Automatic differentiation‐based quadrature method of moments for solving population balance equations

The quadrature method of moments (QMOM) is a promising tool for the solution of population balance equations. QMOM requires solving differential algebraic equations (DAEs) consisting of ordinary differential equations related to the evolution of moments and nonlinear algebraic equations resulting from the quadrature approximation of moments. The available techniques for QMOM are computationally expensive and are able to solve for only a few moments due to numerical robustness deficiencies. In this article, the use of automatic differentiation (AD) is proposed for solution of DAEs arising in QMOM. In the proposed method, the variables of interest are approximated using high‐order Taylor series. The use of AD and Taylor series gives rise to algebraic equations, which can be solved sequentially to obtain high‐fidelity solution of the DAEs. Benchmark examples involving different mechanisms are used to demonstrate the superior accuracy, computational advantage, and robustness of AD‐QMOM over the existing state‐of‐the‐art technique, that is, DAE‐QMOM. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

基于微分代数积分矩量法的聚并器超细颗粒聚团研究

研究聚并器内布朗聚团和湍流聚团引起超细颗粒聚团,特别考虑颗粒之间近程力（范德华引力、静电斥力）和颗粒与气体之间的流体力学作用力对颗粒聚团的影响。基于FLUENT软件UDF功能自定义聚团核,考虑颗粒之间近程力和流体力学作用力对聚并率的影响,引入碰撞效率α对聚团核进行修正,得到修正湍流聚并模型并将该模型与理想湍流聚并模型进行比较。应用群体平衡模型（population balance model,PBM）耦合CFD对颗粒聚团过程进行数值模拟,并采用微分代数积分矩量法（DAE-QMOM）求解群体平衡方程。结果表明：理想湍流聚并模型与实验结果误差为8.92%,而修正改进的湍流聚团模型与实验结果误差仅为3.35%,更加符合实际情况;微分代数积分矩量法具有较高的效率,而且误差较小,相比PD积分矩量法有明显的优势,稳定性也比较突出。     

Experimental investigation of local bubble properties: Comparison to the sectional quadrature method of moments

In recent years, population balance models coupled to computational fluid dynamics have been used as a tool to model bubbly flows. In this work, we established a setup to measure the local size distribution, the velocity and orientation of bubbles in different zones of a bubble column. As a population balance model, we used the sectional quadrature method of moments for the first time in OpenFOAM to simulate the local change in the bubble size distribution and compared its results to the quadrature method of moments as well as to the experimental results. A satisfactory agreement between local experimental values of three investigated flow rates and simulations were found and enables the characterization of heterogeneous bubbly regimes as found in industrial reactive bubble columns.     

Dimension reduction of bivariate population balances using the quadrature method of moments

Crystallization models with direction-dependent growth rates give rise to multi-dimensional population balance equations (PBE) that require a high computational cost. We propose a model reduction based on the quadrature method of moments (QMOM). Using this method a two-dimensional population balance is reduced to a system of one-dimensional advection equations. Despite the dimension reduction the method keeps important volume dependent information of the crystal size distribution (CSD). It returns the crystal volume distribution as well as other volume dependent moments of the two-dimensional CSD. The method is applied to a model problem with direction-dependent growth of barium sulphate crystals, and shows good performance and convergence in these examples. We also compare it on numerical examples to another model reduction using a normal distribution ansatz approach. We can show that our method still gives satisfactory results where the other approach is not suitable.     

The quadrature method of moments and its relationship with the method of weighted residuals

Population balance models are generally computationally intensive, so in many practical applications only a few moments of the density function are computed, minimizing the computational costs. Nevertheless, the moment formulation contains an excess of unknowns with respect to equations denoting a closure problem. One possible solution to this closure problem might be to apply a numerical quadrature approximation.In this work, the relationship between the quadrature approximation and the well-known method of weighted residuals (MWR) is discussed. An important result obtained in this work is that the problem of reconstruction of the density function is avoided using the MWR version of the quadrature approximation. Numerical experiments are performed in order to elucidate the advantages and disadvantages of the quadrature approximations.     

Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems

In this work the quadrature method of moments (QMOM) is implemented in a commercial computational fluid dynamics (CFD) code (FLUENT) for modeling simultaneous aggregation and breakage. Turbulent and Brownian aggregation kernels are considered in combination with different breakage kernels (power law and exponential) and various daughter distribution functions (symmetric, erosion, uniform). CFD predictions are compared with experimental data taken from other work in the literature and conclusions about CPU time required for the simulations and the advantages of this approach are drawn.     

