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A CFD model coupled with population balance equations is developed to systematically study the effects of column internals on specific interfacial area generated in a pulsed sieve plate column (PSPC). The model has been validated against experimental data on holdup and Sauter mean drop diameter in laboratory scale PSPCs of different geometries. The column internals comprised of sieve plates with varying hole diameter, fractional open area and interplate spacing. Typically the standard sieve plate cartridge is characterised by sieve plates (3.2 mm diameter sieve holes with an fractional open area of 23%) placed 50 mm apart. A systematic response surface methodology has been adopted to determine the relative importance of these parameters on specific interfacial area. 3 factor 3 level Box Behnken design has been adopted. Specific interfacial area decreases with an increase in sieve hole diameter, interplate spacing and open area. Sieve hole diameter is found to be the most significant parameter affecting specific interfacial area. Important hydrodynamic variables like flow velocities in continuous and dispersed phases and how column internals affect them are also discussed. • 2D CFD-PB model of PSPC to predict specific interfacial area. • Model extensively validated against experimental data for PSPC. • Box-Behnken study to numerically study effect of internals on specific interfacial area. • Hole diameter is the most influence parameter affecting specific interfacial area. [ABSTRACT FROM AUTHOR]

Systematic changes in interfacial forces, turbulence models for the continuous and dispersed phases, lift force coefficient, and bubble size distribution were used to evaluate, in terms of complexity, the performance of nine computational models in the simulation of the time-averaged axial liquid velocity and holdup profiles in a bubble column. The oscillation period of the gas plume was also included in this evaluation. The mesh used for the simulations was chosen after a convergence analysis carried out with four meshes. The finest mesh displayed the highest divergence from the experimental data. Such behavior indicates that the proper mesh for a simulation of a multiphase flow must consider other factors in addition to element size and the quality indicators. The standard, RNG, and realizable κ-ε models were used for the liquid phase, while an algebraic and the standard κ-ε were the models for the dispersed phase. The isotropic RNG model for the liquid phase achieved accurate predictions in the anisotropic region near the gas distributor. The algebraic approach for the gas phase produced data similar to those calculated with the standard model. The simultaneous presence of the lift and virtual mass forces led to reliable predictions, and a lift coefficient of 0.2 displayed better performance than 0.14. This coefficient produced an excessive dampening of the lift force and caused the simulations to lose accuracy. The population balances did not improve the predictions. [Display omitted] • Systematic assessment of computational models for simulations of bubble columns. • Significant role of the lift and virtual mass forces in the quality of simulations. • The lift force is part of the mechanisms producing the oscillation of the gas plume. • Additional factors to the grid convergence for choosing the right mesh. • Algebraic model of the dispersed phase simplifies the simulations. [ABSTRACT FROM AUTHOR]

Many active pharmaceutical ingredients (APIs) are purified by crystallization and most of such crystals are needle crystals whose size distribution is subject to stringent regulations. Predicting the needle size distribution upon crystallization, filtration, drying and milling is essential for smooth scaleup and interruption free operations. In this work we report an enabling technology for biopharmaceutical manufacturing that consists of a Monte Carlo based population balance solver coupled with standard computational fluid dynamics calculations. We have demonstrated the use of this to develop a scaleup tool that can predict needle crystal size distribution for a 200x scale up. The great advantage of this approach is that the scaleup protocol only needs to be calibrated at the laboratory scale where the quantities of the expensive API required are small and manageable and process analytical tools are robust. It can then be deployed to pilot or even plant scale with confidence. The protocol can predict crystal size D90 with less than 10% deviation from the measured values for a 200x scaleup through crystallization, filtration, drying and milling operations. • Novel PAT for size prediction of needle crystals proposed. • Simple mathematical model trained on laboratory scale. • Model takes inputs from wide variety of lab scale PATs and CFD. • Model solved using Monte Carlo based population balance solver. • 200x scaleup demonstrated. [ABSTRACT FROM AUTHOR]

Emulsions preparation using static mixers is investigated in this work. First, it is demonstrated that a fully continuous emulsification process is experimentally feasible through the direct pumping of the two immiscible phases individually into a set of static mixers. This process is advantageous, especially in the case of highly viscous fluids, compared to that employing a pre-emulsification step in stirred tanks which creates inhomogeneity in the sample in terms of the droplet size. Second, a population balance model is employed to predict the droplet size distribution at the process outlet. To do so, a recently modified breakage kernel of Coulaloglou and Tavlarides, accounting for droplet breakage within the inertial and dissipation subranges of isotropic turbulence, is used. In this work, we propose to calculate the second-order longitudinal structure function using semi-empirical formulas. Moreover, the kernel considers the droplet cohesive force due to the viscosity of the dispersed. While this term is sometimes neglected in the literature, we found that this is certainly due to limited variation of the dispersed phase viscosity in the set used for parameter identification. The developed model is validated against a wide range of experimental data and showed very good prediction capabilities. [Display omitted] • Interest of continuous emulsification using static mixers is demonstrated. • The emulsification process is efficient even at high continuous phase viscosity. • A population balance model is proposed and validated for DSD prediction. • The kernel assumes breakage in both inertial and dissipative ranges of turbulence. • Model simplifications are made and allow implementation without CFD simulations. [ABSTRACT FROM AUTHOR]

Two different approaches for a two dimensional population balance modeling strategy are used and compared in this work to describe the flocculation of PCC particles induced by polyelectrolytes, the fixed pivot technique model and the model proposed by Hounslow (1988) and Spicer and Pratsinis (1996b). The median size diameter of the flocs, obtained by LDS, is compared with those obtained by the application of the two presented models. The models efficiency, when compared to the experimental data, is confirmed through a Goodness of Fit (GoF) above 98 % in all cases. Optimized fitting parameters are obtained for the collision efficiency factor, the fragmentation rate and the flocs restructuring parameter and related to the polyelectrolytes characteristics. The flocculation kinetics and scattering exponent data (related to the flocs structure) evolution with time, obtained from modelling, are discussed in relation with the flocculation process characteristics, including polyelectrolytes properties, and the different conditions considered. • Monitoring of the flocs size (median diameter) with time obtained by LDS. • Two property dimension population balance models to simulate flocculation kinetics. • Assessment of the fixed pivot technique and Hounslow models in the simulation. • Collision efficiency, fragmentation rate and flocs restructuring parameters optimized. • The models efficiency was confirmed through a Goodness of fit above 98 %. [ABSTRACT FROM AUTHOR]

A model for a continuous gravitational separation process of a liquid-liquid dispersion is presented, which is formulated as a population balance equation (PBE) that describes the coalescence and the hindered polydisperse sedimentation of droplets. This is important because this separation process is critical in the operation of many industries as discussed later. This study is based on previous findings by Garcia and Betancourt (2019) where a formulation for a batch process was presented. This analysis shows the improvements and differences between the model presented in this study and the formulation by Garcia and Betancourt (2019). The modeling framework comes from the Kynch's theory that has been employed to model solid particle sedimentation processes; however, in this case the principles are applied to an organic phase-aqueous phase liquid mixture. For the numerical solution of the model, the PBE is discretized in terms of droplet volume. Since the population balance model is subject to mass loss due to droplet coalescence, we propose a formula to compute the source term of the largest droplet class. Model validation used experimental data from literature for a continuous process. Numerical results explored the variation of parameters such as feed flowrate, feed oil volume fraction and the standard deviation of the initial and feed droplet volumes. [Display omitted] • A model that combines coalescence with polydisperse sedimentation is presented. • A formula is proposed in order to conserve the mass of the dispersed phase. • The model was validated through experimental data of an oil-water system. • Effects of standard deviation of initial and feed droplet volumes were investigated. [ABSTRACT FROM AUTHOR]

