Recent Advances in Population Balance Modeling

A special issue of Processes (ISSN 2227-9717). This special issue belongs to the section "Process Control and Monitoring".

Deadline for manuscript submissions: closed (31 December 2018) | Viewed by 53556

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Department of Chemical and Biochemical Engineering, Technical University of Denmark, Søltofts Plads, Building 229, 2800 Kgs. Lyngby, Denmark
Interests: industrial fermentation technology; scale-up/scale-down; resource recovery; continuous production processes; mathematical modeling; process analytical technology (PAT)
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BIOMATH, Department of Data Analysis and Mathematical Modelling, Ghent University, Copure Links 653, 9000 Ghent, Belgium
Interests: model-based design and optimization; computational fluid dynamics (CFD); population balance modelling (PBM)
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Department of Chemical and Biochemical Engineering, Technical University of Denmark, 2800 Kongens Lyngby, Denmark
Interests: process system engineering; process control and optimization; downstream process development; chemical and biochemical process intensification
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TUM School of Life Sciences Weihenstephan, Technical University of Munich, Gregor-Mendel-Str. 4, 85354 Freising, Germany
Interests: modeling and simulation; multiscale modeling; population balance modeling; particulate processes; crystallization

Special Issue Information

Dear Colleagues,

Population Balance Modelling is a powerful modelling framework that allows predicting the dynamics of distributed properties of a population of individuals at the mesoscale. This is of particular interest when such a property is a critical quality attribute of a production system (e.g., particle size distribution, particle composition, etc.). The framework finds its roots in chemical engineering in the 1960s and boomed in the late 1990s with increasing computational power. It is now gaining ground in other application fields, such as pharmaceutical engineering and biotechnology.

Population balance models come in different forms. They can be formulated taking into account different continuous and discrete mechanisms such as growth, aggregation and breakage. For these mechanisms, process rates or kernels need to be defined. Calibration and validation of these kernels based on experimental data is of particular interest to secure the model’s predictive power and, hence, successful use in scenario analysis for process operational and design optimization.

Moreover, PBMs can include one or more distributed properties and either be embedded in a Computational Fluid Dynamics framework or spatial compartments to include the effect of spatial heterogeneities. Specific numerical and computational burden challenges arise when doing so.

The latest research in this intriguing field of research is being shown and discussed at the 6th International Conference on Population Balance Modelling ( PBM2018) held in Ghent, Belgium on 7–9 May, 2018. The issue is a reflection of high-quality papers presented at PBM2018. This Special Issue on “Population Balance Modeling” aims at showing the most recent advances in formulation, solution methods and application areas of population balance modelling.

All the authors of accepted contributions at PBM2018 are invited to submit manuscripts.

Prof. Dr. Krist V. Gernaey
Prof. Dr. Ingmar Nopens
Dr. Seyed Soheil Mansouri
Prof. Dr. Heiko Briesen
Guest editors

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Keywords

  • population balance model
  • formulation
  • numerical solution method
  • spatial heterogeneity
  • calibration
  • validation

Published Papers (13 papers)

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Editorial

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4 pages, 176 KiB  
Editorial
Special Issue on “Recent Advances in Population Balance Modeling”
by Seyed Soheil Mansouri, Heiko Briesen, Krist V. Gernaey and Ingmar Nopens
Processes 2021, 9(1), 122; https://doi.org/10.3390/pr9010122 - 08 Jan 2021
Cited by 2 | Viewed by 1641
Abstract
Population Balance Modeling (PBM) is a powerful modeling framework that allows the prediction of the dynamics of distributed properties of a population of individuals at the mesoscale [...] Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)

