Search

The main purpose of this article is to model an industrial spiral-type gravitational coalescer by applying population balance as the governing equation. In this model, considering the two-dimensional movement of crude oil flow in the coalescer, the mechanisms of differential sedimentation, laminar shear, and Brownian motion are considered to determine the collision rate of droplets. Then, the population balance equation was solved to determine the droplet size distribution, dehydration, and desalination efficiency using the class method. The simulation results were in good match with the gathered industrial data. Finally, the influence of interfacial tension of water and oil, the pressure drop of the mixing valve, oil temperature, oil density, and washing water flow rate on the process efficiency was analyzed. The modeling results indicated that increasing the interfacial tension of water and oil from 30 to 36 mN/m and the oil density from 800 to 900 kg/m3 will decrease the water removal efficiency by 4.35% and 14.18%, respectively. Also, increasing the temperature of oil from 300 to 340 K and the volume fraction of washing water from 3% to 15% improves the efficiency of separating water from crude oil by 9.55% and 9.95%, respectively. [ABSTRACT FROM AUTHOR]

Precipitation during the viral inactivation, neutralization and depth filtration step of a monoclonal antibody (mAb) purification process can provide quantifiable and potentially significant impurity reduction. However, robust commercial implementation of this unit operation is limited due to the lack of a representative scale‐down model to characterize the removal of impurities. The objective of this work is to compare isoelectric impurity precipitation behavior for a monoclonal antibody product across scales, from benchtop to pilot manufacturing. Scaling parameters such as agitation and vessel geometry were investigated, with the precipitate amount and particle size distribution (PSD) characterized via turbidity and flow imaging microscopy. Qualitative analysis of the data shows that maintaining a consistent energy dissipation rate (EDR) could be used for approximate scaling of vessel geometry and agitator speeds in the absence of more detailed simulation. For a more rigorous approach, however, agitation was simulated via computational fluid dynamics (CFD) and these results were applied alongside a population balance model to simulate the trajectory of the size distribution of precipitate. CFD results were analyzed within a framework of a two‐compartment mixing model comprising regions of high‐ and low‐energy agitation, with material exchange between the two. Rate terms accounting for particle formation, growth and breakage within each region were defined, accounting for dependence on turbulence. This bifurcated model was successful in capturing the variability in particle sizes over time across scales. Such an approach enhances the mechanistic understanding of impurity precipitation and provides additional tools for model‐assisted prediction for process scaling. [ABSTRACT FROM AUTHOR]

In the present study, an analytical model is developed to investigate the dehydration efficiency of a NATCO Dual Polarity® electrostatic treater at steady state conditions using population balance equations (PBE). To investigate the effect of frequency on dehydration performance and electric field strength between electrodes, a detailed electrical model is considered. In the developed electrical model, electrodes are virtualized as simple ideal capacitors, with emulsion acting as dielectric. Proving the model's accuracy, gathered industrial data were compared with simulation results. In addition, current frequency as the main electrical parameter besides power supply electrical potential was proved to be effective on the strength of the applied electric field, the electrical potential profile on electrodes, and the outlet water cut in the treated crude oil. The results indicated that increasing the current frequency from 50 to 500 Hz, by enhancing direct current field strength, reduces the water cut by 0.03%. Although high temperatures and frequencies increase the electrical conductivity of the emulsion, electrical power lost during the disconnection of the power supply on electrodes will be retrieved by increasing frequency. Finally, the performance of different electric fields on crude oil dehydration was compared. [ABSTRACT FROM AUTHOR]

AbstractThere are widespread occurrence and application of solid-liquid sedimentation processes among different industries. Therefore, it becomes important to understand the hydrodynamic inside the process equipment and particle agglomeration characteristics. In the present work, solid-liquid sedimentation is analyzed, which will be helpful for the design of continuous process equipment in plutonium (Pu) reconversion. Experiments were carried out in a batch settler to understand solid sedimentation in suspension in terms of varying the overall solid fractions. Euler-Euler two-fluid simulations were performed to investigate the local and overall solid phase volume fraction distributions, position of the active interface (AI) (settling curve), axial solid phase velocity, and pressure distributions during settling, and selected data were compared with the measured data. Further, the discrete population balance model (PBM) with different agglomeration kernels was used with the two-fluid computational fluid dynamics (CFD) model to understand and further improve predictions in terms of the AI position. The variation in number density of the different particles in the settler with time was investigated. The predicted results show that agglomeration is dominant during the sedimentation process and application of the discrete PBM with the CFD model enhances the predictive capability in comparison with the predictions obtained from only the two-fluid model. The results reported using CFD+PBM will aid in the design of continuous process equipment (thickener/precipitator) for Pu reconversion. [ABSTRACT FROM AUTHOR]

