Solution of population balances by high order moment-conserving method of classes: reconstruction of a non-negative density distribution

A method is proposed for reconstruction of a non-negative piecewise continuous density distribution from a solution obtained by the high order moment conserving method of classes (HMMC), presented in our earlier papers [Alopaeus, V., Laakkonen, M., Aittamaa, J., 2006]. Solution of population balances with breakage and agglomeration by high order moment-conserving method of classes. Chemical Engineering Science 61, 6732-6752; Alopaeus, V., Laakkonen, M., Aittamaa, J., 2007. Solution of population balances with growth and nucleation by high order moment-conserving method of classes. Chemical Engineering Science 62, 2277-2289.]. The resulting distribution is strictly non-negative, and the overall distribution moments obtained by HMMC are conserved very accurately. The distribution is reconstructed by using the positive region of HMMC as an initial estimate for the continuous density distribution. This estimate is then refined by transforming the abscissas and scaling the ordinates of the density distribution so that the distribution moments are as close to the original solution as possible. The method is tested with a number of numerical examples. These cases are chosen due to their tendency to produce oscillations in HMMC, or due to their nature of producing multimodal distributions. It is shown that when the reconstruction algorithm is applied to the HMMC solution, the analytical density distribution is captured extremely well. This is true even in cases where the width of the analytical distribution is only a fraction of the original HMMC grid size. Multimodal population density distributions can also be reconstructed accurately from a HMMC solution with a reasonably small number of grid points, without a need to assume any basis functions for the reconstruction.     

Solution of population balances with breakage and agglomeration by high-order moment-conserving method of classes

A general high-order method of classes framework for numerical solution of population balances is developed and tested. It is based on conservation of an arbitrary number of moments in the numerical discretization of the integrodifferential population balance equation. Optimal construction of the product tables for agglomeration and breakage events are discussed separately. It is shown that if more than two moments are set to be conserved in the numerical scheme, accuracy can be improved by several orders of magnitude compared to the state-of-the-art methods. Several test cases related to pure breakage or agglomeration, and also to simultaneous breakage and agglomeration, are shown. The results show that the present method is extremely accurate for all tested cases already with a limited number of size categories.     

On the solution of population balance equations (PBE) with accurate front tracking methods in practical crystallization processes

The PBE (population balance equation) containing birth, growth, agglomeration and breakage kinetics is described by a conservation law with a moving source term. For the solution of the PBE, we compare two accurate front tracking methods such as a modified method of characteristics (MOC) and a finite difference method with the weighted essentially non-oscillatory (WENO) scheme. Both methods are applied to a potassium sulfate crystallization problem (K₂SO₄-H₂O system) with a discontinuous initial condition. Parameters of agglomeration and breakage kinetics are estimated on the basis of the experimental data of the K₂SO₄-H₂O system.Owing to moving axis along a crystal growth rate (i.e. elimination of the growth term), the modified MOC is able to provide a highly accurate solution even at discontinuous points without numerical diffusion error. However, in the case of stiff nucleation which can commonly appear in practical crystallization processes, it is necessary to adaptively determine time levels to add a mesh of the nuclei size. For solving PBEs involving agglomeration and breakage terms, the MOC can take more long computational time than the spatial discretization methods like the WENO scheme. It is pointed out that the MOC is not available to solve more than two coupled PBEs in general.WENO schemes for spatial discretization are firstly addressed in this study for the dynamic simulation of batch crystallization processes. The WENO schemes show improvements of accuracy and stability over conventional discretization methods (e.g., backward, central or common upwinding schemes). However the WENO schemes on fixed meshes show, to some extent, the numerical diffusion error near discontinuities or steep moving fronts like other finite difference methods. Hence, they require spatially-adaptive mesh techniques in order to track more accurately the moving fronts. Even though the WENO schemes are less accurate than the MOC, they are of practical use for solving complex PBEs owing to a short computational time and little limitation to use.     