Research

Jump to: Editorial

16 pages, 1580 KiB  
Article
Calibration of Discrete-Element-Method Parameters for Cohesive Materials Using Dynamic-Yield-Strength and Shear-Cell Experiments
by Subhodh Karkala, Nathan Davis, Carl Wassgren, Yanxiang Shi, Xue Liu, Christian Riemann, Gary Yacobian and Rohit Ramachandran
Processes 2019, 7(5), 278; https://doi.org/10.3390/pr7050278 - 13 May 2019
Cited by 24 | Viewed by 4581
Abstract
This study tested the effectiveness of using dynamic yield strength (DYS) and shear-cell experiments to calibrate the following discrete-element-method (DEM) parameters: surface energy, and the coefficients of sliding and rolling friction. These experiments were carried out on cohesive granules, and DEM models were [...] Read more.
This study tested the effectiveness of using dynamic yield strength (DYS) and shear-cell experiments to calibrate the following discrete-element-method (DEM) parameters: surface energy, and the coefficients of sliding and rolling friction. These experiments were carried out on cohesive granules, and DEM models were developed for these experiment setups using the JKR cohesion contact model. Parameter-sensitivity analysis on the DYS model showed that the DYS results in the simulations were highly sensitive to surface energy and were also impacted by the values of the two friction coefficients. These results indicated that the DYS model could be used to calibrate the surface energy parameter once the friction coefficients were fixed. Shear-cell sensitivity analysis study found that the influence of surface energy on the critical-state shear value cannot be neglected. It was inferred that the shear-cell model has to be used together with the DYS model to identify the right set of friction parameters. Next, surface energy was calibrated using DYS simulations for a chosen set of friction parameters. Calibrations were successfully conducted for simulations involving experimentally sized particles, scaled-up particles, a different shear modulus, and a different set of friction parameters. In all these cases, the simulation DYS results were found to be linearly correlated with surface energy and were within 5% of the experimental DYS result. Shear-cell simulations were then used to compare calibrated surface-energy values for the scaled-up particles with the experimentally sized particles. Both the simulations resulted in similar critical-state shear values. Finally, it was demonstrated that a combination of DYS and shear-cell simulations could be used to compare two sets of friction parameters and their corresponding calibrated surface energy values to identify the set of parameters that better represent the flow behavior demonstrated by the experimental system. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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10 pages, 1225 KiB  
Article
Simulating Stochastic Populations. Direct Averaging Methods
by Vu Tran and Doraiswami Ramkrishna
Processes 2019, 7(3), 132; https://doi.org/10.3390/pr7030132 - 04 Mar 2019
Cited by 3 | Viewed by 2814
Abstract
A method of directly computing the average behavior of stochastic populations is established, which obviates the time-consuming process of generating detailed sample paths. The method relies on suitably discretized time intervals in which nonlinearities are quasi-linearized to produce random variables with known expectations [...] Read more.
A method of directly computing the average behavior of stochastic populations is established, which obviates the time-consuming process of generating detailed sample paths. The method relies on suitably discretized time intervals in which nonlinearities are quasi-linearized to produce random variables with known expectations and variances. The pair of equations is directly solved to obtain the average behavior of the system at the end of a time interval based on its knowledge at the beginning of the interval. The sample path requirement for this process is considerably lower than that for the process over the entire simulation period. The efficiency of the method is demonstrated on the transfer of antibiotics resistance between two bacterial species which is a problem of mounting concern in fighting disease. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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15 pages, 1329 KiB  
Article
An Analysis of Uncertainty Propagation Methods Applied to Breakage Population Balance
by Satyajeet Bhonsale, Dries Telen, Bard Stokbroekx and Jan Van Impe
Processes 2018, 6(12), 255; https://doi.org/10.3390/pr6120255 - 08 Dec 2018
Cited by 10 | Viewed by 3697
Abstract
In data-driven empirical or hybrid modeling, the experimental data influences the model parameters and thus also the model predictions. The experimental data has some variability due to measurement noise and due to the intrinsic stochastic nature of certain pharmaceutical processes such as aggregation [...] Read more.
In data-driven empirical or hybrid modeling, the experimental data influences the model parameters and thus also the model predictions. The experimental data has some variability due to measurement noise and due to the intrinsic stochastic nature of certain pharmaceutical processes such as aggregation or breakage. To use predictive models, it is imperative that the accuracy of the predictions is known. To this extent, various uncertainty propagation techniques applied to a predictive breakage population balance model are studied. Three uncertainty propagation techniques are studied: linearization, sigma point, and polynomial chaos. These are compared to the uncertainty obtained from Monte Carlo simulations. Linearization performs the worst in the given scenario, while sigma point and polynomial chaos methods have similar performance in terms of accuracy. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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17 pages, 7438 KiB  
