Showing papers on "Population balance equation published in 2014"

Population Balance Model-Based Multiobjective Optimization of a Multisegment Multiaddition (MSMA) Continuous Plug-Flow Antisolvent Crystallizer

Abstract: Crystallization is a major separation process in the pharmaceutical industry. Most crystallizations are performed batchwise, but there is great incentive for converting them to continuous operations. This paper investigates the modeling, simulation, and optimization of a special antisolvent plug-flow crystallizer: the multisegmented, multiaddition plug-flow crystallizer (MSMA-PFC). The MSMA-PFC accepts multiple antisolvent flows along its length, permitting finer control of supersaturation. A steady-state population balance equation was applied for tracking the crystal size distribution, and a mass balance equation was used to track the depletion of dissolved solute (flufenamic acid). A multiobjective optimization framework was applied to determine the antisolvent flow rates into each segment that simultaneously maximize the average crystal size, and minimize the coefficient of variation. The set of coupled differential equations was solved, depending on circumstance, with either the method-of-moments (MO...

Modeling and simulation of turbulent polydisperse gas-liquid systems via the generalized population balance equation

Abstract: This article reviews the most critical issues in the simulation of turbulent polydisperse gas-liquid systems. The discussion is here limited to bubbly flows, where the gas appears in the form of separate individual bubbles. First the governing equations are presented with particular focus on the Generalized Population Balance Equation (GPBE). Then the mesoscale models defining the evolution of the gas-liquid system (e.g. interface forces, mass transfer, coalescence and breakup) are introduced and critically discussed. Particular attention is devoted to the choice of the drag model to properly simulate dense gas-liquid systems in the presence of microscale turbulence. Finally the different solution methods, namely Lagrangian and Eulerian, are presented and discussed. The link between mixture, two- and multi-fluid models and the GPBE is also analyzed. Eventually the different methodologies to account for polydispersity, with focus on Lagrangian or Direct Simulation Monte Carlo (DSMC) methods and Eulerian Quadrature-based Moment Methods (QBMM), are also presented. A number of practical examples are discussed and the review is concluded by presenting advantages and disadvantages of the different methods and the corresponding computational costs

Dispersion of oil droplets in rotor–stator mixers: Experimental investigations and modeling

Abstract: Forming emulsions by mixing of immiscible liquids, water and silicone oils was carried out by applying the in-line high-shear rotor–stator mixers. In experimental part investigations were carried out using experimental rig consisting of two tanks and an in-line Silverson rotor-stator (150/250) MS; the system was operated in a multiple pass (MP) mode, which can be compared with a single pass (SP) mode experiments. Emulsification of 1 wt.% silicone oils (Dow Corning 200 fluid) with viscosities of 9.4, 48 and 339 mPa s was investigated. Emulsions were stabilized by adding 0.5 wt.% of sodium laureth sulphate. Effects of rotor speed, number of passes and the drop viscosity on the drop size were investigated. Numerical simulations were carried out using the k–ɛ model of turbulence and the multiple reference frame method (MRF) linked to the population balance equation. The population balance was expressed and solved using the quadrature method of moments (QMOM). The breakage kernel for drops whose diameter falls within the inertial subrange of turbulence was defined based on the multifractal model of intermittent turbulence.

Multimodal particle size distributions of fine-grained sediments: mathematical modeling and field investigation

Abstract: Multimodal particle size distributions (PSDs) of fine-grained cohesive sediments are common in marine and coastal environments. The curve-fitting software in this study decomposed such multimodal PSDs into subordinate log-normal PSDs. Four modal peaks, consisting of four-level ordered structures of primary particles, flocculi, microflocs, and macroflocs, were identified and found to alternately rise and sink in a flow-varying tidal cycle due to shear-dependent flocculation. The four modal PSD could be simplified further into two discrete size groups of flocculi and flocs. This allowed the development of a two-class population balance equation (TCPBE) model with flocculi and flocs to simulate flocculation involving multimodal PSDs. The one-dimensional vertical (1-DV) TCPBE model further incorporated the Navier-Stokes equation with the k-e turbulence closure and the sediment mass balance equations. Multimodal flocculation as well as turbulent flow and sediment transport in a flow-varying tidal cycle could be simulated well using the proposed model. The 1-DV TCPBE was concluded to be the simplest model that is capable of simulating multimodal flocculation in the turbulent flow field of marine and coastal zones.

Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems

Abstract: A complex interplay between aggregation and coalescence occurs in many colloidal polymeric systems and determines the morphology of the final clusters of primary particles. To describe this process, a 2D population balance equation (PBE) based on cluster mass and fractal dimension is solved, employing a discretization method based on Gaussian basis functions. To prove the general reliability of the model and to show its potential, parametric simulations are performed employing both diffusion-limited-cluster aggregation (DLCA) and reaction-limited-cluster-aggregation (RLCA) kernels and different coalescence rates. It turns out that in both DLCA and RLCA regimes, a faster coalescence leads to smaller sized and more compact clusters, whereas a slow coalescence promotes the formation of highly reactive fractals, resulting in larger aggregates.

Ultrasound‐facilitated particle breakage: Estimation of kinetic parameters using population balance modelling

Abstract: Ultrasound-facilitated particle breakage has been proven to be a good technique for size reduction. However, because of the difficulties associated with the solution of population balance equation (PBE), there has been little attention on the estimation of kinetics of this process. In this paper an attempt has been made to develop a generic approach for solving PBE with a given particle size distribution (PSD) data to estimate the breakage kinetics. For this purpose, discretization technique along with gPROMS parameter estimation tool was used. This method was found to be a comprehensive method for the estimation of breakage kinetics in general and was subsequently used for the estimation of ultrasound-facilitated breakage kinetics, using a simple model system. Potassium nitrate (KNO3) particles suspended in toluene was used as model system, and the effect of induced power on breakage strength was studied. The data obtained were used for the estimation of specific breakage rate with different particle size distribution function. The distribution function, for which simulated PSD was found to match with the experimental PSD, was considered for deriving energy-size reduction law.

Impact of Mixing for the Production of CuO Nanoparticles in Supercritical Hydrothermal Synthesis

Abstract: The mixing process of metal salt solutions and supercritical water is essential to supercritical water hydrothermal synthesis (SWHS) to produce nanoparticles. A computational fluid dynamics (CFD) model was developed for predicting mixing efficiency and crystallization kinetics in SWHS mixers. The mixing efficiency was calculated from the micromixing model, and the kinetics model of crystallization was built from the population balance equation. The effects of the fluid dynamics of mixing (Reynolds number and mixer diameter) on the mixing efficiency and crystallization kinetics (nucleation rate and growth rate) were investigated. The results showed that faster mixing can lead to a higher crystallization rate and the production of smaller particles with a narrower particle size distribution.

Coupling of CFD and PBE Calculations to Simulate the Behavior of an Inclusion Population in a Gas-Stirring Ladle

Abstract: Gas-stirring ladle treatment of liquid metal has been pointed out for a long time as the processing stage is mainly responsible for the inclusion population of specialty steels. A steel ladle is a complex three-phase reactor, where strongly dispersed inclusions are transported by the turbulent liquid metal/bubbles flow. We have coupled a population balance model with CFD in order to simulate the mechanisms of transport, aggregation, flotation, and surface entrapment of inclusions. The simulation results, when applied to an industrial gas-stirring ladle operation, show the efficiency of this modeling approach and allow us to compare the respective roles of these mechanisms on the inclusion removal rate. The comparison with literature reporting data emphasizes the good prediction of deoxidating rate of the ladle. On parallel, a simplified zero-dimensional model has been set-up incorporating the same kinetics law for the aggregation rate and all the removal mechanisms. A particular attention has been paid on the averaging method of the hydrodynamics parameters introduced in the flotation and kinetics kernels.

