Multidimensional Population Balance Model Research Articles

A dynamic compartment model for spatially heterogeneous reactors: Scalar and Monte-Carlo particle mixing

The coupling of multidimensional population balance models with transport equations is often necessary for the simulation of spatially heterogeneous reactors. Because of numerical intractability, a compromise on both the hydrodynamics and the population description has to be made. In this work, the Monte-Carlo method is used to calculate the local integral source term due to the dispersed phase (isodense particles) and a compartment model is used to describe the fluid motion in the reactor. The number density function n(x,t,ξ) is approximated by the number of Monte-Carlo particles per compartment and their movement in the space of compartment uses a stochastic transport algorithm adapted from existing works. The numerical tool is thoroughly and successfully validated in the context of mixing and first order reaction in a 70 L stirred tank equipped with a Rushton turbine. The dynamic method makes use of time resolved CFD simulations to built time dependent flow matrices reflecting the periodic nature of the flow field. The dynamic approach is finally compared to other usual compartment model approaches in terms of complexity, accuracy and rapidity which reveals the benefit of considering the flow unsteadiness within a compartiment model approach for stirred tanks.

Multi-dimensional population balance modeling of sonocrystallization of pyrazinamide with systematic estimation of kinetic parameters based on uncertainty and sensitivity analyses

Application of ultrasound has several beneficial effects on both crystallization process and product crystal properties. This work focuses on experimental study and population balance modeling of ultrasound-assisted unseeded batch cooling crystallization of an important polymorphic pharmaceutical. The δ-polymorph of pyrazinamide, a drug for mycobacterium tuberculosis, exhibits plate-like morphology, and such crystals are best described by atleast two dimensions: length and width. A multi-dimensional population balance model for batch cooling sonocrystallization of pyrazinamide from its 1,4-dioxane solution is developed and validated. A series of experiments are performed with different ultrasound amplitudes (silent, 10%, and 50%) to study the effect of ultrasound on nucleation rate, crystal size, and polymorphism. The ultrasound is applied till the point of nucleation and an enhanced nucleation rate is introduced in the model in the presence of ultrasound irradiation. The various kinetic parameters are estimated following a systematic methodology based on uncertainty quantification and sensitivity analysis using experimental observations related to solute concentration and crystal size distribution. The Polynomial Chaos Expansion (PCE) is used to quantify the uncertainties in model prediction due to kinetic parameter uncertainties and sensitivity analysis is performed using total order Sobol’s sensitivity indices derived from PCE model. The kinetic parameters are estimated in two steps. First, a global search technique is used to simultaneously estimate all the kinetic parameters. Next, the less sensitive parameters are kept fixed and the more significant parameters are fine-tuned using a gradient-based local optimizer. Out of total nine kinetic parameters, only four parameters appeared to be significant for both silent and sonicated cases. The high-resolution finite volume scheme is used to solve the multi-dimensional population balance model and the simulation results agree very well with experimental data related to both concentration and crystal size distributions. The results revealed that compared to silent cases, sonication increased the nucleation rate by 102 times and 104 times for 10% and 50% amplitude, respectively. The growth rate was also enhanced significantly with increase in ultrasonic amplitude. All the experiments yielded the desired δ-polymorph of pyrazinamide.

Model development and calibration of two-dimensional population balance model for twin-screw wet granulation based on particle size distribution and porosity

Population balance models (PBMs) have been widely studied for the simulation of continuous twin-screw wet granulation. Due to the limited data availability and computational efforts, one-dimensional PBM is the most popular application to the pharmaceutical industry. While multi-dimensional PBMs can predict multiple granule quality attributes, model development and calibration has been the big challenges. This paper presents a two-dimensional compartmental PBM which compute granule size and porosity. Granulation experiments were performed for two formulations and two process settings, followed by granule characterizations. In PBM development, consolidation was newly introduced, whereas the aggregation and breakage kernels were based on the previous model. Calibration results proved that the model can simulate granule size and porosity simultaneously for different formulations and process settings. Some cases show porosity increase that needs to be considered in the future. Further experiments and model development can increase the applicability of the two-dimensional PBM for the pharmaceutical industry.

