Discretized Population Balance Equation Research Articles

Kinetic parameter estimation for cooling crystallization process based on cell average technique and automatic differentiation

Abstract In this paper, a cell average technique (CAT) based parameter estimation method is proposed for cooling crystallization involved with particle growth, aggregation and breakage, by establishing a more efficient and accurate solution in terms of the automatic differentiation (AD) algorithm. To overcome the deficiency of CAT that demands high computation cost for implementation, a set of ordinary differential equations (ODEs) entailed from CAT based discretized population balance equation (PBE) are solved by using the AD based high-order Taylor expansion. Moreover, an AD based trust-region reflective (TRR) algorithm and another interior-point (IP) algorithm are established for estimating the kinetic parameters associated with particle growth, aggregation and breakage. As a result, the estimation accuracy can be further improved while the computation cost can be significantly reduced, compared to the existing algorithms. Benchmark examples from the literature are used to illustrate the accuracy and efficiency of the AD-based CAT, TRR and IP algorithms in comparison with the existing algorithms. Moreover, seeded batch cooling crystallization experiments of β form L-glutamic acid are performed to validate the proposed method.

Selection function in breakage processes: PBM and Monte Carlo modeling

Breakage, i.e., formation of smaller fragments from larger initial particles, is an important phenomenon – wanted or unwanted – in many particulate processes. In this work, the volume and time dependent selection function for pure breakage process is modeled based on the population balance modeling approach. It is found that the selection function is directly correlated with the stressing frequency, probability of successful events, and some integral properties of the number density function. A discrete population balance equation is applied to compute the total number of particles and particle size distribution numerically. Moreover, an event-driven constant number Monte Carlo simulation algorithm is presented, and the simulation results are used as an alternative to experimental results. The volume dependency of the selection function is incorporated successfully in the Monte Carlo simulation while selecting particles for stressing event. Some important properties of any particulate process, such as the total number of particles and the size distribution of particles are validated successfully using the Monte Carlo results. This offers new insights into the estimation and interpretation of breakage kinetics.

Analytical time-stepping solution of the discretized population balance equation

The prediction of the particle-size distribution (PSD) of the particulate systems in chemical engineering is very important in a variety of different contexts, such as parameter identification, troubleshooting, process control, design, product quality, production economics etc. The time evolution of the PSD can be evaluated by means of the population balance equation (PBE), which is a complex integro-differential equation, whose solution in practical cases always requires sophisticated numerical methods that may be computationally tedious. In this work, we propose a novel technique that tackles this issue by using an analytical time-stepping procedure (ATS) to resolve the PSD time dependency. The ATS is an explicit time integrator, taking advantage of the linear or almost linear time dependency of the discretized population balance equation. Thus, linear approximation of the source term is a precondition for the ATS simulations. The presented technique is compared with a standard variable step time integrator (MATLAB ODE15s stiff solver), for practical examples e.g. emulsion, aging cellulose process, cooling crystallization, reactive dissolution, and liquid-liquid extraction. The results show that this advancing in time procedure is accurate for all tested practical examples, allowing reproducing the same results given by standard time integrators in a fraction of the computational time.

A discrete population balance equation for binary breakage

SummaryThe numerical solution of the population balance equation is frequently achieved by means of discretization, ie, by the method of classes. An important concern of discrete formulations is the preservation of 2 chosen moments of the distribution, eg, numbers and mass, while remaining flexible on the grid and kernels applied. Existing formulations for breakage are either able to perserve only one moment or restricted by the choice of grid and kernels. Two types of kernel functions for the description of breakup rate exist, ie, total breakup model and partial breakup model. The first type states the total breakup rate of a mother particle and requires a daughter size distribution function. The other type gives the breakup rate between a mother and a daughter particle directly. Existing formulations are known to work well for the former type but inefficiently for the latter one due to the need of additional numerical integrations. The particular focus of the present work lies in developing an efficient formulation for this type of kernels. A discrete formulation of the breakup terms due to binary breakage is proposed, which allows a direct implementation of both types of breakup kernels and an efficient solution of the population balance equation, making it favorable for the coupling to computational fluid dynamics codes. The results for a pure breakage process obtained by using the new formulation and various kernels are compared with those by a reference formulation as well as analytical solutions. Good agreement is achieved in all test cases, and numerical efficiency of the new formulation for partial breakup models is evidenced.