Article
Dual Population Balance Monte Carlo Simulation of Particle Synthesis by Flame Spray Pyrolysis
by Ivan Skenderović, Gregor Kotalczyk and Frank Einar Kruis
Processes 2018, 6(12), 253; https://doi.org/10.3390/pr6120253 - 06 Dec 2018
Cited by 14 | Viewed by 4428
Abstract
The Dual Population Balance Monte Carlo Method (DPBMC) takes into account the full size spectrum of the droplet and particle phase. Droplet and particle size distributions are rendered by weighted simulation particles. This allows for an accurate description of particle nucleation and coagulation [...] Read more.
The Dual Population Balance Monte Carlo Method (DPBMC) takes into account the full size spectrum of the droplet and particle phase. Droplet and particle size distributions are rendered by weighted simulation particles. This allows for an accurate description of particle nucleation and coagulation and droplet combustion, simultaneously. Internal droplet properties such as temperature and concentrations fields are used to define criteria for the onset of droplet breakage in the framework of weighted Monte Carlo droplets. We discuss the importance of droplet polydispersity on particle formation in metal oxide particle synthesis, which is shown to strongly affect particle formation and growth. The method is applied to particle synthesis from metal nitrate precursor solutions with flame spray pyrolysis (FSP) and compared to experiments from literature. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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17 pages, 3595 KiB  
Article
Modeling and Simulation Studies of a Novel Coupled Plug Flow Crystallizer for the Continuous Separation of Conglomerate-Forming Enantiomers
by Aniruddha Majumder
Processes 2018, 6(12), 247; https://doi.org/10.3390/pr6120247 - 01 Dec 2018
Cited by 12 | Viewed by 4114
Abstract
Separation of enantiomers is a major concern in pharmaceutical industries due to the different therapeutic activities exhibited by the enantiomers. Preferential crystallization is an attractive means to separate the conglomerate-forming enantiomers. In this work, a simulation study is presented for a proposed novel [...] Read more.
Separation of enantiomers is a major concern in pharmaceutical industries due to the different therapeutic activities exhibited by the enantiomers. Preferential crystallization is an attractive means to separate the conglomerate-forming enantiomers. In this work, a simulation study is presented for a proposed novel preferential crystallization configuration that involves coupled plug flow crystallizers (PFCs). The PFCs are coupled through liquid phase exchange which helps the enrichment of the preferred enantiomer in the liquid phase. A set of coupled population balance equations (PBEs) are used to describe the evolution of the crystal size distribution (CSD) in the PFCs. The PBEs and the relevant mass balance equations are solved using the high-resolution finite-volume method. The simulation results predict that the proposed configuration has higher productivity compared to the currently used crystallization configurations while maintaining the same level of purity. Moreover, the effect of process variables, such as the extent of liquid phase exchange and the location of the PFC where liquid phase exchange occurs, are studied. The insights obtained from this simulation study will be useful in design, development, and optimization of such novel crystallization platforms. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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14 pages, 1662 KiB  
Article
Parameter Identification For Continuous Fluidized Bed Spray Agglomeration
by Ievgen Golovin, Gerd Strenzke, Robert Dürr, Stefan Palis, Andreas Bück, Evangelos Tsotsas and Achim Kienle
Processes 2018, 6(12), 246; https://doi.org/10.3390/pr6120246 - 30 Nov 2018
Cited by 14 | Viewed by 4364
Abstract
Agglomeration represents an important particle formation process used in many industries. One particularly attractive process setup is continuous fluidized bed spray agglomeration, which features good mixing as well as high heat and mass transfer on the one hand and constant product throughput with [...] Read more.
Agglomeration represents an important particle formation process used in many industries. One particularly attractive process setup is continuous fluidized bed spray agglomeration, which features good mixing as well as high heat and mass transfer on the one hand and constant product throughput with constant quality as well as high flow rates compared to batch mode on the other hand. Particle properties such as agglomerate size or porosity significantly affect overall product properties such as re-hydration behavior and dissolubility. These can be influenced by different operating parameters. In this manuscript, a population balance model for a continuous fluidized bed spray agglomeration is presented and adapted to experimental data. Focus is on the description of the dynamic behavior in continuous operation mode in a certain neighborhood around steady-state. Different kernel candidates are evaluated and it is shown that none of the kernels are able to match the first six minutes with time independent parameters. Afterwards, a good fit can be obtained, where the Brownian and the volume independent kernel models match best with the experimental data. Model fit is improved for identification on a shifted time domain neglecting the initial start-up phase. Here, model identifiability is shown and parameter confidence intervals are computed via parametric bootstrap. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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17 pages, 2864 KiB  
Article