Modeling of aggregation kernel using Monte Carlo simulations of spray fluidized bed agglomeration

Abstract: The present work attempts to consider the microscopic mechanisms of spray fluidized bed agglomeration while modeling the macroscopic kinetics of the process. A microscale approach, constant volume Monte-Carlo simulation, is used to analyze the effects of micro-processes on the aggregation behavior and identify the influencing parameters. The identified variables, namely the number of wet particles, the total number of particles, and the number of droplets are modeled and combined in the form of an aggregation kernel. The proposed kernel is then used in a one-dimensional population balance equation for predicting the particle number density distribution. The only fitting parameters remaining in the population balance system are the collision frequency per particle and a success fraction accounting for the dissipation of kinetic energy. Predictions of the population balance model are compared with the results of Monte-Carlo simulations for a variation of significant operating parameters and found to be in good agreement. © 2014 American Institute of Chemical Engineers AIChE J, 60: 855–868, 2014

Bubble Coalescence Modeling in the Population Balance Framework

Abstract: In the churn-turbulent bubbly flow regime with highly nonuniform bubble size distributions, bubble breakage and coalescence are important processes because they govern the bubble size distribution and consequently directly affect the interfacial mass, momentum, and heat transfer fluxes through the renewal bubble surfaces. At present, accurate prediction of bubble size distributions of dispersed gas–liquid flows by use of the population balance (PB) equation is a difficult task. The modeling of bubble breakup and coalescence rates is very complex and is based on the knowledge of collision and breakup frequencies, breakage daughter size distributions, and probability of coalescence. In this work, we focus on the coalescence phenomenon. The coalescence models are still on an empirical level and the mechanisms are not fully understood. This motivates the analysis of the suitability of the coalescence closures for the prediction of experimental data obtained from coalescence dominated gas–liquid flows. For thi...

Topology Evolution in Polymer Modification

Abstract: A recent numerical method has opened new opportunities in multidimensional population balance modeling. Here, this method is applied to a full three-dimensional population balance model (PBM) describing branching topology evolution driven by chain end to backbone coupling. This process is typical for polymer modification reactions, e.g., in polyethylene, where initially linear polymer chains undergo hydrogen abstraction, and subsequent branching or scission. Topologies are distinguished by chain ends, number of branches, and number of reactive ends. The resulting time dependent trivariate distribution is utilized to extract various distributive properties of the polymer. The results exhibit excellent agreement with data from Monte Carlo simulations.

Population balance model: Breakage kernel parameter estimation to emulsification data

Abstract: Many processes used across, for example the cosmetics, pharmaceutical, food and chemical industries involve two-phase liquid–liquid interactions. The quality of liquid–liquid emulsification systems may be related to the droplet size distribution. The population balance equation (PBE) can be used as a modelling tool when accurate description of the dispersed phase is required. Still, the key challenge with the formulation of predictive population balance (PB) models is experimental determination of unknown breakage and coalescence functions. The complexity in the processes and phenomena governing the changes of dispersed systems makes the derivation of the corresponding models a significant challenge. The present study considers a PBE optimisation problem to allow parameter identification to experimental data. The experimental data are measured for a breakage dominated liquid–liquid emulsification system in a stirred tank. Parameter identifications to the breakage frequency models proposed by Coulaloglou and Tavlarides,[26] Alopaeus et al.[28] and Baldyga and Podgorska[27] are performed. The PBE is numerically solved using the high-order least-squares method. Moreover, the nonlinear parameter identification algorithm is based on the minimisation of the residual between the experimental data and the numerical solution in a least-squares sense. The problem has been implemented in the programming language MATLAB where the fmincon function has been used. Parameter estimation can sometimes be straight forward, for instance when the process and formulated model are relatively simple and sufficient data are available. These conditions are not always met, which may result in difficulties in determining accurate parameter values. A thorough statistical analysis is required in order to explore the actual accuracy of the estimated parameter values. The present study presents a relative comprehensive statistical study of the fit compared to what has been provided in previous PBE parameter estimation studies. Moreover, the optimisation algorithm and challenges associated with parameter estimation are discussed. The present study revealed, by systematically assessing the problem formulation and the fit, that a better understanding of the model and more successful parameter estimation can be achieved, or a limitation of the model is unveiled.