It is in urgent requirement to directly measure the droplet coalescence frequency for the application of the population balance model. In this study, a method was firstly developed to directly measure the droplet coalescence frequency in turbulent flow field by using a specially designed mixing tank and a high‐speed camera. The effects of the rotating speed and holdup fraction on the droplet coalescence frequency were quantitatively investigated. The increasing of rotation speed promotes first and then inhibits the coalescence, while the holdup fraction has little influence on the coalescence frequency function. The droplet collision frequency was also counted and the coalescence efficiency was calculated. The models in the literature were tested with our experimental data and were found failing to predict the coalescence frequency in the stirring tank. Empirical correlation was finally proposed and good agreement was found between the prediction results and the experimental data. [ABSTRACT FROM AUTHOR]

The droplet breakups in an orifice jet mixer are studied to gain a deeper understanding of the breakup mechanism and establish breakup models used as population balance model source terms for accurate predictions. The breakup process and details are captured in situ using a high‐speed photomicrography system. Experiments are performed with different levels of pressure drop and mother droplet size. A complete breakup process consists of several different sub‐breakup events, hence the cascade breakup definition is adopted to determine statistical results. High breakup frequency is achieved under high pressure drop and small mother droplet size. The average number of daughter droplets is 102–105 and is positively correlated with the pressure drop and mother droplet size. The daughter droplet size distribution has a tailing phenomenon and is mainly affected by the pressure drop. New breakup models based on the above features are established and predict well with the experimental results. [ABSTRACT FROM AUTHOR]

This study introduces a novel particle model for biomass fast pyrolysis, incorporating an anisotropic cylindrical particle to address mass and energy transport coupled with aerosol ejection, which previous models have overlooked. The main contribution lies in developing a model that considers aerosol generation in anisotropic cylindrical particles for the first time, addressing bubbling dynamics and bursting within the liquid phase. The population balance equation describes bubble dynamics and aerosol formation, capturing phenomena like nucleation, growth, coalescence, and bursting. The model employs the method of moments with bubble volume as an internal variable, substantially reducing computational costs by eliminating dependence on this variable. Results highlight the significant impact of anisotropy and particle size on aerosol ejection: smaller, less elongated particles experience faster heating, quicker conversion, and the increased accumulation of the liquid intermediate phase. Specifically, 1 mm diameter particles yield higher concentrations of metaplast and bio-oil aerosols, exceeding 15%, compared to concentrations below 11% for 3 mm particles. This model provides insights into aerosol structure (volume, surface area), aiding in understanding aerosol reactivity at the reactor scale. [ABSTRACT FROM AUTHOR]

In this study, a fluidized bed reactor for polyethylene production was employed using a dry mode approach, where the recycle stream may contain components of a nature that cannot be condensed through standard cooling. To analyze the behavior of the fluidized bed reactors during the copolymerization of ethylene with butene, a dynamic population balance model was employed. The study includes sensitivity analyses through computer simulations to examine the variations in reactor temperature, molecular weights, catalyst feed rate, and monomer/comonomer concentrations in the fluidized bed reactor. It is noteworthy that the reactor exhibits instability under normal operational conditions and is sensitive to changes in the catalyst feed rate and coolant temperature of the heat exchanger. The findings also highlight challenges such as temperature fluctuations above the polymer melting point. This underscores the importance of implementing a temperature control system to prevent issues like reactor shutdown due to elevated temperatures. Dynamic instabilities were observed under specific circumstances and were successfully controlled using Proportional Integral Derivative (PID) control strategies. The population balance model is essential for understanding the complexity of transient polymerization reactions. It enables researchers to simulate and optimize polymerization processes by utilizing the detailed kinetics of the reaction. [ABSTRACT FROM AUTHOR]