Systematic observability and detectablity analysis of industrial batch crystallizers

Motivated by the lack of hardware analysers for particle size distribution (PSD) and solute concentration measurements in industrial crystallizers, this work investigates the feasibility of designing alternative monitoring tools based on state observers. The observability and detectability properties of the discretized population balance equation accounting for crystal growth, attrition and agglomeration coupled with energy and solute mass balances are studied. A systematic methodology for sensor selection based on nonlinear observability and detectability principles is proposed and applied. Results are corroborated by a machine learning technique (the self-organizing map), leading to the fact that the solute concentration is distinguishable with temperature measurements, while the PSD is not. The results represent the starting point for future detector design where temperature measurements are used to infer composition, while the estimation of the PSD is done in "open loop" fashion.

Population Balance Modeling of Barium Sulfate Nanoparticle Synthesis via Inverse Microemulsion Including Coagulation Effect

A deterministic model based on the discrete population balance equation (PBE) is used to examine the precipitation of barium sulfate nanoparticles in nonionic inverse microemulsion systems. It is shown that coagulation can have a significant effect at low initial reactants concentration. The simulation results show that a bimodal particle size distribution (PSD) observed in experimental analysis reported in the literature cannot be predicted by neglecting the coagulation effect. A coagulation kernel is proposed that takes into account the effect of interparticle forces through the Fuchs’ stability ratio. The effect of electrolyte concentration on the surfactant headgroup area is also investigated. It is shown that at lower electrolyte concentration, the nucleation and growth rates increase because of higher interdroplet exchange rate. The simulation results show a qualitatively good agreement with experimental data regarding PSD prediction and dynamic evolution of mean particle diameter.

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

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.

Modeling differential scanning calorimetry of thermally degrading thermoplastics

Differential scanning calorimetry (DSC) of thermally degrading thermoplastics is modeled using a discrete population balance equation (PBE). The PBE allows for species mole change due to pyrolysis and bubbling mass loss. Efficient solution is achieved by lumping non-volatile species into a single, notional “polymer” species. Thermodynamic properties are calculated by group additivity techniques, and empirical corrections are made to account for the changes in specific heat capacity and enthalpy upon mixing. Simulation results for high-density polyethylene (HDPE) are compared to literature data for DSC at heating rates of 5, 10, and 20K/min. The model predicts peak energy absorption rates to within 4% at the lowest heating rate. The heat of decomposition was found to be within 17% of literature values at all three heating rates. The predicted pyrolysis gas species distribution agrees well with literature gas chromatography–mass spectrometry (GC–MS) data.

High shear granulation of dolomite – II: Effect of amount of binder liquid on process kinetics

This paper, the second of a series, analyzes the effect of binder amount on the kinetics of wet granulation process. Granulation experiments were conducted in batch, lab-scale high shear mixer with formulation system that was initially studied in first part of a series. First, we identify the effect of binder content on the net granulation behaviour during early stage of the process. Single entity property (its size) was accounted only within this research. The results indicate that binder content strongly promotes growth of dolomite entities. Secondly, 1-D discretized population balance equation (PBE) and Equi-Partition of Kinetic Energy (EKE) theory were used to simulate the net granulation process. Tested “coalescence-only” models provide good prediction of real dolomite granule size distributions (GSDs) during early stage of the process for each binder content value. By using modelling procedure granule growth rates were quantified. Ultimate goal of relating the coalescence rates of dolomite entities with binder amount variable is provided. This will result in a better perspective of the meso-scale of the dolomite granulation process.

Understanding mild acid pretreatment of sugarcane bagasse through particle scale modeling

Sugarcane bagasse is an abundant and sustainable resource, generated as a by-product of sugarcane milling. The cellulosic material within bagasse can be broken down into glucose molecules and fermented to produce ethanol, making it a promising feedstock for biofuel production. Mild acid pretreatment hydrolyses the hemicellulosic component of biomass, thus allowing enzymes greater access to the cellulosic substrate during saccharification. A particle-scale mathematical model describing the mild acid pretreatment of sugarcane bagasse has been developed, using a volume averaged framework. Discrete population-balance equations are used to characterise the polymer degradation kinetics, and diffusive effects account for mass transport within the cell wall of the bagasse. As the fibrous material hydrolyses over time, variations in the porosity of the cell wall and the downstream effects on the reaction kinetics are accounted for using conservation of volume arguments. Non-dimensionalization of the model equations reduces the number of parameters in the system to a set of four dimensionless ratios that compare the timescales of different reaction and diffusion events. Theoretical yield curves are compared to macroscopic experimental observations from the literature and inferences are made as to constraints on these "unknown" parameters. These results enable connections to be made between experimental data and the underlying thermodynamics of acid pretreatment. Consequently, the results suggest that data-fitting techniques used to obtain kinetic parameters should be carefully applied, with prudent consideration given to the chemical and physiological processes being modeled.

