An efficient numerical technique for solving population balance equation involving aggregation, breakage, growth and nucleation

A new discretization for simultaneous aggregation, breakage, growth and nucleation is presented. The new discretization is an extension of the cell average technique developed by the authors [J. Kumar, M. Peglow, G. Warnecke, S. Heinrich, and L. Mörl. Improved accuracy and convergence of discretized population balance for aggregation: The cell average technique. Chemical Engineering Science 61 (2006) 3327-3342.]. It is shown that the cell average scheme enjoys the major advantage of simplicity for solving combined problems over other existing schemes. This is done by a special coupling of the different processes that treats all processes in a similar fashion as it handles the individual process. It is demonstrated that the new coupling makes the technique more useful by being not only more accurate but also computationally less expensive. At first, the coupling is performed for combined aggregation and breakage problems. Furthermore, a new idea that considers the growth process as aggregation of existing particle with new small nuclei is presented. In that way the resulting discretization of the growth process becomes very simple and consistent with first two moments. Additionally, it becomes easy to combine the growth discretization with other processes. The new discretization of pure growth is a little diffusive but it predicts the first two moments exactly without any computational difficulties like appearance of negative values or instability etc. The numerical scheme proposed in this work is consistent only with the first two moments but it can easily be extended to the consistency with any two or more than two moments. Finally, the discretization of pure and coupled problems is tasted on several analytically solvable problems.     

Simultaneous determination of nucleation and crystal growth kinetics of gypsum

The chemical treatment of industrial effluent, containing sulphuric acid, is typically completed by neutralising it with lime through solid/liquid separation. The mineral precipitation processes developed by Veolia Environnement use sludge recirculation. The production of a high sludge density characterises these processes, at a minimum of at least 25% weight in solids. The first laboratory tests studied the precipitation reaction, confirming the production of gypsum (CaSO₄·2H₂O). Then the precipitation mechanisms, nucleation and crystal growth kinetics, are determined in order to establish a model which predicts population density in reactor in accordance with fluid dynamics. The determined nucleation and growth kinetics are then used to feed a reaction model, coupling computational fluid dynamics (CFD) and population balance modelling to simulate the precipitation process.     

A three-dimensional population balance model of granulation with a mechanistic representation of the nucleation and aggregation phenomena

A comprehensive model is discussed for wet granulation based on a three-dimensional population balance, as an attempt to capture particle-level phenomena and their influence on the population-level behaviour. The three dimensions of population distribution are the particle size, binder content, and porosity of the granules. In formulating the population balance, these three particle traits are represented in terms of three equivalent traits, namely, the solid volume, liquid volume and gas volume of the granules. The model accounts for wetting, nucleation, aggregation and consolidation phenomena. Mechanistic kernels are derived for aggregation and nucleation, employing theories on these particle-level microscale phenomena that have already been validated in previous studies. The three-dimensional population balance is solved numerically using a finite volume-based decomposition algorithm also called the hierarchical two-tier solution strategy [Pinto, M.A., Immanuel, C.D., Doyle III, F.J., 2007. A feasible solution technique for higher-dimensional population balance models. Computers and Chemical Engineering, 31, 1242-1256].     

Investigation of soot formation in turbulent flames with a pseudo-bivariate population balance model

Soot inception and subsequent aggregation, surface growth and oxidation are described through a new pseudo-bivariate population balance model solved with the direct quadrature method of moments (DQMOM) and implemented in a commercial computational fluid dynamics (CFD) code. This modelling strategy, that in this work is presented in its Reynolds-averaged Navier-Stokes (RANS) equation formulation, has the advantage, over conventional approaches based on the solution of a single transport equation for the soot volume fraction, to overcome the assumption of mono-dispersed soot particle size distribution. On the contrary the pseudo-bivariate approach presented in this work is able to represent the evolution of the soot particle size distribution with good accuracy and affordable computational costs, especially when compared with other multi-variate formulations previously developed. This new pseudo-bivariate model is firstly formulated and presented, then predictions obtained with different soot inception models are compared with some recent experimental data from the literature and the role played by the different phenomena involved (e.g., turbulence, oxidation and radiation) is investigated.     

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.     