Efficient numerical schemes for multidimensional population balance models

Multidimensional population balance models (PBMs) describe chemical and biological processes having a distribution over two or more intrinsic properties (such as size and age, or two independent spatial variables). The incorporation of additional intrinsic variables into a PBM improves its descriptive capability and can be necessary to capture specific features of interest. As most PBMs of interest cannot be solved analytically, computationally expensive high-order finite difference or finite volume methods are frequently used to obtain an accurate numerical solution. We propose a finite difference scheme based on operator splitting and solving each sub-problem at the limit of numerical stability that achieves a discretization error that is zero for certain classes of PBMs and low enough to be acceptable for other classes. In conjunction to employing specially constructed meshes and variable transformations, the scheme exploits the commutative property of the differential operators present in many classes of PBMs. The scheme has very low computational cost — potentially as low as just memory reallocation. Multiple case studies demonstrate the performance of the proposed scheme.

Simulation and 3-D Visualization of Crystallization Process Based on Cellular Automata

Crystallization is widely used for separation and purification. During this process, the different growth conditions, such as temperature and supersaturation, would result in different size and morphology, which not only affect the product quality, but also affect the subsequent process operations. Through the simulation of the crystallization process, a better understanding of crystallization process dynamics can be extracted. The population balance (PB) model has been used to simulate industrial crystallization processes, while the multi-dimensional population balance (MPB) model can be used to obtain the information of size and morphology distribution. However, the morphological distribution information provided by numerical solution of MPB is macroscopic, and the specific morphology of a crystal particle cannot be displayed at the microscopic level, so the difference among crystals morphology cannot be clearly displayed. Cellular automata (CA) is a method that simulates a process by considering simple cell changes and cell-to-cell interactions. It can be employed to conduct cross-scale simulation, i.e., both macro and micro. In this work, a CA method is introduced into an MPB model to simulate the batch cooling crystallization process from solution. The rule of crystal growth comes from classical diffusion theory, and whether the solution is transformed into a crystal is realized by the Monte Carlo method. The crystallization of potassium dihydrogen phosphate (KDP) is taken as a case-study and the results is verified with the results in the literature. Through CA simulation, not only the macroscopic crystal size and morphology distribution information can be obtained, but also the crystal morphology in microscopic can be exhibited. At the same time, the crystallization process can be visualized in 3-D environment, with size and morphology distribution being presented intuitively. In addition, the calculation time is reduced under certain accurate conditions. This result provides a theoretical reference for modeling, analysis and controlling of the crystallization process in the future.

Accelerating multi-dimensional population balance model simulations via a highly scalable framework using GPUs

The solution of high-dimensional PBMs using CPUs are often computationally intractable. This study focuses on the development of a scalable algorithm to parallelize the nested loops inside the PBM via a GPU framework. The developed PBM is unique since it adapts to the size of the problem and uses the GPU cores accordingly. This algorithm was parallelized for NVIDIA® GPUs as it was written in CUDA® and C/C++. The major bottleneck of such algorithms is the communication time between the CPU and the GPU. In our studies, communication time contributed to less than 1% of the total run time and a maximum speedup of about 12 over the serial CPU code was achieved. The GPU PBM achieved a speedup of about two times compared to the PBM’s multi-core configuration on a desktop computer. The speed improvements are also reported for various CPU and GPU architectures and configurations.

Multidimensional Population Balance Modeling for Pathological Calcium Pyrophosphate Dihydrate Crystals in the Presence of Amino Acids

AbstractA multidimensional population balance model was applied to elaborate the effects of amino acids on the crystallization mechanisms of calcium pyrophosphate dihydrate crystals. Breakage and growth terms defined in the model were expressed in two dimensions depending on the sizes. The exponential hyperbolic function based on the particle sizes was selected for expressing the growth. The model was resolved by using the finite differences method. It was detected that in the non‐additive media, the growth along the length‐1 is extremely dominant against both the growth along the length‐2 and the breakage. However, it was discovered that the amino acids made the crystals more fragile, and the breakage at certain size ranges and certain concentrations of amino acids was the dominant mechanism. In any case, it was proven that both growth and breakage are suppressed considerably in the amino acid‐containing media.

