Population Balance Model Research Articles

CFD–PBM coupled modeling of liquid–liquid interphase mass transfer behaviors inside the Kenics static mixer

Kenics static mixers (KSM) are commonly used in various industries for its excellent mass transfer performance. It is essential to systematically analyze the relationship between fluid parameters and mass transfer parameters to better understand the mass transfer behaviors between liquid–liquid phases and optimize the mass transfer efficiency of the KSM. In this work, a computational fluid dynamic–population balance model coupled method is utilized and three mass transfer models are compared and analyzed. Further simulations investigate the mass transfer behaviors inside the KSM under different superficial velocities, dispersed phase volume fractions, and aspect ratios. The results show that the increase of superficial velocity leads to the 2.23–14.15–fold increase of volumetric mass transfer coefficient and the 1.71–4.56–fold increase of mass transfer rate. What's more, increasing the dispersed phase volume fraction leads to the significant increase of 15.51 times in volumetric mass transfer coefficient and 34.92 times in mass transfer rate. Moreover, the increase of aspect ratio could result in the reduction of 4.19–1.72 times in volumetric mass transfer coefficient and 1.81–1.17 times in mass transfer rate. This work provides valuable theoretical insights for the optimal design of the KSM and the enhancement of mass transfer performance.

Collisional breakage population balance equation: An analytical approach

This work presents a unique semi-analytical approach based on the homotopy analysis method (HAM), called accelerated HAM, recently proposed in (Hussain et al., “Semi-analytical methods for solving non-linear differential equations: A review.”, JMAA, 2023), to solve the collisional breakage population balance model, which is an integro-partial differential equation. We compare our findings with those obtained using the Adomian decomposition method, a well-known technique for solving various forms of differential equations. By decomposing the non-linear operator, we investigate how to utilize the convergence control parameter to expedite the convergence of the HAM solution towards its precise value in accelerated HAM. The other objective of the article is to examine the theoretical convergence analysis of the two proposed methods. Additionally, we conduct theoretical research on the error estimates for both the techniques. To validate our schemes, several numerical examples are considered and the numerical simulations demonstrate that the suggested techniques provide accurate estimates for the solution and moments of the collisional breakage equation.

Strengthened oil-water separation by swirl vane hydrocyclone based on short-circuit flow regulation

Hydrocyclones make it possible to separate oil/water efficiently. Short-circuit flow (SCF), which is a local secondary vortex in hydrocyclones, significantly restricts efficient separation. This study proposes a novel swirl vane hydrocyclone to regulate the trajectory of the SCF to reduce the SCF rate. The separation process in the hydrocyclone field is calculated using computational fluid dynamics coupled with the population balance modeling. The strengthened separation mechanism by the swirl vane is revealed, and the response of structural parameters, such as the angle, numbers, and width of the swirl vane, to the SCF rate and separation efficiency is investigated using the response surface optimization method. The results show that the swirl vane structure prevents the direct generation of SCF, strengthens the outward migration of the heavy-phase droplets, and reduces the probability of droplet breakage. The optimal structural parameters of the swirl vane are a swirl vane angle of 31°, vane number of 6 pcs, and vane width of 2 mm, which has the lowest SCF rate of 1.62 %. The separation efficiency increases by 21.53 %, and the SCF rate decreases by 85.33 % compared with a conventional hydrocyclone. This study provides theoretical guidance and data support for SCF regulation and the development of oil-water separation hydrocyclone.

