Size-Independent Nucleation and Growth Model of Potassium Sulfate from Supersaturated Solution Produced by Stirred Crystallization.

This paper explores the kinetics of the crystallization of potassium sulfate in a stirred bed crystallizer through experimental investigation. Employing classical nucleation theory, the homogeneous and heterogeneous nucleation mechanisms of potassium sulfate were investigated. The induction time and critical nucleation parameters, including the surface tension (γ), critical nucleation radius (r*), critical nucleation free energy (ΔG*), and critical nucleation molecule number (i*), were meticulously determined under varying temperatures and supersaturation ratios. The experimental findings revealed that as the temperature and supersaturation ratio increased, the induction time, critical nucleation free energy, critical nucleation radius, and critical molecule number decreased whereas the nucleation rate increased. The crystalline shape remains relatively unaltered with respect to temperature and supersaturation ratio, yet the particle size (D10, D50, D90) increases as the supersaturation and temperature increase. The variations in the measured nucleation parameters align well with the predictions of classical nucleation theory. Furthermore, the kinetic equations of crystal nucleation and the growth rate in a stirred crystallization system were fitted using population balance equations. The results demonstrate that the growth rate increases with increasing supersaturation and stirring rates. Additionally, the effects of the parameters in the nucleation rate equation suggested that the suspension density exerted the greatest influence, followed by the supersaturation ratio and stirring rate. This extensive research provides invaluable theoretical guidance for optimizing the crystallization process and designing industrial crystallizers.

A detailed model and Monte Carlo simulation for predicting DIP genome length distribution in baculovirus infection of insect cells.

Baculoviruses have enormous potential for use as biopesticides to control insect pest populations without the adverse environmental effects posed by the widespread use of chemical pesticides. However, continuous baculovirus production is susceptible to DNA mutation and the subsequent production of defective interfering particles (DIPs). The amount of DIPs produced and their genome length distribution are of great interest not only for baculoviruses but for many other DNA and RNA viruses. In this study, we elucidate this aspect of virus replication using baculovirus as an example system and both experimental and modeling studies. The existing mathematical models for the virus replication process consider DIPs as a lumped quantity and do not consider the genome length distribution of the DIPs. In this study, a detailed population balance model for the cell-virus culture is presented, which predicts the genome length distribution of the DIP population along with their relative proportion. The model is simulated using the kinetic Monte Carlo algorithm, and the results agree well with the experimental results. Using this model, a practical strategy to maintain the DIP fraction to near to its maximum and minimum limits has been demonstrated.

The Influence of Internal Intermittency, Large Scale Inhomogeneity, and Impeller Type on Drop Size Distribution in Turbulent Liquid-Liquid Dispersions.

The influence of the impeller type on drop size distribution (DSD) in turbulent liquid-liquid dispersion is considered in this paper. The effects of the application of two impellers, high power number, high shear impeller (six blade Rushton turbine, RT) and three blade low power number, and a high efficiency impeller (HE3) are compared. Large-scale and fine-scale inhomogeneity are taken into account. The flow field and the properties of the turbulence (energy dissipation rate and integral scale of turbulence) in the agitated vessel are determined using the k-ε model. The intermittency of turbulence is taken into account in droplet breakage and coalescence models by using multifractal formalism. The solution of the population balance equation for lean dispersions (when the only breakage takes place) with a dispersed phase of low viscosity (pure system or system containing surfactant), as well as high viscosity, show that at the same power input per unit mass HE3 impeller produces much smaller droplets. In the case of fast coalescence (low dispersed phase viscosity, no surfactant), the model predicts similar droplets generated by both impellers. In the case of a dispersed phase of high viscosity, when the mobility of the drop surface is reduced, HE3 produces slightly smaller droplets.

ISD3: a particokinetic model for predicting the combined effects of particle sedimentation, diffusion and dissolution on cellular dosimetry for in vitro systems.