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]

To liberate copper from malachite, the ore has to be reduced to appropriate size using comminution processes. Ball mills are commonly used to obtain fine particles in grinding circuits. Population balance equations can be used to model grinding kinetics in a ball mill. The aim of this study was to understand the mineralogical characteristics of malachite copper ore and identify its grinding kinetics. To achieve this, chemical analysis and petrographic studies on thin slides were carried out. Additionally, grinding tests were carried out in a ball mill using five mono-sized fractions. Short grinding cycles of 0.5 minutes were used to calculate breakage distribution functions. Longer grinding cycles up to 30 mis were used to determine specific breakage rates. The experimental results were fitted into population balance models to obtain breakage parameters that characterize the material. The parameters were used to predict particle distributions. The malachite bearing copper was found to exists among other minerals. The average concentration of copper was found to be 4.7%. The specific rate of breakage increased as particle size increased to a maximum value at particle size of 2 mm. The parameters of grinding kinetics computed parameters were applied in prediction of particle size distribution. The particle distribution produced a high degree of agreement between the simulated and experimental results. The parameters can be used in the design of ore flowsheets for processing malachite ore. [ABSTRACT FROM AUTHOR]

The research aims to verify the universal relationship between vessel shape and the risk of hemolysis using a rheological model of blood reflecting the physiological processes related to blood for any blood vessel. Blood is a multi‐component fluid, the rheology of which depends on many factors, such as the concentration of red blood cells and local shear stress, which significantly affect the process of hemolysis. Blood rheology models used so far cannot be used for all flows and geometries. Therefore, a new rheology model has been introduced suitable for modeling hemolytic flows observed in arteries with atherosclerotic lesions in the in vivo environment. The previously presented model also has advantages in modeling local viscosity in stenosis. Geometries of the blood vessels from computed tomography scans and simplified models of the actual arteries observed during medical procedures were used in the calculations. Population Balance Based Rheology model predicts the concentration of single, deagglomerated red blood cells and the concentration and size of red blood cell agglomerates, which affect blood rheology and hemolysis. Based on the simulations carried out, a correlation was found between the shape of the vessel cavity and the risk of hemolysis. Presented results can be used in the future to create a correlation between the shape of the atherosclerotic lesions and the risk of hemolysis in the blood to make an initial risk assessment for a given patient. [ABSTRACT FROM AUTHOR]

Describing spatiotemporal evolution and characteristics of dispersed systems using the population balance equation (PBE), examples including sectional and moment methods are fraught with numerous issues. Hence, this study develops an accurate method by combining computational fluid dynamics and population balance‐Monte Carlo method (CFD‐PBMC) with a moderate computational cost. An efficient sub‐model for particle migration was proposed to simulate the convection and diffusion processes of particulate flows. A graphics processing unit (GPU)‐based parallel computation was performed to accelerate the high‐dimensional CFD‐PBMC. Several classical cases with analytical or benchmark solutions were simulated, and a comprehensive comparison was made using the classical weighted random walk method. Good agreements were obtained, except in the case of radial migration, the reasons for which are explained in detail. The measured speedups on the GPU showed a factor of ~450 for pure migration and ~50 for the CFD‐PBMC method when compared with a standard high‐performance computer. [ABSTRACT FROM AUTHOR]

In the present study, a recently proposed extended population balance equation (PBE) model for aggregation and sintering is incorporated into a large eddy simulation-probability density function (LES-PDF) modelling framework to investigate synthesis of silica nanoparticles in a turbulent diffusion flame. The stochastic field method is employed to solve the LES-PBE-PDF equations, characterising the influence of the unresolved sub-grid scale motions and accounting for the interactions between turbulence, chemistry and particle dynamics. The models for gas-phase chemistry and aerosol dynamics are the same as those recently used by the authors to simulate silica synthesis in a laminar flame (Tsagkaridis et al. in Aerosol Sci Technol 57(4):296–317, 2023). Thus, by retaining the same kinetics without any adjustments in parameters, we focus on the modelling issues arising in silica flame synthesis. The LES results are compared with experimental in-situ small-angle X-ray scattering (SAXS) data from the literature. Good agreement is found between numerical predictions and experimental data for temperature. However, the LES model underestimates the SAXS data for the primary particle diameter by a factor of two. Possible reasons for this discrepancy are discussed in view of the previous laminar flame simulations. [ABSTRACT FROM AUTHOR]