In this article, we present a fully Eulerian framework for modelling the envelope flame around a burning, spatially resolved aluminum particle, including detailed descriptions of the particle's reactive surface, the gas phase composition and the polysized oxide droplets condensing from gaseous precursors. The size distribution of this oxide smoke influences deposition and emission mechanisms and plays a decisive role in the design of oxide particle extraction techniques, for example, as part of the exhaust fume treatment for metal dust flames. In order to describe the reacting gas-droplet dispersion, we combine the population balance equation governing the size distribution of oxide smoke fines with the balance equations for gas phase species and the dispersion's mass, momentum and enthalpy. In this regard, the droplets' bulk velocity and temperature are assumed to equilibrate instantaneously with those of the carrier gas. Additionally, a detailed kinetic framework governing the rates at which gas phase and surface reactions as well as droplet nucleation, growth/dissociation and coagulation occur is augmented by the transport parameters governing species differential diffusion, droplet diffusion and thermophoresis. A major novelty of our physical model lies with the prediction of the oxide smoke size distribution at every location in the flow domain, the accommodation of size-sensitive kinetic and transport processes as well as the prediction of possible NO x pollutants. Considering the steady combustion of a single aluminum particle in an O 2 /Ar mixture, the droplet formation and interaction kinetics are adjusted in such a way that fair agreement with existing experimental measurements is achieved. This calibration shows not only that the interplay of nucleation and dissociation affects the envelope flame through the chemical equilibrium between aluminum suboxides and oxide smoke, but also that the shape of the smoke halo is mainly determined by the droplets' sizes and their diffusional mobility. Concomitantly, we are led to the conclusion that coagulation does not occur in the flame zone, possibly due to electrostatic droplet repulsion induced by thermionic emissions and ionization reactions. Following the consolidation of the droplet formation kinetics, the combustion of an aluminum particle in air is investigated as validation case. For the consolidated droplet formation kinetics, the predicted radial AlO and oxide smoke profiles compare qualitatively very well with the available experimental data, while deviations in the temperature profile remain. [ABSTRACT FROM AUTHOR]

This contribution develops the model predictive control for an unstable chemostat reactor. Initially, we analyze the system's model — a nonlinear first-order hyperbolic partial integro-differential equation (PIDE) — and carry the model linearization around an unstable operating condition. Employing the structure-preserving Cayley–Tustin transformation, we obtain a discrete-time model representation of the continuous model. Subsequently, we solve the operator Ricatti equations in the discrete-time setting to derive a full state feedback controller that stabilizes the closed-loop and design a Luenberger observer for state reconstruction given the system output measures. Finally, we formulate a dual-mode MPC ensuring constraint satisfaction and optimality, integrating the gain-based unconstrained full-state feedback optimal control obtained from the Ricatti equation. This dual-mode strategy describes an optimization problem where the predictive controller acts only if constraints become active within the control horizon. Simulation studies validate the controller performance, where the MPC only takes action if the constraints are predicted to be active within the control horizon while also guaranteeing closed-loop stabilization under only output feedback. This type of controller can be easily implemented with other control strategies and significantly decreases the computational costs of solving the optimal control problems when compared to other MPC approaches. • Modeling and Optimal Constrained Control of a Chemostat Reactor as a Distributed Parameter System. • Analysis of the Stability and possibility of multiple Steady-states of the open-loop system. • The observer design for the system allows the use of only output feedback. • The dual-mode controller used guarantees closed-loop stability and constraints satisfaction. [ABSTRACT FROM AUTHOR]

• Population balance predictions are analyzed in gas–liquid Stirred Tanks (ST). • Different flow regimes are investigated starting from validated flow fields. • Current limitations and modeling challenges of the PBM are identified and discussed. • Common coalescence and breakup kernels fail in predicting DSD in the impeller zone. • A workflow for simulating gas-–liquid ST in different flow regimes is proposed. In this work we analyze the challenges of the adoption of a population balance approach in aerated stirred tanks operating at high gas volume fractions, in different flow regimes and when segregation of the phases is present, i.e. due to the formation of aerated cavities behind flat blades. A decoupled solution of the equations governing the fluid flow and of the population balance equation (PBE) is adopted to obtain preliminary information on the bubble size distribution and to identify the specific issues that need to be addressed to extend the applicability of the population balance model to different regimes. A workflow is proposed for the simulation of aerated stirred tanks with geometries for which established correlations for the estimation of bubble size are not available and a simplified approach to determine whether a coupled CFD PBE solution may be required is presented. [ABSTRACT FROM AUTHOR]

[Display omitted] • A discretization method to solve the steady state population balance is presented. • Nucleation, size independent growth and loose agglomeration are included. • A fixed point acceleration algorithm is applied with an optimal initial vector. • The crossed-secant acceleration method ensures the algorithm convergence. • The algorithm is suitable for coupling with multicompartment model and CFD. The present work focuses on the development of a performing numerical methodology to solve the steady state Population Balance Equation (PBE) including nucleation, independent size growth and loose agglomeration as crystallization mechanisms. The methodology is based on the solution of two PBEs: one for the isolated crystallites and one describing the loose agglomerates formation. Both are solved by a discretization method and only the last one is reformulated as a fixed point problem. The algorithm solving PBE for agglomeration includes the crossed-secant algorithm as a fixed point acceleration method. The numerical PBE solution method is first validated by comparison to analytical solutions and then applied to the neodymium oxalate precipitation in order to compare to experimental results in a wide range of operating conditions. The methodology is tested under highly restrictive numerical conditions: narrow tolerances, a large amount of points in the discretization scheme and a zero vector as initial condition. The crossed-secant method demonstrates to improve the robustness of the standard fixed point iterations by ensuring the convergence of the agglomerates PBE when penalizing conditions are applied and by reducing the number of iterations otherwise. In all cases, the developed methodology predicts accurately the crystal size distribution under the experimental uncertainty in a reasonable computation time and number of iterations. [ABSTRACT FROM AUTHOR]

• Flocculation can be studied applying logistic growth models. • These models reduce to well-known linear interpolations at onset of flocculation. • Studies of flocculation kinetics can be performed over long periods of time. • The dependence on relevant parameters is discussed. The model based on logistic growth theory, that was introduced in Chassagne, Claire, and Zeinab Safar. "Modelling flocculation: Towards an integration in large-scale sediment transport models." Marine Geology 430 (2020): 106361 , can be used to model the time evolution of either the particle (floc) size or the concentration of particles of a given size. In the present article, we show how this model can easily be linked to the many studies performed over the years about the flocculation kinetics at the onset of flocculation experiments. Flocculation experiments done on kaolinite suspensions destabilized by addition of salt are used as examples. Both perikinetic and orthokinetic flocculation are considered. By fitting the experimental data over the whole experiment period, it is shown that the floc size L (t) time evolution follows the relation dL / dt = L / t b − L 2/(L eq t b) where t b is a characteristic time and L eq a characteristic size, which are obtained from the fits. [ABSTRACT FROM AUTHOR]

• Population balance equation is used to model the population dynamics of microalgae. • Number of daughter cells, size-specific growth and division rates are evaluated. • Prediction of number concentration and size distribution in the linear phase. • Calculation of the life-cycle time of microalgae. • Description of the methodology combining experiments and the numerical method. Microalgae are versatile microorganisms with applications in food, energy and fine chemicals, among others. Modelling the dynamics of microalgae inside a photobioreactor is a convenient and inexpensive way to determine the concentration of cells over time. Numerous models have been developed in the literature, but only a few are able to give relevant biological information. Such information can then be used to further improve the production process. The objective of this work was to develop a model for the determination of microalgal dynamics inside a photobioreactor as a function of the environmental conditions, to retrieve the size-specific growth and division rates as well as the number of daughter cells. The results demonstrate how to evaluate the time needed for microalgae to complete a full life-cycle. Inexpensive laboratory-based procedures and mathematical modelling are combined for the determination of relevant biological parameters. [ABSTRACT FROM AUTHOR]