反应沉淀法制备亚微米粒子的粒度分布模拟与实验验证

采用基于粒数衡算方程及物料衡算方程的数值模拟方法模拟了包括反应、成核、生长及凝并的反应沉淀制备亚微米粒子的过程,其中颗粒间的凝并过程采用分级模型来模拟,产物颗粒的粒度分布（PSD）由离散化的粒数衡算方程求得.以BaSO₄的反应沉淀体系作为研究体系,将实验测量数据与数值模拟结果进行对比,验证了所构建的数学模型的正确性与适用性.应用此模型进行分析,发现沉淀时间、产物过饱和度及凝并对粒度及粒度分布有着显著的影响.在此基础上,提出了反应沉淀法制备亚微米粒子过程中颗粒粒度及粒度分布的控制方法.     

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.     

ANALYSIS OF BATCH CRYSTALLIZATION PROCESSES

The analysis of batch crystallization processes normally requires the consideration of the time-dependent, batch conservation equations (e.g., population, mass, and energy balances), together with appropriate nucleation and growth kinetic equations. The solution of these integro-differential equations is relatively difficult, even by numerical techniques. This review outlines the advances that have been made in the experimental techniques and data analyses which can be used to study the crystallization kinetics and the crystal size distribution (CSD) in batch suspension crystallizers. Several simple and useful methods are discussed which include characterization of CSD maximum, cumulative CSD approach, SSBCR crystallizer, thermal response technique, maximum allowable growth rate, and desupersaturation curve technique.     

ANALYSIS OF BATCH CRYSTALLIZATION PROCESSES

The analysis of batch crystallization processes normally requires the consideration of the time-dependent, batch conservation equations (e.g., population, mass, and energy balances), together with appropriate nucleation and growth kinetic equations. The solution of these integro-differential equations is relatively difficult, even by numerical techniques. This review outlines the advances that have been made in the experimental techniques and data analyses which can be used to study the crystallization kinetics and the crystal size distribution (CSD) in batch suspension crystallizers. Several simple and useful methods are discussed which include characterization of CSD maximum, cumulative CSD approach, SSBCR crystallizer, thermal response technique, maximum allowable growth rate, and desupersaturation curve technique.     

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.     

On the solution of population balances for nucleation, growth, aggregation and breakage processes

This work is concerned with the modeling and simulation of population balance equations (PBEs) for combined particulate processes. In this study a PBE with simultaneous nucleation, growth, aggregation and breakage processes is considered. In order to apply the finite volume schemes (FVS) a reformulation of the original PBE is introduced. This reformulation not only help us to treat the aggregation and breakage processes in a manner similar to the growth process in the FVS but also in deriving a stable numerical scheme. Two numerical methods are proposed for the numerical approximation of the resulting reformulated PBE. The first method combines a method of characteristics (MOC) for growth process with an FVS for aggregation and breakage processes. The second method purely uses a semidiscrete FVS for all processes. Both schemes use the same FVS for aggregation and breakage processes. The numerical results of the schemes are compared with each other and with the available analytical solutions. The numerical results were found to be in good agreement with analytical solutions.     

On the stochastic simulation of particulate systems

The stochastic chemical kinetics approach provides one method of formulating the stochastic crystallization population balance equation (PBE). In this formulation, crystal nucleation and growth are modelled as sequential additions of solubilized ions or molecules (units) to either other units or an assembly of any number of units. Monte Carlo methods provide one means of solving this problem. In this paper, we assess the limitations of such methods by both (1) simulating models for isothermal and nonisothermal size-independent nucleation, growth and agglomeration; and (2) performing parameter estimation using these models. We also derive the macroscopic (deterministic) PBE from the stochastic formulation, and compare the numerical solutions of the stochastic and deterministic PBEs. The results demonstrate that even as we approach the thermodynamic limit, in which the deterministic model becomes valid, stochastic simulation provides a general, flexible solution technique for examining many possible mechanisms. Thus the stochastic simulation permits the user to focus more on modelling issues as opposed to solution techniques.     