Airlift reactors (ALRs) are widely used in the chemical, petrochemical biological industry. A fundamental understanding of the flow field in these airlift reactors are necessary for efficient design and scaling up. In this work, the behavior of the flow field is investigated using the Euler–Eulerian approach. The liquid phase is modeled as continuous and the gas phase is dispersed in the form of bubbles. Three dimensional (3 D) transient computational fluid dynamics (CFD) simulations are performed to characterize flow behavior in ALR. The spatio-temporal variations in the flow field are quantified and an optimum liquid level in the ALR is determined. Various gas source locations are chosen and their effects on bubble plume motion are analyzed to find an optimum gas injection point that supports plume oscillation. Further, CFD simulations are performed to identify the prevailing flow regime in ALR for various gas source locations, and it is compared with experimental observations. The homogeneous and heterogeneous flow regimes are observed at lower and higher flow rates, respectively. The bubble size distribution is predicted using population balance equations through bubble coalescence and breakage models with interphase force formulations. This is computed through the discrete method of moments. The bubble size distribution is found to be narrow at lower gas flow rates and wider at higher gas flow rates. These predictions provide a unified description to characterize flow regimes in ALR. [ABSTRACT FROM AUTHOR]

This study introduces a novel particle model for biomass fast pyrolysis, incorporating an anisotropic cylindrical particle to address mass and energy transport coupled with aerosol ejection, which previous models have overlooked. The main contribution lies in developing a model that considers aerosol generation in anisotropic cylindrical particles for the first time, addressing bubbling dynamics and bursting within the liquid phase. The population balance equation describes bubble dynamics and aerosol formation, capturing phenomena like nucleation, growth, coalescence, and bursting. The model employs the method of moments with bubble volume as an internal variable, substantially reducing computational costs by eliminating dependence on this variable. Results highlight the significant impact of anisotropy and particle size on aerosol ejection: smaller, less elongated particles experience faster heating, quicker conversion, and the increased accumulation of the liquid intermediate phase. Specifically, 1 mm diameter particles yield higher concentrations of metaplast and bio-oil aerosols, exceeding 15%, compared to concentrations below 11% for 3 mm particles. This model provides insights into aerosol structure (volume, surface area), aiding in understanding aerosol reactivity at the reactor scale.

In this study, a fluidized bed reactor for polyethylene production was employed using a dry mode approach, where the recycle stream may contain components of a nature that cannot be condensed through standard cooling. To analyze the behavior of the fluidized bed reactors during the copolymerization of ethylene with butene, a dynamic population balance model was employed. The study includes sensitivity analyses through computer simulations to examine the variations in reactor temperature, molecular weights, catalyst feed rate, and monomer/comonomer concentrations in the fluidized bed reactor. It is noteworthy that the reactor exhibits instability under normal operational conditions and is sensitive to changes in the catalyst feed rate and coolant temperature of the heat exchanger. The findings also highlight challenges such as temperature fluctuations above the polymer melting point. This underscores the importance of implementing a temperature control system to prevent issues like reactor shutdown due to elevated temperatures. Dynamic instabilities were observed under specific circumstances and were successfully controlled using Proportional Integral Derivative (PID) control strategies. The population balance model is essential for understanding the complexity of transient polymerization reactions. It enables researchers to simulate and optimize polymerization processes by utilizing the detailed kinetics of the reaction.

This paper aims to present a new model for an industrial desalination system including a mixing valve and an electrostatic coalescer. This model uses a one-dimensional population balance equation in steady-state conditions. The effect of hydrodynamic, diffusion and electrostatic mechanisms on the collision or breaking of water droplets dispersed in crude oil was considered. Also, using the film drainage model, the droplet coalescence rate has been determined. The developed model can predict the outlet droplet size distribution as well as the desalination and dehydration efficiency. The simulation results showed good agreement with the industrial data. Finally, the effect of some important parameters on the electrostatic desalination process, such as mixing valve pressure drop, electric field intensity, crude oil viscosity, crude oil temperature, diluting water flow rate, and crude oil API on the efficiency of the process was analyzed. The results indicated that by increasing the crude oil temperature from 314 to 334 K, the efficiency of dehydration improves from 97.95% to 99.02%. Also, reducing crude oil API from 33 to 30 and crude oil viscosity from 4 to 2 mPa.s caused a decrease in water separation by 2.88% and 1.16%, respectively. [ABSTRACT FROM AUTHOR]