An analytical treatment to spatially inhomogeneous population balance model

In modern liquid–liquid contact components, there is an increasing use of droplet population balance models. These components include differential and completely mixed contractors. These models aim to explain the complex hydrodynamic processes occurring in the dispersion phase. The hydrodynamics of these interacting dispersions include droplet breaking, coalescence, axial dispersion, and both entry and exit events. The resulting equations for population balance are represented as integro-partial differential equations, which rarely have analytical solutions, especially when spatial dependency is apparent. Consequently, the pursuit predominantly lies in seeking numerical solutions to resolve these complex equations. In this study, we have devised analytical solutions for inhomogeneous breakage and coagulation by employing the population balance equation (PBEs) applicable to both batch and continuous flow systems. The innovative approaches for solving PBEs in these systems leverage the Adomian decomposition method (ADM) and the homotopy analysis method (HAM). These semi-analytical methodologies effectively tackle the significant challenges related to numerical discretization and stability, which have often plagued previous solutions of the homogeneous PBEs. Our findings across all test examples demonstrate that the approximated particle size distributions utilizing these two methods converge to the analytical solutions continuously.

On the effect of axial inlet angle on the droplet size distribution and mixing uniformity for swirl jet mixer at high phase ratios

The “short pass” problem of liquid-liquid two-phase mixing at high phase ratios limited the development and applications of jet mixer in industrial process. In this study, the effect of the axial inlet angle on the mixing characteristics of a swirl jet mixer was investigated through laser particle size analysis and computational fluid mechanics (CFD) coupled with population balance model (PBM). The result was shown that as the axial inlet angle increased, the dispersed phase droplet sizes decreased, and the droplet size distribution was widened for the fixed mixer configuration. At Re ≤ 3.000×104, tangential inlet was more conducive to improving mixing uniformity. Conversely, the axial inlet angle of 90° was optimal at Re ≥ 3.750×104. Numerical simulations revealed that the tangential inlet effectively intensified liquid-liquid interactions, including streamlines extension, increased vorticity, improved droplet size uniformity. Additionally, the increase of the axial inlet angle resulted in increased turbulence stress and shortened decay period, the reduction in droplet size, and a wide droplet size distribution. The research findings can serve as theoretical basis for the hydraulic design and promoted applications of swirl jet mixers.

Capture of the Morphology Transfer Process in a High-Momentum Centrally Aerated Bubble Column with a Hybrid Multifield Two-Fluid Model

The role of pool scrubbing in attenuating radioactivity release after severe accidents has been explored extensively. It is known that scrubbing efficiency is largely determined by the hydrodynamic phenomenology in pools, which is still insufficiently understood. Aerosol gas forms large globules at the nozzle exit, which subsequently break up into a swarm of stable bubbles, where the change of bubble size can reach more than two orders depending on the injection rate. Furthermore, with the increase of flow rates, the injection regime changes from globule to jet characterized by a continuous gas structure. The flow field in the pool can be divided into two zones, injection and rise (swarm), according to the gas-liquid interface morphology. In different zones, scrubbing is governed by different mechanisms such as inertial impact, diffusion, and gravity, whereby bubble rise velocity is one major influential parameter. So far, numerical analysis of pool scrubbing is routinely based on system codes, which rely on empirical correlations for the determination of hydrodynamic parameters as well as scrubbing zones. More recently, owing to the increasing availability of computational resources, knowledge is improved through three-dimensional computational fluid dynamics simulations. Nevertheless, morphology and regime change still present a challenge for both two-fluid and interface-tracking models. Because of the limitation of closures, the conventional two-fluid model (TFM) is generally effective for bubble size smaller than cell size while interface-tracking (capturing) methods demand dozens of cells per bubble size. Combining the two kinds of approaches into one simulation is not straightforward. The present work aims to present a hybrid multifield TFM with extensions for capturing the transfer between different morphologies and for considering the effect of mesh resolution in momentum exchange. The developments are validated by simulating the complex hydrodynamic process in a pool-scrubbing column, which operates in the globule regime with an injection Weber number between 103 and 105. By comparison with experimental data, the results are shown to be promising, which provides a versatile framework for investigation of particle scrubbing in the future. The most attractive feature of the model is that compared to one-fluid interface-tracking models, it reliably predicts the bubble rise velocity. Furthermore, it is able to capture the lateral gas distribution without the need of applying a population balance model since most large bubbles are resolved.