Population balance equations (PBEs) play a significant role in describing the dynamics of star formation in astrophysics, cheese manufacturing in dairy sciences, depolymerization in chemical engineering, and granule preparation in the pharmaceutical industry. However, obtaining analytical solutions for these equations remains a formidable challenge due to complex integral terms. In this work, we propose a generalized approach based on He's variational iteration method which enables the efficient estimation of series solutions for breakage and aggregation PBEs for shorter time domains. To extend the solutions for longer time periods and larger size domains, the series solutions obtained by the coupled method are further expanded using powerful Padé approximants. Moreover, a thorough convergence analysis is conducted in the Banach space. The testing of the new approach is done by considering several gelling and non gelling kernels, and the results are validated against the existing analytical solutions. [Display omitted] • Coupled approach is developed to tackle the linear breakage and nonlinear aggregation equation. • Coupled approach comprises of He's variational iteration method and Pade approximant. • The convergence analysis of the new approach is conducted using the contractive mapping in the Banach space. • We effectively overcome the constraints associated with existing methods for achieving solutions within a reduced timeframe. • Lesser relative errors and higher efficiency are advantages of the proposed approach over the existing methods. [ABSTRACT FROM AUTHOR]

Wet granulation is a multiphase process utilised to produce aggregate particles with defined properties from fine powders. Simulating this process on the microscale is challenging because of the substantial number of particles involved, which differ widely in both size and material properties. Macroscale methods, which track only the particle bulk properties, are efficient but do not resolve disperse particle properties such as the particle size distribution, which is key information for downstream processing. These deficiencies are addressed by mesoscale methods, like population balance models, which track distributed properties such as the particle size by adding them as internal variables to the macroscale model. However, most mesoscale methods are either inaccurate (method of moments when closed by cutting off moments) or computationally expensive (Monte Carlo, class methods). Recently a new closure for the method of moments, the quadrature method of moments, was introduced to allow accurate moment tracking of a particle size distribution with low computational effort. The drawbacks associated with this method, such as potential instabilities, can be effectively mitigated through non-dimensionalisation. In this study we show our insights gained by non-dimensionalising the quadrature method of moments equations for wet granulation processes, which model the particle size distribution evolution via growth, aggregation and breakage kernels. Relevant theoretical and numerical issues as well as limitations are discussed. Using constant kernels, the non-dimensionalised model is verified and validated thereafter on certain special cases. The effect of non-constant kernels on the non-dimensionalisation is discussed. Furthermore, we show that the non-dimensionalised model fails to accurately predict the moments of an experimental distribution using a volume-based population balance model, whereas a length-based population balance model can successfully predict the moments by setting non-constant kernels to fit the Sauter mean diameter. [Display omitted] • Non-dimensionalise a quadrature method of moments for wet granulation processes. • Discussion of relevant numerical and theoretical issues/limitations of the model. • Verification for cases with constant growth, aggregation and breakage kernels. • Validation for cases with analytical solutions to the population balance equations. • Demonstration of the model on a twin-screw wet granulation experiment. [ABSTRACT FROM AUTHOR]

• Experimental study of biomass ash agglomeration has been carried out in a fluidized bed. • Effect of temperature rise on the fluidization number has been taken into account. • Defluidization temperature was monitored by pressure measurements. • Population balance equation was used to predict the particle size distribution at defluidization. • First-order growth rate for agglomerates was assumed and its parameters were calculated by fitting to experimental results. Ash agglomeration as a potential problem in fluidized bed gasifiers, leading to defluidization, was investigated in this study in a lab-scale fluidized bed. The experiments were carried out in batch mode in the temperature range of 600–950 °C and atmospheric pressure. The defluidization temperature was determined from the total pressure drop across the bed. The effect of superficial gas velocity on the defluidization temperature was also investigated. With increasing the gas velocity, the defluidization temperature increases up to about 960 °C, beyond which it remained almost constant. The particle size distribution at the end of the tests (defluidization state) was estimated from the population balance equation. Assuming a first order growth rate of agglomerates, the parameters of the rate equation were determined by fitting the population balance equation to the size of the agglomerates at the moment of defluidization. The results showed that as the gas velocity increases, the frequency factor of the growth rate equation initially decreases up to the fluidization number 3, after which it remains constant. [ABSTRACT FROM AUTHOR]

Particulate attrition is of interest for novel carbon-capture processes such as chemical looping combustion because the makeup cost of oxygen carrier is a significant portion of operating cost. Thus, models to study and predict attrition of various oxygen carriers in fluidized bed systems are being developed. One of the regions of concern in fluidized bed systems is the high-velocity jet region near gas distributors in a fluid bed. This work studies the attrition of hematite particles using a modified ASTM apparatus to measure the particle size distribution throughout the experiment. Bed weight and gas velocity were varied. Hematite particles above the corresponding threshold value had a severe variation of particle size distributions which decreased with time. Weight fractions of the sieves over time were fit to a linear, time-variant population balance model to offer insight into particle attrition. The first-order rate constant was modified as a decaying exponential. Unlabelled Image • Attrition of hematite particles studied using a modified ASTM apparatus. • Particle size distribution recorded throughout the experiment. • Data fit to a linear, time-variant population balance model for attrition insight. • First-order PBM rate constant was modified to a decaying exponential term. [ABSTRACT FROM AUTHOR]

The major focus of this study, the second part of a series, is to simulate the net granulation process of TiO 2 powder using 1-D discretized population balance modelling (PBM) and Size-Independent Kernel (SIK) coalescence model. In this, sequential and non-sequential modelling procedures have been used. Tested one-dimensional population balance model and coalescence-only, size-independent kernel provide a good prediction of real TiO 2 particle size distributions (PSDs). The coalescence rate is quantified for both modelling procedures and formulation system used. Reduced-order population balance equation (PBE) and quite simple, fully empirical coalescence-only kernel successfully describe and predict real TiO 2 PSDs. This paper reveals the meso-scale study of the wet granulation process for the not previously granulated powder system. Unlabelled Image • Granulation process of TiO 2 powder has been simulated using 1-D PBM and SIK rate kernel. • Sequential and non-sequential modelling procedures were tested to efficiently track PSDs. • Coalescence-only models successfully describe and predict real TiO 2 PSDs. • Coalescence rates were quantified for the not previously granulated powder system. [ABSTRACT FROM AUTHOR]

Laminar diffusion flames present an elementary configuration for investigating soot formation and validating kinetic models before these are transferred to turbulent combustors. In the present article, we present a joint experimental and modelling investigation of soot formation in a laminar co-flow burner. The diffusion flames are analysed with the aid of laser diagnostic techniques, including elastic light scattering (ELS), planar laser-induced fluorescence of OH (OH-PLIF) and line-of-sight attenuation (LOSA), to measure the spatial distribution of soot, gas phase species and the line-of-sight integrated soot volume fraction (ISVF), respectively. The experimental dataset is supplemented by location-specific TEM images of thermophoretically sampled soot particles. The simulation of the sooting flames is carried out with a recently developed discretisation method for the population balance equation (Liu and Rigopoulos, 2019, Combust. Flame 205, 506-521) that accomplishes an accurate prediction of the particle size distribution, coupled with an in-house CFD code. By minimising numerical errors, we ensure that the discrepancies on the modelling side are mainly due to kinetics and are able to carry out an investigation of alternative models. We include a complete set of soot kinetics for PAH-based nucleation and condensation, HACA-based surface growth and oxidation as well as size-dependent aggregation, and consider three different gas phase reaction mechanisms (ABF, BBP and KM2). Based on predictions of the gas phase composition and particle size distribution of soot, modelled counterparts of the laser diagnostic signals are computed and compared with the experimental measurements. The approach of directly predicting signals circumvents the difficulties of explicitly representing the OH concentration in terms of the measured OH-PLIF data and avoids using 'hybrid' modelled and measured values to approximate the OH concentration. Moreover, the LOSA signal is directly converted to the line-of-sight ISVF instead of a measure of local soot volume fraction to avoid tomographic inversion errors. Lastly, the predicted ELS signal is computed in terms of the particle size distribution resolved by the population balance model, thus circumventing the approximation of an integral soot property using a presumed size distribution. While we cannot obtain quantitative agreement between experiments and simulations, the accuracy of the numerical approach and the direct prediction of experimental signals allow us to conduct sensitivity analyses of key empirical parameters and investigate the importance of the PAH chemistry and its influence on the competition between nucleation, condensation and surface growth. [ABSTRACT FROM AUTHOR]

