A new wavelet-based adaptive method for solving population balance equations

A new wavelet-based adaptive framework for solving population balance equations (PBEs) is proposed in this work. The technique is general, powerful and efficient without the need for prior assumptions about the characteristics of the processes. Because there are steeply varying number densities across a size range, a new strategy is developed to select the optimal order of resolution and the collocation points based on an interpolating wavelet transform (IWT). The proposed technique has been tested for size-independent agglomeration, agglomeration with a linear summation kernel and agglomeration with a nonlinear kernel. In all cases, the predicted and analytical particle size distributions (PSDs) are in excellent agreement. Further work on the solution of the general population balance equations with nucleation, growth and agglomeration and the solution of steady-state population balance equations will be presented in this framework.     

Theoretical prediction of flow regime transition in bubble columns by the population balance model

Theoretical prediction of flow regime transition in bubble columns was studied based on the bubble size distribution by the population balance model (PBM). Models for bubble coalescence and breakup due to different mechanisms, including coalescence due to turbulent eddies, coalescence due to different bubble rise velocities, coalescence due to bubble wake entrainment, breakup due to eddy collision and breakup due to large bubble instability, were proposed. Simulation results showed that at relatively low superficial gas velocities, bubble coalescence and breakup were relatively weak and the bubble size was small and had a narrow distribution; with an increase in the superficial gas velocity, large bubbles began to form due to bubble coalescence, resulting in a much wider bubble size distribution. The regime transition was predicted to occur when the volume fraction of small bubbles sharply decreased. The predicted transition superficial gas velocity was about 4 cm/s for the air-water system, in accordance with the values obtained from experimental approaches.     

Experimental characterization and population balance modelling of the dense silica suspensions aggregation process

Concentrated suspensions of nanoparticles subjected to transport or shear forces are commonly encountered in many processes where particles are likely to undergo processes of aggregation and fragmentation under physico-chemical interactions and hydrodynamic forces. This study is focused on the analysis of the behavior of colloidal silica in dense suspensions subjected to hydrodynamic forces in conditions of destabilization.A colloidal silica suspension of particles with an initial size of about 80 nm was used. The silica suspension concentration was varied between 3% and 20% of weight. The phenomenon of aggregation was observed in the absence of any other process such as precipitation and the destabilization of the colloidal suspensions was obtained by adding sodium chloride salt.The experiments were performed in a batch agitated vessel. The evolution of the particle size distributions versus time during the process of aggregation was particularly followed on-line by acoustic spectroscopy in dense conditions. Samples were also analyzed after an appropriate dilution by laser diffraction. The results show the different stages of the silica aggregation process whose kinetic rates depend either on physico-chemical parameters or on hydrodynamic conditions. Then, the study is completed by a numerical study based on the population balance approach. By the fixed pivot technique of Kumar and Ramkrishna [1996. On the solution of population balance equations by discretization—I. A fixed pivot technique. Chemical Engineering Science 51 (8), 1311-1332], the hypothesis on the mechanisms of the aggregation and breakage processes were justified. Finally, it allows a better understanding of the mechanisms of the aggregation process under flowing conditions.     

A volume-based multi-dimensional population balance approach for modelling high shear granulation

A volume-based multi-dimensional population balance model based on the approach used by Verkoeijen et al. [2002. Population balances for particulate processes—a volume approach. Chemical Engineering Science 57, 2287-2303], is further developed and applied to a wet granulation process of pharmaceutically relevant material, performed in a high shear mixer. The model is improved by a generalization that accounts for initial non-uniformly distributed liquid and air among the different particle size classes. Only the wet massing period of the granulation process has been modelled and it is experimentally found that the pores in the granules are fully saturated by liquid, i.e., no air is present in the granules during this period. Hence, an alternative model formulation is used as no model for the air in the granules is needed. Particle volume distribution, liquid saturation, liquid-to-solid ratio and porosity of the granules can all be modelled, as these properties can all be expressed as combinations of three model parameters, i.e., the volume fraction of solid material, total liquid fraction and the liquid fraction inside the granules. The model is also improved by introducing a new coalescence kernel and by increasing the number of size classes used. The simulated results are compared to measurements from a series of five designed experiments where impeller speed and water content are varied. It is found that the evolution of the volume, liquid saturation and porosity distributions could all be explained by fitting the compaction and coalescence rate constants.     

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.     

Simulation of polydisperse multiphase systems using population balances and example application to bubbly flows

In this work the relationship between multiphase computational fluid dynamics models and population balance models is illustrated by deriving the main governing equations from the generalized population balance equation. The resulting set of equations, consisting of the well known two-fluid model coupled with a bivariate population balance model, is then implemented in the CFD code OpenFOAM. The implementation is used to simulate a particular multiphase problem: bubbly flow in a rectangular column. Results show that, although the different mesoscale models for drag force, coalescence, breakup and mass transfer, can be improved, the agreement with experiments is nevertheless good. Moreover, although the problem investigated is quite complex, as the evolution of bubbles is solved in real-space, time and phase-space (i.e. bubble size and composition) the resulting computational costs are reasonable. This is due to the fact that the bivariate population balance model is solved here with the so-called conditional quadrature method of moments, that very efficiently deals with these problems. The overall approach is demonstrated to be efficient and robust and is therefore suitable for the simulation of many polydisperse multiphase flows.     