Feedback Control for the Size and Shape Evolution of Needle-like Crystals in Suspension. III. Wet Milling

Several operating and control strategies with the purpose of manipulating the size and shape of crystals exhibiting needle-like morphology are presented using an ex situ wet mill as the actuator. A model-based operating policy and different model-free controllers are discussed. These controllers were coupled with an online particle size and shape monitoring tool. First, the attainable region in the size and shape space for a wet milling operation was evaluated in a simulation framework using a multidimensional population balance model describing the breakage phenomenon. Second, the operating and control strategies were tested experimentally in a lab-scale setup using two different milling configurations and two different compounds, namely, β l-glutamic acid and γ d-mannitol. The goal of the experiments was to drive various seed populations of the two compounds to several target average lengths in the size and shape space. It was observed that the model-based operating policy, run without any feedback acti...

A framework for model reliability and estimability analysis of crystallization processes with multi-impurity multi-dimensional population balance models

The development of reliable mathematical models for crystallization processes may be very challenging due the complexity of the underlying phenomena, the inherent Population Balance Models (PBMs) and the large number of parameters that need to be identified from experimental data. Due to the poor information content of the experiments, the structure of the model itself and correlation between model parameters, the mathematical model may contain more parameters than can be accurately and reliably identified from the available experimental data. A novel framework for parameter estimability for guaranteed optimal model reliability is proposed then validated by a complex crystallization process. The latter is described by a differential algebraic system which involves a multi-dimensional population balance model that accounts for the combined effects of different crystal growth modifiers/impurities on the crystal size and shape distribution of needle-like crystals. Two estimability methods were combined: the first is based on a sequential orthogonalization of the local sensitivity matrix and the second is Sobol, a variance-based global sensitivities technic. The framework provides a systematic way to assess the quality of two nominal sets of parameters: one obtained from prior knowledge and the second obtained by simultaneous identification using global optimization. A cut-off value was identified from an incremental least square optimization procedure for both estimability methods, providing the required optimal subset of model parameters. The implemented methodology showed that, although noisy aspect ratio data were used, the 8 most influential and least correlated parameters could be reliably identified out of twenty-three, leading to a crystallization model with enhanced prediction capability.

Hypothesis-driven compartment model for stirred bioreactors utilizing computational fluid dynamics and multiple pH sensors

The ability to understand and predict mixing behaviour in stirred bioreactors through the use of mathematical models is an effective strategy to improve or ensure the performance of a process. Historically, compartment models have been developed to facilitate the prediction of hydrodynamics and in the last decade, computational fluid dynamics (CFD) has become the standard tool to assess this in detail. Despite the continuous increase in computational capacity, performing CFD simulations of bioreactors remains a challenge in industry, where the need for simplified models is apparent due to the complex phenomena occurring at the industrial scale. The combination of CFD and compartment models has been investigated in this work using a pilot scale stirred bioreactor equipped with three Rushton disc turbines and multiple pH sensors as a case study. A hypothesis-driven compartmentalization strategy, comprised of five steps, has been developed and applied to the pilot scale stirred bioreactor. The compartmentalization resulted in 56 compartments with unidirectional flow between adjacent compartments. The performance of the methodology was evaluated against a data-driven compartment approach and the full CFD simulation, in terms of its ability to recreate transient tracer profiles following top and bottom feeding of the tracer. The data-driven compartment model proved to be the most accurate of the three investigated methods, whereas the hypothesis-driven compartment method had a 10–12% higher error on prediction. Nevertheless, the developed methodology should be considered a viable alternative to conventional CFD methods, especially when models of complex phenomena – for example multi-dimensional population balance models – are to be incorporated in the final model as well.

A parallel unidirectional coupled DEM-PBM model for the efficient simulation of computationally intensive particulate process systems

The accurate modeling of the physics underlying particulate processes is complicated and requires significant computational capabilities to solve using particle-based models. In this work, a unidirectional multi-scale approach was used to model the high shear wet granulation process. A multi-dimensional population balance model (PBM) was developed with a mechanistic kernel, which in turn obtained collision data from the discrete element modeling (DEM) simulation. The PBM was parallelized using a hybrid OpenMP+MPI approach. The DEM simulations were performed using LIGGGHTS, which was parallelized using MPI. Speedups of approximately 14 were obtained for the PBM simulations and approximately 12 for the DEM simulations. The uni-directional coupling of DEM to PBM was performed using middle-ware components (RADICAL-Pilot) that did not require modifications of the DEM or PBM codes, yet supported flexible execution on high-performance platforms. Results demonstrate scaling from 1 to 128 cores for the PBM and up to 256 cores for the DEM. The proposed method, implementations and middle-ware enable the modeling of high shear wet granulation process faster than existing approaches in literature.