High shear granulation of dolomite – I: Effect of shear regime on process kinetics

Wet granulation of previously unreported formulation system is presented. Dolomite powder is granulated under different shear regimes by using three-component binder formulation, water-molasses-polyvinylpyrrolidone. 1-D discretized population balance equation (PBE) and Equi-Partition of Kinetic Energy (EKE) coalescence kernel are applied to modelling granulation in a high shear mixer. Process modelling is focused to simulation of changes of the property of a group of entities, granule size distribution (GSD). The GSD predictions indicate the presence of coalescence growth as a dominant mechanism in the dolomite granulation process. Minor deviations between simulated and real GSDs signify the probability of other granulation mechanism(s) existence. A posteriori approach by integral method was used for coalescence rate constant estimation. This research highlights discrepancy in the coalescence rate of dolomite granulation process, between its early and later stages. Moreover, kinetic analysis of the high shear granulation process provides quantification of the macroscopic variable (impeller speed) influence on regarded property of a group of granules in terms of values of growth rate parameter.

Algorithm for the accurate numerical solution of PBE for drop breakup and coalescence under high shear rates

Multiphase flows play an important role in the chemical and process industries and significant strides have recently been achieved in the design of such systems using the population balance models. However, some uncertainties still remain concerning the stability and accuracy of the numerical solution of such integro-differential equations. This paper proposes a new methodology for solving the discretized population balance equation by minimizing the finite domain errors that often arise while discretizing the drop size domain. It relies on the use of the size distribution sampling approach combined with a moving grid technique. In addition, an enhanced solution stability algorithm was proposed and which relies on monitoring the onset of errors in the various birth and death terms encountered in PBE. It consequently allows for corrective action to be undertaken before the errors propagate in an uncontrollable fashion, and was found to improve the stability and robustness of the solution method even under very high shear rate conditions. The proposed algorithm was tested using the model of Coulaloglou and Tavlarides (1977) under breakage and coalescence dominated conditions in low, moderate and high energy dissipation regions, and was found to provide a stable solution that accurately predicts the quasi-equilibrium Sauter mean diameter.

Fast Monte Carlo methodology for multivariate particulate systems—I: Point ensemble Monte Carlo

A fast Monte Carlo methodology for particulate processes is introduced. The proposed methodology combines concepts from discrete population balance equations and dynamic Monte Carlo simulations of chemical kinetics to construct a new jump Markov model that approximates the population balance dynamics. The Markov model consists of a definition of a new type of reaction channel, in which the reaction product is a stochastic process by itself. One feature of this model is that, although a coarse view of the process is taken, it still conserves the history of individual particles. This is a very important aspect for effective modeling of multivariate models, especially when part of the goal is to study the evolution of the internal states of the particles (i.e., composition, phase behavior, etc.). Numerical experiments show that this algorithm can improve the computational load of the exact method by orders of magnitude without sacrificing computational accuracy. The methodology is useful especially in stochastic optimization applications where many function calls (simulations) are required. Possible applications are optimization and dynamic optimization using an artificial chemical process algorithm, genetic algorithm, or simulated annealing among others.