In the present paper, a mathematical model is built and implemented to describe the trajectories of mass inventories, pressures and polymer properties with emphasis on final particle size distributions of industrial scale poly(vinyl chloride) suspension polymerization reactors. The model comprises the mass balances, statistical moment balances, equilibrium relationships and population balance equations. A discretization scheme is employed to transform the integro-differential equations resulting from the population balance model into a system of differential equations. The obtained results show, for the first time, that classical breakage and coalescence kernels described in the literature can provide very good fittings of actual industrial scale data when coupled with proper parameter estimation procedures, so that the proposed model is able to represent the available operation data with good accuracy at distinct conditions. Particularly, it is also shown that the use of a top condenser for control of the reactor temperature can lead to changes of parameters that control the particle size distributions. [ABSTRACT FROM AUTHOR]

Multistaged crystallization systems are used in the production of many chemicals. In this article, employing the population balance framework, we develop a model for a column crystallizer where particle agglomeration is a significant growth mechanism. The main part of the model can be reduced to a system of integrodifferential equations (IDEs) of the Volterra type. To solve this system simultaneously, we examine two numerical schemes that yield a direct method of solution and an implicit Runge–Kutta type method. Our numerical experiments show that the extension of a Hermite predictor–corrector method originally advanced in Khanh (1994) for a single IDE is effective in solving our model. The numerical method is presented for a generalization of the model which can be used to study and simulate a number of possible operating profiles of the column. [ABSTRACT FROM AUTHOR]

Flow reactors are widely used to study the formation of various nanoparticles, such as carbon black, soot, nickel, titania, and silica. Such reactors provide well-controlled conditions, making them a favored laboratory tool to investigate the details of particle formation. Here we present NanoParticle Flow Reactor (NanoPFR), a detailed model to simulate nanoparticle synthesis in flow reactors. The model predicts the agglomerate fractal-like morphology and size distribution with a 2-variable sectional population balance model coupled with gas-phase chemistry. The particle formation processes employed in the code are tested using detailed discrete element modeling simulations and then used to predict carbon black formation from ethylene pyrolysis experiments from the literature. The code is a robust flow reactor predictive tool with a strong foundation that can serve as a basis for further development, including the simulation of other nanoparticles formation. [ABSTRACT FROM AUTHOR]

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

The agglomeration dynamics within colloidal boehmite suspensions is crucial to understand the formation of a porous boehmite network during the manufacturing of γ-alumina catalyst carriers. Such carriers are frequently used in petroleum hydrotreating processes and must have specific textural characteristics. The method described in this work enables to model the three-dimensional morphology of colloidal agglomerates of boehmite for different conditions of pH in the colloidal mixture. These agglomerates are then used to generate a boehmite grain, whose textural properties can be numerically estimated. The coagulation kinetics has been studied with experimental Dynamic Light Scattering and Lagrangian model including Brownian dynamics and DLVO interaction potential. The adjustment of the Brownian aggregation kernel of a population-balance model enables to estimate the agglomerates size distribution and fractal dimension.

Population balance models mathematically describe the particle size distribution based on modeling physical phenomena that influence the distribution, such as aggregation, growth, and breakage. Due to the wide range of mechanisms present, several models are presented in the literature since several hypotheses are considered. In the current work, the Approximate Bayesian Computational statistical technique was used to select four different models of population balance and estimate their parameters. Three strategies were applied to the drawing of parameters, evaluating the correlation between the parameters of the models. An adaptive tolerance in each population and a stopping criterion, based on Morozov's uncertainty principle, were used for the algorithm. The technique obtained reasonable estimates for the phenomenological rates of the models. The algorithm correctly selected the model used for generating measurements, and the three draw strategies demonstrated good applicability. The results obtained showed that the algorithm presented accuracy and precision in estimating the parameters and properly selected the models analyzed.