An efficient approximate moment method for multi-dimensional population balance models – Application to virus replication in multi-cellular systems

Many particulate processes in process and bioprocess engineering can be described with multi-dimensional population balances. Approximate moment methods are frequently used for their solution. In the present paper a new approach is presented, which is particular efficient when the number of internal coordinates is high. It combines the direct quadrature method of moments with monomial cubatures. With the new method the computational effort increases only polynomially, in the simplest case even only linearly with the number of internal coordinates, compared to an exponential increase for the well known Gausssian cubatures. The technique is evaluated for a five dimensional benchmark problem describing virus replication in continuous cell cultures. Furthermore, the algorithm is applied to analyze influenza virus replication in genetically modified cell lines.

Graphical processing unit (GPU) acceleration for numerical solution of population balance models using high resolution finite volume algorithm

Abstract Population balance modeling is a widely used approach to describe crystallization processes. It can be extended to multivariate cases where more internal coordinates i.e., particle properties such as multiple characteristic sizes, composition, purity, etc. can be used. The current study presents highly efficient fully discretized parallel implementation of the high resolution finite volume technique implemented on graphical processing units (GPUs) for the solution of single- and multi-dimensional population balance models (PBMs). The proposed GPU-PBM is implemented using CUDA C++ code for GPU calculations and provides a generic Matlab interface for easy application for scientific computing. The case studies demonstrate that the code running on the GPU is between 2–40 times faster than the compiled C++ code and 50–250 times faster than the standard MatLab implementation. This significant improvement in computational time enables the application of model-based control approaches in real time even in case of multidimensional population balance models.

Analyzing the Impact of Heterogeneity in Genetically Engineered Cell Lines for Influenza Vaccine Production Using Population Balance Modeling

Abstract: Engineering of novel cell lines for biotechnological processes, e.g. influenza virus vaccine production, can be achieved by the genetic modification of host cell gene expression. Therefore, versatile genome editing methods such as lentiviral transduction can be applied to improve the production process. However, due to random integration of lentiviral-delivered genes in the host cell genome, nonuniform, i.e. heterogeneous, gene expression within the host cell population is expected. Within this contribution we investigate the influence of this cell-to-cell variability on important process variables like the maximum virus yield. Therefore, a multi dimensional population balance model is proposed which, on the one hand comprises a detailed description of the intracellular viral replication cycle and, on the other, also accounts for the expected heterogeneity in the host cell population. The results indicate that the overall vaccine production process can be improved by enhancement or inhibition of certain steps in the viral replication cycle. Furthermore, the achieved improvements show robustness against moderate degrees of cell-to-cell variability from genetic modification of host cells via transduction.

Multidimensional Population Balance Model Development and Validation of a Reactive Detergent Granulation Process

Reactive granulation is a complex process which brings about a physical and chemical change in the raw materials. It is extensively used for the production of detergents through the “dry neutralization” reaction. In this paper, the dynamics of the reactive granulation process is captured using a coupled multidimensional population balance model (PBM) for the size enlargement and a diffusion-reaction (DR) equation for the kinetics of the reaction. This framework can capture the experimental observations that have also been reported in Schöngut et al. (Ind. Eng. Chem. Res. 2011, 50, 11576). There are a few empirical parameters involved in the modeling framework for which a dynamic sensitivity analysis was employed to shortlist the parametric quantities that have the largest influence on the critical quality attributes. The model is calibrated against experimental results via the Nelder–Mead simplex algorithm for estimating the empirical parameters, and the model predictions were compared with experimental results. Both the parameter estimation and prediction results showed good agreement with the experimental results. The development in terms of a comprehensive mathematical model and the results as presented in this paper suggest the ability of this approach to make suitable model predictions which can significantly reduce the number of experimental trials required for process optimization and control.