New quadrature‐based moment method for the mixing of inert polydisperse fluidized powders in commercial CFD codes

To describe the behavior of polydisperse multiphase systems in an Eulerian framework, we solved the population balance equation (PBE), letting it account only for particle size dependencies. To integrate the PBE within a commercial computational fluid dynamics code, we formulated and implemented a novel version of the quadrature method of moments (QMOM). This no longer assumes that the particles move with the same velocity, allowing the latter to be size‐dependent. To verify and test the model, we simulated the mixing of inert polydisperse fluidized suspensions initially segregated, validating the results experimentally. Because the accuracy of QMOM increases with the number of moments tracked, we ran three classes of simulations, preserving the first four, six, and eight integer moments of the particle density function. We found that in some cases the numerics corrupts the higher‐order moments and a corrective algorithm, designed to restore the validity of the moment set, has to be implemented. © 2012 American Institute of Chemical Engineers AIChE J, 58: 3054–3069, 2012     

Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds

Most of today's computational fluid dynamics (CFD) calculations for gas-solid flows are carried out assuming that the solid phase is monodispersed, whereas it is well known that in many applications, it is characterized by a particle size distribution (PSD). In order to properly model the evolution of a polydisperse solid phase, the population balance equation (PBE) must be coupled to the continuity and momentum balance equations. In this work, the recently formulated direct quadrature method of moments (DQMOM) is implemented in a multi-fluid CFD code to simulate particle aggregation and breakage in a fluidized-bed (FB) reactor. DQMOM is implemented in the code by representing each node of the quadrature approximation as a distinct solid phase. Since in the multi-fluid model, each solid phase has its own momentum balance, the nodes of the DQMOM approximation are convected with their own velocities. This represents an important improvement with respect to the quadrature method of moments (QMOM) where the moments are tracked using an average solid velocity. Two different aggregation and breakage kernels are tested and the performance of the DQMOM approximation with different numbers of nodes are compared. These results show that the approach is very effective in modeling solid segregation and elutriation and in tracking the evolution of the PSD, even though it requires only a small number of scalars.     

Bubble size distribution modeling in stirred gas–liquid reactors with QMOM augmented by a new correction algorithm

Local gas hold‐up and bubbles size distributions have been modeled and validated against experimental data in a stirred gas–liquid reactor, considering two different spargers. An Eulerian multifluid approach coupled with a population balance model (PBM) has been employed to describe the evolution of the bubble size distribution due to break‐up and coalescence. The PBM has been solved by resorting to the quadrature method of moments, implemented through user defined functions in the commercial computational fluid dynamics code Fluent v. 6.2. To overcome divergence issues caused by moments corruption, due to numerical problems, a correction scheme for the moments has been implemented; simulation results prove that it plays a crucial role for the stability and the accuracy of the overall approach. Very good agreements between experimental data and simulations predictions are obtained, for a unique set of break‐up and coalescence kinetic constants, in a wide range of operating conditions. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

基于集成建模方法的乙二醇全流程模拟

乙二醇生产流程是一个包含十几个精馏塔和固定床催化反应器的长流程。由于流程长且反应催化剂影响不易确定等因素,乙二醇生产过程的全流程模拟十分困难。基于ASPEN PLUS软件平台,集成支持向量机、优化算法、机理建模等多种方法,建立了乙二醇的全流程模型。通过采集并处理实际生产的工业数据,采用优化算法进行模型的智能校正,进一步完善了模型的性能。乙二醇全流程模型模拟结果与实际工业过程数据相符,能够满足工业建模的要求。     

各向同性颗粒系统数量平衡方程直接矩积分求解法的研究

随着颗粒系统微观行为研究的深入,数量平衡方程越来越受到重视。一般数量平衡方程由于复杂难以得到解析解。比较数量平衡方程的求解方法—分组方法(CM)、矩方法(MOM)和在矩方法基础上发展起来的直接矩积分方法(DQMOM),发现DQMOM具有计算量小,容易和计算流体力学(CFD)耦合等优点。运用直接矩积分方法对各向同性系统的颗粒长大、聚并和破碎等微观行为进行模拟,对各向同性液固系统的颗粒长大行为模拟的结果与文献的解析解吻合较好,误差小于10-5,对于各向同性液固系统的颗粒聚并和破碎行为模拟的结果与CM方法的精确解相比误差在2%以内。可见直接矩积分求解法在液固多相系统是有效的,具有良好的应用前景。     

碳酸铵直接沉淀法制备纳米氧化镁的研究

以六水氯化镁和碳酸铵为原料,通过改变温度、浓度、煅烧时间、表面活性剂等条件,用直接沉淀法制备氧化镁的前驱体,将前驱体在空气中焙烧制备氧化镁。用激光粒度仪测定了不同反应条件下产品的粒度分布,总结出不同反应条件与产品粒径的变化规律。用TG-DSC曲线确定了前驱体的分解温度,用X射线衍射仪（XRD）和扫描电镜 （SEM）对制备的产品进行表征。实验结果表明：在最佳反应条件下所得的产品的平均粒径为50 nm左右,其形貌为哑铃状。     