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]

A better estimation of the density of cells has great relevance in the design of harvesting units. In the case of microalgae, the density is a function of the internal composition, which in turn is affected by external environmental conditions. The density of microalgae is often regarded as a constant or a generic value is retrieved from literature. This study proposes a procedure to evaluate the density of Chlorococcum sp. with simple sedimentation and centrifugation experiments coupled with the population balance equation (PBE), which is solved numerically. The density of cells is not constant; instead, it is a function of the size of particles, which in turn changes with the cells' phase of their life cycle. The calculated cellular density ranged between 1000 and 1100 kg m−3 in function of the cell size in both the sedimentation and centrifugation tests. The method can be extended to other microalgae species as well as to other types of cells. [ABSTRACT FROM AUTHOR]

The paper evaluates the behavior of a red-mud solid fraction in a thickener feeder cup, aiming to identify the main characteristics of particle distribution in the flocculation zone and to determine the dependencies affecting the further process taking place in the particle-free sedimentation zone in the thickener-thickening unit. This work used mathematical and numerical modeling to study the influence of such parameters as the flow rate of the feed pulp in the thickener, the flow rate of the flocculant, the density of pulp at the inlet to the unit, and the viscosity and temperature of the pulp on the particle-size distribution from under the feeder cup. The results and dependencies obtained are intended to be used as nominal values in the red-mud thickening process performed on a lab-scale unit. [ABSTRACT FROM AUTHOR]

Haematococcus pluvialis can produce significant amounts of industrially important compounds belonging to lipids and starch classes, including various specific pigments such as β-carotene, lutein and astaxanthin, as well as lipids, carbohydrates and proteins. Their production can vary depending on environmental stress conditions like nutrient starvation. However, stress conditions lead also to undesired phenomena such as cell lysis, which is likely to be related to products loss. The microorganism develops towards smaller single cell volumes during the growth process, and eventually, more likely towards lysis when fission (i.e. cell division) slows down. The lysis process takes place simultaneously with nutrient depletion, so both growth and lysis are linked to the change of environmental conditions. In this work, we develop a novel multiscale segregated-structured model based on Population Balance Equations (PBEs) to describe the photoautotrophic growth of H.pluvialis, in particular cell growth, and lysis, making possible the description of the relationship between cell volume/transition, cell loss, and metabolic product availability. Cell volume is the internal coordinate of the population balance model, and its link with intrinsic concentrations is also presented. The model parameters are fitted against experimental data, extensive sensitivity analysis is performed and the model predictive capabilities are tested in terms of cell density distributions, as well as 0th and 1st order moments.

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]

This paper explores the hyperbolic heat transfer effects in processes involving high heating rates. The behavior of the model is analyzed in detail under different boundary conditions and the circumstances under which a non-Fourier law could be used to describe thermal conduction processes established from physical mathematical analysis. Finally, the model developed here is coupled to a previous population balance framework to predict the bubbling phenomenon that occurs during the fast pyrolysis of biomass. We found that a transient overheating occurs in the central zone of the generated liquid phase due to the high heating rates that take place during that process. [ABSTRACT FROM AUTHOR]

In this paper, we establish an adsorption process model considering particle size distribution based on the population balance model to analyze the effects of particle size distribution of spherical activated carbon adsorbent on its adsorption performance. By comparing the prediction results of using particle size distribution and average particle size, it is proved that the influence of particle size distribution is significant. The adsorption amount predicted based on average particle size is lower than that based on particle size distribution in the early stage of adsorption and gets higher when it is close to equilibrium. It shows that ignoring the influence of particle size distribution will cause underestimation of the required time to reach expected adsorption amount. In the studied case the time taken to reach 97% of the equilibrium amount is underestimated by 13. 9%. When it is required to accurately predict the performance of adsorbent, the influence of particle size distribution needs to be considered. [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]

The mathematical models were used to predict the conversion, molecular weights, and polydispersity in atom transfer radical polymerization (ATRP) of styrene with mono- and bi-functional initiators. The moment and population balance models can predict monomer conversion, averaged-number molecular weight which increased linearly and narrow polydispersity index of synthesized polystyrene. The modeling calculating results were in good agreement with the experimental results. Comparison of the results of these two methods with the experimental data showed that the accuracy of population balance method for prediction of polymer polydispersity index in low conversions is more than the moment method. In addition, the new numerical simulation basis of population balance model can predict the full molecular weight distribution of the polystyrene in all the reaction time. [ABSTRACT FROM AUTHOR]