• Novel closure schemes for moment-transformed population balance are introduced. • The full distribution can be reconstructed in a stepwise manner. • The introduced method provides an easy access to pointwise values of PDF. • The method enables modelling of dissolution and Ostwald ripening using MOM. • First stages of batch silica precipitation are simulated. Ostwald ripening is a main phenomenon responsible for growth of particles manufactured using colloidal methods. This paper considers mathematical description of the process using population balance approach solved by method of moments. However, in the case of particles' dissolution which is a part of the process a closure problem is generated. A novel closure method for the moment – transformed population balance equations based on Gram – Charlier expansions is introduced. Due to its simplicity it allows to reconstruct the full particle size distribution in a stepwise manner giving an access to the pointwise values of the density function required to close the model. The introduced numerical scheme is further used to simulate several carefully chosen test cases including dissolution of solid particles and ageing of particles' population. The mathematical model presented in the paper is then applied to interpret experimental results for batch precipitation of silica. [ABSTRACT FROM AUTHOR]

• CFD model of dense oil–water emulsion separation verified with detailed pilot data. • Eulerian multiphase model with population balance predicts droplet size and phase distributions. • Critical effect of emulsion viscosity incorporated to predict the dense emulsion layer thickness. • Emulsion viscosity model dependent on water fraction and droplet size. The multi-fluid Eulerian multiphase model coupled with the inhomogeneous population balance model is applied to predict the gas, oil, and water separation behavior and emulsion destabilization in a high-pressure horizontal three-phase separator. The population balance model is applied to predict the evolving droplet size distribution due to droplet coalescence in the crude oil emulsion. The effect of water fraction on the emulsion viscosity was incorporated to predict the existence of the dense packed layer or zone (DPZ) – a high water-fraction emulsion layer formed at the interface between the oil and water phases. The novel emulsion viscosity model incorporates a dependence on the local droplet size. Parameter estimation for the droplet coalescence kernel in the population balance equation is determined from experimental measurements of emulsion separation kinetics. A proportional integral derivative (PID) controller logic is used to maintain the gas/oil and oil/water interface levels by automatically adjusting the oil and water outlet pressure, respectively. The pilot separator experimental measurements include inlet and outlet flowrates and density, and electrical capacitance measurement of the vertical water concentration profile. The CFD model determines the flow patterns, phase distributions and the dispersed water droplet size distribution in the emulsion in the separator. The model results are in good agreement with the data for the separation efficiency from the water flow rate in and out of the separator, the phase distributions and the DPZ at different flow rates and water fractions. This study demonstrates the importance in multiphase separator design and troubleshooting of including the influence of water fraction on emulsion viscosity and the droplet size distribution with droplet coalescence phenomena. The methodology presented has significant advantages over previously published methods and improves on the value and potential impact toward research and industry practice. [ABSTRACT FROM AUTHOR]

• Detailed experimental study of the breakage of preformed wet granules. • Well-defined flow conditions in which no agglomeration occurs (breakage-only). • Results for effects of different system parameters on the breakage of granules. • Development of a population balance model to describe breakage process. In high-shear wet granulation, breakage and attrition play an important role but are the least understood among all rate processes. To understand these phenomena, experiments were performed in a breakage-only granulator using pre-made granules. Non-granulating wet sand was used as the shearing medium in the granulator. A size analysis of the granules was done before and after the experiment for each trial in order to get a rough estimate of the size range of the granules after the breakage experiment. The number of granules broken and the granules mass change (before and after the experiment) were recorded after each experiment, in order to measure the extent of granule breakage in a granulator. Experimental results showed that formulation properties and operating parameters significantly influence the breakage behaviour. Population balance simulations were performed for breakage-only granulation and the simulation results were found to be in reasonable agreement with the experimental results. [ABSTRACT FROM AUTHOR]

In the present work, a model was proposed to simulate the growth in an industrial multi-chamber fluidized bed granulator under steady-state condition. To simulate the growth, it was assumed the coating is the only growth mechanism. Mass balance, as well as particle number balance (PNB), was considered in the simulation. Despite previous works, it was not assumed that particles in each chamber are at the same age. Consequently, the growth was simulated along the granulator length, from its beginning to the end. In this model, most of the influencing parameters are collected in one parameter called C 1. Using finite difference method, equations were solved and results were compared with samples from an industrial granulator and very good agreements were observed. Sensitivity of the product normalized mass density function to the parameter C 1 was also studied. It was found that for a specific inlet seeds diameter distribution, d 50 of the product is not a linear function of C 1. Unlabelled Image • PNB approach was used to model the growth of urea particles in multi-chamber industrial granulators. • Results were compared with data from a real industrial granulator. • The change of diameter along that granulator length was simulated for different particles. • The effect of parameter C 1 on the product normalized mass density function was studied. • For a fixed seeds diameter distribution, d S0 d 50 is not a linear function of n. [ABSTRACT FROM AUTHOR]

• A carbon capture strategy is integrated with crystallization process of CaCO 3. • Energy for CO 2 decomposition is avoided by production of value-added chemical. • Using aqueous solutions of sodium hydroxide for carbon capture and valorization. • Demonstration of calcium carbonate crystallization as a by-product of CO 2 capture. • Population balance modeling for evaluating calcium carbonate growth rate constant. A laboratory scale integration of carbon capture and utilization (CCU) technique operating based on highly alkaline aqueous hydroxide sorbents with a semibatch reactive crystallization process of calcium carbonate is presented. The study was conducted with a focus on investigating the kinetics of the crystallization process for CCU applications, where the production of micron-sized calcium carbonate particles occurs by direct addition of CO 2 -loaded solution to a crystallizer containing calcium chloride. Within this system, the need for high temperature heat consumption above 800 °C (109.4 kJ/mol CO 2) for CO 2 decomposition from Na 2 CO 3 is avoided by production of value-added chemical from carbonate reach solution. The current set-up enables potential pre-treatment of carbon emission sources due to separate absorption and liquid-liquid crystallization processes. This leads to production of high-purity solid particles in comparison to conventional industrial practices. Experiments were carried out at several operating conditions at ambient temperature and pressure and effect of the absorbent solution concentration on CO 2 capture efficiency and subsequently the precipitation yield was analyzed. Based on coupling the standard method of moments approach of the population balance formulation with power-law kinetic expressions, a correlation was extracted between the growth rate constant and feed flow rate for the adopted process. Moreover, characterization of the precipitated sample indicates the vaterite polymorphs of calcium carbonate as the dominant crystals in the pH range of 7.9–10.4. These types of demonstrations could contribute to circular economy processes using waste materials and can be considered as a potential approach toward an efficient carbon capture and valorization technique. [ABSTRACT FROM AUTHOR]