Mechanisms of agglomerate growth in green pelletization

The mechanisms responsible for the formation and growth of agglomerates in balling or granulation can be described as nucleation, coalescence, abrasion transfer, breakage, and snowballing. These mechanisms have been identified by means of tracer studies using two calcites which have similar green pelletizing behavior but different fluorescent characteristics. Precise definitions of the growth mechanisms, so as to be useful for mathematical modeling of agglomeration processes, are given. Further, the batch balling behavior of particulate materials in terms of the growth mechanisms is discussed.     

Agglomeration and breakage of nanoparticles in stirred media mills—a comparison of different methods and models

The increasing industrial demand for nanoparticles challenges the application of stirred media mills to grind in the sub-micron size range. It was shown recently [Mende et al., 2003. Mechanical production and stabilization of submicron particles in stirred media mills. Powder Technology 132, 64-73] that the grinding behavior of particles in the sub-micron size range in stirred media mills and the minimum achievable particle size is strongly influenced by the suspension stability and thus the agglomeration behavior of the suspension. Therefore, an appropriate modeling of the process must include a superposition of the two opposing processes in the mill i.e., breakage and agglomeration which can be done by means of population balance models. Modeling must now include the influence of colloidal surface forces and hydrodynamic forces on particle aggregation and breakup. The superposition of the population balance models for agglomeration and grinding with the appropriate kernels leads to a system of partial differential equations, which can be solved in various ways numerically. Here a modified h-p Galerkin algorithm which is implemented in the commercially available software package PARSIVAL developed by CiT (CiT GmbH, Rastede, Germany) and the moment methodology according to [Diemer and Olsen, 2002a. A moment methodology for coagulation and breakage problems: Part I—analytical solution of the steady-state population balance. Chemical Engineering Science 57 (12), 2193-2209; Diemer and Olsen, 2002b. A moment methodology for coagulation and breakage problems: Part II—moment models and distribution reconstruction. Chemical Engineering Science 57 (12), 2211-2288] are used and compared to explicit data on alumina. This includes a comparison of the derived particle size distributions, moments and its accuracy depending on the starting particle size distribution and the used agglomeration and breakage kernels. Finally, the computational effort of both methods in comparison to the prior mentioned parameters is evaluated in terms of practical application.     

Towards a Complete Population Balance Model for Fluidized-Bed Spray Agglomeration

This article concerns the simultaneous processes of agglomeration and drying. In order to predict temperatures and moisture content in gas and particle phase, heat and mass transfer mechanism and particle size enlargement has been considered in one model. The model takes heat and mass transfer phenomena between particle phase, suspension gas, and bypass gas into account. The disperse phase is modeled by a three-dimensional population balance (PBE), which can be reduced to a set of three one-dimensional PBEs. The latter are coupled with heat and mass transfer balances of the gas phase. Furthermore, some simulation and experimental results are presented.     

Computationally efficient solution of population balance models incorporating nucleation, growth and coagulation: application to emulsion polymerization

A computationally efficient solution technique is presented for population balance models accounting for nucleation, growth and coagulation (aggregation) (with extensions for breakage). In contrast to earlier techniques, this technique is not based on approximating the population balance equation, but is based on employing individual rates of nucleation, growth and coagulation to update the PSD in a hierarchical framework. The method is comprised of two steps. The first step is the calculation of the rates of nucleation, growth and coagulation by solving an appropriate system of equations. This information is then used in the second step to update the PSD. The method effectively decomposes the fast and the slow kinetics, thereby eliminating the stiffness in the solution. In solving the coagulation kernel, a semi-analytical solution strategy is adapted, which substantially reduces the computational requirement, but also ensures the consistency of properties such as the number and mass of particles.     

A Critical Review of the Application of Drop-Population Balances for the Design of Solvent Extraction Columns: I. Concept of Solving Drop-Population Balances and Modelling Breakage and Coalescence

Population-balances are a powerful method to predict the population behavior of drops in chemical-engineering equipment such as solvent extraction columns. In such columns a complex interaction of different phenomena, namely drop sedimentation, mass transfer, drop breakage and coalescence as well as axial dispersion occurs. In this article the concept of drop-population balances is discussed in detail as well as possible solution methods. Also, a critical review of existing models accounting for breakage and coalescence taking place in extraction columns is presented. Future parts of this series will be devoted to modelling mass-transfer and sedimentation as well as on application of single-drop based modelling.     