Simulation of oscillatory baffled column: CFD and population balance

In the present work, an attempt has been made to combine population balance and a CFD approach for simulating the flow in oscillatory baffled column (OBC). Three-dimensional Euler-Euler two-fluid simulations are carried out for the experimental data of Oliveira and Ni [2001. Gas hold-up and bubble diameter in a gassed oscillatory baffled column. Chemical Engineering Science 56, 6143-6148]. The experimental data include the average hold-up profile and bubble size distribution in the OBC. All the non-drag forces (turbulent dispersion force, lift force) and the drag force are incorporated in the model. The coalescence and breakage effects of the gas bubbles are modeled according to the coalescence by the random collision driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Predicted liquid velocity and averaged gas hold-up are compared with the experimental data. The profile of the mean bubble diameter in the column and its variation with the superficial gas velocity is studied. Bubble size distribution obtained by the model is compared with the experimental data.     

A new set of population balance equations for microbial and cell cultures

Population balance equations for microbial or cell cultures contain a state-dependent fission intensity function which is such that the product of this function, evaluated at a given state, and a differential increment of time is the fraction of cells of the given state that divide in that increment of time. Ways to determine experimentally how the fission intensity function depends on cell state have been proposed but, so far as is known to the authors, no one has yet proposed a model that would predict what the state dependence of the fission intensity function is for any population. If one wants to take account of the existence of cell cycle phases, one will have to write a population balance equation for each phase, a transition intensity function for each phase will have to be introduced, and the problem of making models for these functions will arise again, and in multiplied fashion. In this paper, we describe a new and different approach which circumvents the necessity of having intensity functions for transitions between cell cycle phases, and for which the fission intensity function is state-independent.     

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.     

A new discretization of space for the solution of multi-dimensional population balance equations

In this work, a novel radial grid is combined with the framework of minimal internal consistency of discretized equations of Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112-4125] to solve n-dimensional population balance equations (PBEs) with preservation of (n+1) instead of 2ⁿ properties required in direct extension of the 1-d fixed pivot technique of Kumar and Ramkrishna [1996a. On the solutions of population balance equation by discretization-I. A fixed pivot technique. Chemical Engineering Science 51, 1311-1332]. The radial grids for the solution of 2-d PBEs are obtained by intersecting arbitrarily spaced radial lines with arcs of arbitrarily increasing radii. The quadrilaterals obtained thus are divided into triangles to represent a non-pivot particle in 2-d space through three surrounding pivots by preserving three properties, the number and the two masses of the species that constitute the newly formed particle. Such a grid combines the ease of generating and handling a structured grid with the effectiveness of the framework of minimal internal consistency. A new quantitative measure to supplement visual comparison of two solutions is also introduced. The comparison of numerical and analytical solutions of 2-d PBEs for a number of uniform and selectively refined radial grids shows that the quality of solution obtained with radial grids is substantially better than that obtained with the direct extension of the 1-d fixed pivot technique to higher dimensions for both size independent and size dependent aggregation kernels. The framework of Chakraborty and Kumar combined with the proposed 2-d radial grid, which offers flexibility and achieves both reduced numerical dispersion and the ease of implementation, appears as an effective extension of the widely used 1-d fixed pivot technique to solve 2-d PBEs.     

Numerical solution of the bivariate population balance equation for the interacting hydrodynamics and mass transfer in liquid-liquid extraction columns

A comprehensive model for predicting the interacting hydrodynamics and mass transfer is formulated on the basis of a spatially distributed population balance equation in terms of the bivariate number density function with respect to droplet diameter and solute concentration. The two macro- (droplet breakage and coalescence) and micro- (interphase mass transfer) droplet phenomena are allowed to interact through the dispersion interfacial tension. The resulting model equations are composed of a system of partial and algebraic equations that are dominated by convection, and hence it calls for a specialized discretization approach. The model equations are applied to a laboratory segment of an RDC column using an experimentally validated droplet transport and interaction functions. Aside from the model spatial discretization, two methods for the discretization of the droplet diameter are extended to include the droplet solute concentration. These methods are the generalized fixed-pivot technique (GFP) and the quadrature method of moments (QMOM). The numerical results obtained from the two extended methods are almost identical, and the CPU time of both methods is found acceptable so that the two methods are being extended to simulate a full-scale liquid-liquid extraction column.     