BACKGROUND: The development of particokinetic models describing the delivery of insoluble or poorly soluble nanoparticles to cells in liquid cell culture systems has improved the basis for dose-response analysis, hazard ranking from high-throughput systems, and now allows for translation of exposures across in vitro and in vivo test systems. Complimentary particokinetic models that address processes controlling delivery of both particles and released ions to cells, and the influence of particle size changes from dissolution on particle delivery for cell-culture systems would help advance our understanding of the role of particles and ion dosimetry on cellular toxicology. We developed ISD3, an extension of our previously published model for insoluble particles, by deriving a specific formulation of the Population Balance Equation for soluble particles. RESULTS: ISD3 describes the time, concentration and particle size dependent dissolution of particles, their delivery to cells, and the delivery and uptake of ions to cells in in vitro liquid test systems. We applied the model to calculate the particle and ion dosimetry of nanosilver and silver ions in vitro after calibration of two empirical models, one for particle dissolution and one for ion uptake. Total media ion concentration, particle concentration and total cell-associated silver time-courses were well described by the model, across 2 concentrations of 20 and 110 nm particles. ISD3 was calibrated to dissolution data for 20 nm particles as a function of serum protein concentration, but successfully described the media and cell dosimetry time-course for both particles at all concentrations and time points. We also report the finding that protein content in media affects the initial rate of dissolution and the resulting near-steady state ion concentration in solution for the systems we have studied. CONCLUSIONS: By combining experiments and modeling, we were able to quantify the influence of proteins on silver particle solubility, determine the relative amounts of silver ions and particles in exposed cells, and demonstrate the influence of particle size changes resulting from dissolution on particle delivery to cells in culture. ISD3 is modular and can be adapted to new applications by replacing descriptions of dissolution, sedimentation and boundary conditions with those appropriate for particles other than silver.

Numerical rate function determination in partial differential equations modeling cell population dynamics.

This paper introduces a method to solve the inverse problem of determining an unknown rate function in a partial differential equation (PDE) based on discrete measurements of the modeled quantity. The focus is put on a size-structured population balance equation (PBE) predicting the evolution of the number distribution of a single cell population as a function of the size variable. Since the inverse problem at hand is ill-posed, an adequate regularization scheme is required to avoid amplification of measurement errors in the solution method. The technique developed in this work to determine a rate function in a PBE is based on the approximate inverse method, a pointwise regularization scheme, which employs two key ideas. Firstly, the mollification in the directions of time and size variables are separated. Secondly, instable numerical data derivatives are circumvented by shifting the differentiation to an analytically given function. To examine the performance of the introduced scheme, adapted test scenarios have been designed with different levels of data disturbance simulating the model and measurement errors in practice. The success of the method is substantiated by visualizing the results of these numerical experiments.

On the importance of temporal floc size statistics and yield strength for population balance equation flocculation model.

Many aquatic environments contain cohesive sediments that flocculate and create flocs with a wide range of sizes. The Population Balance Equation (PBE) flocculation model is designed to predict the time-dependent floc size distribution and should be more complete than models based on median floc size. However, a PBE flocculation model includes many empirical parameters to represent important physical, chemical, and biological processes. We report a systematic investigation of key model parameters of the open-source PBE-based size class flocculation model FLOCMOD (Verney, Lafite, Claude Brun-Cottan and Le Hir, 2011) using the measured temporal floc size statistics reported by Keyvani and Strom (2014) at a constant turbulent shear rate S. Results show that the median floc size d50, in terms of both the equilibrium floc size and the initial floc growth, is insufficient to constrain the model parameters. A comprehensive error analysis shows that the model is capable of predicting three floc size statistics d16, d50 and d84, which also reveals a clear trend that the best calibrated fragmentation rate (inverse of floc yield strength) is proportional to the floc size statistics considered. Motivated by this finding, the importance of floc yield strength is demonstrated in the predicted temporal evolution of floc size by modeling the floc yield strength as microflocs and macroflocs giving two corresponding fragmentation rates. The model shows a significantly improved agreement in matching the measured floc size statistics.

Further developments of the extended quadrature method of moments to solve population balance equations.

Developing numerical methods to solve polydispersed flows using a Population Balance Equation (PBE) is an active research topic with wide engineering applications. The Extended Quadrature Method of Moments (EQMOM) approximates the number density as a positive mixture of Kernel Density Functions (KDFs) that allows physical source terms in the PBEs to compute continuous or point-wise form according to the moments. The moment-inversion procedure used in EQMOM has limitations such as the inability to calculate certain roots even if it is defined, absence of consistent result when multiple roots exist or when the roots are nearly equal. To address these limitations, the study proposes a modification of the moment-inversion procedure to solve the PBE based on the proposed Halley-Ridder (H-R) method. Although there is no significant improvement in the extent of variability relative to the mean of the tested shape parameter σ values, an increase in the number of floating point operations (FLOPS) is observed which the proposed algorithm responds in limitations mentioned above. The total number of FLOPS for all the kernels used for the approximation increased by around 30%. This is an improvement towards the development of a more reliable and robust moment-inversion procedure.

Measuring and modeling nanoparticle transport by foam in porous media.