• A model that merges the coalescence with the hindered sedimentation is presented. • Two terms are incorporated into the model in order to conserve the mass of the dispersed phase. • The model was calibrated and validated through an experiment with an oil-water system. • Effects of standard deviation of the initial droplet volumes and coalescence frequency are investigated. Population balance equations (PBE) provide a suitable a framework for dealing with drop breakage and coalescence. In addition, modeling of the solid particle sedimentation process based on Kynch's theory has been successfully used and validated in mineral processing and wastewater treatment. In this work, we present a model that merges the coalescence process with hindered polydisperse sedimentation. The PBE model is projected onto a partial differential equation (PDE) system by discretizing the droplet volume. Because there is loss of mass in the system when the daughter droplets are greater than the larger species considered for the numerical solution, two terms that produce the conservation of the mass or total volume of the dispersed phase are incorporated into the PDE system. The resulting PDE system is split into two systems: homogeneous PDEs and ordinary differential equations (ODEs). The homogeneous PDEs and the ODEs are discretized using a first-order central differencing scheme and the second-order, two-stage Runge-Kutta method, respectively. The model predicts the vertical variation of the composition of the dispersed phase layer that forms at the top or bottom of the gravity settler. The proposed model was calibrated and validated through an experiment with an oil and water system. In particular, simulations illustrate the effects of: the continuity of the dispersion (oil-in-water and water-in-oil) and the standard deviation of the initial droplet volumes on phase separation quality, as well as the influence of the coalescence frequency on the average droplet volume. [ABSTRACT FROM AUTHOR]

• Formation of dilute water in oil emulsion in a horizontal pipe was simulated. • Droplet shear dispersion phenomenon was taken into account. • An effect of flow velocity on droplet dispersion and stratification was considered. • An effect of a size fraction number on computational results was illustrated. Dispersion of two immiscible fluids in a horizontal turbulent pipe flow is simulated with the developer's version of the STAR-CCM+® CFD code of Siemens PLM. Formation of a stable (no coalescence) dilute water in oil emulsion is analyzed. The Reynolds stress turbulence model in combination with the Daly & Harlow transport model are employed for modeling a three-dimensional flow. The CFD - population balance method, available within the code, is a basis for computations. A model of the droplet shear dispersion phenomenon that we incorporated into the code allows to properly account for formation of an enhanced droplet concentration area in the pipe bottom vicinity. In the computations conducted, each droplet size fraction is characterized by its own velocity, whereas majority of 3-D population balance computations known from open literature have been done by a simplified approach assuming a single average velocity. The major peculiarities of droplet dispersion in a horizontal pipe flow are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Natural talc is a very interesting filler for plastic and rubber materials and paints because it enables polymer matrix properties to be improved. Nano-sized particles are required to get a good dispersion of the solid in the polymer matrix. However, at the moment the main drawback in the use of natural talc is that, with a conventional milling process, nanometric particle sizes are difficult and expensive to obtain. A process of magnesium silicate synthesis made by ultrasound-assisted precipitation has been developed in two steps. The first step consists of the synthesis of amorphous magnesium silicate by precipitation from sodium metasilicate and acid magnesium chloride. In the second step, the amorphous solid is transformed in a crystalline solid under pressure. In this study an ultrasound-assisted synthesis of amorphous magnesium silicate is proposed. Some process parameters (ultrasound power, initial reactants molalities and reactants flow rate) have effects on the particle-size distribution. The population balance is modelled and solved by moments method in the steady state to identify nucleation and growth rates and agglomeration kernel as a function of operating conditions and to understand how these conditions affect the particle-size distribution. Unlabelled Image • Ultrasound-assisted precipitation permit to obtain smaller amorphous particles. • The nucleation rate is increasing and growth rate is decreasing by ultrasounds. • Ultrasounds limit agglomeration, because of the decreasing of agglomeration kernel. • Effects are intensificated by increasing ultrasound power. [ABSTRACT FROM AUTHOR]

• Experimentally-validated model to predict size and distribution of nanoparticles. • Model couples time-resolved fluid dynamic simulations with population balance. • Increasing mixing rate, increases overall nanoparticle synthesis but not local rate. • Combination of fast mixing and high rate leads to a burst of nuclei formation. Achieving size control during nanomaterial synthesis is critical for their deployment in numerous applications due to their strong size-properties relationships. However, it remains a challenge in the field due to multi-step mechanisms in nanoparticle synthesis. In this paper, we present an experimentally-validated model able to predict the particle size and distribution of silver nanoparticles synthesised in flow reactors by coupling time-resolved fluid dynamic simulations with population balance. The model reveals fundamental correlations between reactor design and the corresponding size of the nanoparticles. It shows that increasing mixing rate, increases the overall rate of nanoparticle formation, where nanoparticle synthesis takes place homogeneously in the whole of the reaction volume. In contrast, when there is a distinguishable interface between reactant streams (i.e. low mixing rate), nucleation and growth take place in a small fraction of the reaction volume in the interface between the inlet streams. The combination of fast mixing and high average nucleation rate can lead to a burst of nuclei formation translated in small size and narrow size distribution. This work presents a new approach for the use of computational-guided design of reactors for the synthesis of nanomaterials to move the field away from current trial-and-error strategies. [ABSTRACT FROM AUTHOR]

The mechanism of dehydration of a water-in-oil emulsion is complex and involves the growth of an emulsion of polydispersed water droplets in oil, under an electric field, and their gravitational settling and phase separation. In this work, a simple configuration is chosen to understand the mechanism of this complex process. A population balance-based model is developed to predict the performance of an electrocoalescer. The model explicitly accounts for the coalescence of water drops by the action of dipolar forces and gravity settling. A distributed parameter plug flow and a tank-in-series model are developed to account for the gravity settling. The dynamics of electrocoalescence are captured by analyzing the drop size distribution and mean diameter along with the height of the coalescer and time. A simplified model - considering that monodispersity prevails throughout the process is also developed, and the separation curves are compared with the polydispersed system. The effect of operational parameters, such as the strength of the electric field, water cut, viscosity of oil, the height of the coalescer, and polydispersity in the initial drop size distribution, is investigated. The main insight from this work is that there are essentially three stages of electrocoalescence, a slow initial incubation stage where droplets grow to around 1 mm in size without much settling, followed by a rapid settling of these droplets to set in the fast coalescence stage where most water is removed to result in a lean emulsion, followed by a slow simultaneous electrocoalescence and separation stage of this lean emulsion to result in a final separation. A scaled masterplot for the performance of an electrocoalescer, encompassing the effect of the process parameters is obtained. The model also suggests that electrocoalescence essentially leads to a polydisperse system and a population balance approach is critical to its analysis. • Dehydration efficiency of W/O emulsion is predicted using population balance model. • Model accounts for droplet coalescence due to dipolar forces & gravity settling. • Effect of electric field, water cut, oil viscosity & other factors is investigated. • Scaled masterplot for performance, encompassing process parameters, is constructed. [ABSTRACT FROM AUTHOR]

The computational cost of the discrete element method (DEM)-population balance model (PBM) coupled framework is predominantly attributed to DEM simulations. To overcome this challenge, coarse-grained (CG) particles have been introduced in the DEM-PBM coupled framework. In this study, we proposed a new CG-enabled DEM-PBM coupled framework that builds upon the previous work of Das et al. (Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 478 (2261) (2022) 20220076). By incorporating the CG technique, the particle number density is reduced, resulting in fewer collisions compared to the resolved system. To address this issue, a scaling law has been developed to derive the collision frequency of the resolved system from the CG system. The verification of the new scaling law has been demonstrated through various simulation studies. Furthermore, the entire DEM-PBM coupled framework has been modified using the proposed methodology. The efficiency of the CG–DEM–PBM coupled simulation method has been successfully demonstrated through simulations of rotating drum and continuous mixing technology (CMT). Compared to the resolved simulation approach, the newly proposed CG-enabled DEM-PBM coupled framework maintains accuracy in terms of particle size distribution and other essential findings while significantly reducing simulation time. [ABSTRACT FROM AUTHOR]