Numerical solution of population balance equations for nucleation, growth and aggregation processes

This article focuses on the derivation of numerical schemes for solving population balance models (PBMs) with simultaneous nucleation, growth and aggregation processes. Two numerical methods are proposed for this purpose. The first method combines a method of characteristics (MOC) for growth process with a finite volume scheme (FVS) for aggregation process. For handling nucleation terms, a cell of nuclei size is added at a given time level. The second method purely uses a semi-discrete finite volume scheme for nucleation, growth and aggregation of particles. Note that both schemes use the same finite volume scheme for aggregation process. On one hand, the method of characteristics offers a technique which is in general a powerful tool for solving linear growth processes, has the capability to overcome numerical diffusion and dispersion, is computationally efficient, as well as give highly resolved solutions. On the other hand, the finite volume schemes which were derived for a general system in divergence form, are applicable to any grid to control resolution, and are also computationally not expensive. In the first method a combination of finite volume scheme and the method of characteristics gives a highly accurate and efficient scheme for simultaneous nucleation, growth and aggregation processes. The second method demonstrates the applicability, generality, robustness and efficiency of high-resolution schemes. The proposed techniques are tested for pure growth, simultaneous growth and aggregation, nucleation and growth, as well as simultaneous nucleation, growth and aggregation processes. The numerical results of both schemes are compared with each other and are also validated against available analytical solutions. The numerical results of the schemes are in good agreement with the analytical solutions.     

A control oriented study on the numerical solution of the population balance equation for crystallization processes

The population balance equation provides a well established mathematical framework for dynamic modeling of numerous particulate processes. Numerical solution of the population balance equation is often complicated due to the occurrence of steep moving fronts and/or sharp discontinuities. This study aims to give a comprehensive analysis of the most widely used population balance solution methods, namely the method of characteristics, the finite volume methods and the finite element methods, in terms of the performance requirements essential for on-line control applications. The numerical techniques are used to solve the dynamic population balance equation of various test problems as well as industrial crystallization processes undergoing simultaneous nucleation and growth. The time-varying supersaturation profiles in the latter real-life case studies provide more realistic scenarios to identify the advantages and pitfalls of a particular numerical technique.The simulation results demonstrate that the method of characteristics gives the most accurate numerical predictions, whereas high computational burden limits its use for complex real crystallization processes. It is shown that the high order finite volume methods in combination with flux limiting functions are well capable of capturing sharp discontinuities and steep moving fronts at a reasonable computational cost, which facilitates their use for on-line control applications. The finite element methods, namely the orthogonal collocation and the Galerkin's techniques, on the other hand may severely suffer from numerical problems. This shortcoming, in addition to their complex implementation and low computational efficiency, makes the finite element methods less appealing for the intended application.     

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.     

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.     

Solution of population balance equation with pure aggregation in a fully developed turbulent pipe flow

This paper presents our preliminary effort in predicting particle size distributions in particulate processes in turbulent flow systems. The focus has been on processes of pure aggregation, occurring in a turbulent environment. A remarkably simple strategy has been used to solve the population balance equation (PBE) for spatially dependent pure aggregation with insignificant diffusive transport of particles in turbulent flow systems. The method makes use of the solution of a batch PBE through a mathematical transformation linking time to spatial variables. Furthermore, we investigate the self-similar solution of batch aggregation to show scaling behavior of particle size distributions in such flow systems using spatially dependent average particle sizes. Average particle sizes across the pipe cross section have been computed using both averaged frequencies as well as spatially varying frequencies. Comparison of the two solutions shows significant differences between them, establishing the sheer inappropriateness of the use of average aggregation frequencies in the prediction of absolute particle size distribution as done in the past.     

Two state estimators for the barium sulfate precipitation in a semi-batch reactor

The on-line determination of particle property distributions by direct measurements is often difficult, because the measurement equations are not invertible or because the inverse problem is ill-posed. If the process is observable, one can use state estimation techniques in order to reconstruct unmeasurable internal states of the process. This is discussed here for a semi-batch precipitation reactor. A square root unscented Kalman filter and state estimation by online minimisation are studied for the case of a measurable average particle size. Both estimators use a one-dimensional population balance model. The two approaches are compared in simulations.     