The dispersed phase holdup and drop size in solvent extraction columns vary along the column height and this affects the mass transfer coefficient and interfacial area. In this article, mass transfer study was performed experimentally using a 25 mm diameter agitated pulsed column. The axial distribution of mass transfer coefficient was determined by coupling population balance equation and axial dispersion model by taking the longitudinal variation in hydrodynamic performance into consideration. Feasibility of different mass transfer models in predicting concentration profiles was evaluated and a novel correlation based on effective diffusivity was developed. The results showed that both overall and volumetric mass transfer coefficients have significant change along the column height and greatly depends on the agitation speed and pulsation intensity. Increasing dispersed phase velocity also augments the overall mass transfer coefficient. The maximum number of transfer unit was measured to be 10 m−1 at agitation speed of 1000 rpm. [ABSTRACT FROM AUTHOR]

In this work, we outline the development of a thermodynamically consistent microscopic model for a suspension of aggregating particles under arbitrary, inertia-less deformation. As a proof-of-concept, we show how the combination of a simplified population-balance-based description of the aggregating particle microstructure along with the use of the single-generator bracket description of nonequilibrium thermodynamics, which leads naturally to the formulation of the model equations. Notable elements of the model are a lognormal distribution for the aggregate size population, a population balance-based model of the aggregation and breakup processes and a conformation tensor-based viscoelastic description of the elastic network of the particle aggregates. The resulting example model is evaluated in steady and transient shear forces and elongational flows and shown to offer predictions that are consistent with observed rheological behavior of typical systems of aggregating particles. Additionally, an expression for the total entropy production is also provided that allows one to judge the thermodynamic consistency and to evaluate the importance of the various dissipative phenomena involved in given flow processes. [ABSTRACT FROM AUTHOR]

In this work, the Markov Chain Monte Carlo is applied to estimate parameters that represent mechanisms that describe particles' dynamics in particulate systems from the literature's proposed models. Initially, the reduced sensitivity coefficient is evaluated to verify which parameters could be estimated simultaneously. The technique is then applied to estimate the models' parameters in different numerical scenarios to determine the rates that influence population dynamics. After the analyzes are performed, the estimates show good precision, accuracy, and a good fit between the measured and estimated state variables. The results show that the Markov chain Monte Carlo can determine the rates of population balance phenomenon.

Statistical descriptions of particulate phases and granular media based on the notion of a property distribution are particularly advantageous for dense, many-particle dispersions since they support the direct evaluation of Eulerian property statistics, permit a kinematically simple treatment of particle collisions and feature a model complexity that is independent of the number of physical particles. On the minus side, the property distribution is governed by a high-dimensional population balance equation (PBE) whose moment reduction is accompanied by a challenging closure problem and engenders intricate dynamical nonlinearities. Circumventing both the curse of dimensionality and the moment closure problem, we present a Eulerian Monte Carlo (EMC) solution method that harnesses a representation of the property distribution in terms of an ensemble of property samples and is based on a statistical enforcement of the spatially and temporally discretized PBE. After application of a kinetic finite volume scheme, the PBE is subjected to a fractional steps decomposition that separates spatial transport from transport in property space. Conceptually, spatial transport is emulated by assembling the ensemble in one spatial grid cell from samples drawn from the ensembles in neighbouring cells at the previous time point. In this regard, we propose a two-level sampling scheme that is sensitive to large-scale features of the marginal velocity distribution and effectively reduces the instantaneous sampling error. The property transport step, by contrast, is based on the method of characteristics and involves the propagation of individual samples as property characteristics in time. While the sampling error afflicting the property statistics needs to be controlled using a top-level time or ensemble averaging, the EMC method remains well-conditioned for an arbitrary number of samples, is easy to implement and involves a computational expense that scales only linearly with the number of particle properties. Based on two benchmark examples, we analyze the spatial and statistical convergence properties of the EMC method and demonstrate its capacity to capture multiple crossings of particle trajectories. A third example involves the two-way coupled transport of size-polydispersed inertial particles in an inhomogeneous flow field. • We present a Eulerian Monte Carlo (EMC) method for solving multivariate population balances. • The particle property distribution is represented in terms of the total number density and an ensemble of property samples. • In a spatial grid cell, the samples evolve as jump-drift processes with neighbour and mean-field interactions. • While closure problems do not arise, a statistical approximation error occurs. • The EMC method is well-suited for capturing crossing trajectories in inertial particulate flows. [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]

