Multidimensional Population Balance 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.

Process Optimization of Para-xylene Crystallization Separation Process via Morphology Approach, Multi-dimensional Population Balance Equation, and Equation-Oriented Models.

An activity coefficient-based model was proposed to predict pertinent saturated concentrations in organic solid-liquid equilibrium, and the binary parameters of xylene mixtures were experimentally obtained. Also, a novel monocular 3D reconstruction technique was developed to measure crystal size and applied to derive the kinetics of nucleation and growth of para-xylene crystals. Subsequently, a multi-dimensional population balance equation was used to predict the particle size distribution in the crystallizer and an algorithm was designed to simulate and optimize the economic benefit of the crystallization separation process. Consequently, it became possible to predict the optimal coolant flowrate and inlet temperature, as well as the feed flowrate for a crystallization process with given operating conditions and device parameters.

Modeling and Optimization of the Para-Xylene Continuous Suspension Crystallization Separation Process via a Morphology Technique and a Multi-Dimensional Population Balance Equation

In this study, we carried out a para-xylene crystallization experiment at constant temperature and concentration levels. Throughout the process, the kinetics of nucleation, growth, breakage, and aggregation of para-xylene particles were measured and built using a morphological approach. An additional a three-stage continuous suspension crystallization separation experiment was carried out, the process for which was simulated using the population balance model based on correlated kinetic equations. The population balance equation was solved using an extended moment of classes algorithm, and the solving process was implemented in MATLAB. In this case, the predicted particle size distribution of the products matched well with the experiment. In order to provide references for the optimization of the industrial para-xylene crystallization process, a three-stage suspension crystallization separation experiment was designed and conducted, in which each crystallizer had a distinct operating temperature and mean residence time. The effects of operating parameters on the final product were investigated further. The proposed models and algorithms can also be applied in other cases and provide an alternative approach for optimizing continuous crystallization processes.

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.

Characterization of Simultaneous Evolution of Size and Composition Distributions Using Generalized Aggregation Population Balance Equation.

The application of multi-dimensional population balance equations (PBEs) for the simulation of granulation processes is recommended due to the multi-component system. Irrespective of the application area, numerical scheme selection for solving multi-dimensional PBEs is driven by the accuracy in (size) number density prediction alone. However, mixing the components, i.e., the particles (excipients and API) and the binding liquid, plays a crucial role in predicting the granule compositional distribution during the pharmaceutical granulation. A numerical scheme should, therefore, be able to predict this accurately. Here, we compare the cell average technique (CAT) and finite volume scheme (FVS) in terms of their accuracy and applicability in predicting the mixing state. To quantify the degree of mixing in the system, the sum-square parameter is studied to observe the deviation in the amount binder from its average. It has been illustrated that the accurate prediction of integral moments computed by the FVS leads to an inaccurate prediction of the parameter for a bicomponent population balance equation. Moreover, the cell average technique (CAT) predicts the moments with moderate accuracy; however, it computes the mixing of components parameter with higher precision than the finite volume scheme. The numerical testing is performed for some benchmarking kernels corresponding to which the analytical solutions are available in the literature. It will be also shown that both numerical methods equally well predict the average size of the particles formed in the system; however, the finite volume scheme takes less time to compute these results.

Sonofragmentation of Rectangular Plate-like Crystals: Bivariate Population Balance Modeling and Experimental Validation

The use of multidimensional population balance equations in predicting ultrasound assisted sonocrystalization processes are curbed due to the unavailability of physically motivated macroscopic math...

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.

Hybrid finite volume and Monte Carlo method for solving multi-dimensional population balance equations in crystallization processes

A novel hybrid framework that combines our recently proposed event driven constant number Monte Carlo (MC) method and a finite volume discretization scheme is developed to solve population balance equation (PBE) involving nucleation, growth, and breakage events for a batch crystallization process. At each time step, the nucleation and growth terms of the PBE are solved using high resolution finite volume method (HRFV), and then the breakage term is solved using the MC approach. This integrated approach is especially advantageous for solving multi-dimensional PBE involving nucleation, growth, and breakage events. The efficiency of the proposed approach is demonstrated by solving both one-dimensional (1D) and two-dimensional (2D) PBE from published literature. Also, a set of seeded batch cooling crystallization of plate-like pyrazinamide crystals were performed in presence of ultrasound (US) and the proposed hybrid solution scheme was used successfully to model and simulate this system characterised by 2D PBE.

Multi-dimensional population balance modelling of pharmaceutical formulations for continuous twin-screw wet granulation: Determination of liquid distribution

Two-dimensional population balance model (PBM) is developed in order to model pharmaceutical granules formation in a twin-screw wet granulator. Granule size and liquid content are considered as internal coordinates, while axial length of granulator is considered as external coordinate. Two types of initial liquid distribution are considered for the model development, i.e. constant and linear distributions. The main focus is on modeling and validation of liquid content distribution of granules. Regime-separated approach was used in order to capture the non-homogeneity of the granulator. The plug flow regime is considered for the conveying zone, while well-mixed regime is assumed for the kneading zone of twin-screw granulator. Aggregation and breakage are considered as the main mechanisms for granule formation and size control. Cell average method is used for solution of the PBM based on lumped parameter approach. In order to determine experimentally the distribution of liquid, liquid binder by dye addition was used in the process. The model findings are calibrated and validated by comparing with measured liquid content in each size fraction. The measured data is collected on a 12 mm twin-screw wet granulator using microcrystalline cellulose (MCC) and water soluble dye plus water as binder. The model indicated to be valid for MCC and needs to be validated with further excipients. The results revealed that increasing screw speed led to more uniform liquid distribution. Finally, the model findings indicated that 2D PBM is capable of predicting liquid distribution, and can be used as predictive tool in pharmaceutical continuous granulation.

Morphological population balance modelling of the effect of crystallisation environment on the evolution of crystal size and shape of para-aminobenzoic acid

A current morphological population balance (MPB) modelling methodology, which integrates crystal morphology, facet growth kinetics with multi-dimensional population balance, is overviewed and demonstrated, hence providing an attractive approach for modelling crystallisation processes. MPB modelling is applied to simulate the batch crystallisation of the alpha-form of para-aminobenzoic acid from ethanolic solutions as a function of the crystallisation environment including cooling rate, seeding temperature and seed conditions (loading, size and shape). The evolution of crystal shape/size and their distributions revealed that higher loading led to smaller and less needle-like crystals with similar yields, hence potentially being an important parameter for process control. Examination of the development of the fracture surface for broken seeds, mimicking the seed conditions after milling in practice in the simulated processes, demonstrated that these faces grew fast and then rapidly disappeared from the external crystal morphology. Restriction and challenges inherent in the current model are also highlighted.

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.

A deterministic model for positional gradients in copolymers

A deterministic modelling approach is developed to predict the internal structure of gradient copolymer chains. A key innovation of the modelling approach is the introduction of a positional variable that gives direct access to quantitative gradient characteristics: the ensemble average composition and the gradient deviation. This positional variable is used to develop multi-dimensional population balance equations that can be solved numerically to calculate gradient quality measures. The methodology is illustrated using the gradient copolymerisation of ethylene and 1-octene via coordinative chain transfer mechanism, which is representative of a variety of polymerisation schemes for gradient copolymers. Simulation results are validated with those obtained by stochastic simulations which, until now, were the only means of predicting detailed gradient quality.

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.