A new discretization of space for the solution of multi-dimensional population balance equations: Simultaneous breakup and aggregation of particles

In this work, we show that straight forward extensions of the existing techniques to solve 2-d population balance equations (PBEs) for particle breakup result in very high numerical dispersion, particularly in directions perpendicular to the direction of evolution of population. These extensions also fail to predict formation of particles of uniform composition at steady state for simultaneous breakup and aggregation of particles, starting with particles of both uniform and non-uniform compositions. The straight forward extensions of 1-d techniques preserve 2ⁿ properties of non-pivot particles, which are taken to be number, two masses, and product of masses for the solution of 2-d PBEs. Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112-4125] have recently proposed a new framework of minimal internal consistency of discretization which requires preservation of only (n+1) properties. In this work, we combine a new radial grid [proposed in 2008. part I, Chemical Engineering Science 63, 2198] with the above framework to solve 2-d PBEs consisting of terms representing breakup of particles. Numerical dispersion with the use of straight forward extensions arises on account of formation of daughter particles of compositions different from that of the parent particles. The proposed technique eliminates numerical dispersion completely with a radial distribution of grid points and preservation of only three properties: number and two masses. The same features also enable it to correctly capture mixing brought about by aggregation of particles. The proposed technique thus emerges as a powerful and flexible technique, naturally suited to simulate PBE based models incorporating simultaneous breakup and aggregation of particles.     

A two-level discretisation algorithm for the efficient solution of higher-dimensional population balance models

The increasing range of complex applications that call for population balances based models motivates the development of efficient and robust solution techniques. Several contemporary applications mandate the consideration of multi-dimensional population balances, accounting for the non-homogeneous and/or wide distribution of particle populations among several state variables (internal coordinates). This article presents an efficient and novel solution technique for multi-dimensional population balance models, inspired by the underlying physics of the problem and the dictates of the relevant rate processes in the population balance in terms of the accuracy of the solutions.     

Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.     

Simulation and analysis of industrial crystallization processes through multidimensional population balance equations. Part 1: a resolution algorithm based on the method of classes

In order to obtain constant solid properties with particles exhibiting a low order of symmetry, it is necessary to monitor and to control several distributed parameters characterising the crystal shape and size. A bi-dimensional population balance model was developed to simulate the time variations of two characteristic sizes of crystals. The nonlinear population balance equations were solved numerically over the bi-dimensional size domain using the so-called method of classes. An effort was made to improve usual simulation studies through the introduction of physical knowledge in the kinetic laws involved during nucleation and growth phenomena of complex organic products. The performances of the simulation algorithm were successfully assessed through the reproduction of two well-known theoretical and experimental features of ideal continuous crystallization processes: the computation of size-independent growth rates from the plot of the steady-state crystal size distribution and the possibility for MSMPR crystallizers to exhibit low-frequency oscillatory behaviours in the case of insufficient secondary nucleation.     

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.     

Implementation of the population balance equation in CFD codes for modelling soot formation in turbulent flames

The simulation of soot formation in turbulent diffusion flames is carried out within a CFD code, by coupling kinetics and fluid dynamics computations with the solution of the population balance equation via the Direct Quadrature Method of Moments, a novel and efficient approach based on a quadrature approximation of the size distribution of soot particles. A turbulent non-premixed ethylene-air flame is used as the test case for validation of the model. Simplified kinetic expressions are employed for modelling nucleation, molecular growth and oxidation of particles, along with a Brownian aggregation kernel. A recently proposed approach for modelling the evolution of fractal dimension is used with a monovariate population balance to predict the morphological properties of aggregates.     

A population balance framework for nucleation, growth, and aggregation

Nucleation, growth, and aggregation for particulate systems are explored by distribution kinetics and population balances to build a new framework for understanding a range of natural and manufacturing phenomena. Nucleation is assumed to follow classical homogeneous theory or to be caused by heterogeneous nuclei added to the solution. Growth due to monomer addition from solution to clusters, and aggregation between clusters are both represented by integrals of the cluster distribution. When growth and aggregation rate coefficients are independent of cluster size, the population balance equations are readily solved by the moment method. Equations for steady-state well-mixed flow and unsteady-state closed (batch) vessels have relatively straightforward solutions. By incorporating solute (monomer) depletion, the results afford reasonable behavior for the cluster number and mass concentration. The monomer addition terms are shown to be consistent with (and a generalization of) conventional differential growth and growth dispersion expressions.     

On the use of bi-variate population balance equations for modelling barium titanate nanoparticle precipitation

In this work barium titanate hydrothermal synthesis is studied from the modelling point of view. In fact, a mathematical model can be used to investigate the origin of the typical multi-crystalline particles obtained during this precipitation process. Their origin is not completely clear, since it is not understood which is the controlling mechanism (i.e., secondary nucleation and/or aggregation). Previous works on this subject tried to retrieve the kinetics underlying this particulate process but being based on mono-varied population balance equations failed to give a definitive answer. In this work a bi-variate model is presented, discussed and solved. Results clearly show that such an approach can overcome the limitations of previous modelling works and provides an useful tool for more detailed kinetic parameters estimation. Moreover the model shows that secondary nucleation is indeed very important but aggregation cannot be neglected.