compared to the experiments; simulations with the empirical model from Alopaeus et al. yield better agreement. By perturbing the classical HIT spectrum, it is shown that the breakage time in the empirical model corresponds to a non‐isotropic energy spectrum. Such spectra have been observed in the non‐isotropic near field in a model A Sonolator, which provides a plausible explanation of why the empirical model performs better than the HIT‐based model.     

Population balance modeling of non-linear effects in milling processes

Specific breakage rate (selection) and breakage distribution functions are used to describe the birth and death terms in population balance models (PBMs) for milling processes. Traditional PBMs for milling processes are inherently linear because the breakage rate is assumed first-order. The specific breakage rate is independent of the population density while it depends on particle size and possibly on time. Even though the linear theory has been applied with some success to the modeling, optimization, and design of various mills in the last 50 years, many researchers have indicated its restrictions and subjected it to serious criticism. In this paper, we first categorize the experimentally observed deviations from the linear theory and suggest the multi-particle interactions as the origin of these deviations. To account for the peculiar non-linear effects, a phenomenological theory has been proposed via multiplicative decomposition of the specific breakage into a size-dependent apparent breakage rate and a population density dependent functional. The proposed theory recovers the first-order breakage kinetics in the limit, yet it is sufficiently general to explain all experimentally observed deviations. Numerical simulations of a batch milling process have demonstrated the potential of the non-linear theory.     

A numerical method for solving the transient multidimensional population balance equation using an Euler-Lagrange formulation

A numerical method was developed to solve the population balance equation for transient multidimensional problems including particle-particle interactions. The population balance equation was written in a mixed Euler-Lagrange formulation which was solved using the discretization method that represents the number density function by impulse functions, an operator splitting method and a remeshing procedure for the internal variable that conserves the mass and the number of particles.This method was successfully tested against analytical and semi-analytical solutions for pure breakage, pure coalescence, breakage and coalescence, pure advection, advection with absorption, advection with binary uniform breakage and with constant or linear absorption. The method was also applied to a free-boundary transient one-dimensional gas-phase model in a bubble column reactor with simplified hydrodynamics. Accurate solutions were obtained for several simulation conditions for the bubble column, including gas absorption, bubble breakage, bubble coalescence and variable gas density effects. The results showed that the numerical method is adequate and robust for solving transient population balance problems with spatial dependence and particle-particle interactions.     

Population balance model for nucleation,growth, aggregation,and breakage of hydrate particles in turbulent flow

This article describes a computational model for the size evolution of hydrate particles in a pipeline‐pump system with turbulent flow. The model is based on the population balance principle, and the simulation results were validated with data from an experimental study of a flow loop containing hydrate particles reported in the literature. It is found that the particle size is significantly influenced by aggregation and breakage, related to shear in the flow, and that these effects are comparable to those of growth and nucleation, related to diffusional processes. Two different approaches for hydrate growth and nucleation, one of continuous nucleation during the process and one of only an initial nucleation‐pulse, were used. This was done to compare the aggregation and breakage parameters which come out when fitting the models output to experiment. These two approaches are found to give rise to similar aggregation/breakage parameters, lending credence to the pbm‐based modeling. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

基于群体平衡理论的管内水合物浆流动特性数值模拟

水合物浆的流动特性对深水油气管道的流动安全保障和水合物法气体储运的工业推广具有重要意义。目前已有的水合物浆液流动特性数值模拟方法存在较多的缺陷及不足,为此本文引入了基于水合物颗粒聚集动力学的群体平衡模型。该模型重点考虑水合物颗粒在流动过程中的碰撞频率、聚并效率、破碎频率及破碎后子颗粒的粒径分布函数,可较好地描述管内水合物颗粒的流动行为。根据文献中的实验装置建立三维几何模型,利用Fluent 14.5软件对上述群体平衡模型和相关固液两相流模型进行联合求解,借此模拟流速及水合物体积分数对水合物浓度分布、水合物颗粒粒径分布和流动压降等水合物浆流动特性的影响。本文模拟结果与文献中相关实验数据吻合良好,可为水合物浆技术的规模应用提供借鉴。     

Energy and population balances in comminution process modelling based on the informational entropy

The results of theoretical and experimental studies of a comminution process are presented. There are two random functions: the selection function and the breakage function in the stochastic model based on a population balance. This model enables prediction of particle size distributions of comminution products after determination of both random functions. Maximum entropy method is used in the entropy model for determining the breakage function. Two cases are analysed, based on continuous and discrete particle size distribution functions of the fed material. Apart from mass balance, the energy balance of comminution process is also used. Searched form of breakage function is determined with the application of methodology of calculus of variations. The results of experimental identification of both models are presented. The parameters that occur in the discrete form of the selection and breakage functions were the identification objects. The results of experimental investigations of quartz sand single comminution in a laboratory jet mill provided an identification base. The experimentally identified results of the entropy model confirmed the adequacy of the theoretical analysis and demonstrated the possibility of adequate prediction of particle size distributions resulting from single comminution.     