Influence of Thermal Conditions on Particle Properties in Fluidized Bed Layering Granulation
by Christoph Neugebauer, Andreas Bück, Stefan Palis, Lisa Mielke, Evangelos Tsotsas and Achim Kienle
Processes 2018, 6(12), 235; https://doi.org/10.3390/pr6120235 - 22 Nov 2018
Cited by 16 | Viewed by 4794
Abstract
Fluidized bed layering granulation is frequently used to formulate particles of high quality. From previous studies, it is well known that the dynamic behavior of the process, as well as the product properties depend on operating parameters. The process is characterized by heat [...] Read more.
Fluidized bed layering granulation is frequently used to formulate particles of high quality. From previous studies, it is well known that the dynamic behavior of the process, as well as the product properties depend on operating parameters. The process is characterized by heat and mass transfer between fluidized particles and the surrounding fluidization medium. To investigate the mutual influence between particle phase and fluidization medium, a dynamic model is introduced. The model comprises two parts: a population balance model to describe the evolution of the particle sizes and a system of ordinary differential equations to account for thermal conditions. For the first time, the dynamic model considers the bidirectional coupling of particles and fluidization medium in fluidized bed layering granulation. By means of simulations, it is shown that the derived model is capable of reproducing the experimental findings. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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24 pages, 5155 KiB  
Article
Local Fixed Pivot Quadrature Method of Moments for Solution of Population Balance Equation
by Junwei Su, Wang Le, Zhaolin Gu and Chungang Chen
Processes 2018, 6(11), 209; https://doi.org/10.3390/pr6110209 - 31 Oct 2018
Cited by 1 | Viewed by 3045
Abstract
A local fixed pivot quadrature method of moments (LFPQMOM) is proposed for the solution of the population balance equation (PBE) for the aggregation and breakage process. First, the sectional representation for aggregation and breakage is presented. The continuous summation of the Dirac Delta [...] Read more.
A local fixed pivot quadrature method of moments (LFPQMOM) is proposed for the solution of the population balance equation (PBE) for the aggregation and breakage process. First, the sectional representation for aggregation and breakage is presented. The continuous summation of the Dirac Delta function is adopted as the discrete form of the continuous particle size distribution in the local section as performed in short time Fourier transformation (STFT) and the moments in local sections are tracked successfully. Numerical simulation of benchmark test cases including aggregation, breakage, and aggregation breakage combined processes demonstrate that the new method could make good predictions for the moments along with particle size distribution without further assumption. The accuracy in the numerical results of the moments is comparable to or higher than the quadrature method of moment (QMOM) in most of the test cases. In theory, any number of moments can be tracked with the new method, but the computational expense can be relatively large due to many scalar equations that may be included. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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13 pages, 632 KiB  
Article
Mathematical Modelling and Simulation of a Spray Fluidized Bed Granulator
by Gurmeet Kaur, Mehakpreet Singh, Jitendra Kumar, Thomas De Beer and Ingmar Nopens
Processes 2018, 6(10), 195; https://doi.org/10.3390/pr6100195 - 18 Oct 2018
Cited by 26 | Viewed by 6307
Abstract
In this present work, a study of the modelling and simulation for a top-sprayed fluidized bed granulator (SFBG) is presented, which is substantially used by the pharmaceutical industry to prepare granules. The idea is to build a number-based mathematical model using the notion [...] Read more.
In this present work, a study of the modelling and simulation for a top-sprayed fluidized bed granulator (SFBG) is presented, which is substantially used by the pharmaceutical industry to prepare granules. The idea is to build a number-based mathematical model using the notion of population balances by dividing a top SFBG into two different zones, namely the wet zone and dry zone. To solve a two-compartment model, an existing accurate and efficient finite volume scheme is implemented. In order to validate the compartmental model, a new class of analytical moments is derived corresponding to various combinations of aggregation and breakage kernels. To verify the accuracy of a modified finite volume scheme, the zeroth and first order moments computed using the finite volume scheme are compared with the newly-derived analytical results. Moreover, the stability of the compartmental model and the numerical scheme is tested by varying the size of the wet zone. It is also shown that the relative errors in both order moments increase with the increase in the size of the wet zone. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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16 pages, 1675 KiB  
Article
A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations
by Robert Dürr and Steffen Waldherr
Processes 2018, 6(10), 187; https://doi.org/10.3390/pr6100187 - 10 Oct 2018
Cited by 4 | Viewed by 3212
Abstract
Multicellular systems play an important role in many biotechnological processes. Typically, these exhibit cell-to-cell variability, which has to be monitored closely for process control and optimization. However, some properties may not be measurable due to technical and financial restrictions. To improve the monitoring, [...] Read more.