Combination of limited measurement information and multidimensional population balance models

Cell to cell variability with respect to intracellular composition as measured by flow cytometry is a core feature of many bioprocesses.Models of the underlying processes are often available from small scale or single cell observations but generally fail to account for heterogeneity within the cell culture. This heterogeneity can be incorporated using population balance equations that have to be adapted to the flow cytometric measurements.However, in general not the whole intracellular composition can be determined simultaneously for technical or economical reasons with flow cytometry. Furthermore, in many cases only integral quantities of certain compounds are measurable (e.g. overall amount of biomass in a cell culture).In this contribution we will present a procedure for combination of these limited measurement informations with complex models. Application will be shown for production of biopolymer poly(3-hydroxybutyrate) (PHB) in bacteria cultures. Starting from a single cell description a corresponding population balance model which describes the heterogeneity on a macroscopic scale will be derived. Afterwards an efficient approximate moment method and the Method of Characteristics will be used to combine population balance models and limited measurement data.

Multi-dimensional population balance model development and validation for a twin screw granulation process

In this study, a novel multi-component population balance model (PBM) for the twin-screw granulation (TSG) process was developed, taking into account the rate processes of aggregation, breakage, liquid addition, and consolidation. Interactions between multiple solid components (e.g. active pharmaceutical ingredient and excipient) and the amount of liquid were accounted for in quantifying the aggregation and breakage rates. Experimental data was obtained for the TSG process, whereby the effect of initial particle size distribution and liquid-to-solid ratios on key granule properties was studied. The data was used to estimate adjustable parameters in the proposed mechanistic kernels and to validate the calibrated process model as a predictive tool. The simulation results showed a good agreement with experimental data.

Systematic classification of unseeded batch crystallization systems for achievable shape and size analysis

The purpose of the current work is to develop a systematic classification scheme for crystallization systems considering simultaneous size and shape variations, and to study the effect of temperature profiles on the achievable final shape of crystals for various crystallization systems. A classification method is proposed based on the simultaneous consideration of the effect of temperature profiles on nucleation and growth rates of two different characteristic crystal dimensions. Hence the approach provides direct indication of the extent in which crystal shape may be controlled for a particular system class by manipulating the supersaturation. A multidimensional population balance model (PBM) was implemented for unseeded crystallization processes of four different compounds. The effect between the nucleation and growth mechanisms on the final aspect ratio (AR) was investigated and it was shown that for nucleation dominated systems the AR is independent of the supersaturation profile. The simulation results confirmed experimentally also show that most crystallization systems tend to achieve an equilibrium shape hence the variation in the aspect ratio that can be achieved by manipulating the supersaturation is limited, in particular when nucleation is also taken into account as a competing phenomenon.

Modification of Crystal Shape through Deep Temperature Cycling

The evolution of particle shape is an important consideration in many industrial crystallizations. This article describes the design of temperature-cycling experiments (between alternating positive and negative supersaturations) to substantially change crystal shape with only a small number of cycles. The growth and dissolution of monosodium glutamate crystals of varying shapes were monitored using in-process attenuated total reflection–Fourier transform infrared spectroscopy (ATR-FTIR), focused beam reflectance measurement (FBRM), particle vision and measurement (PVM), and off-line optical microscopy. The growth and dissolution kinetics were estimated in a multidimensional population balance model based on solute concentration and crystal dimension measurements. This model fitted the experimental data with a limited number of parameters of small uncertainty. In addition, with the estimated kinetic parameters, the model predicted the crystal size and shape distribution in a different temperature-cycling e...

Application of stochastic weighted algorithms to a multidimensional silica particle model

This paper presents a detailed study of the numerical behaviour of stochastic weighted algorithms (SWAs) using the transition regime coagulation kernel and a multidimensional silica particle model. The implementation in the SWAs of the transition regime coagulation kernel and associated majorant rates is described. The silica particle model of Shekar et al. [S. Shekar, A.J. Smith, W.J. Menz, M. Sander, M. Kraft, A multidimensional population balance model to describe the aerosol synthesis of silica nanoparticles, Journal of Aerosol Science 44 (2012) 83–98] was used in conjunction with this coagulation kernel to study the convergence properties of SWAs with a multidimensional particle model. High precision solutions were calculated with two SWAs and also with the established direct simulation algorithm. These solutions, which were generated using large number of computational particles, showed close agreement. It was thus demonstrated that SWAs can be successfully used with complex coagulation kernels and high dimensional particle models to simulate real-world systems.