Gray-Box Modeling of the Molecular Weight Distribution in a Batch Polymerization Reactor

We introduce a gray-box approach for modeling the molecular weight distribution in step-growth polymerization reactions using the aggregation population balance equation. The approach is based on extracting a data-based kernel function from in-process measurements of the molecular weight distribution. The method is applied to historical data from an industrial batch polymerization reactor. The resulting model is used for decision support in production by predicting the reaction endpoint corresponding to a target molecular weight. The accuracy of the predictions proved to be sufficient for the deployment of the method.     

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.     

Experimental and numerical investigation of the precipitation of barium sulfate in a rotating liquid film reactor

Precipitation of nanosized barium sulfate in a rotating liquid film reactor (RLFR) has been investigated experimentally and through simulations based on the computational fluid dynamics technique including the population balance equation coupled with the Navier–Stokes equations, renormalization group k–ε model equations, and species transport equations. A comparative experiment was carried out involving conventional precipitation in a flask. The structure of the precipitate was identified by powder X‐ray diffraction (PXRD), which showed that the crystals obtained using the RLFR were smaller in size than those obtained in the flask. Transmission electron microscopy (TEM) images demonstrated that the crystals produced by the two different processes had different morphologies. Further detailed experiments involving varying the operating parameters of the RLFR were performed to investigate the effects on crystal size distribution (CSD). Increasing the speed of the rotor in the RLFR in the range 1000–5000 rpm or increasing the rotor‐stator gap in the range 0.1–0.5 mm resulted in a decrease in particle size and narrower particle size distributions. The simulation results suggested that turbulent effects and reaction processes in the effective reactor space were directly related to rotor speed and rotor‐stator gap. The simulated volume weighted mean diameter and CSD of particles of barium sulfate were almost identical to the corresponding experimental results obtained using TEM and laser particle size analyzer. The effects of other parameters such as the Kolmogorov scale and competition between induction time and mixing time are also discussed. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

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.     

氯化物三价铬镀铬槽持续出现沉淀故障的处理

氯化物三价铬镀铬槽中盐酸加入量不够时,碱式硫酸铬分子中的氢氧根不能与盐酸充分反应,导致在生产中三价铬水解,持续生成沉淀物。向镀槽中加浓盐酸10 mL/L调节pH至1.8,24 h后镀液pH升高至2.73,镀槽中没有出现沉淀物,恢复正常生产。     

Flow behaviors of a large spout-fluid bed at high pressure and temperature by 3D simulation with kinetic theory of granular flow

Flow behaviors of a large spout-fluid bed (I.D. 1.0 m) at high pressure and temperature were investigated by Eulerian simulation. The gas phase was modeled with k − ε turbulent model and the particle phase was modeled with kinetic theory of granular flow. The development of an internal jet, gas-solid flow patterns, particle concentrations, particle velocities and jet penetration depths at high pressure and temperature at different operating conditions were simulated. The results show that the bed operated at an initial bed height larger than the maximum spoutable bed height resembles the flow patterns of jetting fluidized beds. The radial profiles of particle velocities and concentrations at high temperature and pressure have the similar characteristic shapes to those at ambient pressure and temperature. The particle concentrations and velocities appear to depend on the bed heights when increasing pressure while keeping the gas velocities and temperature constant. The particle velocities in the lower region of the bed increase with increasing pressure, while they tend to decrease in the middle and upper regions of the bed. The particle concentrations have an opposite dependency with increasing pressure. They decrease in the lower region of the bed but increase in the middle and upper regions of the bed. Besides, the jet penetration depths are found to increase with increasing pressure.     

3种定值方法用于维生素D3纯度标准物质的研究

通过定性分析、定值分析、均匀性检验、稳定性考察和不确定度评定,研制了维生素D3纯度标准物质。其中采用质量平衡法(包括液相色谱法、水分、灰分、挥发性物质和无机元素分析)、定量核磁共振法与差示扫描量热法3种不同原理的方法对维生素D3纯度标准物质进行了定值分析。准确的纯度定值结果为99.6%,扩展不确定度为0.6%。对于维生素D3的准确测定及相关疾病的正确诊断治疗具有重要意义。