In this paper, an experimental study of nanoparticle transport by foam is presented. Bubbles made of N2-gas were stabilized with either a cationic surfactant (Cetyl Trimethyl Ammonium Bromide, CTAB), silica nanoparticles, or a combination of them. The concentrations of the surface active materials were selected upon foamability and stability tests. Column-flood tests were run until steady-state changing nanoparticle concentration, foam quality (fg), and flow rate. A synergistic behaviour of surfactant and nanoparticles help the formation of a strong foam. The measurements were used to validate a mechanistic model, presented in our earlier work (Li and Prigiobbe, 2020), which couples foam and nanoparticles transport with agglomeration and extended-DLVO theory. The model agrees well with the measurements and results show that an high-quality (ca. 90% gas fraction) can be used to carry nanoparticles and the efficient increases with flow velocity. This opens the opportunity for the application of foam as a carrier of nanoparticles in subsurface applications such as the remediation of contaminated sites and makes the model a valuable tool to design and predict such operations.

Modeling of Mixing-Precipitation Processes: Agglomeration.

A comprehensive description of the barium sulfate precipitation process in a wide range of supersaturations is presented. By using an additive to stabilize the particles, the decoupling of the primary from the secondary processes, as well as the agglomeration from aggregation was possible. By being able to study the two processes independently, a model describing the agglomeration of barium sulfate in the range of high supersaturations was validated experimentally for the first time. The proposed model has proven to describe the experiments with a high degree of accuracy in the whole range of supersaturations investigated. Additionally, by comparing agglomeration kernels of various complexity, ranges where simplifications are possible were identified, thus enabling the future development of models with better performance.

Mechanistic modelling of fluid bed granulation, Part I: Agglomeration in pilot scale process.

The present study aims to develop a mechanistic model to predict the performance of a fluid bed granulation process. Therefore, the behavior of the bed was investigated experimentally for various operating conditions. It was observed that the granule Loss on Drying (LoD) and granule size are strongly interrelated. In detail, the maximum final granule size was observed at an intermediate final LoD. Consequently, there is an optimum spray rate and inlet temperature with respect to the granule size. Besides, it was demonstrated that the experiments delivering lower LoD result in a more elongated final granule. Aimed at enabling the prediction of the bed performance numerically, a single-compartment, population-balance-based model was developed and validated against experimental data. The model parameters associated with the growth rate of granule were estimated and mechanistically correlated to the relevant operating conditions. Detailed analysis of the experimental results suggested that these model parameters may be partially connected to the granule LoD. Subsequently, in order to examine the accuracy of the developed model, a simulation was performed for a new set of operating conditions not previously accounted for in the correlations. The comparison of the simulated bed performance, when compared to the experimental results, proved with reasonable accuracy the reliability of the developed model in predicting the temporal evolution of granule size. Therefore, this study can be a step forward in developing a stand-alone granulation model, via modeling heat and mass transfer, to simulate evaporation and drying in a fluid bed granulator.

Numerical Methods for the Solution of Population Balance Equations Coupled with Computational Fluid Dynamics.

This review article discusses the solution of population balance equations, for the simulation of disperse multiphase systems, tightly coupled with computational fluid dynamics. Although several methods are discussed, the focus is on quadrature-based moment methods (QBMMs) with particular attention to the quadrature method of moments, the conditional quadrature method of moments, and the direct quadrature method of moments. The relationship between the population balance equation, in its generalized form, and the Euler-Euler multiphase flow models, notably the two-fluid model, is thoroughly discussed. Then the closure problem and the use of Gaussian quadratures to overcome it are analyzed. The review concludes with the presentation of numerical issues and guidelines for users of these modeling approaches.

Mechanistic modelling of fluid bed granulation, Part II: Eased process development via degree of wetness.

The performance of a fluid bed granulator was investigated through experimental and numerical study to develop a stand-alone fluid bed granulation model. The single-compartment model proposed in part I (for agglomeration modeling) was extended to account for i) evaporation of freely-flowing droplets, and ii) particle drying. This model enables us to predict the granule liquid content and temperature besides the granule size. Accurately, the equations of heat and mass conservation were solved in parallel to the population balance calculation of the agglomeration. In the same manner as for the agglomeration model, the model parameters associated with the drying model were estimated and correlated to the relevant quantities. The analysis of the experimental results revealed the significant contribution of the system "degree of wetness" to the bed performance, i.e., granule size and loss on drying (LoD). As the agglomeration model parameters were partially correlated to LoD in Part I, the presented model was revisited by inclusion of the degree of wetness. The reliability of the developed model in predicting the temporal evolution of granule size, liquid content, and temperature was proven through comparing the bed performance between simulation and experiment. Subsequently, to lowering the costs associated with experimental run, an approach was proposed based on the degree of wetness, aimed at reducing the number of experiments required for the design of experiment (DoE). The results of our simulation using reduced experiments demonstrated that the degree of wetness can be a promising indicator for the performance of the fluid bed granulator as well as for more efficient design of experiment.