The population balance is an indispensable tool for studying colloidal, aerosol, and, in general, particulate systems. The need to incorporate spatial variation (imposed by flow) to it invokes the reduction of its complexity and degrees of freedom. It has been shown in the past that the method of moments and, in particular, the log-normal approximation can serve this purpose for certain phenomena and mechanisms. However, it is not adequate to treat gravitational deposition. In the present work, the ability of the particular method to treat diffusional and convective diffusional depositions relevant to colloid systems is studied in detail. [ABSTRACT FROM AUTHOR]

The drop size distributions (DSDs) of a dilute immiscible liquid‐liquid mixture were measured in a fully turbulent stirred tank operating at different impeller speeds. The results were used to infer the best daughter distribution function (DDF) leading to the best reproduction of the shape of the DSD. Bell‐shaped, U‐shaped, M‐shaped, and uniform statistical DDFs were studied, producing from two to four daughters from each breakup event. A simplified approach from the literature was adopted to solve the population balance equation that considers the spectrum of the turbulence inside the tank obtained from computational fluid dynamics simulations. The U‐shaped distribution producing four fragments better reproduces the shape of the experimental DSD in the studied system. [ABSTRACT FROM AUTHOR]

< p>Hydrogen in spite of having a chain transfer agent role is one of the most important factors which directly affecting on kinetic of propylene polymerization. Hydrogen causes to dramatically increase the percentage of activated sites in proportion to being total potential sites on the catalyst surface. On the other hand, the chain transfer agent role of hydrogen gives rise to changing some vital indices of final product properties. This study has attempted to present a validated mathematical model that able to predicting profile polymerization rate and also calculating some of the vital indices of the final product, properties to be derived from kinetic equations. Furthermore, in this paper, we present a developed model that calculates fraction activated sites catalyst via hydrogen concentration based on dormant site theory and determining the best process condition. The modeling approach is based on the polymer moment balance method, i.e. population balance technique, and validated by experimental works. The main aim of this study is assigned to investigate the behavior change of polymerization rate to hydrogen. The experimental data and model outputs were compared and concluded that the global errors were in the acceptable range.

In this paper, a dynamic model for a rotary drum granulation loop with external product separator is developed. A population balance is used to capture dynamic particle size distribution in the 3-compartment rotary drum granulator model. Particle agglomeration along with particle growth due to layering are assumed as granulation mechanisms in the rotary drum. The model of the granulation loop includes models of the drum, screens and a crusher. Simulations using the developed model provide valuable data on dynamic fluctuations in the inlet and the outlet particle size distribution for the rotary drum. Simulation results showed that at smaller crusher gap spacings, the instabilities of the drum granulation loop occur, and damped oscillations are observed. Above the critical crusher gap spacing, sustained periodic oscillations are observed. The reason for oscillations is the off-spec particle flow that is recycled back to the granulator.

Three-dimensional Eulerian-Eulerian transient simulations were conducted to represent the gas-liquid flow of a heterogeneous bubble column. Different drag closures, breakup and coalescence models were evaluated in order to verify their influence on the model prediction. Numerical simulations were compared to experimental data, with industrial conditions of gas superficial velocities: 20cm/s and 40cm/s, in order to select the most suitable models to describe the bubble’ dynamics in the heterogeneous flow. The standard k − ε model for both phases was set for turbulence. 12 combinations of breakup and coalescence models were compared and analyzed. In the case of coalescence, Models of Prince and Blanch, and Luo presented similar behavior and good agreement with experimental data, while for breakup, a breakage forming three daughter bubbles appeared to be the best choice. Simulations presented relative errors around 7.7% and 14.0%, for 20cm/s and 40cm/s respectively, for the gas axial velocity, and around 14% and 21.9%, for gas holdup. For drag force, density and viscosity were accounted by an average of the phases, which resulted in an improvement about 7% on model validation