Dynamic Modeling and Optimization of Batch Crystallization of Sugar Cane under Uncertainty

Abstract: This work presents a study on the agitation rate effects on the average diameter (% volume D(4,3)) in the batch crystallization of sugar cane in pilot-scale process. The mathematical model presented in this work includes the population balance equation (PBE), the mass and energy balances, and the kinetics equations of nucleation and growth rate. The kinetic parameters were calculated from optimization using experimental data obtained from a pilot-scale process. An uncertainty analysis was performed and used to specify robust agitation trajectories that minimize the variations of crystal size from batch to batch. Four cases studies are presented to obtain 920, 1000, 1200, and 1300 μm of D(4,3) subject to a constraint in the formed crystal mass (FCM) of 4700 g under uncertainty in the kinetic parameters. The resulting robust agitation trajectories were implemented in the pilot-scale process. Comparisons between experimental data and the model predictions are presented.

Statistical charge distribution over dust particles in a non-Maxwellian Lorentzian plasma

Abstract: On the basis of statistical mechanics and charging kinetics, the charge distribution over uniform size spherical dust particles in a non-Maxwellian Lorentzian plasma is investigated. Two specific situations, viz., (i) the plasma in thermal equilibrium and (ii) non-equilibrium state where the plasma is dark (no emission) or irradiated by laser light (including photoemission) are taken into account. The formulation includes the population balance equation for the charged particles along with number and energy balance of the complex plasma constituents. The departure of the results for the Lorentzian plasma, from that in case of Maxwellian plasma, is graphically illustrated and discussed; it is shown that the charge distribution tends to results corresponding to Maxwellian plasma for large spectral index. The charge distribution predicts the opposite charging of the dust particles in certain cases.

Solution of the dynamic population balance equation describing breakage–coalescence systems in agitated vessels: The least‐squares method

Abstract: A variety of processes used across, for example the cosmetics, pharmaceutical and chemical industries involve two-phase liquid–liquid interactions. The quality of liquid–liquid emulsification systems may be importantly related to the droplet size distribution. The population balance equation (PBE) can be used to describe complex processes where the accurate prediction of the dispersed phase plays a major role for the overall behaviour of the system. In recent years, the high-order least-squares method has been applied to approximate the solution to population balance (PB) problems. From the chemical engineering point of view, the least-squares method is associated with complex algebra. Moreover, in previous chemical engineering publications the method has been outlined using rather compact mathematical notations. For this reason, in this study, details of the least-squares algebra and implementation issues are revealed. The solution strategy is illustratively applied to a test problem: a liquid–liquid emulsification system with breakage and coalescence events in a stirred tank.

Numerical Simulation of Turbulence-Induced Flocculation and Sedimentation in a Flocculant-Aided Sediment Retention Pond

Abstract: A model combining multi-dimensional discretized population balance equations with a computational fluid dynamics simulation (CFD-DPBE model) was developed and applied to simulate turbulent flocculation and sedimentation processes in sediment retention basins. Computation fluid dynamics and the discretized population balance equations were solved to generate steady state flow field data and simulate flocculation and sedimentation processes in a sequential manner. Up-to-date numerical algorithms, such as opera tor splitting and LeVeque flux-corrected upwind schemes, were applied to cope with the computational demands caused by complexity and nonlinearity of the population balance equations and the instability caused by advection-dominated transport. In a modeling and simulation study with a two-dimensional simplified pond system, applicability of the CFD-DPBE model was demonstrated by tracking mass balances and floc size evolutions and by examining particle/floc size and solid concentration distributions. Thus, the CFD-DPBE model may be used as a valuable simulation tool for natural and engineered flocculation and sedimentation systems as well as for flocculant-aided sediment retention ponds.