Bivariate Population Balance Model Research Articles

Multivariate population balance modeling and simulation of ultrasound-assisted crystallization of a plate-type pharmaceutical: Nucleation, growth, and breakage

The application of ultrasound is recently gaining significant interest in crystallization because of its significant impact on nucleation and growth rates, particle breakage, crystal habit, polymorphism, and crystal size distribution. Pyrazinamide, an essential drug for the treatment of mycobacterium tuberculosis, has four different polymorphic forms. The metastable δ-polymorph exhibit plate-type habit which is desirable for downstream operations compared to commercially available needle-type α-polymorph. This contribution presents a detail experimental and computational study on the effect of ultrasound and its amplitude on nucleation, growth, and breakage of crystals for unseeded batch cooling sonocrystallization of δ-polymorph of pyrazinamide from its 1,4-dioxane solution. A series of sonocrystallization experiments are conducted with different cooling rates and ultrasonic amplitudes and a bivariate population balance model involving crystal nucleation, growth, and breakage is developed and fully validated. The model simulations give important insights on time evolution of mean crystal size and aspect ratio of plate-type crystals.

A population balance model for the flow-induced preparation of Pickering emulsions

In this work we studied the flow-induced preparation of oil-in-water Pickering emulsions by means of a bivariate population balance model. We characterized oil droplets by their size and surface coverage by solid particles, and we took into account all phenomena responsible for their change, namely coalescence and breakup of the droplets, and particle-droplet collisions leading to surface coverage. We studied the population dynamics in a uniform shear flow and in a turbulent flow field, and stabilization was observed in both cases. Under shear, the population evolved by coalescence phenomena, which, together with particle adsorption, led to a prompt and full stabilization of the emulsion droplets. An inverse scaling between particle load and droplets Sauter diameter was found, and seen to well compare with previously reported experimental data for arrested coalescence. In turbulence, stabilization was also obtained. In this case, the droplet size evolved through a peculiar path, showing a size undershoot before equilibrium conditions were reached. We explained this as a consequence of the coalescence phenomena, that alongside breakup, affected the transient of the process. Results shed a new light onto the dynamics of preparation of Pickering emulsions and aim at constituting a reference for the set-up and interpretation of experiments.

Biomineral Flocculation of Kaolinite and Microalgae: Laboratory Experiments and Stochastic Modeling

AbstractIn estuaries and coastal waters, the flocculation of cohesive sediments is significantly affected by extracellular polymeric substances secreted by microalgae. In this study, laboratory experiments are carried out to explore the role of two typical microalgae (Skeletonema costatum and Cyclotella meneghiniana) on suspended fine‐grained sediments under various turbulent shear and environmental conditions. The results show that the shear rate is a dominant factor in controlling the biomineral flocculation processes, and the existence of microalgae generally enhances the aggregation of mineral sediments with microalgae. It is also found that floc size distributions (FSDs) and mean sizes are clearly influenced by different algal species and concentrations, especially under intensive turbulent environments. Moreover, a quasi‐Monte Carlo (QMC) based bivariate population balance model is developed to simulate the FSDs of sediment‐algae flocs. This model is reasonably validated by two analytical solutions and laboratory observations, and indicates a potential extension for tracking the properties of microalgae‐associated fine sediment flocs due to the discrete nature of the QMC method.

Optimal Control of Crystal Size and Shape in Batch Crystallization Using a Bivariate Population Balance Modeling

Abstract An optimal control framework was employed to obtain optimal supersaturation/temperature policies for controlling the crystal mass, size, and shape that meet target product specifications. It uses a bivariate population balance model that includes crystal nucleation, growth, dissolution, and disappearance. The optimal control scheme, solving a dynamic optimization problem, was applied to the batch cooling crystallization of potassium dihydrogen phosphate. The population balance model was evaluated in open-loop experiments, showing good prediction for the mean characteristic lengths and number of particles, both for the supersaturation and undersaturation zones. The deterministic optimal control simulations demonstrated the application of the control action policies to produce crystals of desired mass and average shape for different control targets.