A tri-modal flocculation model coupled with TELEMAC for estuarine muds both in the laboratory and in the field.

Estuarine and coastal regions are often characterized by a high variability of suspended sediment concentrations in their waters, which influences dredging projects, contaminant transport, aquaculture and fisheries. Although various three-dimensional open source software are available to model the hydrodynamics of coastal water with a sediment module, the prediction of the fate and transport of cohesive sediments is still far from satisfied due to the lack of an efficient and robust flocculation model to estimate the floc settling velocity and the deposition rate. Single-class and sometimes two-class flocculation models are oversimplified and fail to examine complicated floc size distributions, while quadrature-based or multi-class based flocculation models may be too complicated to be coupled with large scale estuarine or ocean models. Therefore, a three-class population balance model was developed to track the sizes and number concentrations of microflocs, macroflocs and megaflocs, respectively. With the assumption of a fixed size of microflocs and megaflocs, only four tracers are needed when coupled with the open-source TELEMAC system. It enables better settling flux estimates and better addresses the occurrence and concentration of larger megaflocs. This tri-modal flocculation model was validated with two experimental data sets: (1) 1-D settling column tests with the Ems mud and (2) in-situ measurements at the WZ Buoy station on the Belgian coast. Results show that the flocculation properties of cohesive sediments can be reasonably simulated in both environments. It is also found that the number of macroflocs created, when a larger macrofloc breaks up, is a statistical mean value and may not be an integer when applying the model in the field.

Control of molecular weight distribution in synthesis of poly(2-hydroxyethyl methacrylate) using ultrasonic irradiation.

Poly(2-hydroxyethyl methacrylate) (PHEMA) was synthesized using ultrasonic irradiation without any chemical initiator. The effect of the ultrasonic power intensity on the time course of the conversion to polymer, the number average molecular weight, and the polydispersity were investigated in order to synthesize a polymer with a low molecular weight distribution (i.e., low polydispersity). The conversion to polymer increased with time. A higher ultrasonic power intensity resulted in a faster reaction rate. The number average molecular weight increased during the early stage of the reaction and then gradually decreased with time. A higher ultrasonic intensity resulted in a faster degradation rate of the polymer. The polydispersity decreased with time. This was because the degradation rate of a polymer with a higher molecular weight was faster than that of a polymer with a lower molecular weight. A polydispersity below 1.3 was obtained under ultrasonic irradiation. By changing the ultrasonic power intensity during the reaction, the number average molecular weight can be controlled while maintaining low polydispersity. When the ultrasonic irradiation was halted, the reactions stopped and the number average molecular weight and polydispersity did not change. On the basis of the experimental results, a kinetic model for synthesis of PHEMA under ultrasonic irradiation was constructed considering both polymerization and polymer degradation. The kinetic model was in good agreement with the experimental results for the time courses of the conversion to polymer, the number average molecular weight, and the polydispersity for various ultrasonic power intensities.

Inverse problem analysis of pluripotent stem cell aggregation dynamics in stirred-suspension cultures.

The cultivation of stem cells as aggregates in scalable bioreactor cultures is an appealing modality for the large-scale manufacturing of stem cell products. Aggregation phenomena are central to such bioprocesses affecting the viability, proliferation and differentiation trajectory of stem cells but a quantitative framework is currently lacking. A population balance equation (PBE) model was used to describe the temporal evolution of the embryonic stem cell (ESC) cluster size distribution by considering collision-induced aggregation and cell proliferation in a stirred-suspension vessel. For ESC cultures at different agitation rates, the aggregation kernel representing the aggregation dynamics was successfully recovered as a solution of the inverse problem. The rate of change of the average aggregate size was greater at the intermediate rate tested suggesting a trade-off between increased collisions and agitation-induced shear. Results from forward simulation with obtained aggregation kernels were in agreement with transient aggregate size data from experiments. We conclude that the framework presented here can complement mechanistic studies offering insights into relevant stem cell clustering processes. More importantly from a process development standpoint, this strategy can be employed in the design and control of bioreactors for the generation of stem cell derivatives for drug screening, tissue engineering and regenerative medicine.