Multicellular systems play an important role in many biotechnological processes. Typically, these exhibit cell-to-cell variability, which has to be monitored closely for process control and optimization. However, some properties may not be measurable due to technical and financial restrictions. To improve the monitoring, model-based online estimators can be designed for their reconstruction. The multicellular dynamics is accounted for in the framework of population balance models (PBMs). These models are based on single cell kinetics, and each cellular state translates directly into an additional dimension of the obtained partial differential equations. As multicellular dynamics often require detailed single cell models and feature a high number of cellular components, the resulting population balance equations are often high-dimensional. Therefore, established state estimation concepts for PBMs based on discrete grids are not recommended due to the large computational effort. In this contribution a novel approach is proposed, which is based on the approximation of the underlying number density functions as the weighted sum of Gaussian distributions. Thus, the distribution is described by the characteristic properties of the individual Gaussians, like the mean and covariance. Thereby, the complex infinite dimensional estimation problem can be reduced to a finite dimension. The characteristic properties are estimated in a recursive approach. The method is evaluated for two academic benchmark examples, and the results indicate its potential for model-based online reconstruction for multicellular systems. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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22 pages, 1118 KiB  
Article
Modeling the Separation of Microorganisms in Bioprocesses by Flotation
by Stefan Schmideder, Christoph Kirse, Julia Hofinger, Sascha Rollié and Heiko Briesen
Processes 2018, 6(10), 184; https://doi.org/10.3390/pr6100184 - 06 Oct 2018
Cited by 10 | Viewed by 4112
Abstract
Bioprocesses for the production of renewable energies and materials lack efficient separation processes for the utilized microorganisms such as algae and yeasts. Dissolved air flotation (DAF) and microflotation are promising approaches to overcome this problem. The efficiency of these processes depends on the [...] Read more.
Bioprocesses for the production of renewable energies and materials lack efficient separation processes for the utilized microorganisms such as algae and yeasts. Dissolved air flotation (DAF) and microflotation are promising approaches to overcome this problem. The efficiency of these processes depends on the ability of microorganisms to aggregate with microbubbles in the flotation tank. In this study, different new or adapted aggregation models for microbubbles and microorganisms are compared and investigated for their range of suitability to predict the separation efficiency of microorganisms from fermentation broths. The complexity of the heteroaggregation models range from an algebraic model to a 2D population balance model (PBM) including the formation of clusters containing several bubbles and microorganisms. The effect of bubble and cell size distributions on the flotation efficiency is considered by applying PBMs, as well. To determine the sensitivity of the results on the model assumptions, the modeling approaches are compared, and suggestions for their range of applicability are given. Evaluating the computational fluid dynamics (CFD) of a dissolved air flotation (DAF) system shows the heterogeneity of the fluid dynamics in the flotation tank. Since analysis of the streamlines of the tank show negligible back mixing, the proposed aggregation models are coupled to the CFD data by applying a Lagrangian approach. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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20 pages, 415 KiB  
Article
Population Balance Modeling and Opinion Dynamics—A Mutually Beneficial Liaison?
by Michael Kuhn, Christoph Kirse and Heiko Briesen
Processes 2018, 6(9), 164; https://doi.org/10.3390/pr6090164 - 11 Sep 2018
Cited by 5 | Viewed by 4184
Abstract
In this contribution, we aim to show that opinion dynamics and population balance modeling can benefit from an exchange of problems and methods. To support this claim, the Deffuant-Weisbuch model, a classical approach in opinion dynamics, is formulated as a population balance model. [...] Read more.
In this contribution, we aim to show that opinion dynamics and population balance modeling can benefit from an exchange of problems and methods. To support this claim, the Deffuant-Weisbuch model, a classical approach in opinion dynamics, is formulated as a population balance model. This new formulation is subsequently analyzed in terms of moment equations, and conservation of the first and second order moment is shown. Exemplary results obtained by our formulation are presented and agreement with the original model is found. In addition, the influence of the initial distribution is studied. Subsequently, the Deffuant-Weisbuch model is transferred to engineering and interpreted as mass transfer between liquid droplets which results in a more flexible formulation compared to alternatives from the literature. On the one hand, it is concluded that the transfer of opinion-dynamics problems to the domain of population balance modeling offers some interesting insights as well as stimulating challenges for the population-balance community. On the other hand, it is inferred that population-balance methods can contribute to the solution of problems in opinion dynamics. In a broad outlook, some further possibilities of how the two fields can possibly benefit from a close interaction are outlined. Full article
(This article belongs to the Special Issue Recent Advances in Population Balance Modeling)
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