A novel simplified multivariate PBE solution method for mass transfer problems

Interphase mass transfer estimation may require not only the accurate knowledge of the interfacial area, which depends on the information about the size of each dispersed element, but also on the driving force, that can be different if the elements of the disperse phase have different chemical composition. To take into account such polydispersity, bivariate (or multivariate) population balance model (PBM) are formulated according to physical phenomena occurring in the investigated mass transfer problem. This often includes aggregation, breakage, advection, mass transfer of the chemical species and chemical reactions of the transferring components. In this work we propose a novel and simplified method to solve the bivariate/multivariate population balance equation for a mass transfer problem, based on the high-order moment-conserving method of classes (HMMC) (Alopaeus et al., 2006). The proposed method is based on the idea of deriving additional material balance equations for the concentration of the droplets belonging to each size class, reducing significantly the total number of unknown variables with respect to true bivariate/multivariate method of classes. This modeling approach is compared with two other possible solution methods for a test case in which mass transfer and chemical reactions occur in a system with two immiscible liquid phases. In the first the traditional approach is used, where a single material balance is formulated for the disperse phase along with PBM, while in the second a true bivariate/multivariate solution method is used. The results of this comparison show that the proposed method is robust and accurate, capable of properly describing the multidimensional droplet size-composition distribution needed to evaluate the mass transfer rates, in a fraction of the computational time compared with more accurate methods.

Non-isothermal population balance model of the formation and dissociation of gas hydrates

Data reduction processes based on laboratory-scale experiments related to hydrate formation and dissociation kinetics usually employ isothermal models where the hydrate particles and continuous phase are at the same constant temperature despite the large latent heat in this phase change phenomenon. In this study, we tested this hypothesis by developing a non-isothermal model of methane hydrate formation and dissociation in a three-phase agitated vessel with a cooling jacket. The hydrate particle mass and energy comprised the distribution variables in a bivariate population balance model, which included particle nucleation, growth, breakage, and aggregation. The mass and energy balances in the gas and liquid phases completed the model. The heat transfer between the liquid phase and the coolant in the jacket was also modeled. Under the conditions analyzed, the results showed that the particle temperatures were quite close to that of the liquid phase but large changes occurred during hydrate formation or dissociation due to the slow heat transfer to the cooling fluid. Therefore, hydrate formation or dissociation occur in a environment with a changing temperature, which must be considered to obtain meaningful kinetic data.

Charging and coagulation of radioactive and nonradioactive particles in the atmosphere

Abstract. Charging and coagulation influence one another and impact the particle charge and size distributions in the atmosphere. However, few investigations to date have focused on the coagulation kinetics of atmospheric particles accumulating charge. This study presents three approaches to include mutual effects of charging and coagulation on the microphysical evolution of atmospheric particles such as radioactive particles. The first approach employs ion balance, charge balance, and a bivariate population balance model (PBM) to comprehensively calculate both charge accumulation and coagulation rates of particles. The second approach involves a much simpler description of charging, and uses a monovariate PBM and subsequent effects of charge on particle coagulation. The third approach is further simplified assuming that particles instantaneously reach their steady-state charge distributions. It is found that compared to the other two approaches, the first approach can accurately predict time-dependent changes in the size and charge distributions of particles over a wide size range covering from the free molecule to continuum regimes. The other two approaches can reliably predict both charge accumulation and coagulation rates for particles larger than about 0.04 micrometers and atmospherically relevant conditions. These approaches are applied to investigate coagulation kinetics of particles accumulating charge in a radioactive neutralizer, the urban atmosphere, and an atmospheric system containing radioactive particles. Limitations of the approaches are discussed.