Teaching old laws new tricks: Specific energy requirements for breakage based on modern kernels and a population balance model

A general expression for specific energy requirements in particle breakage is derived from a population balance model. The model employs power-law rate kernels, self-similar fragment distributions and an energy dependence based on the Vogel-Peukert master curve. This model connects the empirical constants and exponents of size in older empirical energy laws to the parameters of the population balance model, the nature of the particles, and certain moments of the self-similar particle size distribution predicted by the model. The resulting model is used to extract values of the order of the power-law rate kernel from literature data. The extracted order values are consistent with known expressions for individual particle breakage. The model developed here may be coupled with experimental determination of the material parameter needed in the Vogel-Peukert master curve to enable extraction of energy utilization efficiency values from experimentally determined values of the size-independent portion of the selection function.

Population balance modeling-assisted prediction of oxygen mass transfer coefficients with optical measurements

Prediction of bubble size distributions (BSD) is challenging but necessary to develop a more advanced bubble population-based model for oxygen mass transfer with increased data quality while reducing the required experimental effort. In this paper, we experimentally investigated bubble sizes in a pilot-scale setup using a submersible in-situ flow-microscope. This technique enables bubble size measurements in dense bubbly flows and at airflow rates of up to 40 sL/min above the diffuser, where former measurement methods were limited to a range below 8 slpm with a comparable diffuser configuration. The data obtained were used to study coalescence and breakup behavior and to predict bubble size distributions using population balance modeling (PBM). We also investigated the prediction of the volumetric oxygen mass transfer coefficient based on measurement of bubble size at only one height in combination with PBM to provide reliable estimates of the apparent mass transfer rate with less experimental effort. A mass transfer rate estimate was obtained with deviations from the experimentally determined mass transfer rate of <10 %.

Mechanism and kinetic modeling study on the crystallization of concomitant polymorphs

AbstractConcomitant polymorphs routinely observed in fine chemical industry could impact product purity and consistency; however, both molecular mechanism and process kinetics of concomitant crystallization remain elusive. Herein, we developed a population balance model to understand process kinetics of concomitantly dipolymorphic crystallization using DL‐methionine as a model compound. Kinetic parameters were estimated from induction time measurements and unseeded crystallization experiments. Experimental and simulation results demonstrate that the stable β form has a comparable nucleation rate with α form thanks to their close nucleation barrier leading to the concurrent nucleation. Several solution chemistry techniques were utilized to examine the speciation of solute molecules, together revealing the solutes' self‐association and the formation of micelle‐like aggregates driven by hydrophobic interactions, not hydrogen bonds. These aggregates show dynamic nature against conventional thoughts of classical nucleation kinetics. Finally, the molecular mechanism of concomitant crystallization was uncovered and the implications for polymorph selection and control were discussed.

Numerical investigation of the influence of the heat exchange structure on the gas-liquid dispersion in a large-scale fermenter

A combination of computational fluid dynamics (CFD) and population balance model (PBM) was used to study the gas-liquid mixing conditions in a fermenter. The effects of the heat exchange structure on the flow structure, gas holdup, agitator power, bubble size distribution and interfacial area were investigated by numerical simulations. The simulation results showed that radial gas dispersion and liquid penetration are strongly hindered by the spiral heat exchanger, as there is no gap between the annular tubes at the circumference. An unstable flow structure, gas accumulation around the shaft and gas leakage, as well as large bubbles around the heat exchange tubes are due to the poor gas distribution. By increasing the spacing of heat exchange tube bundles in circumferential, the gas-liquid mixing in the fermenter is gradually improved with the uniform row heat exchanger, the compact row heat exchanger and the bundle type heat exchanger. The bundle heat exchanger acts as a baffle to suppress the large bubbles and effectively reduce the bubbles sizes and promote gas dispersion, which exhibits the maximum interfacial area and interfacial area per unit power. Compared to the fermenter with a spiral heat exchanger, the interfacial area and interfacial area per unit power of the fermenter with a bundle heat exchanger increase by 36.5 % and 9.1 %, respectively.