• A unified Monte Carlo approach for simultaneous coagulation and breakage problems. • Global reconstruction of stochastic weighted particles. • High-precision predictions of PSDs and moments with good computational efficiency. • Low statistical noise for long-time simulations. This paper proposes a unified Monte Carlo approach to model simultaneous particle coagulation and breakage processes. Based on the reconstruction of stochastic weighted particles, the present approach treats coagulation and breakage in a unified sense and resamples the weighted particle population globally. The prediction results are validated through several benchmark coagulation-breakage cases with known analytical solutions, and the performances are compared with those of conventional combination approaches. It is confirmed that, in terms of resolving particle size distribution and moments, the proposed reconstruction approach generally yields higher computational accuracy and efficiency, with especially reduced statistical noise for long-time simulations. The simple implementation, the unified treatment for different dynamic processes, and the well-balanced accuracy and efficiency make this new approach a promising candidate for solving general population balance problems of particles. [ABSTRACT FROM AUTHOR]

• hp-adaptive quadrature increases the accuracy of numerical integration. • Better error control for the population balance source terms. • Irregular integrands are efficiently dealt with. • Combination of piece wise Gaussian quadrature with weighted residual methods. The solution of the population balance equation requires the integration of several source terms. In the numerical weighted residuals methods, Gaussian quadrature is a natural candidate for numerical integration. Previous works using the weighted residuals methods for solving the population balance equation did use a fixed grid of quadrature points. This work shows that the use of adaptive quadrature points for the numerical integration can lead to more efficient and accurate solutions of the equation. For cases where the integrand shows a high degree of irregularity, the hp-optimization method distributes the quadrature points such that the method becomes more efficient than with a fixed grid. An additional improvement is that the amount of quadrature points changes to fit the need for each integral present, rather than having one set of quadrature points for all cases. A simple population balance model demonstrates the use of the adaptive quadrature approach. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Existing model extended to include novel layering mechanism. • Novel compartment residence time model developed. • New breakage model reflects screw element geometry effects. • Model captures the liquid distributing/growth promoting effects of kneading elements. Abstract In this work, a framework for modelling twin-screw granulation processes with variable screw configurations using a high-dimensional stochastic population balance method is presented. A modular compartmental approach is presented and a method for estimating residence times for model compartments based on screw element geometry is introduced. The model includes particle mechanisms for nucleation, primary particle layering, coalescence, breakage, and consolidation. A new twin-screw breakage model is introduced, which takes into account the differing breakage dynamics between two types of screw element. Additionally, a new sub-model for the layering of primary particles onto larger agglomerates is presented. The resulting model is used to simulate a twin-screw system with a number of different screw configurations and the predictive power of the model is assessed through comparison with an existing experimental data set in the literature. For most of the screw configurations simulated, the model predicts the product particle size distribution at large particle sizes with a reasonable degree of accuracy. However, the model has a tendency to over-predict the amount of fines in the final product. Nevertheless, the model qualitatively captures the reduction in fines associated with an increase in the number of kneading elements, as observed experimentally. Based on model results, a number of key areas for future model development are identified and discussed. [ABSTRACT FROM AUTHOR]

Shear aggregation of micrometric particles is an important phenomenon occurring in agitated chemical reactors. The resulting clusters of particles are porous and more or less elongated. It has previously been proved that the equivalent ellipsoid computed from the inertia tensor describes well the colliding clusters [F.Gruy and P. Nortier « Statistics about collisions of ellipsoids under shear flow » Colloids and Surfaces A: Physicochemical and Engineering Aspects, 558(2018)250-262]. The aim of the current paper is to consider chosen kinetic and morphological aspects of the complete aggregation dynamics. Monte Carlo simulations are used as a means of investigation. Four representations of the collision event have been compared: the classical collision of oriented clusters of sphere (A), the collision of oriented equivalent ellipsoids (B), the orientation-averaged collision of equivalent ellipsoids (C), the use of approximate expressions for C (D). For each case, the probability density functions of various morphological parameters of the cluster population are calculated as a function of the time. The comparison of the generated data shows that the approximate expressions for the aggregation kernel and the ellipsoid parameters allow for fast and representative simulations of the shear aggregation of suspensions. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present a finite volume method for the discretisation of the population balance equation with coagulation, growth and nucleation that combines: (a) accurate prediction of the distribution with a small number of sections, (b) conservation of the first moment (or any other single moment) in a coagulation process, (c) applicability to an arbitrary non-uniform grid, and (d) speed and robustness that make it suitable for combining with a CFD code for solving problems such as soot formation in flames. The conservation of the first moment of a distribution with respect to particle volume is of particular importance for two reasons: it is an invariant during a coagulation process and it represents conservation of mass. The method is based on a geometric evaluation of the double integrals arising from the finite volume discretisation of the coagulation terms and an exact balance of coagulation source and sink terms to ensure moment conservation. Extensive testing is performed by comparison with analytical solutions and direct numerical solutions of the discrete PBE for both theoretical and physically important coagulation kernels. Finally, the method is applied to the simulation of a laminar co-flow diffusion sooting flame, in order to assess its potential for coupling with CFD, chemical kinetics, transport and radiation models. The results show that accurate solutions can be obtained with a small number of sections, and that the PBE solution requires less than one fourth of the time of the complete simulation, only half of which is spent on the discretisation (the remaining being for the evaluation of the temperature dependence of the coagulation kernel). [ABSTRACT FROM AUTHOR]

Highlights • A pragmatic and efficient poly-disperse method for bubbly flow simulation is presented. • Closure is achieved using the log-normal presumed number density function. • Good agreement is found with reference data for bubbly flow in the mono-dispersed limit. • Poly-dispersity and poly-celerity are shown to play an important role in bubbly flow. • Good agreement is found between our moment method and an accurate sectional approach. Abstract The bubble size distribution plays an important role in interfacial mass, momentum and energy transfer between bubbles and their carrier liquid in bubbly flow. Accurate modeling of the size distribution is therefore key. A Log-Normal presumed Number Density Function (LNpNDF) approach is proposed, which is embedded into the two-fluid model. Two additional moment transport equations are formulated which are shown to be consistent with the two-fluid model. From the moments, the size distribution can be fully reconstructed using the assumption that its underlying shape is log-normal. This methodology offers closure for the modeling of processes such as bubble coalescence, break-up and bubble poly-celerity. Special attention is paid to the concept of poly-celerity, which is shown to play an important role in the evolution of finite-width size distributions. A new average diameter, which is based on the fifth and third moment of the size distribution, is proposed, and it is shown that this diameter is a more suitable quadrature node for the modeling of bubble–liquid Stokes-like drag. The paper lays the mathematical foundation for a pragmatic, computationally efficient and effective poly-dipsersed method for the modeling of dispersed two-phase flow. [ABSTRACT FROM AUTHOR]