CFD-population balance modeling and simulation of coupled hydrodynamics and mass transfer in liquid extraction columns

A hierarchical approach for modeling and simulation of coupled hydrodynamics and mass transfer in liquid extraction columns using detailed and reduced bivariate population balance models is presented. The hierarchical concept utilizes a one-dimensional CFD model with detailed bivariate population balances. This population balance model is implemented in the PPBLAB software which is used to optimize the column hydrodynamics. The optimized droplet model parameters (droplet breakage and coalescence) are then used by a two-dimensional CFD reduced population balance model. As a reduced bivariate population balance model, OPOSPM (One Primary and One Secondary Particle Method) is implemented in the commercial FLUENT software to predict the coupled hydrodynamics and mass transfer of an RDC extraction column with 88 compartments. The simulation results show that the coupled two-dimensional-OPOSPM model produces results that are very close to that of the one-dimensional PPBLAB detailed population balance model. The advantages of PPBLAB are the ease of model setup, implementation and the reduced simulation time (order of minutes), when compared to the computational time (order of weeks) and computational resources using FLUENT software. The advantages of the two-dimensional CFD model is the direct estimation of the turbulent energy dissipation using the k–ε model and the local resolution of continuous phase back mixing.

Batch Cooling Crystallization of Plate-like Crystals: A Simulation Study

In the paper, batch cooling crystallization of plate-like crystals is investigated by numerical experimentation. The population balance approach is used to describe a seeded crystallization process, developing a bivariate morphological population balance model including nucleation and size dependent crystal growth. The population balance equation is reduced to a system of equations for the bivariate mixed moments of internal variables, which are solved by means of the quadrature method of moments. A sensitivity analyses is carried out to investigate the effects of the process parameters on dynamic evolution of particulate properties including the characteristic mean particle size and shape.

On the role of gas-phase and surface chemistry in the production of titania nanoparticles in turbulent flames

Combustion-based synthesis is the prominent technique for large-scale production of commercial-grade nanoparticles, such as titanium dioxide (titania or TiO2). Both time and economic constraints have led to an increase in the sophistication of the models for such chemical processes. State-of-the-art models for combustion-based nanoparticle synthesis incorporate highly detailed gas-phase kinetic models to describe the effects of the complex chemical reactions on particle formation and growth. Accurate models for particle evolution must be coupled with the detailed gas-phase kinetics in order to predict the particle properties. In this work, a bivariate population balance model for titania nanoparticle produced in flame reactors is used to investigate the role of gas-phase and surface chemistry in the determination of particle properties. The model considers all relevant particle evolution events including nucleation, surface growth, aggregation and sintering. In order to focus on the relative importance of the gas-phase mechanism, the flow field is modeled using a simple multi-environment plug-flow reactor model. Both one-step and detailed chemistry for Ti oxidation from the precursor, TiCl4, are compared for two different flame configurations. The simulation results demonstrate the importance of the location of nuclei formation in the flame, which depends strongly on the gas-phase and surface growth kinetic models, and their effect on the final product properties. These results suggest that detailed gas-phase chemical kinetics combined with a detailed surface growth model are required to accurately describe the combustion-based synthesis of nanoparticles.

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

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

Mathematical modeling of high-pressure oil-splitting reactor using a reduced population balance model

A reduced non-equilibrium bivariate population balance model is developed to model simultaneous reversible chemical reaction and extraction of an industrial scale high-pressure oil-splitting reactor with a first-order reversible kinetics. The model includes details about the reacting mixture by considering the discrete (particulate) nature of the dispersed phase. This, however; increases the mathematical complexity of the model and results in a coupled system of partial differential equations (PDE) of hyperbolic type with nonlinear source terms. The characteristics analysis shows that these hyperbolic PDEs have two dominant and distinct characteristic speeds: The mean droplet speed and the mean continuous phase (oil) speed. This generates a set of contact waves moving along the reactor height in opposite directions. Therefore, modern numerical methods based on CFD literature are applicable to provide a numerical solution of the developed model. As a first step in the numerical investigation, a finite volume method with a first-order upwind scheme is developed and implemented. Also, sufficient conditions of stability are derived and implemented. The numerical model is validated against analytical solution of a special case, where numerical convergence in the sense of l2-norm is established. Numerical simulation results showed that the unreacted oil concentration profile decreased exponentially along the reactor height. On the other hand, the glycerine concentration in the oil phase passes through a maximum due to the simultaneous reaction and extraction processes. Three operating parameters: temperature, mean droplet diameter and excess water flow rate were studied to highlight their effect on the reactor performance. The most important parameter was the reactor operating temperature, where 240 °C was found sufficient for complete oil conversion. This agrees with the operating temperature in industrial practice.