基于分阶段的LSSVM发酵过程建模

发酵过程建模是研究微生物发酵的重要课题,基于模型可实现被测参量的软测量、系统的优化控制。鉴于引入混合核函数的最小二乘支持向量机在过程建模中具有优良表现,采用基于混合核函数的最小二乘支持向量机建模。但由于发酵过程周期较长,最小二乘支持向量机的全局模型预测精度难以保证,算法复杂度很高,因此提出一种分阶段建模方法。首先,选择表征阶段特性的辅助变量,利用模糊C均值聚类算法对样本数据聚类,将发酵过程分成不同的阶段,然后为各个阶段分别建立最优混合核最小二乘支持向量机局部模型,最后将局部模型合成构成过程的完整模型。将此方法应用于青霉素发酵过程和重组大肠杆菌发酵过程中,验证了该方法的有效性。     

Novel model for the sintering of ceramics with bimodal pore size distributions: Application to the sintering of lime

A mathematical model for the sintering of ceramics with bimodal pore size distributions at intermediate and final stages is developed. It considers the simultaneous effects of coarsening by surface diffusion, and densification by grain boundary diffusion and lattice diffusion. This model involves population balances for the pores in different zones determined by each porosimetry peak, and is able to predict the evolution of pore size distribution function, surface area, and porosity over time. The model is experimentally validated for the sintering of lime and it is reliable in predicting the so called “initial induction period” in sintering, which is due to a decrease in intra‐aggregate porosity offset by an increase inter‐aggregate porosity. In addition, a novel methodology for determination of mechanisms based on the analysis of the pore size distribution function is proposed, and with this, it was demonstrated that lattice diffusion is the controlling mechanism in the CaO sintering. © 2016 American Institute of Chemical Engineers AIChE J, 63: 893–902, 2017     

Solution of the population balance equation using the sectional quadrature method of moments (SQMOM)

A discrete framework is introduced for simulating the particulate physical systems governed by population balance equations (PBE) with particle splitting (breakage) and aggregation based on accurately conserving (from theoretical point of view) an unlimited number of moments associated with the particle size distribution. The basic idea is based on the concept of primary and secondary particles, where the former is responsible for distribution reconstruction while the latter is responsible for different particle interactions such as splitting and aggregation. The method is found to track accurately any set of low-order moments with the ability to reconstruct the shape of the distribution. The method is given the name: the sectional quadrature method of moments (SQMOM) and has the advantage of being not tied to the inversion of large sized moment problems as required by the classical quadrature method of moments (QMOM). These methods become ill conditioned when a large number of moments are needed to increase their accuracy. On the contrary, the accuracy of the SQMOM increases by increasing the number of primary particles while using fixed number of secondary particles. Since the positions and local distributions for two secondary particles are found to have an analytical solution, no large moment inversion problems are anymore encountered. The generality of the SQMOM is proved by showing that all the related sectional and quadrature methods appearing in the literature for solving the PBE are merely special cases. The method has already been extended to bivariate PBEs.     

Simulation of the compressible flow with mass transfer of semi-continuous mixtures using the direct quadrature method of moments

We developed a method for the solution of the compressible flow with mass transfer of semi-continuous mixtures and it is based on the quadrature method of moments (QMoM) for continuous thermodynamics. The method extends the adaptive characterization of the continuous mixture to field problems and solves the mass transport equation for the continuous component. The method is referred as direct QMoM, or DQMoM, for continuous thermodynamics. It was implemented in the OpenFOAM^® as a compressible ideal gas flow solver. The DQMoM was applied to the mixing flow of two gas streams with different compositions of a mixture with 57 hydrocarbons diluted in nitrogen. We showed that 4 adaptive components reproduced the mixture properties within 3% accuracy. Furthermore, the DQMoM CFD solution was approximately two times faster than the solution using 58 discrete components.     

Mathematical modeling and experimental study of conveyor belt continuous curing process

A mathematical model was developed to predict the temperature distribution within various layers of conveyor belts during the continuous curing process. The results predicted by the model are found to be in good agreement with the experimental results, hence justifying the capability of the model for simulation of the conveyor belt continuous process. This information was utilized to provide more insight into the curing process in terms of the state of cure (SOC) and/or the degree of conversion, which may, in turn, be utilized for the optimization of the curing process. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 77: 2448–2454, 2000     

粉磨方式对硅酸盐水泥颗粒和早期水化行为的影响

采用同一批熟料,通过辊压机终粉磨、辊压机—球磨机联合粉磨和实验室球磨三种粉磨方式粉磨。试验发现,在细度相同时,不同粉磨方式制成的水泥颗粒特性具有较大的差异。水泥颗粒形状、形貌对水泥的早期水化行为(如:标准稠度、凝结时间、流动度和水化热)影响较大,但1d的水化总放热量却相差不大。