Highlights • A novel continuum theory accurately predicts aggregation in a gas-solid flow. • The continuum theory accurately predicts entrainment of cohesive particles. • Aggregation leads to increased granular temperature and breakup of agglomerates. Abstract In this work, a continuum theory for cohesive particles developed recently (Kellogg et al., 2017) is applied to lightly-cohesive particles in an unbounded, gas-solid riser. The novelty of this recent theory is its fundamental incorporation of the effects of the granular temperature (i.e., continuum measure of impact velocity) and the material and cohesion properties on the rates of aggregation and breakage of agglomerates. Here, we extend this theory to multiphase systems and place particular emphasis on its ability to predict entrainment rate, as past empirical correlations vary by orders of magnitude when applied to the same system (Chew et al., 2015). Specifically, continuum predictions of entrainment rates and agglomerate fraction are compared with Discrete Element Method (DEM) simulations of lightly-cohesive particles in a gas-solid flow. The agreement obtained for these quantities provides preliminary validation for the use of the continuum theory for cohesive particles in gas-solid flows. [ABSTRACT FROM AUTHOR]

Highlights • First time use of moving overset meshes to simulate agitator motion. • Reynolds and Strouhal number equations account for non-linear impact of amplitude. • DSD comparison reveals a closer approximation using volume of fluid multiphase model. • User-defined functions are applied to model oscillatory translational velocity. Abstract This paper presents a CFD model and hydrodynamic study of a moving-baffle oscillatory baffled column (moving-baffle OBC). This work marks the first instance that moving overset meshing was used to simulate agitator motion in a fluid system. Population balance results are validated with experimental data for the inverse-suspension of non-reactive aqueous acrylamide in Isopar oil. A comparison of the droplet size distributions produced via various multiphase simulation methods was performed, resulting in a better overall agreement with the experimental literature for simulations applying the volume of fluid (VOF) multiphase method. Hydrodynamic studies reveal patterns of local flow circulations and centermost axial currents in relation to agitator position. Examination of the dimensionless groups traditionally used to describe flow conditions for moving-baffle OBCs reveal a considerable discrepancy between the previously-defined oscillatory Reynolds number and oscillatory Strouhal number with numbers derived from fluid flow within the column. A numerical correction has been presented to illustrate the nonlinear effect of oscillation amplitude on fluid flow through the system and to provide a more realistic estimation of the Reynolds number and Strouhal number for the modeled moving-baffle OBC. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • A method for analyzing the three-dimensional particle size distribution was proposed. • The accuracy was proved in a numerical simulation of backward-step flow. • The calculated local particle size distribution agree well with the experimental value. • In the low velocity region, the numerical and experiment result disagree slightly. Abstract Aimed at optimizing the resin-molding process, a method for numerically analyzing aggregation and dispersion behavior of the filler in resin composite was proposed. The flow of a resin composite during molding was calculated by using computational fluid dynamics (CFD), and particle-size distribution (PSD) of the cluster in each computational domain of the CFD model was estimated by solving the population balance equation (PBE). The proposed numerical-analysis method is based on the thixotropy model of Usui et al. In the thixotropy model, PSD is calculated by taking into account the aggregation and dispersion rate of the cluster by Brownian coagulation, shear coagulation, and shear breakage. Shear-breakage rate of the cluster is evaluated by solving the energy balance of bonding energy of the primary particles cut at the breakage and the drag applied to the cluster by the flow of the fluid. The composite viscosity was calculated using Krieger and Dougherty's model based on apparent-solid-volume fraction estimated from the calculated PSD. To solve the PBE at low calculation cost, it was discretized using the fixed-pivot technique of Kumar and Ramkrishna. The proposed method was incorporated into the general-purpose CFD software FLOW-3D®, and its accuracy was proved. [ABSTRACT FROM AUTHOR]

Abstract A two-step simulation methodology is presented that allows a detailed particle model to be used to resolve the complex morphology of aggregate nanoparticles synthesised in a stagnation flame. In the first step, a detailed chemical mechanism is coupled to a one-dimensional stagnation flow model and spherical particle model solved using method of moments with interpolative closure. The resulting gas-phase profile is post-processed with a detailed stochastic population balance model to simulate the evolution of the population of particles, including the evolution of each individual primary particle and their connectivity with other primaries in an aggregate. A thermophoretic correction is introduced to the post-processing step through a simulation volume scaling term to account for thermophoretic transport effects arising due to the steep temperature gradient near the stagnation surface. The methodology is evaluated by applying it to a test case: the synthesis of titanium dioxide from titanium tetraisopropoxide (TTIP) precursor. The thermophoretic correction is shown to improve the fidelity of the post-process to the first fully-coupled simulation, and the methodology is demonstrated to be feasible for simulating the morphology of aggregate nanoparticles formed in a stagnation flame, permitting the simulation of quantities that are directly comparable to experimental observations. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Droplets breakage in SMX+ mixers is investigated experimentally and numerically. • A population balance model is developed to predict the droplet size distribution. • The model parameters are identified using experimental results. • The dispersed phase properties and the operating conditions are explored. • The model predictions fit accurately the experimental data in all the cases. Abstract This work deals with the numerical and experimental investigation of droplets breakage in SMX+ static mixers. While steady-state empirical correlations are commonly used for the prediction of the mean droplets size (e.g. Sauter mean diameter) during emulsification in static mixers, in the present study, a population balance equation (PBE)- based model is developed for the prediction of the dynamic evolution of the droplet size distribution (DSD). The system silicon oil-in-water stabilized by Polysorbate 20 (Tween20®) is considered under dilute conditions (<5%vol). Due to the physico-chemical properties of the system and the operating conditions, the droplets breakage process is dominant, while coalescence and Ostwald ripening are negligible. The breakage kernel proposed by Alopaeus et al. (2002) is employed and its parameters are identified and validated for the present system under different operating conditions. The effects of the number of SMX+ elements, viscosity and fraction of the dispersed phase, the DSD at the inlet of the mixers as well as the volume-average energy dissipation rate within the SMX+ elements on the evolution of the DSD are investigated. The model was found to be able of predicting the DSD over a wide range of operating conditions. [ABSTRACT FROM AUTHOR]

Abstract In this work, a study to evaluate the effect of the concentration of nanoparticles of an Al 2 O 3 /water nanofluid on the capillary limit of a heat pipe with a screen mesh wick was performed. For the first time, a numerical model has been developed to couple the hydrodynamical equations that describe the flow and the phase change of the working fluid in the wick of a screen mesh heat pipe, with a population balance which account for the agglomeration of Al 2 O 3 nanoparticles and the deposition of them on the surface of the wick. The model has been validated with experimental data of temperature and capillary limit. Experimental results showed that capillary limit is reached at higher heat input when nanoparticles are added to the working fluid. The best improvement of the capillary limit was around 30–40%. At high nanoparticle concentration, the improvement of the capillary limit was more variable, and the thermal resistance was higher than even the working fluid without nanoparticles. The improvement of the capillary limit has been found to be due to the modification of the wettability of the wick instead of reduction of effective pore radius or enhancing of the thermal conductivity. Numerical results fitted in good agreement the experimental data, but some limitations were found when high nanoparticle concentration was simulated because model can not predict the non-uniform deposition observed at microscopic level. Finally, it was found that there is an optimal concentration of nanoparticles to delay the occurrence of capillary limit. In this work, that optimal concentration was Al 2 O 3 /water 0.5% w/w. Highlights • The wetting angle is reduced by nanoparticles deposition. • Population balance in the model allows for considering wettability variation. • Capillary limit is higher when nanoparticles are added to the working fluid. • There is an optimal concentration of nanoparticles to reduce the thermal resistance. [ABSTRACT FROM AUTHOR]

Abstract A 2D computational model coupling two-phase CFD and population balance equations is used to predict dispersed phase holdup and Sauter mean drop diameter in a pulsed sieve plate column which is relevant for solvent extraction in nuclear fuel cycle. Spatial and temporal variations of drop diameter are captured by the model. Population balance equations consider both drop breakage and coalescence in the column. Standard breakage and coalescence kernels are used to model breakage and coalescence rates. The model is validated using reported experimental data of dispersed phase holdup and Sauter mean drop diameter. In the first round of simulations standard Kumar-Hartland drag model is used to model the interphase momentum exchange. The average absolute relative error in estimation of dispersed phase holdup and Sauter mean diameter are found to be 13.6% and 8.5%, respectively. Optimization of the model constants of Kumar-Hartland drag model is done to reduce the error in dispersed phase holdup prediction to 5.6%. [ABSTRACT FROM AUTHOR]