A bivariate population balance model for the microbial production of poly(3-hydroxybutyrate)

A bivariate population balance model is developed for describing the dynamic evolution of a PHB-producing microbial batch culture. The population balance model is solved with three different numerical methods namely, a continuous finite element method, a discrete-continuous finite element method and a discretized method with different high-resolution finite difference schemes for the calculation of the cell flux term. The various numerical methods are evaluated in terms of accuracy and computational requirements by a direct comparison of the leading moments of the bivariate distribution (i.e., M 10 and M 01) to the respective quantities calculated by a simple homogeneous model for the case of zero product inhibition. Furthermore, the results obtained from the solution of the 2-D population balance model are compared with the respective results of a 1-D PBM for the case of product inhibition of the residual biomass growth rate. It is shown that the 1-D PBM cannot accurately describe the dynamics of the microbial production system when the intracellularly accumulated PHB inhibits the residual biomass growth rate and, hence, a bivariate population balance modeling approach is required to describe the dynamic behavior of the microbial culture.

Advanced prediction of pulsed (packed and sieve plate) extraction columns performance using population balance modelling

A bivariate population balance model (the base of LLECMOD program) for the steady state simulation of liquid extraction columns is extended to simulate pulsed (packed and sieve plate) extraction columns. The model is programmed using visual digital FORTRAN and then integrated into the LLECMOD program. As a case study, LLECMOD is used to simulate the separation performance of a pulsed extraction column. Two chemical test systems recommended by the EFCE (namely toluene–acetone–water and n-butyl acetate–acetone–water) are used in the simulation. Model predictions are successfully validated against steady state experimental data, where good agreements are achieved. The simulated results (holdup, mean droplet diameter and mass transfer profiles) compared to the experimental data show that LLECMOD is a powerful simulation tool, which can efficiently predict the performance of pulsed extraction columns.

Cell population modelling describes intrinsic heterogeneity: a case study for hematopoietic stem cells

The control of stem cell properties during in vitro expansion is of paramount importance for their clinical use. According to Food and Drug Administration (FDA) guidelines, phenotypic heterogeneity is a critical aspect influencing therapeutic response. Even if the authors ability to reduce heterogeneity were limited, the sources from which it arises should be well understood for safe clinical applications. The aim of this work was to describe theoretically the intrinsic cell population heterogeneity that is present even when cells are cultured in a perfectly homogeneous environment. A bivariate population balance model is developed to account for the heterogeneity in the number of receptors and receptor-ligand complexes per cell, and is coupled with a ligand conservation equation. As a case study, the model is applied to the hematopoietic stem cell expansion, considering the c-Kit receptor and stem cell factor pair. Results show the dependence of intrinsic heterogeneity from ligand concentration and the kinetics of its administration. By tracking the cell generations within the total population, the authors highlight intra- and an inter-generational contributions to total population heterogeneity. In terms of dimensionless variables, intrinsic heterogeneity is dependent on the ratio of the characteristic time of cell division to that needed by a newborn cell to reach its single-cell steady state. [Includes supplementary material].