Development of a multi-scale, multi-phase, multi-zone dynamic model for the prediction of particle segregation in catalytic olefin polymerization FBRs

A comprehensive multi-scale, multi-phase, multi-compartment dynamic model is developed to analyze the extent of particle segregation in catalytic, gas-phase ethylene–propylene copolymerization fluidized bed reactors (FBRs). From the numerical solution of the proposed integrated model, the temporal-spatial evolution of the morphological (i.e., particle size distribution, PSD) and molecular (i.e., molecular weight distribution, MWD) polymer properties in a catalytic polymerization FBR can be predicted. In particular, the polymer molecular properties are determined by employing a generalized multi-site, Ziegler–Natta kinetic scheme. To determine the growth of a single catalyst/polymer particle, the random pore polymeric flow model (RPPFM) is utilized. The RPPFM is solved together with a dynamic discretized particle population balance equation (PBE) to calculate the dynamic evolution of PSD in the various compartments of the FBR. Moreover, overall dynamic mass and energy balances are derived in order to assess the dynamic behavior of catalytic gas-phase FBRs. The effects of various fluidized bed operating conditions (e.g., fluidization gas velocity, temperature and catalyst feed rate) on the morphological and molecular distributed polymer properties are thoroughly analyzed.

A Fluidized Bed Biomass Combustion Model with Discretized Population Balance. 1. Sensitivity Analysis

A model of a 1 MWth atmospheric bubbling fluidized combustor burning waste wood fuel (feeding) is presented. The model incorporates both the solid and gas phases. The bed is assumed to consist of two phases, of which the emulsion phase takes both gas and solids into account, while the bubble phase consists only of gas. A wide size distribution of biomass feed, representative of the actual boiler feed, has been assumed. The model calculates the gas composition, velocities, and other important hydrodynamic parameters in both the emulsion and bubble phase. A particle size distribution model is included to calculate elutriation losses of fine char particles. This approach is novel in the sense that a population balance for fine particle class is derived, using the well-known mass balance principles, and a coupled discretized population balance equation, valid for the whole particle size range, is presented. The model takes into account devolatilization, fragmentation, and attrition of the solid phase along with gaseous profiles. It includes nine components for which differential equations have been derived and solved to calculate species concentration at any point along the bed height. In total, 20 particle size classes have been considered, 10 of which are considered as feed and the rest as fine classes. The model aims to assess the effect of different parameters on boiler performance and gaseous emissions. A sensitivity analysis of the gaseous emission profiles with respect to different variables and parameters defining different submodels has been carried out. A homogeneous NOx model has been included with individual kinetic parameters for relevant species. It has been found that gas hydrodynamics play a significant role, and the system can be optimized using these parameters.

Deposition and aggregation of Brownian particles in trickle‐bed reactors

AbstractWhen dilute liquid suspensions contaminated with Brownian fine solids are treated in catalytic trickle‐bed reactors, bed plugging develops and increases the resistance to two‐phase flow until ultimate unit shutdown for bed substitution with pristine catalyst is imposed. One of the important aspects during plugging with Brownian particles is the aggregation of fines and the release of the fine particles and aggregates from pore bodies within the porous bed as a result of the hydrodynamic or colloidal forces. Current physical models linking gas–liquid flow to the filtration process in high‐pressure/temperature trickle beds neglect the possible colloidal particle aggregation and the release of aggregates. This work attempts to fill this gap by developing a Euler–Euler fluid dynamic model based on the volume average mass, momentum, and species balance equations, filtration equations for the Brownian particles and the aggregates, and the discrete population balance equations for the agglomeration of particles. Both monolayer and multilayer depositions were considered for Brownian particles and only the monolayer deposition in the case of the detaching aggregates. The release of fine particles and aggregates from the collector surface was assumed to be induced by the colloidal forces in the case of Brownian particles/aggregates or by the hydrodynamic forces in the case of non‐Brownian aggregates. Brownian particle aggregation was described by the rate at which a certain size aggregate is being formed by smaller aggregates less the rate at which the aggregate combines to form a larger aggregate. © 2006 American Institute of Chemical Engineers AIChE J 2006

Population balance and computational fluid dynamics modelling of ice crystallisation in a scraped surface freezer

Ice crystallisation in a scraped surface freezer is exceedingly complex and there is very limited fundamental understanding of the influences of fluid flow and heat transfer on ice crystal size, number and shape. This paper presents a computer modelling study that combines population balance method and computational fluid dynamics method. Ice crystal nucleation and growth kinetics has been described by discrete population balance equation. Algorithms for solving the coupled equations of fluid flow, heat transfer and discrete population balance of ice crystallisation have been developed and implemented into a commercial code. Demonstrated is a 2-D computer simulation of ice crystallisation in a scraped surface freezer. Although the simulation makes several assumptions including simplified fluid flow conditions, the predicted ice crystal size distribution is comparable to the experimental data. It is shown that with combined computational fluid dynamics and population balance modelling much insight into the interaction of fluid flow, heat transfer and ice crystallisation in the scraped surface freezer can be obtained.