普鲁卡因青霉素的结晶动力学

通过对用生死函数表征粒子聚结和破裂的粒数衡算模型,采用新的分析计算方法来研究普鲁卡因青霉素反应结晶过程中粒子的聚结和破裂对过程的影响机理;采用ＭＳＭＰＲ结晶实验获得了普鲁卡因青霉素反应结晶的结晶动力学。     

Simulation of coalescence and breakage: an assessment of two stochastic methods suitable for simulating liquid-liquid extraction

Industrial problems such as liquid-liquid extraction often require the use of population balances, so a thorough understanding of the advantages and disadvantages of all the available solution techniques is essential. The mathematical literature on stochastic solution methods for the population balance equation is often quite fragmentary and tends to focus mainly on existence proofs rather than examining the applicability of a particular algorithm to a practical problem of interest to researchers or industrialists. In this paper, a practical study is made of two stochastic solution methods for the population balance equation, simulating coalescence and binary breakage. The first algorithm studied is the existing direct simulation algorithm (DSA), proposed in (Eibeck and Wagner, Stoch. Anal. Appl. 18(6) (2000) 921). The second is an extension of the Mass Flow Algorithm (MFA), which was proposed in (Eibeck and Wagner, Ann. Appl. Probab. 11(4) (2001) 1137) for coagulation only and in (Jourdain, Markov Processes and Related Fields 9(1) (2003) 103) for discrete coagulation-fragmentation. MFA (mass flow algorithm) is extended to include breakage in the continuous case, and a binary search method of distribution generation is introduced, leading to improved efficiency. Numerical investigation of the performance of the two algorithms is carried out by applying them both to a test case, for which an analytical solution is calculated. For both algorithms, convergence of the predicted moments to the analytical solution goes as the inverse of the number of stochastic particles, N, except for the zeroth moment predicted by MFA. This exhibits large fluctuations, due to the presence of very small particles, and converges approximately as N^-1/3. The new algorithm, MFA, exhibits significant variance reduction—and therefore improved simulation efficiency—for the prediction of higher moments, but for our test case the zeroth moment (the total number of particles) is predicted with better efficiency by DSA. In many breakage models for liquid-liquid systems however, the introduction of a minimum particle size reduces the advantage held by DSA for predicting the zeroth moment. Depending on the minimum particle size, MFA can perform comparably with DSA for predicting the zeroth moment. Using models for coalescence and breakage from literature, DSA is successfully applied to the case of a laboratory scale rotating disc contactor.     

Crystallization of aluminium hydroxide from the reactive NaAl(OH)4–NaHCO3 solution: Experiment and modeling

The crystallization kinetics of aluminium hydroxide from the sodium aluminate solution reacted with sodium bicarbonate were systematically investigated in a steady-state MSMPR (mixed-suspension mixed-product removal) crystallizer for the first time, and the expressions of the nucleation rate, growth rate and the agglomeration kernel of aluminium hydroxide were successfully regressed. The aluminium hydroxide particles precipitated from the reactive system are identified as gibbsite by XRD and SEM examinations. The volume growth rate order of gibbsite with respect to the relative supersaturation of the solution is above the linear growth rate order, and the spiral growth mechanism for the growth of the basal face of gibbsite in the reactive system was further identified by the growth rate, morphology analysis as well as the calculated surface entropy factor. The secondary nucleation rate of gibbsite from the reactive system is three to four orders of magnitude larger than that from seeded process reported in the only available literature reference. The agglomeration kernel of gibbsite in the reactive system increases linearly with growth rate and residence time, and the positive order about 0.55 of magma density is thoroughly different from the negative order of magma density for gibbsite agglomeration in seeded process presented in the literature.