Bivariate population balance model of ethanol‐fueled spray combustors

AbstractWe present a bivariate population balance‐based formulation of the performance of well‐mixed adiabatic combustors fed by ethanol (EtOH)‐containing sprays of prescribed droplet size distribution (DSD) and composition. Our historically interesting example is the fuel‐cooled V‐2 chemical rocket—using 75 wt % EtOH + H2O solution, and oxidizer O2(L). Of special interest are the predicted combustion “intensity” (GW/m3) and efficiency (EtOH fraction vaporized) at each ratio of combustor mean residence time to feed‐droplet characteristic vaporization time. Our formulation exploits a quasi‐steady, gas‐diffusion‐controlled individual droplet evaporation rate law, and the method‐of‐characteristics to solve the associated first‐order population balance partial differential equation governing the joint distribution function n(m1, m2) of the fuel spray exiting such a chamber, where m1 = EtOH mass/droplet, and m2 = H2O mass/droplet. Besides the combustor efficiency and intensity, this bivariate distribution function enables predictions of corresponding unconditional DSD, and the joint distribution function(diam., droplet temperature)—perhaps measurable. Our numerically exact formulation/results also provide valuable test cases for convenient approximate methods (bivariate moment and spectral/weighted residual) to predict these “correlated” bivariate distribution functions in more complex situations. © 2011 American Institute of Chemical Engineers AIChE J, 2011

Droplet population balance modelling—hydrodynamics and mass transfer

The hydrodynamic and mass transfer behavior of a rotating disc contactor extraction (RDC) column based on a bivariate population balance model is investigated using the generalized fixed-pivot technique for the discretization of droplet internal coordinate. Single-droplet and swarm-droplet studies in small lab-scale devices were used to evaluate breakage and coalescence parameters necessary for column simulations. The breakage probability of single droplets was measured and a new correlation was developed, which also takes viscosity effects into account. Coalescence probability studies resulted in chemical system dependent parameters, which were obtained by an inverse solution of a simplified balance model. In a final study, the hydrodynamic and mass transfer behavior of pilot plant RDC columns have been simulated based on the parameter set derived from the lab-scale units. The simulated mean Sauter diameter, hold-up values and concentration profiles were found to be well predicted at different operating conditions. The relative error for the simulated mean Sauter diameters is about 15%, for the hold-up about 20% and for the concentration profiles about 20%.

MAGNETIC-SEEDING FILTRATION1*

Magnetic-seeding filtration consists of two steps: heterogeneous particle flocculation of magnetic and nonmagnetic particles in a stirred tank and high-gradient magnetic filtration (HGMF). The effects of various parameters affecting magnetic-seeding filtration (HGMF). The effects of various parameters affecting magnetic seeding filtration are theoretically and experimentally investigated. A trajectory model that includes hydrodynamic resistance, van der Waals, and electrostatic forces is developed to calculate the flocculation frequency in a turbulent-shear regime. Fractal dimension is introduced to simulate the open structure of aggregates. A magnetic-filtration model that consists of trajectory analysis, a particle build-up model, a breakthrough model, and a bivariate population-balance model is developed to predict the breakthrough curve of magnetic-seeding filtration. A good agreement between modeling results and experimental data is obtained. The results show that the model developed in this study can be used to predict the performance of magnetic-seeding filtration without using empirical coefficients or fitting parameters. 35 refs., 7 figs., 1 tab.

MAGNETIC-SEEDING FILTRATION*

ABSTRACT Magnetic-seeding filtration consists of two steps: heterogeneous particle flocculation of magnetic and nonmagnetic particles in a stirred tank and high-gradient magnetic filtration (HGMF). The effects of various parameters affecting magnetic-seeding filtration are theoretically and experimentally investigated. A trajectory model that includes hydrodynamic resistance, van der Waals, and electrostatic forces is developed to calculate the flocculation frequency in a turbulent-shear regime. Fractal dimension is introduced to simulate the open structure of aggregates. A magnetic-filtration model that consists of trajectory analysis, a particle build-up model, a breakthrough model, and a bivariate population-balance model is developed to predict the breakthrough curve of magnetic-seeding filtration. A good agreement between modeling results and experimental data is obtained. The results show that the model developed in this study can be used to predict the performance of magnetic-seeding filtration without using empirical coefficients or fitting parameters. * This research was supported by the Efficient Separations and Processing Crosscutting Program, Office of Environmental Management, U.S. Department of Energy, under contract DE-AC05-96OR22464 with Lockheed Martin Energy Research Corp. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. government purposes.