Process Simulation of Twin-Screw Granulation: A Review.

Twin-screw granulation has emerged as a key process in powder processing industries and in the pharmaceutical sector to produce granules with controlled properties. This comprehensive review provides an overview of the simulation techniques and approaches that have been employed in the study of twin-screw granulation processes. This review discusses the major aspects of the twin-screw granulation process which include the fundamental principles of twin-screw granulation, equipment design, process parameters, and simulation methodologies. It highlights the importance of operating conditions and formulation designs in powder flow dynamics, mixing behaviour, and particle interactions within the twin-screw granulator for enhancing product quality and process efficiency. Simulation techniques such as the population balance model (PBM), computational fluid dynamics (CFD), the discrete element method (DEM), process modelling software (PMS), and other coupled techniques are critically discussed with a focus on simulating twin-screw granulation processes. This paper examines the challenges and limitations associated with each simulation approach and provides insights into future research directions. Overall, this article serves as a valuable resource for researchers who intend to develop their understanding of twin-screw granulation and provides insights into the various techniques and approaches available for simulating the twin-screw granulation process.

Enhancing CO2 capture efficiency: Computational fluid dynamics investigation of gas-liquid vortex reactor configurations for process intensification

This study employs non-thermal Computational Fluid Dynamics (CFD) simulations to explore the efficacy of a gas–liquid vortex reactor (GLVR) for intensifying CO2 capture. The investigation concentrates on the multiphase flow and mass transfer behavior in diverse GLVR experimental units. Employing a multiphase Euler-Euler CFD model integrated with a Population Balance Model (CFD-PBM), a mass transfer model, and reaction kinetics, allows us to accurately simulate the reactive absorption of CO2 into aqueous Monoethanolamine (MEA). Experiments are conducted using a 30 wt% MEA solution for CO2 absorption, serving the purpose of model validation. This comprehensive approach enables a simulation of complex dynamics within the GLVR, emphasizing bubble breakage, coalescence, and reactive mass transfer processes. Examining bubble size distribution, pressure drop, CO2 absorption efficiency, and energy input systematically across various reactor geometries and operational conditions, our findings demonstrate that an optimized GLVR configuration significantly enhances CO2 absorption compared to the original design. Furthermore, the optimized GLVR outperforms state-of-the-art process intensification equipment in terms of CO2 absorption rate per unit reactor volume and energy efficiency.

A constrained machine learning surrogate model to predict the distribution of water-in-oil emulsions in electrostatic fields

Accurately describing the evolution of water droplet size distribution in crude oil is fundamental for evaluating the water separation efficiency in dehydration systems. Enhancing the separation of an aqueous phase dispersed in a dielectric oil phase, which has a significantly lower dielectric constant than the dispersed phase, can be achieved by increasing the water droplet size through the application of an electrostatic field in the pipeline. Mathematical models, while being accurate, are computationally expensive. Herein, we introduced a constrained machine learning (ML) surrogate model developed based on a population balance model. This model serves as a practical alternative, facilitating fast and accurate predictions. The constrained ML model, utilizing an extreme gradient boosting (XGBoost) algorithm tuned with a genetic algorithm (GA), incorporates the key parameters of the electrostatic dehydration process, including droplet diameter, voltage, crude oil properties, temperature, and residence time as input variables, with the output being the number of water droplets per unit volume. Furthermore, we modified the objective function of the XGBoost algorithm by incorporating two penalty terms to ensure the model’s predictions adhere to physical principles. The constrained model demonstrated accuracy on the test set, with a mean squared error of 0.005 and a coefficient of determination of 0.998. The efficiency of the model was validated through comparison with the experimental data and the results of the population balance mathematical model. The analysis shows that the initial droplet diameter and voltage have the highest influence on the model, which aligns with the observed behaviour in the real-world process.