精馏塔板相界面积的粒数衡算模型

通过对气液湍流系统中气泡的动力学行为进行分析推导出气泡破碎速率与聚并速率的表达式,在此基础上建立了描述气泡分散特性的粒数衡算模型.求解粒数衡算方程可以计算精馏塔板上气泡的粒度分布以及气液相界面积,模拟计算的结果与实验测量的数据相当一致,证明可以利用粒数衡算模型较准确地预测气液湍流系统的分散性质.     

A Population Balance Model for Disperse Systems： Particle Size Distribution in Suspension Polymerization

On the basis of population balance a mathematical model is developed to describe the formation of polymer partlcle in styrene suspension polymerization. The characteristics of coalescence and breakage of droplets and the gel effect are analyzed in particular. Parameters of the models are esthnztted by experimental. on reaction conversion and particle size distributlon. The results show that the model is suitable for predicting polymerization processes.     

采用粒数衡算模型研究沉淀过程中粒子的聚结和破裂

通过采用MSMPR反应沉淀器,研究了普鲁卡因青霉素反应沉淀过程中粒子的聚结和破裂对过程的影响机理,并对用生死函数表征聚结和破裂的粒数衡算模型采用新的分析方法。结果表明,将二元破裂模型应用于该粒数衡算模型是可行的。     

采用粒数衡算模型研究沉淀过程中粒子的聚结和破裂

通过采用MSMPR反应沉淀器,研究了普鲁卡因青霉素反应沉淀过程中粒子的聚结和破裂对过程的影响机理,并对用生死函数表征聚结和破裂的粒数衡算模型采用新的分析方法。结果表明,将二元破裂模型应用于该粒数衡算模型是可行的。     

A rigorous breakage matrix methodology for characterization of multi-particle interactions in dense-phase particle breakage

Broadbent and Calcott's breakage matrix methodology has been used for more than 50 years to model various comminution processes and to determine breakage functions from experimental data. The methodology assumes first-order law of breakage and neglects mechanical multi-particle interactions that are especially prevalent in dense-phase comminution processes and breakage tests. Although several researchers severely criticized this aspect of the methodology, Baxter et al. (2004, Powder Technol. 143–144:174–178) were the first to modify the methodology toward determining the elements of a feed-dependent breakage matrix. However, no non-linear breakage matrix has ever been constructed from experimental data using the modified approach. In this study, a critical analysis of this modified approach has been performed, and the non-linear breakage matrix was fundamentally derived from a non-linear population balance model. Different approaches were proposed to identify the breakage functions based on the nature of available breakage tests on multiple mono-dispersed feed samples and at least one poly-dispersed sample. Using the derived equations, available experimental data on the breakage of a binary mixture of coarse and fine limestone particles in uniaxial compression test were fitted to quantify the multi-particle interactions. Superior fitting capability of rational approximation to the effectiveness factor was demonstrated.     

AGGLOMERATION,BREAKAGE, POPULATION BALANCE,AND CRYSTALLIZATION KINETICS OF REACTIVE PRECIPITATION PROCESS

The agglomeration and breakage of particles play a significant role in determining final particle size distribution (PSD) and other qualities such as filtering characteristics and impurity content. In reactive precipitation processes, especially during the precipitation of fine particles, the agglomeration and breakage of particles normally cannot be neglected. In this study, the agglomeration and breakage of particles during the reactive precipitation process of procaine penicillin has been investigated experimentally through a continuous steady MSMPR crystallizer. Based on the population balance theory, a crystallization kinetics model including agglomeration and breakage is established, in which the breakage of particles is expressed by a two-body equal-volume birth function and a two-body power-law death function. The crystallization kinetics model is shown to be more suitable than size-dependent growth models as ASL and MJ2.     

AGGLOMERATION, BREAKAGE, POPULATION BALANCE, AND CRYSTALLIZATION KINETICS OF REACTIVE PRECIPITATION PROCESS

The agglomeration and breakage of particles play a significant role in determining final particle size distribution (PSD) and other qualities such as filtering characteristics and impurity content. In reactive precipitation processes, especially during the precipitation of fine particles, the agglomeration and breakage of particles normally cannot be neglected. In this study, the agglomeration and breakage of particles during the reactive precipitation process of procaine penicillin has been investigated experimentally through a continuous steady MSMPR crystallizer. Based on the population balance theory, a crystallization kinetics model including agglomeration and breakage is established, in which the breakage of particles is expressed by a two-body equal-volume birth function and a two-body power-law death function. The crystallization kinetics model is shown to be more suitable than size-dependent growth models as ASL and MJ2.