Population balances provide an economic means of describing dense particulate phases with particle-particle interactions. Here, we focus on particle coagulation and address the evaluation of the corresponding integral source terms within the scope of a conservative finite volume formulation. Based on one kernel evaluation for every pair of parent cells, efficient analytical formulas are derived that obviate the explicit decomposition of the integration domains into elementary geometrical shapes and are valid for arbitrary volume-grids. Considering the volume-weighted cell averages as degrees of freedom, the formulas are presented for both cell-wise constant and linear reconstructions of the particle volume distribution. We find the latter to be effective at mitigating the unphysically heavy tails that are characteristic of solutions obtained with piecewise constant reconstructions. For Brownian coagulation, the computational expense is mainly caused by the kernel evaluations, while the cost of the parent-daughter pairing and integration algorithm is almost negligibly small. • We revisit the solution of the coagulation equation using a conservative finite volume approach. • The source terms are closed with the aid of cell-wise constant or linear reconstructions of the particle volume distribution. • For every pair of parent cells, the coagulation kernel is approximated by a single value. • We present efficient analytical formulas for evaluating the coagulation double integrals on arbitrary grids. • The linear reconstruction mitigates the appearance of unphysically heavy tails in the solutions. [ABSTRACT FROM AUTHOR]

In this work, five population balance models are applied to the simulation of isothermal bubbly flows, and the results are compared with experimental measurements of pipe flows. The models cover a wide range of different modeling approaches and include several of the most frequently used models for bubbly flow. We find that in the investigated cases, the sectional approach, despite being significantly more computationally expensive, is generally not more accurate than the most accurate moment method included in this work, which is based on a log-normal size distribution. Other moment methods have even lower computational costs, but give unrealistic representations of the bubble size distribution. Additionally, they offer less flexibility and accuracy for modeling break-up and coalescence. This work can be used to make better-informed choices when selecting a population balance model for future multiphase flow research, based on specific requirements in terms of accuracy, computational costs and model complexity. • Five population balance models are compared in three different flow configurations. • Their strengths and weaknesses are identified based on numerical simulations. • Significant differences in computational time and accuracy are observed. • The underlying causes of these large differences are identified. • The choice of a population balance model is facilitated for future researchers. [ABSTRACT FROM AUTHOR]

Hydrate agglomeration in oil-water systems could lead to pipeline hydrate plugging. Dynamic modeling of pipeline hydrate particle agglomeration is of great importance to hydrate management. For this purpose, a dynamic model of hydrate agglomeration was established based on population balance equation, which took both hydrate agglomeration and hydrate breakage into consideration. The calculation methods of collision frequency, agglomeration efficiency, breakage frequency and particle size distribution function of the sub particles resulting from particle breakage were given and fully discussed in the dynamic model. Then, combined with several traditional solid-liquid flow models, the dynamic model was solved by the CFD software FLUENT 14.5 to simulate the influences of flow rate and hydrate volume fraction on pipeline hydrate slurry flow behavior in oil-water systems. The simulation results show that hydrate particle size distribution in pipeline flowing systems can be well described by log-normal distribution. Hydrate particle diameters and hydrate volume concentrations in the near wall region are larger than that in the center region of the pipeline. The proportion of large hydrate particles and the mean hydrate particle diameter in pipeline increase with the increase of flow rate and hydrate volume fraction. Hydrate particle diameter and pressure gradient in oil-water systems are larger than that in water systems. [ABSTRACT FROM AUTHOR]

The interaction of bubbles with flow boundaries has been of high interest for marine engineering; for example, when a propeller is cavitating or air is entrained in the wake of a maneuvering ship, the strong interaction with the boundary layer will lead to the formation of a bubbly wake. To be able to develop the best mitigation strategy, a deep understanding of the associated physics is required. Only a few articles published in open literature address this two-phase fluid system. In the present study, a 3D numerical simulation has been performed to model a bubbly two-phase flow around the DARPA SUBOFF, in which exhaust gas is discharged into the flow around the object to provide a platform for investigating the distribution of the bubbles around a curved body. The two-phase flow is modelled using the Eulerian-Eulerian approach coupled with the MUSIG model. The bubble distribution is characterized based on different gas discharge configurations. It has been found that the boundary layer flow has a strong effect on bubble formation process, particularly encourages the bubble fragmentation. As a result, many small bubbles will be trapped at the aft of the vehicle. [ABSTRACT FROM AUTHOR]

Oil sands mature fine tailings are stable suspensions of clay particles and residual bitumen in an alkaline aqueous medium. They are formed when bitumen is extracted from oil sands using the Clark hot water process. Polymer flocculants are widely used to flocculate and subsequently dewater these tailings, but our knowledge is limited to empirical observations on how they work. A fundamental mathematical model would help monitor and design flocculation and dewatering processes more efficiently. In this study, we used a population balance model to describe the flocculation of mature fine tailings (MFT) with poly(vinylbenzyl trimethylammonium chloride). A time varying function was defined to account for the aggregate size evolution during MFT flocculation induced by polymer chains relaxation and rearrangement on the particle surfaces. The validity of the model was tested by varying the shear rate, mixing time and flocculant dosage using focused beam reflectance measurements (FBRM). The proposed model is a first step towards a more rational and quantitative approach to monitor and control treatment processes for oil sands tailings. [ABSTRACT FROM AUTHOR]

The event-driven acceptance rejection (AR) method is a computationally very advantageous Monte Carlo (MC) simulation technique for the solution of population balance equations (PBE) of coagulating systems. In the scope of the event-driven simulation approach, the simulation time is stepwise increased by a simulation time step r, which is given be the simulated particle properties. Within this time step τ, exactly one coagulation event takes place. The method is therefore not applicable in situations, for which specific time points have to be reached by the simulation, or the time step has to be reset to a smaller value. We propose a methodology termed 'fractional MC step' which allows to reset the simulation time step of the AR method to any arbitrary smaller value than the one initially proposed. The proposed method is validated by simulations of coagulation for different initial conditions and comparison with results gained from the discrete sectional method. The potential increase of the stochastic noise is investigated by comparisons with the results from conventional MC simulation techniques. The advantages of a parallel implementation are briefly discussed. [ABSTRACT FROM AUTHOR]

The flame spray pyrolysis (FSP) is a well-known process to produce nanoparticles and presents several advantages when compared to others, mainly regarding final product purity and operational flexibility. The correct modeling of the process is fundamental for the applicability of such techniques and, given that the temperature and chemical composition throughout the reactor are essential for the development of the nanoparticles, the correct representation of the chemical reactions is necessary. In this work, the production of zirconia (ZrO 2 ) nanoparticles via FSP is modeled and the combustion of the precursor-solvent mixture is described through seven sets of different chemical reaction mechanisms to analyse their influence on flame temperature and particle evolution within the process. The reacting turbulent multiphase flow is described by an Eulerian-Lagrangian approach and ZrO 2 nanoparticle growth is estimated by solving the population balance equations from a monodisperse model based on coagulation and sintering. Temperatures and primary particle diameters obtained from simulations are within 9% and 6% accuracies of experimental values, respectively. Although unmatching temperature profiles are found for the different mechanisms considered (mainly in the lower regions of the reactor), a small variation of primary particle diameter is observed when cases are compared. [ABSTRACT FROM AUTHOR]