Mass transfer and bubble hydrodynamics in stirred tank with multiple properties fluid via a CFD‐PBM method

AbstractThe performance of a stirred bioreactor was evaluated in this study in terms of the bubble hydrodynamics and the mass transfer efficiency, using a non‐viscous Newtonian fluid of water, a viscous non‐Newtonian fluid of xanthan, and a viscous non‐Newtonian fluids of xanthan with dispersed soybean powder, respectively. The computational fluid dynamics–population balance model (CFD‐PBM) method coupled with the viscosity model and the mass transfer model is established to simulate the gas–liquid mass transfer process and bubble size distribution in the stirred bioreactor. The results demonstrate that the rheological properties of the fluid play an important role in determining the gas holdup, the mass transfer efficiency, and the bubble size distribution. Viscosity of the fluid exhibits a negative impact on gas–liquid mass transfer rate and gas holdup. Moreover, by properly adjusting the operating conditions such as the stirrer speed, it is possible to modulate the gas dispersion and mass transfer rate in the reactor.

Predicting biomass comminution: Physical experiment, population balance model, and deep learning

An extended population balance model (PBM) and a deep learning-based enhanced deep neural operator (DNO+) model are introduced for predicting particle size distribution (PSD) of comminuted biomass through a large knife mill. Experimental tests using corn stalks with varied moisture contents, mill blade speeds, and discharge screen sizes are conducted to support model development. A novel mechanism in the extended PBM allows for including additional input parameters such as moisture content, which is not possible in the original PBM. The DNO+ model can include influencing factors of different data types such as moisture content and discharge screen size, which significantly extends the engineering applicability of the standard DNO model that only admits feed PSD and outcome PSD. Test results show that both models are remarkably accurate in the calibration or training parameter space and can be used as surrogate models to provide effective guidance for biomass preprocessing design.

Mechanistic modeling of twin screw wet granulation for pharmaceutical formulations: Calibration, sensitivity analysis, and model-driven workflow

Wet granulation, a particle size enlargement process, can significantly enhance the critical quality attributes of powders and improve the ability to form tablets in pharmaceutical manufacturing. In this study, a mechanistic-based population balance model is applied to twin screw wet granulation. This model incorporated a recently developed breakage kernel specifically designed for twin screw granulation, along with nucleation, layering, and consolidation. Calibration and validation were performed on Hydrochlorothiazide and Acetaminophen formulations, which exhibit different particle size and wettability characteristics. Utilizing a compartmental experimental dataset, a comprehensive global sensitivity analysis identified critical inputs impacting quality attributes. The study revealed that the nucleation rate process model, effectively represented particle size distributions for both formulations. Adjustments to nucleation and breakage rate parameters, influenced by material properties and screw configuration, improved the model’s accuracy. A model-driven workflow was proposed, offering step-by-step guidelines and facilitating PBM model usage, providing essential details for future active pharmaceutical ingredient (API) formulations.

A population balance model for the kinetics of covalent organic framework synthesis.

This study presents a population balance model for the kinetics of nucleation and growth in covalent organic framework (COF) synthesis. The model incorporates second-order nucleation and first-order growth rates, consistent with proposals in the literature. Despite having non-linear terms, an implicit analytic solution is derived and then converted to explicit solutions for the monomer concentration and size distribution of COF flakes as a function of time. For experimental definitions of the induction time and the initial growth rate based on yield (y) vs time (t) curves, the model predicts power-law relationships: tind=0.409kN-1/3kG-2/3cA0-1 and dy/dtmax=0.965kN1/3kG2/3cA0, respectively. We discuss the implications for the interpretation of Arrhenius plots. We also discuss key discrepancies with experiments, including the predicted attainment of 100% yield instead of 30%-40% as observed and the value of the yield at the inflection point in the yield vs time curve. We suggest extensions to the model, including nucleation and growth kinetics with equilibrium solubility limitations and two-dimensional nucleation for the formation of multilayer COF particles.