Adjustable discretized population balance equations: Numerical simulation and parameter estimation for fractal aggregation and break-up

Improved adjustable discretized population balance equations (PBEs) are proposed in this study. The authors extended an improved particle coagulation model previously developed to an adjustable geometric size interval ( q), where q is a volume ratio of class k + 1 particle to class k particle (υ k+1 /υ k = q). This model was verified with the time derivative of the zero and first moments to show mass conservation and compared with previous analytical and numerical solutions. Also, the self-preserving distribution test was conducted by using size-independent and size-dependent kernels. After direct numerical simulations (DNS), the proposed model was found to have excellent agreement with the analytical and continuous numerical solutions. In addition, this proposed model was converted to a dimensionless form to enhance computational efficiency in order to be coupled with computational fluid dynamic solutions in the future. Furthermore, a parameter estimation scheme was created to computationally determine the two key parameters, the collision efficiency ( α) and the break-up coefficient ( K B), from orthokinetic experimental data. To verify the estimation procedure the fractal aggregate orthokinetic experimental data of Li and Zhang (a primary d o of 2.8 μm in dia., D f of 2.0, a fluid strain-rate of 15 (1/s)) was used [X. Li, J. Zhang, Numerical simulation and experimental verification of particle coagulation dynamics for a pulsed input, J. Coll. Interf. Sci. 262 (2003) 149–161]. Considering fractal aggregation and break-up, two major parameters were found to be collision efficiency α of 0.3938 and aggregate break-up coefficient K B of 4.4105 using a parameter estimation scheme coupled with an improved discretized population balance equation. This parameter estimation scheme was able to compute the coefficients in the coagulation model rapidly, especially in particle systems having a fractal aggregate structure.

Multi‐Phase‐Multi‐Size‐Group Model for the Inclusion of Population Balances into the CFD Simulation of Gas‐Liquid Bubbly Flows

AbstractA CFD model for the simulation of gas‐liquid bubbly flow is developed. In the model, the multi‐phase flow is simulated by an Eulerian‐Eulerian approach using several phase definitions (from 3 to 10). The bubble size distribution is simulated by a solution of the discretized population balance equation with coalescence and break‐up of bubbles. The number of the discretized population balance equations in the model is larger than the number of the phases used in the flow field simulation. A desired accuracy in the simulation can be achieved by choosing a suitable number of phases as a compromise between accuracy and computational cost. With this model, more detailed flow hydrodynamics and bubble size distribution can be obtained. The model was tested with different operating conditions and for different numbers of dispersed phases in a bubble column, and was verified with a bubble size distribution obtained experimentally.

Dynamics of cells attachment, aggregation, growth and detachment in trickle-bed bioreactors

Excessive biomass formation in two-phase flow trickle-bed bioreactors induces biological clogging and leads to the progressive obstruction of the bed that is accompanied with a build-up in pressure drop and flow channeling. One of the important aspects during biological clogging in trickle-bed bioreactors is the aggregation of cells and the detachment of cells and aggregates from pore bodies within the porous bed. Current theoretical models describing the transient behaviour of biomass accumulation and the biological clogging in trickle-bed bioreactors for wastewater treatment neglect the cell aggregation process and the aggregates detachment. An attempt has been made with this contribution in which the authors strived to develop an Euler–Euler two-fluid dynamic model based on the volume-average mass, momentum equations, species balance equations, biomass dynamics equation, filtration equations for the cells and the aggregates and discrete population balance equations for the cells agglomeration to describe two-phase flow and space-time evolution of biological clogging in trickle-bed bioreactors for wastewater treatment. Phenol biodegradation by Pseudomonas putida as the predominant species immobilized on activated carbon was chosen as a case study to illustrate the consequences of formation of excessive amounts of biomass. Cells aggregation was described by the rate at which a certain size aggregate is being formed by smaller aggregates minus the rate at which the aggregate combines to form a larger aggregate. The detachment of the cells or aggregates from the collector surface was supposed to be induced by the colloidal forces in the case of Brownian cells/aggregates or by the hydrodynamic forces in the case of non-Brownian aggregates.