Modeling and simulation of bubble condensation using polydisperse approach with bubble collapse model

An accurate modeling of the bubble condensation phenomena in sub-cooled water is developed in this work, using a two-fluid model with one disperse phase and one Interfacial Area Transport Equation (IATE). For this, the aspects of some models are investigated, such as the choice of population balance models, Nusselt closure model and bubble collapse model in IATE. The standard method of moments is formulated using two different bubble size distribution functions among all, namely Dirac and quadratic laws. Therefore, different models for mass and energy interfacial transfers, interfacial forces and source terms of the IATE are developed using both functions. While limited differences are noticeable for the mass and energy transfers obtained by both functions, the results are largely improved using quadratic function for the drag force term and for the IATE source terms. Moreover, the simulations results show that the widely used Ranz-Marshall correlation clearly underestimates the condensation rate. However, this work shows that Chen–Mayinger correlation is relevant to simulate this type of flow, regardless the distribution function. Afterwards, a model is introduced to take into account the effect of bubble collapse by condensation in the IATE. The numerical results obtained using the quadratic function, the collapse model in the IATE and the Chen–Mayinger correlation are comparable to those obtained by the inhomogeneous MUltiple Size Group (iMUSIG) approach, while being less time-consuming.

Experimental and numerical investigation on the pyrolysis of oil shale particles in a bubbling fluidized bed

The efficient utilization of oil shale resource is beneficial to environment and the resource diversity. In this work, a bubbling fluidized bed (BFB) reactor was first used to pyrolyze Huadian oil shale experimentally, mainly focusing on the influence of fluidization flow rates. The yield of shale oil, char and non-condensable gas varies little at different fluidization flow rates, but the components in shale oil are different due to the secondary cracking, such as the fractions and species of aliphatic hydrocarbons, aromatic hydrocarbons and non-hydrocarbons. Then, the dense discrete phase model (DDPM) was used to track the trajectories of oil shale particles with different diameters in a bubbling fluidized bed. Population balance model (PBM) coupled Eulerian model was used to well describe the bubbling status of bed materials to characterize the hydrodynamic behaviors. Furthermore, a segmented kinetic model was compiled by UDF module to illuminate the pyrolysis of oil shale particles. The multi-phase reactive flow modeling clearly indicated the distribution of oil shale particles and gaseous products in BFB reactor, as well as pyrolysis time and the secondary cracking at different fluidization flow rates. In summary, this work provides a reference basis for complete pyrolysis of oil shale particles in a bubbling fluidized bed and generating shale oil with high yield and good quality.

CFD-PBM coupled modeling of the liquid-liquid dispersion characteristics and structure optimization for Kenics static mixer

Kenics static mixers (KSM) are extensively used in industrial mixing–reaction processes by virtue of high mixing efficiency, low power homogenization and easy continuous production. Resolving liquid droplet size and its distribution and thus revealing the dispersion characteristics are of great significance for structural optimization and process intensification in the KSM. In this work, a computational fluid dynamics–population balance model (CFD–PBM) coupled method is employed to systematically investigate the effects of operating conditions and structural parameters of KSM on droplet size and its distribution, to further reveal the liquid–liquid dispersion characteristics. Results indicate that higher Reynolds numbers or higher dispersed phase volume fractions increase energy dissipation, reducing Sauter mean diameter (SMD) of dispersed phase droplets and with a shift in droplet size distribution (DSD) towards smaller size. Smaller aspect ratios, greater blade twist and assembly angles amplify shear rate, leading to smaller droplet size and a narrower DSD in the smaller range. The degree of impact exerted by the aspect ratio is notably greater. Notably, mixing elements with different spin enhance shear and stretching efficiency. Compared to the same spin, SMD becomes 3.7–5.8 times smaller in the smaller size range with a significantly narrower distribution. Taking into account the pressure drop and efficiency in a comprehensive manner, optimized structural parameters for the mixing element encompass an aspect ratio of 1–1.5, a blade twist angle of 180°, an assembly angle of 90°, and interlaced assembly of adjacent elements with different spin. This work provides vital theoretical underpinning and future reference for enhancing KSM performance.